Reorganize the Lean backend

4 files changed, 753 insertions, 715 deletions

diff --git a/backends/lean/Base/Primitives.lean b/backends/lean/Base/Primitives.lean
index 1a0c665d..91823cb6 100644
--- a/backends/lean/Base/Primitives.lean
+++ b/backends/lean/Base/Primitives.lean
@@ -1,715 +1,3 @@
-import Lean
-import Lean.Meta.Tactic.Simp
-import Init.Data.List.Basic
-import Mathlib.Tactic.RunCmd
-import Mathlib.Tactic.Linarith
-
-namespace Primitives
-
---------------------
--- ASSERT COMMAND --Std.
---------------------
-
-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:\n" ++ _stx[1])
-      throwError ("Expression reduced to false:\n"  ++ _stx[1])
-    pure ())
-
-#eval 2 == 2
-#assert (2 == 2)
-
--------------
--- PRELUDE --
--------------
-
--- Results & monadic combinators
-
-inductive Error where
-   | assertionFailure: Error
-   | integerOverflow: Error
-   | divisionByZero: Error
-   | arrayOutOfBounds: Error
-   | maximumSizeExceeded: Error
-   | panic: Error
-deriving Repr, BEq
-
-open Error
-
-inductive Result (α : Type u) where
-  | ret (v: α): Result α
-  | fail (e: Error): Result α
-  | div
-deriving Repr, BEq
-
-open Result
-
-instance Result_Inhabited (α : Type u) : Inhabited (Result α) :=
-  Inhabited.mk (fail panic)
-
-instance Result_Nonempty (α : Type u) : Nonempty (Result α) :=
-  Nonempty.intro div
-
-/- HELPERS -/
-
-def ret? {α: Type u} (r: Result α): Bool :=
-  match r with
-  | ret _ => true
-  | fail _ | div => false
-
-def div? {α: Type u} (r: Result α): Bool :=
-  match r with
-  | div => true
-  | ret _ | fail _ => false
-
-def massert (b:Bool) : Result Unit :=
-  if b then ret () else fail assertionFailure
-
-def eval_global {α: Type u} (x: Result α) (_: ret? x): α :=
-  match x with
-  | fail _ | div => by contradiction
-  | ret x => x
-
-/- DO-DSL SUPPORT -/
-
-def bind {α : Type u} {β : Type v} (x: Result α) (f: α -> Result β) : Result β :=
-  match x with
-  | ret v  => f v 
-  | fail v => fail v
-  | div => div
-
--- Allows using Result in do-blocks
-instance : Bind Result where
-  bind := bind
-
--- Allows using return x in do-blocks
-instance : Pure Result where
-  pure := fun x => ret x
-
-@[simp] theorem bind_ret (x : α) (f : α → Result β) : bind (.ret x) f = f x := by simp [bind]
-@[simp] theorem bind_fail (x : Error) (f : α → Result β) : bind (.fail x) f = .fail x := by simp [bind]
-@[simp] theorem bind_div (f : α → Result β) : bind .div f = .div := by simp [bind]
-
-/- CUSTOM-DSL SUPPORT -/
-
--- Let-binding the Result of a monadic operation is oftentimes not sufficient,
--- because we may need a hypothesis for equational reasoning in the scope. We
--- rely on subtype, and a custom let-binding operator, in effect recreating our
--- own variant of the do-dsl
-
-def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
-  match o with
-  | ret x => ret ⟨x, rfl⟩
-  | fail e => fail e
-  | div => div
-
-@[simp] theorem bind_tc_ret (x : α) (f : α → Result β) :
-  (do let y ← .ret x; f y) = f x := by simp [Bind.bind, bind]
-
-@[simp] theorem bind_tc_fail (x : Error) (f : α → Result β) :
-  (do let y ← fail x; f y) = fail x := by simp [Bind.bind, bind]
-
-@[simp] theorem bind_tc_div (f : α → Result β) :
-  (do let y ← div; f y) = div := by simp [Bind.bind, bind]
-
-----------------------
--- MACHINE INTEGERS --
-----------------------
-
--- We redefine our machine integers types.
-
--- 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
--- of a subtype is less discoverable by the simplifier or tactics like
--- library_search." So, we will not add refinements on the return values of the
--- operations defined on Primitives, but will rather rely on custom lemmas to
--- invert on possible return values of the primitive operations.
-
--- Machine integer constants, done via `ofNatCore`, which requires a proof that
--- the `Nat` fits within the desired integer type. We provide a custom tactic.
-
-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.
-
--- The "structured" bounds
-def Isize.smin : Int := - (HPow.hPow 2 (size_num_bits - 1))
-def Isize.smax : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
-def I8.smin    : Int := - (HPow.hPow 2 7)
-def I8.smax    : Int := HPow.hPow 2 7 - 1
-def I16.smin   : Int := - (HPow.hPow 2 15)
-def I16.smax   : Int := HPow.hPow 2 15 - 1
-def I32.smin   : Int := -(HPow.hPow 2 31)
-def I32.smax   : Int := HPow.hPow 2 31 - 1
-def I64.smin   : Int := -(HPow.hPow 2 63)
-def I64.smax   : Int := HPow.hPow 2 63 - 1
-def I128.smin  : Int := -(HPow.hPow 2 127)
-def I128.smax  : Int := HPow.hPow 2 127 - 1
-def Usize.smin : Int := 0
-def Usize.smax : Int := HPow.hPow 2 size_num_bits - 1
-def U8.smin    : Int := 0
-def U8.smax    : Int := HPow.hPow 2 8 - 1
-def U16.smin   : Int := 0
-def U16.smax   : Int := HPow.hPow 2 16 - 1
-def U32.smin   : Int := 0
-def U32.smax   : Int := HPow.hPow 2 32 - 1
-def U64.smin   : Int := 0
-def U64.smax   : Int := HPow.hPow 2 64 - 1
-def U128.smin  : Int := 0
-def U128.smax  : Int := HPow.hPow 2 128 - 1
-
--- The "normalized" bounds, that we use in practice
-def I8.min    := -128
-def I8.max   := 127
-def I16.min  := -32768
-def I16.max  := 32767
-def I32.min  := -2147483648
-def I32.max  := 2147483647
-def I64.min  := -9223372036854775808
-def I64.max  := 9223372036854775807
-def I128.min := -170141183460469231731687303715884105728
-def I128.max := 170141183460469231731687303715884105727
-@[simp] def U8.min   := 0
-def U8.max   := 255
-@[simp] def U16.min  := 0
-def U16.max  := 65535
-@[simp] def U32.min  := 0
-def U32.max  := 4294967295
-@[simp] def U64.min  := 0
-def U64.max  := 18446744073709551615
-@[simp] def U128.min := 0
-def U128.max := 340282366920938463463374607431768211455
-@[simp] def Usize.min := 0
-
-def Isize.refined_min : { n:Int // n = I32.min ∨ n = I64.min } :=
-  ⟨ Isize.smin, by
-    simp [Isize.smin]
-    cases System.Platform.numBits_eq <;>
-    unfold System.Platform.numBits at * <;> simp [*] ⟩
-
-def Isize.refined_max : { n:Int // n = I32.max ∨ n = I64.max } :=
-  ⟨ Isize.smax, by
-    simp [Isize.smax]
-    cases System.Platform.numBits_eq <;>
-    unfold System.Platform.numBits at * <;> simp [*] ⟩
-
-def Usize.refined_max : { n:Int // n = U32.max ∨ n = U64.max } :=
-  ⟨ Usize.smax, by
-    simp [Usize.smax]
-    cases System.Platform.numBits_eq <;>
-    unfold System.Platform.numBits at * <;> simp [*] ⟩
-
-def Isize.min := Isize.refined_min.val
-def Isize.max := Isize.refined_max.val
-def Usize.max := Usize.refined_max.val
-
-inductive ScalarTy :=
-| Isize
-| I8
-| I16
-| I32
-| I64
-| I128
-| Usize
-| U8
-| U16
-| U32
-| U64
-| U128
-
-def Scalar.smin (ty : ScalarTy) : Int :=
-  match ty with
-  | .Isize => Isize.smin
-  | .I8    => I8.smin
-  | .I16   => I16.smin
-  | .I32   => I32.smin
-  | .I64   => I64.smin
-  | .I128  => I128.smin
-  | .Usize => Usize.smin
-  | .U8    => U8.smin
-  | .U16   => U16.smin
-  | .U32   => U32.smin
-  | .U64   => U64.smin
-  | .U128  => U128.smin
-
-def Scalar.smax (ty : ScalarTy) : Int :=
-  match ty with
-  | .Isize => Isize.smax
-  | .I8    => I8.smax
-  | .I16   => I16.smax
-  | .I32   => I32.smax
-  | .I64   => I64.smax
-  | .I128  => I128.smax
-  | .Usize => Usize.smax
-  | .U8    => U8.smax
-  | .U16   => U16.smax
-  | .U32   => U32.smax
-  | .U64   => U64.smax
-  | .U128  => U128.smax
-
-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
-
-def Scalar.smin_eq (ty : ScalarTy) : Scalar.min ty = Scalar.smin ty := by
-  cases ty <;> rfl
-
-def Scalar.smax_eq (ty : ScalarTy) : Scalar.max ty = Scalar.smax ty := by
-  cases ty <;> rfl
-
--- "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 => Scalar.min .I32
-  | _ => Scalar.min ty
-
-def Scalar.cMax (ty : ScalarTy) : Int :=
-  match ty with
-  | .Isize => Scalar.max .I32
-  | .Usize => Scalar.max .U32
-  | _ => Scalar.max ty
-
-theorem Scalar.cMin_bound ty : Scalar.min ty ≤ Scalar.cMin ty := by
-  cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
-  have h := Isize.refined_min.property
-  cases h <;> simp [*, Isize.min]
-
-theorem Scalar.cMax_bound ty : Scalar.cMax ty ≤ Scalar.max ty := by
-  cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
-  . have h := Isize.refined_max.property
-    cases h <;> simp [*, Isize.max]
-  . have h := Usize.refined_max.property
-    cases h <;> simp [*, Usize.max]
-
-theorem Scalar.cMin_suffices ty (h : Scalar.cMin ty ≤ x) : Scalar.min ty ≤ x := by
-  have := Scalar.cMin_bound ty
-  linarith
-
-theorem Scalar.cMax_suffices ty (h : x ≤ Scalar.cMax ty) : x ≤ Scalar.max ty := by
-  have := Scalar.cMax_bound ty
-  linarith
-
-structure Scalar (ty : ScalarTy) where
-  val : Int
-  hmin : Scalar.min ty ≤ val
-  hmax : val ≤ Scalar.max ty
-deriving Repr
-
-theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
-  Scalar.cMin ty ≤ x ∧ x ≤ Scalar.cMax ty ->
-  Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty
-  :=
-  λ h => by
-  apply And.intro <;> have hmin := Scalar.cMin_bound ty <;> have hmax := Scalar.cMax_bound ty <;> linarith
-
-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 }
-
--- Tactic to prove that integers are in bounds
--- TODO: use this: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/instance.20with.20tactic.20autoparam
-syntax "intlit" : tactic
-macro_rules
-  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices; decide)
-
-def Scalar.ofInt {ty : ScalarTy} (x : Int)
-  (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by intlit) : Scalar ty :=
-  -- Remark: we initially wrote:
-  --  let ⟨ hmin, hmax ⟩ := h
-  --  Scalar.ofIntCore x hmin hmax
-  -- We updated to the line below because a similar pattern in `Scalar.tryMk`
-  -- made reduction block. Both versions seem to work for `Scalar.ofInt`, though.
-  -- TODO: investigate
-  Scalar.ofIntCore x h.left h.right
-
-@[simp] def Scalar.check_bounds (ty : ScalarTy) (x : Int) : Bool :=
-  (Scalar.cMin ty ≤ x || Scalar.min ty ≤ x) ∧ (x ≤ Scalar.cMax ty || x ≤ Scalar.max ty)
-
-theorem Scalar.check_bounds_prop {ty : ScalarTy} {x : Int} (h: Scalar.check_bounds ty x) :
-  Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by
-  simp at *
-  have ⟨ hmin, hmax ⟩ := h
-  have hbmin := Scalar.cMin_bound ty
-  have hbmax := Scalar.cMax_bound ty
-  cases hmin <;> cases hmax <;> apply And.intro <;> linarith
-
--- 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 Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
-  if h:Scalar.check_bounds ty x then
-    -- If we do:
-    -- ```
-    -- let ⟨ hmin, hmax ⟩ := (Scalar.check_bounds_prop h)
-    -- Scalar.ofIntCore x hmin hmax
-    -- ```
-    -- then normalization blocks (for instance, some proofs which use reflexivity fail).
-    -- However, the version below doesn't block reduction (TODO: investigate):
-    return Scalar.ofInt x (Scalar.check_bounds_prop h)
-  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
-
--- Our custom remainder operation, which satisfies the semantics of Rust
--- TODO: is there a better way?
-def scalar_rem (x y : Int) : Int :=
-  if 0 ≤ x then |x| % |y|
-  else - (|x| % |y|)
-
--- Our custom division operation, which satisfies the semantics of Rust
--- TODO: is there a better way?
-def scalar_div (x y : Int) : Int :=
-  if 0 ≤ x && 0 ≤ y then |x| / |y|
-  else if 0 ≤ x && y < 0 then - (|x| / |y|)
-  else if x < 0 && 0 ≤ y then - (|x| / |y|)
-  else |x| / |y|
-
--- Checking that the remainder operation is correct
-#assert scalar_rem 1 2 = 1
-#assert scalar_rem (-1) 2 = -1
-#assert scalar_rem 1 (-2) = 1
-#assert scalar_rem (-1) (-2) = -1
-#assert scalar_rem 7 3 = (1:Int)
-#assert scalar_rem (-7) 3 = -1
-#assert scalar_rem 7 (-3) = 1
-#assert scalar_rem (-7) (-3) = -1
-
--- Checking that the division operation is correct
-#assert scalar_div 3 2 = 1
-#assert scalar_div (-3) 2 = -1
-#assert scalar_div 3 (-2) = -1
-#assert scalar_div (-3) (-2) = 1
-#assert scalar_div 7 3 = 2
-#assert scalar_div (-7) 3 = -2
-#assert scalar_div 7 (-3) = -2
-#assert scalar_div (-7) (-3) = 2
-
-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
-
--- -- 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 --
--------------
-
-def Vec (α : Type u) := { l : List α // List.length l ≤ Usize.max }
-
--- TODO: do we really need it? It should be with Subtype by default
-instance Vec.cast (a : Type): Coe (Vec a) (List a)  where coe := λ v => v.val
-
-def Vec.new (α : Type u): Vec α := ⟨ [], by apply Scalar.cMax_suffices .Usize; simp ⟩
-
-def Vec.len (α : Type u) (v : Vec α) : Usize :=
-  let ⟨ v, l ⟩ := v
-  Usize.ofIntCore (List.length v) (by simp [Scalar.min, Usize.min]) l
- 
--- This shouldn't be used
-def Vec.push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
-
--- This is actually the backward function
-def Vec.push (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
-  :=
-  let nlen := List.length v.val + 1
-  if h : nlen ≤ U32.max || nlen ≤ Usize.max then
-    have h : nlen ≤ Usize.max := by
-      simp [Usize.max] at *
-      have hm := Usize.refined_max.property
-      cases h <;> cases hm <;> simp [U32.max, U64.max] at * <;> try linarith
-    return ⟨ List.concat v.val x, by simp at *; assumption ⟩
-  else
-    fail maximumSizeExceeded
-
--- This shouldn't be used
-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
-
--- This is actually the backward function
-def Vec.insert (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 := i.val.toNat
-    .ret ⟨ List.set v.val i x, by
-      have h: List.length v.val ≤ Usize.max := v.property
-      simp [*] at *
-    ⟩
-  else
-    .fail arrayOutOfBounds
-
-def Vec.index_to_fin {α : Type u} {v: Vec α} {i: Usize} (h : i.val < List.length v.val) :
-  Fin (List.length v.val) :=
-  let j := i.val.toNat
-  let h: j < List.length v.val := by
-    have heq := @Int.toNat_lt (List.length v.val) i.val i.hmin
-    apply heq.mpr
-    assumption
-  ⟨j, h⟩
-
-def Vec.index (α : Type u) (v: Vec α) (i: Usize): Result α :=
-  if h: i.val < List.length v.val then
-    let i := Vec.index_to_fin h
-    .ret (List.get v.val i)
-  else
-    .fail arrayOutOfBounds
-
--- This shouldn't be used
-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 (α : Type u) (v: Vec α) (i: Usize): Result α :=
-  if h: i.val < List.length v.val then
-    let i := Vec.index_to_fin h
-    .ret (List.get v.val i)
-  else
-    .fail arrayOutOfBounds
-
-def Vec.index_mut_back (α : Type u) (v: Vec α) (i: Usize) (x: α): Result (Vec α) :=
-  if h: i.val < List.length v.val then
-    let i := Vec.index_to_fin h
-    .ret ⟨ List.set v.val i x, by
-      have h: List.length v.val ≤ Usize.max := v.property
-      simp [*] at *
-    ⟩
-  else
-    .fail arrayOutOfBounds
-
-----------
--- MISC --
-----------
-
-@[simp] def mem.replace (a : Type) (x : a) (_ : a) : a := x
-@[simp] def mem.replace_back (a : Type) (_ : a) (y : a) : a := y
-
-/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
- Use with `simp [ aeneas ]` -/
-register_simp_attr aeneas
-
-end Primitives
+import Base.Primitives.Base
+import Base.Primitives.Scalar
+import Base.Primitives.Vec
diff --git a/backends/lean/Base/Primitives/Base.lean b/backends/lean/Base/Primitives/Base.lean
new file mode 100644
index 00000000..db462c38
--- /dev/null
+++ b/backends/lean/Base/Primitives/Base.lean
@@ -0,0 +1,130 @@
+import Lean
+
+namespace Primitives
+
+--------------------
+-- ASSERT COMMAND --Std.
+--------------------
+
+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:\n" ++ _stx[1])
+      throwError ("Expression reduced to false:\n"  ++ _stx[1])
+    pure ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
+-------------
+-- PRELUDE --
+-------------
+
+-- Results & monadic combinators
+
+inductive Error where
+   | assertionFailure: Error
+   | integerOverflow: Error
+   | divisionByZero: Error
+   | arrayOutOfBounds: Error
+   | maximumSizeExceeded: Error
+   | panic: Error
+deriving Repr, BEq
+
+open Error
+
+inductive Result (α : Type u) where
+  | ret (v: α): Result α
+  | fail (e: Error): Result α
+  | div
+deriving Repr, BEq
+
+open Result
+
+instance Result_Inhabited (α : Type u) : Inhabited (Result α) :=
+  Inhabited.mk (fail panic)
+
+instance Result_Nonempty (α : Type u) : Nonempty (Result α) :=
+  Nonempty.intro div
+
+/- HELPERS -/
+
+def ret? {α: Type u} (r: Result α): Bool :=
+  match r with
+  | ret _ => true
+  | fail _ | div => false
+
+def div? {α: Type u} (r: Result α): Bool :=
+  match r with
+  | div => true
+  | ret _ | fail _ => false
+
+def massert (b:Bool) : Result Unit :=
+  if b then ret () else fail assertionFailure
+
+def eval_global {α: Type u} (x: Result α) (_: ret? x): α :=
+  match x with
+  | fail _ | div => by contradiction
+  | ret x => x
+
+/- DO-DSL SUPPORT -/
+
+def bind {α : Type u} {β : Type v} (x: Result α) (f: α -> Result β) : Result β :=
+  match x with
+  | ret v  => f v 
+  | fail v => fail v
+  | div => div
+
+-- Allows using Result in do-blocks
+instance : Bind Result where
+  bind := bind
+
+-- Allows using return x in do-blocks
+instance : Pure Result where
+  pure := fun x => ret x
+
+@[simp] theorem bind_ret (x : α) (f : α → Result β) : bind (.ret x) f = f x := by simp [bind]
+@[simp] theorem bind_fail (x : Error) (f : α → Result β) : bind (.fail x) f = .fail x := by simp [bind]
+@[simp] theorem bind_div (f : α → Result β) : bind .div f = .div := by simp [bind]
+
+/- CUSTOM-DSL SUPPORT -/
+
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
+-- because we may need a hypothesis for equational reasoning in the scope. We
+-- rely on subtype, and a custom let-binding operator, in effect recreating our
+-- own variant of the do-dsl
+
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+  match o with
+  | ret x => ret ⟨x, rfl⟩
+  | fail e => fail e
+  | div => div
+
+@[simp] theorem bind_tc_ret (x : α) (f : α → Result β) :
+  (do let y ← .ret x; f y) = f x := by simp [Bind.bind, bind]
+
+@[simp] theorem bind_tc_fail (x : Error) (f : α → Result β) :
+  (do let y ← fail x; f y) = fail x := by simp [Bind.bind, bind]
+
+@[simp] theorem bind_tc_div (f : α → Result β) :
+  (do let y ← div; f y) = div := by simp [Bind.bind, bind]
+
+----------
+-- MISC --
+----------
+
+@[simp] def mem.replace (a : Type) (x : a) (_ : a) : a := x
+@[simp] def mem.replace_back (a : Type) (_ : a) (y : a) : a := y
+
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
+end Primitives
diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
new file mode 100644
index 00000000..241dfa07
--- /dev/null
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -0,0 +1,507 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Mathlib.Tactic.Linarith
+import Base.Primitives.Base
+
+namespace Primitives
+
+----------------------
+-- MACHINE INTEGERS --
+----------------------
+
+-- We redefine our machine integers types.
+
+-- 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
+-- of a subtype is less discoverable by the simplifier or tactics like
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
+
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
+
+open Result Error
+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.
+
+-- The "structured" bounds
+def Isize.smin : Int := - (HPow.hPow 2 (size_num_bits - 1))
+def Isize.smax : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+def I8.smin    : Int := - (HPow.hPow 2 7)
+def I8.smax    : Int := HPow.hPow 2 7 - 1
+def I16.smin   : Int := - (HPow.hPow 2 15)
+def I16.smax   : Int := HPow.hPow 2 15 - 1
+def I32.smin   : Int := -(HPow.hPow 2 31)
+def I32.smax   : Int := HPow.hPow 2 31 - 1
+def I64.smin   : Int := -(HPow.hPow 2 63)
+def I64.smax   : Int := HPow.hPow 2 63 - 1
+def I128.smin  : Int := -(HPow.hPow 2 127)
+def I128.smax  : Int := HPow.hPow 2 127 - 1
+def Usize.smin : Int := 0
+def Usize.smax : Int := HPow.hPow 2 size_num_bits - 1
+def U8.smin    : Int := 0
+def U8.smax    : Int := HPow.hPow 2 8 - 1
+def U16.smin   : Int := 0
+def U16.smax   : Int := HPow.hPow 2 16 - 1
+def U32.smin   : Int := 0
+def U32.smax   : Int := HPow.hPow 2 32 - 1
+def U64.smin   : Int := 0
+def U64.smax   : Int := HPow.hPow 2 64 - 1
+def U128.smin  : Int := 0
+def U128.smax  : Int := HPow.hPow 2 128 - 1
+
+-- The "normalized" bounds, that we use in practice
+def I8.min    := -128
+def I8.max   := 127
+def I16.min  := -32768
+def I16.max  := 32767
+def I32.min  := -2147483648
+def I32.max  := 2147483647
+def I64.min  := -9223372036854775808
+def I64.max  := 9223372036854775807
+def I128.min := -170141183460469231731687303715884105728
+def I128.max := 170141183460469231731687303715884105727
+@[simp] def U8.min   := 0
+def U8.max   := 255
+@[simp] def U16.min  := 0
+def U16.max  := 65535
+@[simp] def U32.min  := 0
+def U32.max  := 4294967295
+@[simp] def U64.min  := 0
+def U64.max  := 18446744073709551615
+@[simp] def U128.min := 0
+def U128.max := 340282366920938463463374607431768211455
+@[simp] def Usize.min := 0
+
+def Isize.refined_min : { n:Int // n = I32.min ∨ n = I64.min } :=
+  ⟨ Isize.smin, by
+    simp [Isize.smin]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Isize.refined_max : { n:Int // n = I32.max ∨ n = I64.max } :=
+  ⟨ Isize.smax, by
+    simp [Isize.smax]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Usize.refined_max : { n:Int // n = U32.max ∨ n = U64.max } :=
+  ⟨ Usize.smax, by
+    simp [Usize.smax]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Isize.min := Isize.refined_min.val
+def Isize.max := Isize.refined_max.val
+def Usize.max := Usize.refined_max.val
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def Scalar.smin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.smin
+  | .I8    => I8.smin
+  | .I16   => I16.smin
+  | .I32   => I32.smin
+  | .I64   => I64.smin
+  | .I128  => I128.smin
+  | .Usize => Usize.smin
+  | .U8    => U8.smin
+  | .U16   => U16.smin
+  | .U32   => U32.smin
+  | .U64   => U64.smin
+  | .U128  => U128.smin
+
+def Scalar.smax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.smax
+  | .I8    => I8.smax
+  | .I16   => I16.smax
+  | .I32   => I32.smax
+  | .I64   => I64.smax
+  | .I128  => I128.smax
+  | .Usize => Usize.smax
+  | .U8    => U8.smax
+  | .U16   => U16.smax
+  | .U32   => U32.smax
+  | .U64   => U64.smax
+  | .U128  => U128.smax
+
+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
+
+def Scalar.smin_eq (ty : ScalarTy) : Scalar.min ty = Scalar.smin ty := by
+  cases ty <;> rfl
+
+def Scalar.smax_eq (ty : ScalarTy) : Scalar.max ty = Scalar.smax ty := by
+  cases ty <;> rfl
+
+-- "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 => Scalar.min .I32
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Scalar.max .I32
+  | .Usize => Scalar.max .U32
+  | _ => Scalar.max ty
+
+theorem Scalar.cMin_bound ty : Scalar.min ty ≤ Scalar.cMin ty := by
+  cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
+  have h := Isize.refined_min.property
+  cases h <;> simp [*, Isize.min]
+
+theorem Scalar.cMax_bound ty : Scalar.cMax ty ≤ Scalar.max ty := by
+  cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
+  . have h := Isize.refined_max.property
+    cases h <;> simp [*, Isize.max]
+  . have h := Usize.refined_max.property
+    cases h <;> simp [*, Usize.max]
+
+theorem Scalar.cMin_suffices ty (h : Scalar.cMin ty ≤ x) : Scalar.min ty ≤ x := by
+  have := Scalar.cMin_bound ty
+  linarith
+
+theorem Scalar.cMax_suffices ty (h : x ≤ Scalar.cMax ty) : x ≤ Scalar.max ty := by
+  have := Scalar.cMax_bound ty
+  linarith
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty ≤ val
+  hmax : val ≤ Scalar.max ty
+deriving Repr
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty ≤ x ∧ x ≤ Scalar.cMax ty ->
+  Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty
+  :=
+  λ h => by
+  apply And.intro <;> have hmin := Scalar.cMin_bound ty <;> have hmax := Scalar.cMax_bound ty <;> linarith
+
+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 }
+
+-- Tactic to prove that integers are in bounds
+-- TODO: use this: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/instance.20with.20tactic.20autoparam
+syntax "intlit" : tactic
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices; decide)
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by intlit) : Scalar ty :=
+  -- Remark: we initially wrote:
+  --  let ⟨ hmin, hmax ⟩ := h
+  --  Scalar.ofIntCore x hmin hmax
+  -- We updated to the line below because a similar pattern in `Scalar.tryMk`
+  -- made reduction block. Both versions seem to work for `Scalar.ofInt`, though.
+  -- TODO: investigate
+  Scalar.ofIntCore x h.left h.right
+
+@[simp] def Scalar.check_bounds (ty : ScalarTy) (x : Int) : Bool :=
+  (Scalar.cMin ty ≤ x || Scalar.min ty ≤ x) ∧ (x ≤ Scalar.cMax ty || x ≤ Scalar.max ty)
+
+theorem Scalar.check_bounds_prop {ty : ScalarTy} {x : Int} (h: Scalar.check_bounds ty x) :
+  Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by
+  simp at *
+  have ⟨ hmin, hmax ⟩ := h
+  have hbmin := Scalar.cMin_bound ty
+  have hbmax := Scalar.cMax_bound ty
+  cases hmin <;> cases hmax <;> apply And.intro <;> linarith
+
+-- 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 Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  if h:Scalar.check_bounds ty x then
+    -- If we do:
+    -- ```
+    -- let ⟨ hmin, hmax ⟩ := (Scalar.check_bounds_prop h)
+    -- Scalar.ofIntCore x hmin hmax
+    -- ```
+    -- then normalization blocks (for instance, some proofs which use reflexivity fail).
+    -- However, the version below doesn't block reduction (TODO: investigate):
+    return Scalar.ofInt x (Scalar.check_bounds_prop h)
+  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
+
+-- Our custom remainder operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_rem (x y : Int) : Int :=
+  if 0 ≤ x then |x| % |y|
+  else - (|x| % |y|)
+
+-- Our custom division operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_div (x y : Int) : Int :=
+  if 0 ≤ x && 0 ≤ y then |x| / |y|
+  else if 0 ≤ x && y < 0 then - (|x| / |y|)
+  else if x < 0 && 0 ≤ y then - (|x| / |y|)
+  else |x| / |y|
+
+-- Checking that the remainder operation is correct
+#assert scalar_rem 1 2 = 1
+#assert scalar_rem (-1) 2 = -1
+#assert scalar_rem 1 (-2) = 1
+#assert scalar_rem (-1) (-2) = -1
+#assert scalar_rem 7 3 = (1:Int)
+#assert scalar_rem (-7) 3 = -1
+#assert scalar_rem 7 (-3) = 1
+#assert scalar_rem (-7) (-3) = -1
+
+-- Checking that the division operation is correct
+#assert scalar_div 3 2 = 1
+#assert scalar_div (-3) 2 = -1
+#assert scalar_div 3 (-2) = -1
+#assert scalar_div (-3) (-2) = 1
+#assert scalar_div 7 3 = 2
+#assert scalar_div (-7) 3 = -2
+#assert scalar_div 7 (-3) = -2
+#assert scalar_div (-7) (-3) = 2
+
+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)
+
+/- Remark: we can't write the following instance because of restrictions about
+   the type class parameters (`ty` doesn't appear in the return type, which is
+   forbidden):
+
+   ```
+   instance Scalar.cast (ty : ScalarTy) : Coe (Scalar ty) Int where coe := λ v => v.val
+   ```
+ -/
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- -- 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
+
+end Primitives
diff --git a/backends/lean/Base/Primitives/Vec.lean b/backends/lean/Base/Primitives/Vec.lean
new file mode 100644
index 00000000..7851a232
--- /dev/null
+++ b/backends/lean/Base/Primitives/Vec.lean
@@ -0,0 +1,113 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+import Base.IList
+import Base.Primitives.Scalar
+import Base.Arith
+
+namespace Primitives
+
+open Result Error
+
+-------------
+-- VECTORS --
+-------------
+
+def Vec (α : Type u) := { l : List α // List.length l ≤ Usize.max }
+
+-- TODO: do we really need it? It should be with Subtype by default
+instance Vec.cast (a : Type): Coe (Vec a) (List a)  where coe := λ v => v.val
+
+instance (a : Type) : Arith.HasIntProp (Vec a) where
+  prop_ty := λ v => v.val.length ≤ Scalar.max ScalarTy.Usize
+  prop := λ ⟨ _, l ⟩ => l
+
+example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
+  intro_has_int_prop_instances
+  simp_all [Scalar.max, Scalar.min]
+
+example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
+  scalar_tac
+
+def Vec.new (α : Type u): Vec α := ⟨ [], by apply Scalar.cMax_suffices .Usize; simp ⟩
+
+def Vec.len (α : Type u) (v : Vec α) : Usize :=
+  let ⟨ v, l ⟩ := v
+  Usize.ofIntCore (List.length v) (by simp [Scalar.min, Usize.min]) l
+
+def Vec.length {α : Type u} (v : Vec α) : Int := v.val.len
+
+-- This shouldn't be used
+def Vec.push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
+
+-- This is actually the backward function
+def Vec.push (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
+  :=
+  let nlen := List.length v.val + 1
+  if h : nlen ≤ U32.max || nlen ≤ Usize.max then
+    have h : nlen ≤ Usize.max := by
+      simp [Usize.max] at *
+      have hm := Usize.refined_max.property
+      cases h <;> cases hm <;> simp [U32.max, U64.max] at * <;> try linarith
+    return ⟨ List.concat v.val x, by simp at *; assumption ⟩
+  else
+    fail maximumSizeExceeded
+
+-- This shouldn't be used
+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
+
+-- This is actually the backward function
+def Vec.insert (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)
+    .ret ⟨ v.val.update i.val x, by have := v.property; simp [*] ⟩
+  else
+    .fail arrayOutOfBounds
+
+-- TODO: remove
+def Vec.index_to_fin {α : Type u} {v: Vec α} {i: Usize} (h : i.val < List.length v.val) :
+  Fin (List.length v.val) :=
+  let j := i.val.toNat
+  let h: j < List.length v.val := by
+    have heq := @Int.toNat_lt (List.length v.val) i.val i.hmin
+    apply heq.mpr
+    assumption
+  ⟨j, h⟩
+
+def Vec.index (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  match v.val.indexOpt i.val with
+  | none => fail .arrayOutOfBounds
+  | some x => ret x
+
+-- This shouldn't be used
+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 (α : Type u) (v: Vec α) (i: Usize): Result α :=
+  if h: i.val < List.length v.val then
+    let i := Vec.index_to_fin h
+    .ret (List.get v.val i)
+  else
+    .fail arrayOutOfBounds
+
+def Vec.index_mut_back (α : Type u) (v: Vec α) (i: Usize) (x: α): Result (Vec α) :=
+  if h: i.val < List.length v.val then
+    let i := Vec.index_to_fin h
+    .ret ⟨ List.set v.val i x, by
+      have h: List.length v.val ≤ Usize.max := v.property
+      simp [*] at *
+    ⟩
+  else
+    .fail arrayOutOfBounds
+
+end Primitives