Regenerate the translated files for Lean

12 files changed, 1055 insertions, 696 deletions

diff --git a/tests/lean/hashmap/Base/Primitives.lean b/tests/lean/hashmap/Base/Primitives.lean
index 5b64e908..034f41b2 100644
--- a/tests/lean/hashmap/Base/Primitives.lean
+++ b/tests/lean/hashmap/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/hashmap/Hashmap/Clauses/Clauses.lean b/tests/lean/hashmap/Hashmap/Clauses/Clauses.lean
index fad5c11a..197b0a6a 100644
--- a/tests/lean/hashmap/Hashmap/Clauses/Clauses.lean
+++ b/tests/lean/hashmap/Hashmap/Clauses/Clauses.lean
@@ -1,11 +1,11 @@
--- [hashmap]: the decreases clauses
+-- [hashmap]: templates for the decreases clauses
 import Base.Primitives
 import Hashmap.Types
 
 /- [hashmap::HashMap::{0}::allocate_slots]: termination measure -/
 @[simp]
 def hash_map_allocate_slots_loop_terminates (T : Type) (slots : Vec (list_t T))
-  (n : USize) :=
+  (n : Usize) :=
   (slots, n)
 
 /- [hashmap::HashMap::{0}::allocate_slots]: decreases_by tactic -/
@@ -16,7 +16,7 @@ macro_rules
 /- [hashmap::HashMap::{0}::clear]: termination measure -/
 @[simp]
 def hash_map_clear_loop_terminates (T : Type) (slots : Vec (list_t T))
-  (i : USize) :=
+  (i : Usize) :=
   (slots, i)
 
 /- [hashmap::HashMap::{0}::clear]: decreases_by tactic -/
@@ -26,7 +26,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::insert_in_list]: termination measure -/
 @[simp]
-def hash_map_insert_in_list_loop_terminates (T : Type) (key : USize)
+def hash_map_insert_in_list_loop_terminates (T : Type) (key : Usize)
   (value : T) (ls : list_t T) :=
   (key, value, ls)
 
@@ -51,7 +51,7 @@ macro_rules
 /- [hashmap::HashMap::{0}::move_elements]: termination measure -/
 @[simp]
 def hash_map_move_elements_loop_terminates (T : Type) (ntable : hash_map_t T)
-  (slots : Vec (list_t T)) (i : USize) :=
+  (slots : Vec (list_t T)) (i : Usize) :=
   (ntable, slots, i)
 
 /- [hashmap::HashMap::{0}::move_elements]: decreases_by tactic -/
@@ -62,7 +62,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::contains_key_in_list]: termination measure -/
 @[simp]
-def hash_map_contains_key_in_list_loop_terminates (T : Type) (key : USize)
+def hash_map_contains_key_in_list_loop_terminates (T : Type) (key : Usize)
   (ls : list_t T) :=
   (key, ls)
 
@@ -74,7 +74,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::get_in_list]: termination measure -/
 @[simp]
-def hash_map_get_in_list_loop_terminates (T : Type) (key : USize)
+def hash_map_get_in_list_loop_terminates (T : Type) (key : Usize)
   (ls : list_t T) :=
   (key, ls)
 
@@ -86,7 +86,7 @@ macro_rules
 /- [hashmap::HashMap::{0}::get_mut_in_list]: termination measure -/
 @[simp]
 def hash_map_get_mut_in_list_loop_terminates (T : Type) (ls : list_t T)
-  (key : USize) :=
+  (key : Usize) :=
   (ls, key)
 
 /- [hashmap::HashMap::{0}::get_mut_in_list]: decreases_by tactic -/
@@ -96,7 +96,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::remove_from_list]: termination measure -/
 @[simp]
-def hash_map_remove_from_list_loop_terminates (T : Type) (key : USize)
+def hash_map_remove_from_list_loop_terminates (T : Type) (key : Usize)
   (ls : list_t T) :=
   (key, ls)
 
diff --git a/tests/lean/hashmap/Hashmap/Clauses/Template.lean b/tests/lean/hashmap/Hashmap/Clauses/Template.lean
index 7ba079f2..560592c8 100644
--- a/tests/lean/hashmap/Hashmap/Clauses/Template.lean
+++ b/tests/lean/hashmap/Hashmap/Clauses/Template.lean
@@ -6,7 +6,7 @@ import Hashmap.Types
 /- [hashmap::HashMap::{0}::allocate_slots]: termination measure -/
 @[simp]
 def hash_map_allocate_slots_loop_terminates (T : Type) (slots : Vec (list_t T))
-  (n : USize) :=
+  (n : Usize) :=
   (slots, n)
 
 /- [hashmap::HashMap::{0}::allocate_slots]: decreases_by tactic -/
@@ -17,7 +17,7 @@ macro_rules
 /- [hashmap::HashMap::{0}::clear]: termination measure -/
 @[simp]
 def hash_map_clear_loop_terminates (T : Type) (slots : Vec (list_t T))
-  (i : USize) :=
+  (i : Usize) :=
   (slots, i)
 
 /- [hashmap::HashMap::{0}::clear]: decreases_by tactic -/
@@ -27,7 +27,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::insert_in_list]: termination measure -/
 @[simp]
-def hash_map_insert_in_list_loop_terminates (T : Type) (key : USize)
+def hash_map_insert_in_list_loop_terminates (T : Type) (key : Usize)
   (value : T) (ls : list_t T) :=
   (key, value, ls)
 
@@ -52,7 +52,7 @@ macro_rules
 /- [hashmap::HashMap::{0}::move_elements]: termination measure -/
 @[simp]
 def hash_map_move_elements_loop_terminates (T : Type) (ntable : hash_map_t T)
-  (slots : Vec (list_t T)) (i : USize) :=
+  (slots : Vec (list_t T)) (i : Usize) :=
   (ntable, slots, i)
 
 /- [hashmap::HashMap::{0}::move_elements]: decreases_by tactic -/
@@ -63,7 +63,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::contains_key_in_list]: termination measure -/
 @[simp]
-def hash_map_contains_key_in_list_loop_terminates (T : Type) (key : USize)
+def hash_map_contains_key_in_list_loop_terminates (T : Type) (key : Usize)
   (ls : list_t T) :=
   (key, ls)
 
@@ -75,7 +75,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::get_in_list]: termination measure -/
 @[simp]
-def hash_map_get_in_list_loop_terminates (T : Type) (key : USize)
+def hash_map_get_in_list_loop_terminates (T : Type) (key : Usize)
   (ls : list_t T) :=
   (key, ls)
 
@@ -87,7 +87,7 @@ macro_rules
 /- [hashmap::HashMap::{0}::get_mut_in_list]: termination measure -/
 @[simp]
 def hash_map_get_mut_in_list_loop_terminates (T : Type) (ls : list_t T)
-  (key : USize) :=
+  (key : Usize) :=
   (ls, key)
 
 /- [hashmap::HashMap::{0}::get_mut_in_list]: decreases_by tactic -/
@@ -97,7 +97,7 @@ macro_rules
 
 /- [hashmap::HashMap::{0}::remove_from_list]: termination measure -/
 @[simp]
-def hash_map_remove_from_list_loop_terminates (T : Type) (key : USize)
+def hash_map_remove_from_list_loop_terminates (T : Type) (key : Usize)
   (ls : list_t T) :=
   (key, ls)
 
diff --git a/tests/lean/hashmap/Hashmap/Funs.lean b/tests/lean/hashmap/Hashmap/Funs.lean
index 535ac9b2..77b1a157 100644
--- a/tests/lean/hashmap/Hashmap/Funs.lean
+++ b/tests/lean/hashmap/Hashmap/Funs.lean
@@ -5,19 +5,19 @@ import Hashmap.Types
 import Hashmap.Clauses.Clauses
 
 /- [hashmap::hash_key] -/
-def hash_key_fwd (k : USize) : Result USize :=
+def hash_key_fwd (k : Usize) : Result Usize :=
   Result.ret k
 
 /- [hashmap::HashMap::{0}::allocate_slots] -/
 def hash_map_allocate_slots_loop_fwd
-  (T : Type) (slots : Vec (list_t T)) (n : USize) :
+  (T : Type) (slots : Vec (list_t T)) (n : Usize) :
   (Result (Vec (list_t T)))
   :=
-  if h: n > (USize.ofNatCore 0 (by intlit))
+  if h: n > (Usize.ofInt 0 (by intlit))
   then
     do
       let slots0 ← vec_push_back (list_t T) slots list_t.Nil
-      let n0 ← USize.checked_sub n (USize.ofNatCore 1 (by intlit))
+      let n0 ← n - (Usize.ofInt 1 (by intlit))
       hash_map_allocate_slots_loop_fwd T slots0 n0
   else Result.ret slots
 termination_by hash_map_allocate_slots_loop_fwd slots n =>
@@ -26,23 +26,23 @@ decreasing_by hash_map_allocate_slots_loop_decreases slots n
 
 /- [hashmap::HashMap::{0}::allocate_slots] -/
 def hash_map_allocate_slots_fwd
-  (T : Type) (slots : Vec (list_t T)) (n : USize) : Result (Vec (list_t T)) :=
+  (T : Type) (slots : Vec (list_t T)) (n : Usize) : Result (Vec (list_t T)) :=
   hash_map_allocate_slots_loop_fwd T slots n
 
 /- [hashmap::HashMap::{0}::new_with_capacity] -/
 def hash_map_new_with_capacity_fwd
-  (T : Type) (capacity : USize) (max_load_dividend : USize)
-  (max_load_divisor : USize) :
+  (T : Type) (capacity : Usize) (max_load_dividend : Usize)
+  (max_load_divisor : Usize) :
   Result (hash_map_t T)
   :=
   do
     let v := vec_new (list_t T)
     let slots ← hash_map_allocate_slots_fwd T v capacity
-    let i ← USize.checked_mul capacity max_load_dividend
-    let i0 ← USize.checked_div i max_load_divisor
+    let i ← capacity * max_load_dividend
+    let i0 ← i / max_load_divisor
     Result.ret
       {
-        hash_map_num_entries := (USize.ofNatCore 0 (by intlit)),
+        hash_map_num_entries := (Usize.ofInt 0 (by intlit)),
         hash_map_max_load_factor := (max_load_dividend, max_load_divisor),
         hash_map_max_load := i0,
         hash_map_slots := slots
@@ -50,19 +50,19 @@ def hash_map_new_with_capacity_fwd
 
 /- [hashmap::HashMap::{0}::new] -/
 def hash_map_new_fwd (T : Type) : Result (hash_map_t T) :=
-  hash_map_new_with_capacity_fwd T (USize.ofNatCore 32 (by intlit))
-    (USize.ofNatCore 4 (by intlit)) (USize.ofNatCore 5 (by intlit))
+  hash_map_new_with_capacity_fwd T (Usize.ofInt 32 (by intlit))
+    (Usize.ofInt 4 (by intlit)) (Usize.ofInt 5 (by intlit))
 
 /- [hashmap::HashMap::{0}::clear] -/
 def hash_map_clear_loop_fwd_back
-  (T : Type) (slots : Vec (list_t T)) (i : USize) :
+  (T : Type) (slots : Vec (list_t T)) (i : Usize) :
   (Result (Vec (list_t T)))
   :=
   let i0 := vec_len (list_t T) slots
   if h: i < i0
   then
     do
-      let i1 ← USize.checked_add i (USize.ofNatCore 1 (by intlit))
+      let i1 ← i + (Usize.ofInt 1 (by intlit))
       let slots0 ← vec_index_mut_back (list_t T) slots i list_t.Nil
       hash_map_clear_loop_fwd_back T slots0 i1
   else Result.ret slots
@@ -76,22 +76,22 @@ def hash_map_clear_fwd_back
   do
     let v ←
       hash_map_clear_loop_fwd_back T self.hash_map_slots
-        (USize.ofNatCore 0 (by intlit))
+        (Usize.ofInt 0 (by intlit))
     Result.ret
       {
         self
           with
-          hash_map_num_entries := (USize.ofNatCore 0 (by intlit)),
+          hash_map_num_entries := (Usize.ofInt 0 (by intlit)),
           hash_map_slots := v
       }
 
 /- [hashmap::HashMap::{0}::len] -/
-def hash_map_len_fwd (T : Type) (self : hash_map_t T) : Result USize :=
+def hash_map_len_fwd (T : Type) (self : hash_map_t T) : Result Usize :=
   Result.ret self.hash_map_num_entries
 
 /- [hashmap::HashMap::{0}::insert_in_list] -/
 def hash_map_insert_in_list_loop_fwd
-  (T : Type) (key : USize) (value : T) (ls : list_t T) : (Result Bool) :=
+  (T : Type) (key : Usize) (value : T) (ls : list_t T) : (Result Bool) :=
   match h: ls with
   | list_t.Cons ckey cvalue tl =>
     if h: ckey = key
@@ -104,12 +104,12 @@ decreasing_by hash_map_insert_in_list_loop_decreases key value ls
 
 /- [hashmap::HashMap::{0}::insert_in_list] -/
 def hash_map_insert_in_list_fwd
-  (T : Type) (key : USize) (value : T) (ls : list_t T) : Result Bool :=
+  (T : Type) (key : Usize) (value : T) (ls : list_t T) : Result Bool :=
   hash_map_insert_in_list_loop_fwd T key value ls
 
 /- [hashmap::HashMap::{0}::insert_in_list] -/
 def hash_map_insert_in_list_loop_back
-  (T : Type) (key : USize) (value : T) (ls : list_t T) : (Result (list_t T)) :=
+  (T : Type) (key : Usize) (value : T) (ls : list_t T) : (Result (list_t T)) :=
   match h: ls with
   | list_t.Cons ckey cvalue tl =>
     if h: ckey = key
@@ -126,25 +126,24 @@ decreasing_by hash_map_insert_in_list_loop_decreases key value ls
 
 /- [hashmap::HashMap::{0}::insert_in_list] -/
 def hash_map_insert_in_list_back
-  (T : Type) (key : USize) (value : T) (ls : list_t T) : Result (list_t T) :=
+  (T : Type) (key : Usize) (value : T) (ls : list_t T) : Result (list_t T) :=
   hash_map_insert_in_list_loop_back T key value ls
 
 /- [hashmap::HashMap::{0}::insert_no_resize] -/
 def hash_map_insert_no_resize_fwd_back
-  (T : Type) (self : hash_map_t T) (key : USize) (value : T) :
+  (T : Type) (self : hash_map_t T) (key : Usize) (value : T) :
   Result (hash_map_t T)
   :=
   do
     let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
     let inserted ← hash_map_insert_in_list_fwd T key value l
     if h: inserted
     then
       do
-        let i0 ← USize.checked_add self.hash_map_num_entries
-          (USize.ofNatCore 1 (by intlit))
+        let i0 ← self.hash_map_num_entries + (Usize.ofInt 1 (by intlit))
         let l0 ← hash_map_insert_in_list_back T key value l
         let v ← vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
         Result.ret
@@ -156,9 +155,9 @@ def hash_map_insert_no_resize_fwd_back
         Result.ret { self with hash_map_slots := v }
 
 /- [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)
+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)
 
 /- [hashmap::HashMap::{0}::move_elements_from_list] -/
 def hash_map_move_elements_from_list_loop_fwd_back
@@ -182,7 +181,7 @@ def hash_map_move_elements_from_list_fwd_back
 
 /- [hashmap::HashMap::{0}::move_elements] -/
 def hash_map_move_elements_loop_fwd_back
-  (T : Type) (ntable : hash_map_t T) (slots : Vec (list_t T)) (i : USize) :
+  (T : Type) (ntable : hash_map_t T) (slots : Vec (list_t T)) (i : Usize) :
   (Result ((hash_map_t T) × (Vec (list_t T))))
   :=
   let i0 := vec_len (list_t T) slots
@@ -192,7 +191,7 @@ def hash_map_move_elements_loop_fwd_back
       let l ← vec_index_mut_fwd (list_t T) slots i
       let ls := mem_replace_fwd (list_t T) l list_t.Nil
       let ntable0 ← hash_map_move_elements_from_list_fwd_back T ntable ls
-      let i1 ← USize.checked_add i (USize.ofNatCore 1 (by intlit))
+      let i1 ← i + (Usize.ofInt 1 (by intlit))
       let l0 := mem_replace_back (list_t T) l list_t.Nil
       let slots0 ← vec_index_mut_back (list_t T) slots i l0
       hash_map_move_elements_loop_fwd_back T ntable0 slots0 i1
@@ -203,7 +202,7 @@ decreasing_by hash_map_move_elements_loop_decreases ntable slots i
 
 /- [hashmap::HashMap::{0}::move_elements] -/
 def hash_map_move_elements_fwd_back
-  (T : Type) (ntable : hash_map_t T) (slots : Vec (list_t T)) (i : USize) :
+  (T : Type) (ntable : hash_map_t T) (slots : Vec (list_t T)) (i : Usize) :
   Result ((hash_map_t T) × (Vec (list_t T)))
   :=
   hash_map_move_elements_loop_fwd_back T ntable slots i
@@ -212,19 +211,19 @@ def hash_map_move_elements_fwd_back
 def hash_map_try_resize_fwd_back
   (T : Type) (self : hash_map_t T) : Result (hash_map_t T) :=
   do
-    let max_usize ← scalar_cast USize core_num_u32_max_c
+    let max_usize ← Scalar.cast .Usize core_num_u32_max_c
     let capacity := vec_len (list_t T) self.hash_map_slots
-    let n1 ← USize.checked_div max_usize (USize.ofNatCore 2 (by intlit))
+    let n1 ← max_usize / (Usize.ofInt 2 (by intlit))
     let (i, i0) := self.hash_map_max_load_factor
-    let i1 ← USize.checked_div n1 i
+    let i1 ← n1 / i
     if h: capacity <= i1
     then
       do
-        let i2 ← USize.checked_mul capacity (USize.ofNatCore 2 (by intlit))
+        let i2 ← capacity * (Usize.ofInt 2 (by intlit))
         let ntable ← hash_map_new_with_capacity_fwd T i2 i i0
         let (ntable0, _) ←
           hash_map_move_elements_fwd_back T ntable self.hash_map_slots
-            (USize.ofNatCore 0 (by intlit))
+            (Usize.ofInt 0 (by intlit))
         Result.ret
           {
             ntable0
@@ -236,7 +235,7 @@ def hash_map_try_resize_fwd_back
 
 /- [hashmap::HashMap::{0}::insert] -/
 def hash_map_insert_fwd_back
-  (T : Type) (self : hash_map_t T) (key : USize) (value : T) :
+  (T : Type) (self : hash_map_t T) (key : Usize) (value : T) :
   Result (hash_map_t T)
   :=
   do
@@ -248,7 +247,7 @@ def hash_map_insert_fwd_back
 
 /- [hashmap::HashMap::{0}::contains_key_in_list] -/
 def hash_map_contains_key_in_list_loop_fwd
-  (T : Type) (key : USize) (ls : list_t T) : (Result Bool) :=
+  (T : Type) (key : Usize) (ls : list_t T) : (Result Bool) :=
   match h: ls with
   | list_t.Cons ckey t tl =>
     if h: ckey = key
@@ -261,22 +260,22 @@ decreasing_by hash_map_contains_key_in_list_loop_decreases key ls
 
 /- [hashmap::HashMap::{0}::contains_key_in_list] -/
 def hash_map_contains_key_in_list_fwd
-  (T : Type) (key : USize) (ls : list_t T) : Result Bool :=
+  (T : Type) (key : Usize) (ls : list_t T) : Result Bool :=
   hash_map_contains_key_in_list_loop_fwd T key ls
 
 /- [hashmap::HashMap::{0}::contains_key] -/
 def hash_map_contains_key_fwd
-  (T : Type) (self : hash_map_t T) (key : USize) : Result Bool :=
+  (T : Type) (self : hash_map_t T) (key : Usize) : Result Bool :=
   do
     let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ← vec_index_fwd (list_t T) self.hash_map_slots hash_mod
     hash_map_contains_key_in_list_fwd T key l
 
 /- [hashmap::HashMap::{0}::get_in_list] -/
 def hash_map_get_in_list_loop_fwd
-  (T : Type) (key : USize) (ls : list_t T) : (Result T) :=
+  (T : Type) (key : Usize) (ls : list_t T) : (Result T) :=
   match h: ls with
   | list_t.Cons ckey cvalue tl =>
     if h: ckey = key
@@ -289,22 +288,22 @@ decreasing_by hash_map_get_in_list_loop_decreases key ls
 
 /- [hashmap::HashMap::{0}::get_in_list] -/
 def hash_map_get_in_list_fwd
-  (T : Type) (key : USize) (ls : list_t T) : Result T :=
+  (T : Type) (key : Usize) (ls : list_t T) : Result T :=
   hash_map_get_in_list_loop_fwd T key ls
 
 /- [hashmap::HashMap::{0}::get] -/
 def hash_map_get_fwd
-  (T : Type) (self : hash_map_t T) (key : USize) : Result T :=
+  (T : Type) (self : hash_map_t T) (key : Usize) : Result T :=
   do
     let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ← vec_index_fwd (list_t T) self.hash_map_slots hash_mod
     hash_map_get_in_list_fwd T key l
 
 /- [hashmap::HashMap::{0}::get_mut_in_list] -/
 def hash_map_get_mut_in_list_loop_fwd
-  (T : Type) (ls : list_t T) (key : USize) : (Result T) :=
+  (T : Type) (ls : list_t T) (key : Usize) : (Result T) :=
   match h: ls with
   | list_t.Cons ckey cvalue tl =>
     if h: ckey = key
@@ -317,12 +316,12 @@ decreasing_by hash_map_get_mut_in_list_loop_decreases ls key
 
 /- [hashmap::HashMap::{0}::get_mut_in_list] -/
 def hash_map_get_mut_in_list_fwd
-  (T : Type) (ls : list_t T) (key : USize) : Result T :=
+  (T : Type) (ls : list_t T) (key : Usize) : Result T :=
   hash_map_get_mut_in_list_loop_fwd T ls key
 
 /- [hashmap::HashMap::{0}::get_mut_in_list] -/
 def hash_map_get_mut_in_list_loop_back
-  (T : Type) (ls : list_t T) (key : USize) (ret0 : T) : (Result (list_t T)) :=
+  (T : Type) (ls : list_t T) (key : Usize) (ret0 : T) : (Result (list_t T)) :=
   match h: ls with
   | list_t.Cons ckey cvalue tl =>
     if h: ckey = key
@@ -338,28 +337,28 @@ decreasing_by hash_map_get_mut_in_list_loop_decreases ls key
 
 /- [hashmap::HashMap::{0}::get_mut_in_list] -/
 def hash_map_get_mut_in_list_back
-  (T : Type) (ls : list_t T) (key : USize) (ret0 : T) : Result (list_t T) :=
+  (T : Type) (ls : list_t T) (key : Usize) (ret0 : T) : Result (list_t T) :=
   hash_map_get_mut_in_list_loop_back T ls key ret0
 
 /- [hashmap::HashMap::{0}::get_mut] -/
 def hash_map_get_mut_fwd
-  (T : Type) (self : hash_map_t T) (key : USize) : Result T :=
+  (T : Type) (self : hash_map_t T) (key : Usize) : Result T :=
   do
     let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
     hash_map_get_mut_in_list_fwd T l key
 
 /- [hashmap::HashMap::{0}::get_mut] -/
 def hash_map_get_mut_back
-  (T : Type) (self : hash_map_t T) (key : USize) (ret0 : T) :
+  (T : Type) (self : hash_map_t T) (key : Usize) (ret0 : T) :
   Result (hash_map_t T)
   :=
   do
     let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
     let l0 ← hash_map_get_mut_in_list_back T l key ret0
     let v ← vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
@@ -367,7 +366,7 @@ def hash_map_get_mut_back
 
 /- [hashmap::HashMap::{0}::remove_from_list] -/
 def hash_map_remove_from_list_loop_fwd
-  (T : Type) (key : USize) (ls : list_t T) : (Result (Option T)) :=
+  (T : Type) (key : Usize) (ls : list_t T) : (Result (Option T)) :=
   match h: ls with
   | list_t.Cons ckey t tl =>
     if h: ckey = key
@@ -385,12 +384,12 @@ decreasing_by hash_map_remove_from_list_loop_decreases key ls
 
 /- [hashmap::HashMap::{0}::remove_from_list] -/
 def hash_map_remove_from_list_fwd
-  (T : Type) (key : USize) (ls : list_t T) : Result (Option T) :=
+  (T : Type) (key : Usize) (ls : list_t T) : Result (Option T) :=
   hash_map_remove_from_list_loop_fwd T key ls
 
 /- [hashmap::HashMap::{0}::remove_from_list] -/
 def hash_map_remove_from_list_loop_back
-  (T : Type) (key : USize) (ls : list_t T) : (Result (list_t T)) :=
+  (T : Type) (key : Usize) (ls : list_t T) : (Result (list_t T)) :=
   match h: ls with
   | list_t.Cons ckey t tl =>
     if h: ckey = key
@@ -411,33 +410,32 @@ decreasing_by hash_map_remove_from_list_loop_decreases key ls
 
 /- [hashmap::HashMap::{0}::remove_from_list] -/
 def hash_map_remove_from_list_back
-  (T : Type) (key : USize) (ls : list_t T) : Result (list_t T) :=
+  (T : Type) (key : Usize) (ls : list_t T) : Result (list_t T) :=
   hash_map_remove_from_list_loop_back T key ls
 
 /- [hashmap::HashMap::{0}::remove] -/
 def hash_map_remove_fwd
-  (T : Type) (self : hash_map_t T) (key : USize) : Result (Option T) :=
+  (T : Type) (self : hash_map_t T) (key : Usize) : Result (Option T) :=
   do
     let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
     let x ← hash_map_remove_from_list_fwd T key l
     match h: x with
     | Option.none => Result.ret Option.none
     | Option.some x0 =>
       do
-        let _ ← USize.checked_sub self.hash_map_num_entries
-          (USize.ofNatCore 1 (by intlit))
+        let _ ← self.hash_map_num_entries - (Usize.ofInt 1 (by intlit))
         Result.ret (Option.some x0)
 
 /- [hashmap::HashMap::{0}::remove] -/
 def hash_map_remove_back
-  (T : Type) (self : hash_map_t T) (key : USize) : Result (hash_map_t T) :=
+  (T : Type) (self : hash_map_t T) (key : Usize) : Result (hash_map_t T) :=
   do
     let hash ← hash_key_fwd key
     let i := vec_len (list_t T) self.hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ← vec_index_mut_fwd (list_t T) self.hash_map_slots hash_mod
     let x ← hash_map_remove_from_list_fwd T key l
     match h: x with
@@ -448,8 +446,7 @@ def hash_map_remove_back
         Result.ret { self with hash_map_slots := v }
     | Option.some x0 =>
       do
-        let i0 ← USize.checked_sub self.hash_map_num_entries
-          (USize.ofNatCore 1 (by intlit))
+        let i0 ← self.hash_map_num_entries - (Usize.ofInt 1 (by intlit))
         let l0 ← hash_map_remove_from_list_back T key l
         let v ← vec_index_mut_back (list_t T) self.hash_map_slots hash_mod l0
         Result.ret
@@ -458,65 +455,59 @@ def hash_map_remove_back
 /- [hashmap::test1] -/
 def test1_fwd : Result Unit :=
   do
-    let hm ← hash_map_new_fwd UInt64
+    let hm ← hash_map_new_fwd U64
     let hm0 ←
-      hash_map_insert_fwd_back UInt64 hm (USize.ofNatCore 0 (by intlit))
-        (UInt64.ofNatCore 42 (by intlit))
+      hash_map_insert_fwd_back U64 hm (Usize.ofInt 0 (by intlit))
+        (U64.ofInt 42 (by intlit))
     let hm1 ←
-      hash_map_insert_fwd_back UInt64 hm0 (USize.ofNatCore 128 (by intlit))
-        (UInt64.ofNatCore 18 (by intlit))
+      hash_map_insert_fwd_back U64 hm0 (Usize.ofInt 128 (by intlit))
+        (U64.ofInt 18 (by intlit))
     let hm2 ←
-      hash_map_insert_fwd_back UInt64 hm1 (USize.ofNatCore 1024 (by intlit))
-        (UInt64.ofNatCore 138 (by intlit))
+      hash_map_insert_fwd_back U64 hm1 (Usize.ofInt 1024 (by intlit))
+        (U64.ofInt 138 (by intlit))
     let hm3 ←
-      hash_map_insert_fwd_back UInt64 hm2 (USize.ofNatCore 1056 (by intlit))
-        (UInt64.ofNatCore 256 (by intlit))
-    let i ← hash_map_get_fwd UInt64 hm3 (USize.ofNatCore 128 (by intlit))
-    if h: not (i = (UInt64.ofNatCore 18 (by intlit)))
+      hash_map_insert_fwd_back U64 hm2 (Usize.ofInt 1056 (by intlit))
+        (U64.ofInt 256 (by intlit))
+    let i ← hash_map_get_fwd U64 hm3 (Usize.ofInt 128 (by intlit))
+    if h: not (i = (U64.ofInt 18 (by intlit)))
     then Result.fail Error.panic
     else
       do
         let hm4 ←
-          hash_map_get_mut_back UInt64 hm3 (USize.ofNatCore 1024 (by intlit))
-            (UInt64.ofNatCore 56 (by intlit))
-        let i0 ←
-          hash_map_get_fwd UInt64 hm4 (USize.ofNatCore 1024 (by intlit))
-        if h: not (i0 = (UInt64.ofNatCore 56 (by intlit)))
+          hash_map_get_mut_back U64 hm3 (Usize.ofInt 1024 (by intlit))
+            (U64.ofInt 56 (by intlit))
+        let i0 ← hash_map_get_fwd U64 hm4 (Usize.ofInt 1024 (by intlit))
+        if h: not (i0 = (U64.ofInt 56 (by intlit)))
         then Result.fail Error.panic
         else
           do
             let x ←
-              hash_map_remove_fwd UInt64 hm4 (USize.ofNatCore 1024 (by intlit))
+              hash_map_remove_fwd U64 hm4 (Usize.ofInt 1024 (by intlit))
             match h: x with
             | Option.none => Result.fail Error.panic
             | Option.some x0 =>
-              if h: not (x0 = (UInt64.ofNatCore 56 (by intlit)))
+              if h: not (x0 = (U64.ofInt 56 (by intlit)))
               then Result.fail Error.panic
               else
                 do
                   let hm5 ←
-                    hash_map_remove_back UInt64 hm4
-                      (USize.ofNatCore 1024 (by intlit))
+                    hash_map_remove_back U64 hm4 (Usize.ofInt 1024 (by intlit))
                   let i1 ←
-                    hash_map_get_fwd UInt64 hm5 (USize.ofNatCore 0 (by intlit))
-                  if h: not (i1 = (UInt64.ofNatCore 42 (by intlit)))
+                    hash_map_get_fwd U64 hm5 (Usize.ofInt 0 (by intlit))
+                  if h: not (i1 = (U64.ofInt 42 (by intlit)))
                   then Result.fail Error.panic
                   else
                     do
                       let i2 ←
-                        hash_map_get_fwd UInt64 hm5
-                          (USize.ofNatCore 128 (by intlit))
-                      if h: not (i2 = (UInt64.ofNatCore 18 (by intlit)))
+                        hash_map_get_fwd U64 hm5 (Usize.ofInt 128 (by intlit))
+                      if h: not (i2 = (U64.ofInt 18 (by intlit)))
                       then Result.fail Error.panic
                       else
                         do
                           let i3 ←
-                            hash_map_get_fwd UInt64 hm5
-                              (USize.ofNatCore 1056 (by intlit))
-                          if h: not (i3 = (UInt64.ofNatCore 256 (by intlit)))
+                            hash_map_get_fwd U64 hm5
+                              (Usize.ofInt 1056 (by intlit))
+                          if h: not (i3 = (U64.ofInt 256 (by intlit)))
                           then Result.fail Error.panic
                           else Result.ret ()
 
-/- Unit test for [hashmap::test1] -/
-#assert (test1_fwd == .ret ())
-
diff --git a/tests/lean/hashmap/Hashmap/Types.lean b/tests/lean/hashmap/Hashmap/Types.lean
index 9e9e5c03..6eabf7da 100644
--- a/tests/lean/hashmap/Hashmap/Types.lean
+++ b/tests/lean/hashmap/Hashmap/Types.lean
@@ -4,13 +4,13 @@ import Base.Primitives
 
 /- [hashmap::List] -/
 inductive list_t (T : Type) :=
-| Cons : USize -> T -> list_t T -> list_t T
+| Cons : Usize -> T -> list_t T -> list_t T
 | Nil : list_t T
 
 /- [hashmap::HashMap] -/
 structure hash_map_t (T : Type) where
-  hash_map_num_entries : USize
-  hash_map_max_load_factor : (USize × USize)
-  hash_map_max_load : USize
+  hash_map_num_entries : Usize
+  hash_map_max_load_factor : (Usize × Usize)
+  hash_map_max_load : Usize
   hash_map_slots : Vec (list_t T)
 
diff --git a/tests/lean/hashmap_on_disk/Base/Primitives.lean b/tests/lean/hashmap_on_disk/Base/Primitives.lean
index 5b64e908..034f41b2 100644
--- a/tests/lean/hashmap_on_disk/Base/Primitives.lean
+++ b/tests/lean/hashmap_on_disk/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/hashmap_on_disk/HashmapMain/Clauses/Clauses.lean b/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Clauses.lean
index 1b69eb2f..a4dc996a 100644
--- a/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Clauses.lean
+++ b/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Clauses.lean
@@ -1,39 +1,37 @@
--- [hashmap_main]: the decreases clauses
 import Base.Primitives
 import HashmapMain.Types
 
 /- [hashmap_main::hashmap::HashMap::{0}::allocate_slots]: termination measure -/
 @[simp]
 def hashmap_hash_map_allocate_slots_loop_terminates (T : Type)
-  (slots : Vec (hashmap_list_t T)) (n : USize) :=
+  (slots : Vec (hashmap_list_t T)) (n : Usize) :=
   (slots, n)
 
+/- [hashmap_main::hashmap::HashMap::{0}::allocate_slots]: decreases_by tactic -/
 syntax "hashmap_hash_map_allocate_slots_loop_decreases" term+ : tactic
-
 macro_rules
 | `(tactic| hashmap_hash_map_allocate_slots_loop_decreases $slots $n) =>
   `(tactic| sorry)
 
-/- [hashmap_main::hashmap::HashMap::{0}::clear_slots]: termination measure -/
+/- [hashmap_main::hashmap::HashMap::{0}::clear]: termination measure -/
 @[simp]
-def hashmap_hash_map_clear_slots_loop_terminates (T : Type)
-  (slots : Vec (hashmap_list_t T)) (i : USize) :=
+def hashmap_hash_map_clear_loop_terminates (T : Type)
+  (slots : Vec (hashmap_list_t T)) (i : Usize) :=
   (slots, i)
 
-syntax "hashmap_hash_map_clear_slots_loop_decreases" term+ : tactic
-
+/- [hashmap_main::hashmap::HashMap::{0}::clear]: decreases_by tactic -/
+syntax "hashmap_hash_map_clear_loop_decreases" term+ : tactic
 macro_rules
-| `(tactic| hashmap_hash_map_clear_slots_loop_decreases $slots $i) =>
-  `(tactic| sorry)
+| `(tactic| hashmap_hash_map_clear_loop_decreases $slots $i) =>`(tactic| sorry)
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert_in_list]: termination measure -/
 @[simp]
-def hashmap_hash_map_insert_in_list_loop_terminates (T : Type) (key : USize)
+def hashmap_hash_map_insert_in_list_loop_terminates (T : Type) (key : Usize)
   (value : T) (ls : hashmap_list_t T) :=
   (key, value, ls)
 
+/- [hashmap_main::hashmap::HashMap::{0}::insert_in_list]: decreases_by tactic -/
 syntax "hashmap_hash_map_insert_in_list_loop_decreases" term+ : tactic
-
 macro_rules
 | `(tactic| hashmap_hash_map_insert_in_list_loop_decreases $key $value $ls) =>
   `(tactic| sorry)
@@ -44,8 +42,8 @@ def hashmap_hash_map_move_elements_from_list_loop_terminates (T : Type)
   (ntable : hashmap_hash_map_t T) (ls : hashmap_list_t T) :=
   (ntable, ls)
 
+/- [hashmap_main::hashmap::HashMap::{0}::move_elements_from_list]: decreases_by tactic -/
 syntax "hashmap_hash_map_move_elements_from_list_loop_decreases" term+ : tactic
-
 macro_rules
 | `(tactic| hashmap_hash_map_move_elements_from_list_loop_decreases $ntable
 $ls) =>`(tactic| sorry)
@@ -53,12 +51,12 @@ $ls) =>`(tactic| sorry)
 /- [hashmap_main::hashmap::HashMap::{0}::move_elements]: termination measure -/
 @[simp]
 def hashmap_hash_map_move_elements_loop_terminates (T : Type)
-  (ntable : hashmap_hash_map_t T) (slots : Vec (hashmap_list_t T)) (i : USize)
+  (ntable : hashmap_hash_map_t T) (slots : Vec (hashmap_list_t T)) (i : Usize)
   :=
   (ntable, slots, i)
 
+/- [hashmap_main::hashmap::HashMap::{0}::move_elements]: decreases_by tactic -/
 syntax "hashmap_hash_map_move_elements_loop_decreases" term+ : tactic
-
 macro_rules
 | `(tactic| hashmap_hash_map_move_elements_loop_decreases $ntable $slots $i) =>
   `(tactic| sorry)
@@ -66,46 +64,46 @@ macro_rules
 /- [hashmap_main::hashmap::HashMap::{0}::contains_key_in_list]: termination measure -/
 @[simp]
 def hashmap_hash_map_contains_key_in_list_loop_terminates (T : Type)
-  (key : USize) (ls : hashmap_list_t T) :=
+  (key : Usize) (ls : hashmap_list_t T) :=
   (key, ls)
 
+/- [hashmap_main::hashmap::HashMap::{0}::contains_key_in_list]: decreases_by tactic -/
 syntax "hashmap_hash_map_contains_key_in_list_loop_decreases" term+ : tactic
-
 macro_rules
 | `(tactic| hashmap_hash_map_contains_key_in_list_loop_decreases $key $ls) =>
   `(tactic| sorry)
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_in_list]: termination measure -/
 @[simp]
-def hashmap_hash_map_get_in_list_loop_terminates (T : Type) (key : USize)
+def hashmap_hash_map_get_in_list_loop_terminates (T : Type) (key : Usize)
   (ls : hashmap_list_t T) :=
   (key, ls)
 
+/- [hashmap_main::hashmap::HashMap::{0}::get_in_list]: decreases_by tactic -/
 syntax "hashmap_hash_map_get_in_list_loop_decreases" term+ : tactic
-
 macro_rules
 | `(tactic| hashmap_hash_map_get_in_list_loop_decreases $key $ls) =>`(tactic| sorry)
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut_in_list]: termination measure -/
 @[simp]
 def hashmap_hash_map_get_mut_in_list_loop_terminates (T : Type)
-  (ls : hashmap_list_t T) (key : USize) :=
+  (ls : hashmap_list_t T) (key : Usize) :=
   (ls, key)
 
+/- [hashmap_main::hashmap::HashMap::{0}::get_mut_in_list]: decreases_by tactic -/
 syntax "hashmap_hash_map_get_mut_in_list_loop_decreases" term+ : tactic
-
 macro_rules
 | `(tactic| hashmap_hash_map_get_mut_in_list_loop_decreases $ls $key) =>
   `(tactic| sorry)
 
 /- [hashmap_main::hashmap::HashMap::{0}::remove_from_list]: termination measure -/
 @[simp]
-def hashmap_hash_map_remove_from_list_loop_terminates (T : Type) (key : USize)
+def hashmap_hash_map_remove_from_list_loop_terminates (T : Type) (key : Usize)
   (ls : hashmap_list_t T) :=
   (key, ls)
 
+/- [hashmap_main::hashmap::HashMap::{0}::remove_from_list]: decreases_by tactic -/
 syntax "hashmap_hash_map_remove_from_list_loop_decreases" term+ : tactic
-
 macro_rules
 | `(tactic| hashmap_hash_map_remove_from_list_loop_decreases $key $ls) =>
   `(tactic| sorry)
diff --git a/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Template.lean b/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Template.lean
index 753c92ac..33802597 100644
--- a/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Template.lean
+++ b/tests/lean/hashmap_on_disk/HashmapMain/Clauses/Template.lean
@@ -6,7 +6,7 @@ import HashmapMain.Types
 /- [hashmap_main::hashmap::HashMap::{0}::allocate_slots]: termination measure -/
 @[simp]
 def hashmap_hash_map_allocate_slots_loop_terminates (T : Type)
-  (slots : Vec (hashmap_list_t T)) (n : USize) :=
+  (slots : Vec (hashmap_list_t T)) (n : Usize) :=
   (slots, n)
 
 /- [hashmap_main::hashmap::HashMap::{0}::allocate_slots]: decreases_by tactic -/
@@ -18,7 +18,7 @@ macro_rules
 /- [hashmap_main::hashmap::HashMap::{0}::clear]: termination measure -/
 @[simp]
 def hashmap_hash_map_clear_loop_terminates (T : Type)
-  (slots : Vec (hashmap_list_t T)) (i : USize) :=
+  (slots : Vec (hashmap_list_t T)) (i : Usize) :=
   (slots, i)
 
 /- [hashmap_main::hashmap::HashMap::{0}::clear]: decreases_by tactic -/
@@ -28,7 +28,7 @@ macro_rules
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert_in_list]: termination measure -/
 @[simp]
-def hashmap_hash_map_insert_in_list_loop_terminates (T : Type) (key : USize)
+def hashmap_hash_map_insert_in_list_loop_terminates (T : Type) (key : Usize)
   (value : T) (ls : hashmap_list_t T) :=
   (key, value, ls)
 
@@ -53,7 +53,7 @@ $ls) =>`(tactic| sorry)
 /- [hashmap_main::hashmap::HashMap::{0}::move_elements]: termination measure -/
 @[simp]
 def hashmap_hash_map_move_elements_loop_terminates (T : Type)
-  (ntable : hashmap_hash_map_t T) (slots : Vec (hashmap_list_t T)) (i : USize)
+  (ntable : hashmap_hash_map_t T) (slots : Vec (hashmap_list_t T)) (i : Usize)
   :=
   (ntable, slots, i)
 
@@ -66,7 +66,7 @@ macro_rules
 /- [hashmap_main::hashmap::HashMap::{0}::contains_key_in_list]: termination measure -/
 @[simp]
 def hashmap_hash_map_contains_key_in_list_loop_terminates (T : Type)
-  (key : USize) (ls : hashmap_list_t T) :=
+  (key : Usize) (ls : hashmap_list_t T) :=
   (key, ls)
 
 /- [hashmap_main::hashmap::HashMap::{0}::contains_key_in_list]: decreases_by tactic -/
@@ -77,7 +77,7 @@ macro_rules
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_in_list]: termination measure -/
 @[simp]
-def hashmap_hash_map_get_in_list_loop_terminates (T : Type) (key : USize)
+def hashmap_hash_map_get_in_list_loop_terminates (T : Type) (key : Usize)
   (ls : hashmap_list_t T) :=
   (key, ls)
 
@@ -89,7 +89,7 @@ macro_rules
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut_in_list]: termination measure -/
 @[simp]
 def hashmap_hash_map_get_mut_in_list_loop_terminates (T : Type)
-  (ls : hashmap_list_t T) (key : USize) :=
+  (ls : hashmap_list_t T) (key : Usize) :=
   (ls, key)
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut_in_list]: decreases_by tactic -/
@@ -100,7 +100,7 @@ macro_rules
 
 /- [hashmap_main::hashmap::HashMap::{0}::remove_from_list]: termination measure -/
 @[simp]
-def hashmap_hash_map_remove_from_list_loop_terminates (T : Type) (key : USize)
+def hashmap_hash_map_remove_from_list_loop_terminates (T : Type) (key : Usize)
   (ls : hashmap_list_t T) :=
   (key, ls)
 
diff --git a/tests/lean/hashmap_on_disk/HashmapMain/ExternalFuns.lean b/tests/lean/hashmap_on_disk/HashmapMain/ExternalFuns.lean
new file mode 100644
index 00000000..a5103acc
--- /dev/null
+++ b/tests/lean/hashmap_on_disk/HashmapMain/ExternalFuns.lean
@@ -0,0 +1,5 @@
+import Base.Primitives
+import HashmapMain.Types
+import HashmapMain.Opaque
+
+def opaque_defs : OpaqueDefs := by sorry
diff --git a/tests/lean/hashmap_on_disk/HashmapMain/Funs.lean b/tests/lean/hashmap_on_disk/HashmapMain/Funs.lean
index 2be03d98..342c3833 100644
--- a/tests/lean/hashmap_on_disk/HashmapMain/Funs.lean
+++ b/tests/lean/hashmap_on_disk/HashmapMain/Funs.lean
@@ -2,25 +2,23 @@
 -- [hashmap_main]: function definitions
 import Base.Primitives
 import HashmapMain.Types
-import HashmapMain.Opaque
+import HashmapMain.ExternalFuns
 import HashmapMain.Clauses.Clauses
 
-section variable (opaque_defs: OpaqueDefs)
-
 /- [hashmap_main::hashmap::hash_key] -/
-def hashmap_hash_key_fwd (k : USize) : Result USize :=
+def hashmap_hash_key_fwd (k : Usize) : Result Usize :=
   Result.ret k
 
 /- [hashmap_main::hashmap::HashMap::{0}::allocate_slots] -/
 def hashmap_hash_map_allocate_slots_loop_fwd
-  (T : Type) (slots : Vec (hashmap_list_t T)) (n : USize) :
+  (T : Type) (slots : Vec (hashmap_list_t T)) (n : Usize) :
   (Result (Vec (hashmap_list_t T)))
   :=
-  if h: n > (USize.ofNatCore 0 (by intlit))
+  if h: n > (Usize.ofInt 0 (by intlit))
   then
     do
       let slots0 ← vec_push_back (hashmap_list_t T) slots hashmap_list_t.Nil
-      let n0 ← USize.checked_sub n (USize.ofNatCore 1 (by intlit))
+      let n0 ← n - (Usize.ofInt 1 (by intlit))
       hashmap_hash_map_allocate_slots_loop_fwd T slots0 n0
   else Result.ret slots
 termination_by hashmap_hash_map_allocate_slots_loop_fwd slots n =>
@@ -29,25 +27,25 @@ decreasing_by hashmap_hash_map_allocate_slots_loop_decreases slots n
 
 /- [hashmap_main::hashmap::HashMap::{0}::allocate_slots] -/
 def hashmap_hash_map_allocate_slots_fwd
-  (T : Type) (slots : Vec (hashmap_list_t T)) (n : USize) :
+  (T : Type) (slots : Vec (hashmap_list_t T)) (n : Usize) :
   Result (Vec (hashmap_list_t T))
   :=
   hashmap_hash_map_allocate_slots_loop_fwd T slots n
 
 /- [hashmap_main::hashmap::HashMap::{0}::new_with_capacity] -/
 def hashmap_hash_map_new_with_capacity_fwd
-  (T : Type) (capacity : USize) (max_load_dividend : USize)
-  (max_load_divisor : USize) :
+  (T : Type) (capacity : Usize) (max_load_dividend : Usize)
+  (max_load_divisor : Usize) :
   Result (hashmap_hash_map_t T)
   :=
   do
     let v := vec_new (hashmap_list_t T)
     let slots ← hashmap_hash_map_allocate_slots_fwd T v capacity
-    let i ← USize.checked_mul capacity max_load_dividend
-    let i0 ← USize.checked_div i max_load_divisor
+    let i ← capacity * max_load_dividend
+    let i0 ← i / max_load_divisor
     Result.ret
       {
-        hashmap_hash_map_num_entries := (USize.ofNatCore 0 (by intlit)),
+        hashmap_hash_map_num_entries := (Usize.ofInt 0 (by intlit)),
         hashmap_hash_map_max_load_factor :=
           (max_load_dividend, max_load_divisor),
         hashmap_hash_map_max_load := i0,
@@ -56,19 +54,19 @@ def hashmap_hash_map_new_with_capacity_fwd
 
 /- [hashmap_main::hashmap::HashMap::{0}::new] -/
 def hashmap_hash_map_new_fwd (T : Type) : Result (hashmap_hash_map_t T) :=
-  hashmap_hash_map_new_with_capacity_fwd T (USize.ofNatCore 32 (by intlit))
-    (USize.ofNatCore 4 (by intlit)) (USize.ofNatCore 5 (by intlit))
+  hashmap_hash_map_new_with_capacity_fwd T (Usize.ofInt 32 (by intlit))
+    (Usize.ofInt 4 (by intlit)) (Usize.ofInt 5 (by intlit))
 
 /- [hashmap_main::hashmap::HashMap::{0}::clear] -/
 def hashmap_hash_map_clear_loop_fwd_back
-  (T : Type) (slots : Vec (hashmap_list_t T)) (i : USize) :
+  (T : Type) (slots : Vec (hashmap_list_t T)) (i : Usize) :
   (Result (Vec (hashmap_list_t T)))
   :=
   let i0 := vec_len (hashmap_list_t T) slots
   if h: i < i0
   then
     do
-      let i1 ← USize.checked_add i (USize.ofNatCore 1 (by intlit))
+      let i1 ← i + (Usize.ofInt 1 (by intlit))
       let slots0 ←
         vec_index_mut_back (hashmap_list_t T) slots i hashmap_list_t.Nil
       hashmap_hash_map_clear_loop_fwd_back T slots0 i1
@@ -83,23 +81,23 @@ def hashmap_hash_map_clear_fwd_back
   do
     let v ←
       hashmap_hash_map_clear_loop_fwd_back T self.hashmap_hash_map_slots
-        (USize.ofNatCore 0 (by intlit))
+        (Usize.ofInt 0 (by intlit))
     Result.ret
       {
         self
           with
-          hashmap_hash_map_num_entries := (USize.ofNatCore 0 (by intlit)),
+          hashmap_hash_map_num_entries := (Usize.ofInt 0 (by intlit)),
           hashmap_hash_map_slots := v
       }
 
 /- [hashmap_main::hashmap::HashMap::{0}::len] -/
 def hashmap_hash_map_len_fwd
-  (T : Type) (self : hashmap_hash_map_t T) : Result USize :=
+  (T : Type) (self : hashmap_hash_map_t T) : Result Usize :=
   Result.ret self.hashmap_hash_map_num_entries
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert_in_list] -/
 def hashmap_hash_map_insert_in_list_loop_fwd
-  (T : Type) (key : USize) (value : T) (ls : hashmap_list_t T) :
+  (T : Type) (key : Usize) (value : T) (ls : hashmap_list_t T) :
   (Result Bool)
   :=
   match h: ls with
@@ -114,12 +112,12 @@ decreasing_by hashmap_hash_map_insert_in_list_loop_decreases key value ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert_in_list] -/
 def hashmap_hash_map_insert_in_list_fwd
-  (T : Type) (key : USize) (value : T) (ls : hashmap_list_t T) : Result Bool :=
+  (T : Type) (key : Usize) (value : T) (ls : hashmap_list_t T) : Result Bool :=
   hashmap_hash_map_insert_in_list_loop_fwd T key value ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert_in_list] -/
 def hashmap_hash_map_insert_in_list_loop_back
-  (T : Type) (key : USize) (value : T) (ls : hashmap_list_t T) :
+  (T : Type) (key : Usize) (value : T) (ls : hashmap_list_t T) :
   (Result (hashmap_list_t T))
   :=
   match h: ls with
@@ -139,28 +137,28 @@ decreasing_by hashmap_hash_map_insert_in_list_loop_decreases key value ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert_in_list] -/
 def hashmap_hash_map_insert_in_list_back
-  (T : Type) (key : USize) (value : T) (ls : hashmap_list_t T) :
+  (T : Type) (key : Usize) (value : T) (ls : hashmap_list_t T) :
   Result (hashmap_list_t T)
   :=
   hashmap_hash_map_insert_in_list_loop_back T key value ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert_no_resize] -/
 def hashmap_hash_map_insert_no_resize_fwd_back
-  (T : Type) (self : hashmap_hash_map_t T) (key : USize) (value : T) :
+  (T : Type) (self : hashmap_hash_map_t T) (key : Usize) (value : T) :
   Result (hashmap_hash_map_t T)
   :=
   do
     let hash ← hashmap_hash_key_fwd key
     let i := vec_len (hashmap_list_t T) self.hashmap_hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ←
       vec_index_mut_fwd (hashmap_list_t T) self.hashmap_hash_map_slots hash_mod
     let inserted ← hashmap_hash_map_insert_in_list_fwd T key value l
     if h: inserted
     then
       do
-        let i0 ← USize.checked_add self.hashmap_hash_map_num_entries
-          (USize.ofNatCore 1 (by intlit))
+        let i0 ← self.hashmap_hash_map_num_entries +
+          (Usize.ofInt 1 (by intlit))
         let l0 ← hashmap_hash_map_insert_in_list_back T key value l
         let v ←
           vec_index_mut_back (hashmap_list_t T) self.hashmap_hash_map_slots
@@ -180,9 +178,9 @@ def hashmap_hash_map_insert_no_resize_fwd_back
         Result.ret { self with hashmap_hash_map_slots := v }
 
 /- [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)
+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)
 
 /- [hashmap_main::hashmap::HashMap::{0}::move_elements_from_list] -/
 def hashmap_hash_map_move_elements_from_list_loop_fwd_back
@@ -210,7 +208,7 @@ def hashmap_hash_map_move_elements_from_list_fwd_back
 /- [hashmap_main::hashmap::HashMap::{0}::move_elements] -/
 def hashmap_hash_map_move_elements_loop_fwd_back
   (T : Type) (ntable : hashmap_hash_map_t T) (slots : Vec (hashmap_list_t T))
-  (i : USize) :
+  (i : Usize) :
   (Result ((hashmap_hash_map_t T) × (Vec (hashmap_list_t T))))
   :=
   let i0 := vec_len (hashmap_list_t T) slots
@@ -221,7 +219,7 @@ def hashmap_hash_map_move_elements_loop_fwd_back
       let ls := mem_replace_fwd (hashmap_list_t T) l hashmap_list_t.Nil
       let ntable0 ←
         hashmap_hash_map_move_elements_from_list_fwd_back T ntable ls
-      let i1 ← USize.checked_add i (USize.ofNatCore 1 (by intlit))
+      let i1 ← i + (Usize.ofInt 1 (by intlit))
       let l0 := mem_replace_back (hashmap_list_t T) l hashmap_list_t.Nil
       let slots0 ← vec_index_mut_back (hashmap_list_t T) slots i l0
       hashmap_hash_map_move_elements_loop_fwd_back T ntable0 slots0 i1
@@ -233,7 +231,7 @@ decreasing_by hashmap_hash_map_move_elements_loop_decreases ntable slots i
 /- [hashmap_main::hashmap::HashMap::{0}::move_elements] -/
 def hashmap_hash_map_move_elements_fwd_back
   (T : Type) (ntable : hashmap_hash_map_t T) (slots : Vec (hashmap_list_t T))
-  (i : USize) :
+  (i : Usize) :
   Result ((hashmap_hash_map_t T) × (Vec (hashmap_list_t T)))
   :=
   hashmap_hash_map_move_elements_loop_fwd_back T ntable slots i
@@ -242,19 +240,19 @@ def hashmap_hash_map_move_elements_fwd_back
 def hashmap_hash_map_try_resize_fwd_back
   (T : Type) (self : hashmap_hash_map_t T) : Result (hashmap_hash_map_t T) :=
   do
-    let max_usize ← scalar_cast USize core_num_u32_max_c
+    let max_usize ← Scalar.cast .Usize core_num_u32_max_c
     let capacity := vec_len (hashmap_list_t T) self.hashmap_hash_map_slots
-    let n1 ← USize.checked_div max_usize (USize.ofNatCore 2 (by intlit))
+    let n1 ← max_usize / (Usize.ofInt 2 (by intlit))
     let (i, i0) := self.hashmap_hash_map_max_load_factor
-    let i1 ← USize.checked_div n1 i
+    let i1 ← n1 / i
     if h: capacity <= i1
     then
       do
-        let i2 ← USize.checked_mul capacity (USize.ofNatCore 2 (by intlit))
+        let i2 ← capacity * (Usize.ofInt 2 (by intlit))
         let ntable ← hashmap_hash_map_new_with_capacity_fwd T i2 i i0
         let (ntable0, _) ←
           hashmap_hash_map_move_elements_fwd_back T ntable
-            self.hashmap_hash_map_slots (USize.ofNatCore 0 (by intlit))
+            self.hashmap_hash_map_slots (Usize.ofInt 0 (by intlit))
         Result.ret
           {
             ntable0
@@ -266,7 +264,7 @@ def hashmap_hash_map_try_resize_fwd_back
 
 /- [hashmap_main::hashmap::HashMap::{0}::insert] -/
 def hashmap_hash_map_insert_fwd_back
-  (T : Type) (self : hashmap_hash_map_t T) (key : USize) (value : T) :
+  (T : Type) (self : hashmap_hash_map_t T) (key : Usize) (value : T) :
   Result (hashmap_hash_map_t T)
   :=
   do
@@ -278,7 +276,7 @@ def hashmap_hash_map_insert_fwd_back
 
 /- [hashmap_main::hashmap::HashMap::{0}::contains_key_in_list] -/
 def hashmap_hash_map_contains_key_in_list_loop_fwd
-  (T : Type) (key : USize) (ls : hashmap_list_t T) : (Result Bool) :=
+  (T : Type) (key : Usize) (ls : hashmap_list_t T) : (Result Bool) :=
   match h: ls with
   | hashmap_list_t.Cons ckey t tl =>
     if h: ckey = key
@@ -291,23 +289,23 @@ decreasing_by hashmap_hash_map_contains_key_in_list_loop_decreases key ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::contains_key_in_list] -/
 def hashmap_hash_map_contains_key_in_list_fwd
-  (T : Type) (key : USize) (ls : hashmap_list_t T) : Result Bool :=
+  (T : Type) (key : Usize) (ls : hashmap_list_t T) : Result Bool :=
   hashmap_hash_map_contains_key_in_list_loop_fwd T key ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::contains_key] -/
 def hashmap_hash_map_contains_key_fwd
-  (T : Type) (self : hashmap_hash_map_t T) (key : USize) : Result Bool :=
+  (T : Type) (self : hashmap_hash_map_t T) (key : Usize) : Result Bool :=
   do
     let hash ← hashmap_hash_key_fwd key
     let i := vec_len (hashmap_list_t T) self.hashmap_hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ←
       vec_index_fwd (hashmap_list_t T) self.hashmap_hash_map_slots hash_mod
     hashmap_hash_map_contains_key_in_list_fwd T key l
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_in_list] -/
 def hashmap_hash_map_get_in_list_loop_fwd
-  (T : Type) (key : USize) (ls : hashmap_list_t T) : (Result T) :=
+  (T : Type) (key : Usize) (ls : hashmap_list_t T) : (Result T) :=
   match h: ls with
   | hashmap_list_t.Cons ckey cvalue tl =>
     if h: ckey = key
@@ -320,23 +318,23 @@ decreasing_by hashmap_hash_map_get_in_list_loop_decreases key ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_in_list] -/
 def hashmap_hash_map_get_in_list_fwd
-  (T : Type) (key : USize) (ls : hashmap_list_t T) : Result T :=
+  (T : Type) (key : Usize) (ls : hashmap_list_t T) : Result T :=
   hashmap_hash_map_get_in_list_loop_fwd T key ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::get] -/
 def hashmap_hash_map_get_fwd
-  (T : Type) (self : hashmap_hash_map_t T) (key : USize) : Result T :=
+  (T : Type) (self : hashmap_hash_map_t T) (key : Usize) : Result T :=
   do
     let hash ← hashmap_hash_key_fwd key
     let i := vec_len (hashmap_list_t T) self.hashmap_hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ←
       vec_index_fwd (hashmap_list_t T) self.hashmap_hash_map_slots hash_mod
     hashmap_hash_map_get_in_list_fwd T key l
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut_in_list] -/
 def hashmap_hash_map_get_mut_in_list_loop_fwd
-  (T : Type) (ls : hashmap_list_t T) (key : USize) : (Result T) :=
+  (T : Type) (ls : hashmap_list_t T) (key : Usize) : (Result T) :=
   match h: ls with
   | hashmap_list_t.Cons ckey cvalue tl =>
     if h: ckey = key
@@ -349,12 +347,12 @@ decreasing_by hashmap_hash_map_get_mut_in_list_loop_decreases ls key
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut_in_list] -/
 def hashmap_hash_map_get_mut_in_list_fwd
-  (T : Type) (ls : hashmap_list_t T) (key : USize) : Result T :=
+  (T : Type) (ls : hashmap_list_t T) (key : Usize) : Result T :=
   hashmap_hash_map_get_mut_in_list_loop_fwd T ls key
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut_in_list] -/
 def hashmap_hash_map_get_mut_in_list_loop_back
-  (T : Type) (ls : hashmap_list_t T) (key : USize) (ret0 : T) :
+  (T : Type) (ls : hashmap_list_t T) (key : Usize) (ret0 : T) :
   (Result (hashmap_list_t T))
   :=
   match h: ls with
@@ -372,31 +370,31 @@ decreasing_by hashmap_hash_map_get_mut_in_list_loop_decreases ls key
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut_in_list] -/
 def hashmap_hash_map_get_mut_in_list_back
-  (T : Type) (ls : hashmap_list_t T) (key : USize) (ret0 : T) :
+  (T : Type) (ls : hashmap_list_t T) (key : Usize) (ret0 : T) :
   Result (hashmap_list_t T)
   :=
   hashmap_hash_map_get_mut_in_list_loop_back T ls key ret0
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut] -/
 def hashmap_hash_map_get_mut_fwd
-  (T : Type) (self : hashmap_hash_map_t T) (key : USize) : Result T :=
+  (T : Type) (self : hashmap_hash_map_t T) (key : Usize) : Result T :=
   do
     let hash ← hashmap_hash_key_fwd key
     let i := vec_len (hashmap_list_t T) self.hashmap_hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ←
       vec_index_mut_fwd (hashmap_list_t T) self.hashmap_hash_map_slots hash_mod
     hashmap_hash_map_get_mut_in_list_fwd T l key
 
 /- [hashmap_main::hashmap::HashMap::{0}::get_mut] -/
 def hashmap_hash_map_get_mut_back
-  (T : Type) (self : hashmap_hash_map_t T) (key : USize) (ret0 : T) :
+  (T : Type) (self : hashmap_hash_map_t T) (key : Usize) (ret0 : T) :
   Result (hashmap_hash_map_t T)
   :=
   do
     let hash ← hashmap_hash_key_fwd key
     let i := vec_len (hashmap_list_t T) self.hashmap_hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ←
       vec_index_mut_fwd (hashmap_list_t T) self.hashmap_hash_map_slots hash_mod
     let l0 ← hashmap_hash_map_get_mut_in_list_back T l key ret0
@@ -407,7 +405,7 @@ def hashmap_hash_map_get_mut_back
 
 /- [hashmap_main::hashmap::HashMap::{0}::remove_from_list] -/
 def hashmap_hash_map_remove_from_list_loop_fwd
-  (T : Type) (key : USize) (ls : hashmap_list_t T) : (Result (Option T)) :=
+  (T : Type) (key : Usize) (ls : hashmap_list_t T) : (Result (Option T)) :=
   match h: ls with
   | hashmap_list_t.Cons ckey t tl =>
     if h: ckey = key
@@ -426,12 +424,12 @@ decreasing_by hashmap_hash_map_remove_from_list_loop_decreases key ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::remove_from_list] -/
 def hashmap_hash_map_remove_from_list_fwd
-  (T : Type) (key : USize) (ls : hashmap_list_t T) : Result (Option T) :=
+  (T : Type) (key : Usize) (ls : hashmap_list_t T) : Result (Option T) :=
   hashmap_hash_map_remove_from_list_loop_fwd T key ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::remove_from_list] -/
 def hashmap_hash_map_remove_from_list_loop_back
-  (T : Type) (key : USize) (ls : hashmap_list_t T) :
+  (T : Type) (key : Usize) (ls : hashmap_list_t T) :
   (Result (hashmap_list_t T))
   :=
   match h: ls with
@@ -455,18 +453,18 @@ decreasing_by hashmap_hash_map_remove_from_list_loop_decreases key ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::remove_from_list] -/
 def hashmap_hash_map_remove_from_list_back
-  (T : Type) (key : USize) (ls : hashmap_list_t T) :
+  (T : Type) (key : Usize) (ls : hashmap_list_t T) :
   Result (hashmap_list_t T)
   :=
   hashmap_hash_map_remove_from_list_loop_back T key ls
 
 /- [hashmap_main::hashmap::HashMap::{0}::remove] -/
 def hashmap_hash_map_remove_fwd
-  (T : Type) (self : hashmap_hash_map_t T) (key : USize) : Result (Option T) :=
+  (T : Type) (self : hashmap_hash_map_t T) (key : Usize) : Result (Option T) :=
   do
     let hash ← hashmap_hash_key_fwd key
     let i := vec_len (hashmap_list_t T) self.hashmap_hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ←
       vec_index_mut_fwd (hashmap_list_t T) self.hashmap_hash_map_slots hash_mod
     let x ← hashmap_hash_map_remove_from_list_fwd T key l
@@ -474,19 +472,19 @@ def hashmap_hash_map_remove_fwd
     | Option.none => Result.ret Option.none
     | Option.some x0 =>
       do
-        let _ ← USize.checked_sub self.hashmap_hash_map_num_entries
-          (USize.ofNatCore 1 (by intlit))
+        let _ ← self.hashmap_hash_map_num_entries -
+          (Usize.ofInt 1 (by intlit))
         Result.ret (Option.some x0)
 
 /- [hashmap_main::hashmap::HashMap::{0}::remove] -/
 def hashmap_hash_map_remove_back
-  (T : Type) (self : hashmap_hash_map_t T) (key : USize) :
+  (T : Type) (self : hashmap_hash_map_t T) (key : Usize) :
   Result (hashmap_hash_map_t T)
   :=
   do
     let hash ← hashmap_hash_key_fwd key
     let i := vec_len (hashmap_list_t T) self.hashmap_hash_map_slots
-    let hash_mod ← USize.checked_rem hash i
+    let hash_mod ← hash % i
     let l ←
       vec_index_mut_fwd (hashmap_list_t T) self.hashmap_hash_map_slots hash_mod
     let x ← hashmap_hash_map_remove_from_list_fwd T key l
@@ -500,8 +498,8 @@ def hashmap_hash_map_remove_back
         Result.ret { self with hashmap_hash_map_slots := v }
     | Option.some x0 =>
       do
-        let i0 ← USize.checked_sub self.hashmap_hash_map_num_entries
-          (USize.ofNatCore 1 (by intlit))
+        let i0 ← self.hashmap_hash_map_num_entries -
+          (Usize.ofInt 1 (by intlit))
         let l0 ← hashmap_hash_map_remove_from_list_back T key l
         let v ←
           vec_index_mut_back (hashmap_list_t T) self.hashmap_hash_map_slots
@@ -516,79 +514,73 @@ def hashmap_hash_map_remove_back
 /- [hashmap_main::hashmap::test1] -/
 def hashmap_test1_fwd : Result Unit :=
   do
-    let hm ← hashmap_hash_map_new_fwd UInt64
+    let hm ← hashmap_hash_map_new_fwd U64
     let hm0 ←
-      hashmap_hash_map_insert_fwd_back UInt64 hm
-        (USize.ofNatCore 0 (by intlit)) (UInt64.ofNatCore 42 (by intlit))
+      hashmap_hash_map_insert_fwd_back U64 hm (Usize.ofInt 0 (by intlit))
+        (U64.ofInt 42 (by intlit))
     let hm1 ←
-      hashmap_hash_map_insert_fwd_back UInt64 hm0
-        (USize.ofNatCore 128 (by intlit)) (UInt64.ofNatCore 18 (by intlit))
+      hashmap_hash_map_insert_fwd_back U64 hm0 (Usize.ofInt 128 (by intlit))
+        (U64.ofInt 18 (by intlit))
     let hm2 ←
-      hashmap_hash_map_insert_fwd_back UInt64 hm1
-        (USize.ofNatCore 1024 (by intlit)) (UInt64.ofNatCore 138 (by intlit))
+      hashmap_hash_map_insert_fwd_back U64 hm1 (Usize.ofInt 1024 (by intlit))
+        (U64.ofInt 138 (by intlit))
     let hm3 ←
-      hashmap_hash_map_insert_fwd_back UInt64 hm2
-        (USize.ofNatCore 1056 (by intlit)) (UInt64.ofNatCore 256 (by intlit))
-    let i ←
-      hashmap_hash_map_get_fwd UInt64 hm3 (USize.ofNatCore 128 (by intlit))
-    if h: not (i = (UInt64.ofNatCore 18 (by intlit)))
+      hashmap_hash_map_insert_fwd_back U64 hm2 (Usize.ofInt 1056 (by intlit))
+        (U64.ofInt 256 (by intlit))
+    let i ← hashmap_hash_map_get_fwd U64 hm3 (Usize.ofInt 128 (by intlit))
+    if h: not (i = (U64.ofInt 18 (by intlit)))
     then Result.fail Error.panic
     else
       do
         let hm4 ←
-          hashmap_hash_map_get_mut_back UInt64 hm3
-            (USize.ofNatCore 1024 (by intlit))
-            (UInt64.ofNatCore 56 (by intlit))
+          hashmap_hash_map_get_mut_back U64 hm3 (Usize.ofInt 1024 (by intlit))
+            (U64.ofInt 56 (by intlit))
         let i0 ←
-          hashmap_hash_map_get_fwd UInt64 hm4
-            (USize.ofNatCore 1024 (by intlit))
-        if h: not (i0 = (UInt64.ofNatCore 56 (by intlit)))
+          hashmap_hash_map_get_fwd U64 hm4 (Usize.ofInt 1024 (by intlit))
+        if h: not (i0 = (U64.ofInt 56 (by intlit)))
         then Result.fail Error.panic
         else
           do
             let x ←
-              hashmap_hash_map_remove_fwd UInt64 hm4
-                (USize.ofNatCore 1024 (by intlit))
+              hashmap_hash_map_remove_fwd U64 hm4
+                (Usize.ofInt 1024 (by intlit))
             match h: x with
             | Option.none => Result.fail Error.panic
             | Option.some x0 =>
-              if h: not (x0 = (UInt64.ofNatCore 56 (by intlit)))
+              if h: not (x0 = (U64.ofInt 56 (by intlit)))
               then Result.fail Error.panic
               else
                 do
                   let hm5 ←
-                    hashmap_hash_map_remove_back UInt64 hm4
-                      (USize.ofNatCore 1024 (by intlit))
+                    hashmap_hash_map_remove_back U64 hm4
+                      (Usize.ofInt 1024 (by intlit))
                   let i1 ←
-                    hashmap_hash_map_get_fwd UInt64 hm5
-                      (USize.ofNatCore 0 (by intlit))
-                  if h: not (i1 = (UInt64.ofNatCore 42 (by intlit)))
+                    hashmap_hash_map_get_fwd U64 hm5
+                      (Usize.ofInt 0 (by intlit))
+                  if h: not (i1 = (U64.ofInt 42 (by intlit)))
                   then Result.fail Error.panic
                   else
                     do
                       let i2 ←
-                        hashmap_hash_map_get_fwd UInt64 hm5
-                          (USize.ofNatCore 128 (by intlit))
-                      if h: not (i2 = (UInt64.ofNatCore 18 (by intlit)))
+                        hashmap_hash_map_get_fwd U64 hm5
+                          (Usize.ofInt 128 (by intlit))
+                      if h: not (i2 = (U64.ofInt 18 (by intlit)))
                       then Result.fail Error.panic
                       else
                         do
                           let i3 ←
-                            hashmap_hash_map_get_fwd UInt64 hm5
-                              (USize.ofNatCore 1056 (by intlit))
-                          if h: not (i3 = (UInt64.ofNatCore 256 (by intlit)))
+                            hashmap_hash_map_get_fwd U64 hm5
+                              (Usize.ofInt 1056 (by intlit))
+                          if h: not (i3 = (U64.ofInt 256 (by intlit)))
                           then Result.fail Error.panic
                           else Result.ret ()
 
-/- Unit test for [hashmap_main::hashmap::test1] -/
-#assert (hashmap_test1_fwd == .ret ())
-
 /- [hashmap_main::insert_on_disk] -/
 def insert_on_disk_fwd
-  (key : USize) (value : UInt64) (st : State) : Result (State × Unit) :=
+  (key : Usize) (value : U64) (st : State) : Result (State × Unit) :=
   do
     let (st0, hm) ← opaque_defs.hashmap_utils_deserialize_fwd st
-    let hm0 ← hashmap_hash_map_insert_fwd_back UInt64 hm key value
+    let hm0 ← hashmap_hash_map_insert_fwd_back U64 hm key value
     let (st1, _) ← opaque_defs.hashmap_utils_serialize_fwd hm0 st0
     Result.ret (st1, ())
 
@@ -596,6 +588,3 @@ def insert_on_disk_fwd
 def main_fwd : Result Unit :=
   Result.ret ()
 
-/- Unit test for [hashmap_main::main] -/
-#assert (main_fwd == .ret ())
-
diff --git a/tests/lean/hashmap_on_disk/HashmapMain/Opaque.lean b/tests/lean/hashmap_on_disk/HashmapMain/Opaque.lean
index 3531e6e0..d98f431a 100644
--- a/tests/lean/hashmap_on_disk/HashmapMain/Opaque.lean
+++ b/tests/lean/hashmap_on_disk/HashmapMain/Opaque.lean
@@ -7,9 +7,9 @@ structure OpaqueDefs where
   
   /- [hashmap_main::hashmap_utils::deserialize] -/
   hashmap_utils_deserialize_fwd
-    : State -> Result (State × (hashmap_hash_map_t UInt64))
+    : State -> Result (State × (hashmap_hash_map_t U64))
   
   /- [hashmap_main::hashmap_utils::serialize] -/
   hashmap_utils_serialize_fwd
-    : hashmap_hash_map_t UInt64 -> State -> Result (State × Unit)
+    : hashmap_hash_map_t U64 -> State -> Result (State × Unit)
   
diff --git a/tests/lean/hashmap_on_disk/HashmapMain/Types.lean b/tests/lean/hashmap_on_disk/HashmapMain/Types.lean
index 989dd2a9..0509fbbd 100644
--- a/tests/lean/hashmap_on_disk/HashmapMain/Types.lean
+++ b/tests/lean/hashmap_on_disk/HashmapMain/Types.lean
@@ -4,14 +4,14 @@ import Base.Primitives
 
 /- [hashmap_main::hashmap::List] -/
 inductive hashmap_list_t (T : Type) :=
-| Cons : USize -> T -> hashmap_list_t T -> hashmap_list_t T
+| Cons : Usize -> T -> hashmap_list_t T -> hashmap_list_t T
 | Nil : hashmap_list_t T
 
 /- [hashmap_main::hashmap::HashMap] -/
 structure hashmap_hash_map_t (T : Type) where
-  hashmap_hash_map_num_entries : USize
-  hashmap_hash_map_max_load_factor : (USize × USize)
-  hashmap_hash_map_max_load : USize
+  hashmap_hash_map_num_entries : Usize
+  hashmap_hash_map_max_load_factor : (Usize × Usize)
+  hashmap_hash_map_max_load : Usize
   hashmap_hash_map_slots : Vec (hashmap_list_t T)
 
 /- The state type used in the state-error monad -/