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+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+
+-------------
+-- PRELUDE --
+-------------
+
+-- Results & monadic combinators
+
+inductive Error where
+ | assertionFailure: Error
+ | integerOverflow: 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
+deriving Repr, BEq
+
+open Result
+
+/- HELPERS -/
+
+def ret? {α: Type} (r: Result α): Bool :=
+ match r with
+ | Result.ret _ => true
+ | Result.fail _ => false
+
+def massert (b:Bool) : Result Unit :=
+ if b then .ret () else fail assertionFailure
+
+def eval_global {α: Type} (x: Result α) (_: ret? x): α :=
+ match x with
+ | Result.fail _ => by contradiction
+ | Result.ret x => x
+
+/- DO-DSL SUPPORT -/
+
+def bind (x: Result α) (f: α -> Result β) : Result β :=
+ match x with
+ | ret v => f v
+ | fail v => fail v
+
+-- Allows using Result in do-blocks
+instance : Bind Result where
+ bind := bind
+
+-- Allows using return x in do-blocks
+instance : Pure Result where
+ pure := fun x => ret x
+
+/- 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
+
+macro "let" e:term " ⟵ " f:term : doElem =>
+ `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- TODO: any way to factorize both definitions?
+macro "let" e:term " <-- " f:term : doElem =>
+ `(doElem| let ⟨$e, h⟩ ← Result.attach $f)
+
+-- We call the hypothesis `h`, in effect making it unavailable to the user
+-- (because too much shadowing). But in practice, once can use the French single
+-- quote notation (input with f< and f>), where `‹ h ›` finds a suitable
+-- hypothesis in the context, this is equivalent to `have x: h := by assumption in x`
+#eval do
+ let y <-- .ret (0: Nat)
+ let _: y = 0 := by cases ‹ ret 0 = ret y › ; decide
+ let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
+ .ret r
+
+----------------------
+-- 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.
+
+-- USize is designed properly: you cannot reduce `getNumBits` using the
+-- simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really
+-- support, at least officially, 16-bit microcontrollers, so this seems like a
+-- fine design decision for now.)
+
+-- Note from Chris Bailey: "If there's more than one salient property of your
+-- definition then the subtyping strategy might get messy, and the property part
+-- of a subtype is less discoverable by the simplifier or tactics like
+-- library_search." 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.
+
+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`).
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+
+-- Further thoughts: look at what has been done here:
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
+-- and
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
+-- which both contain a fair amount of reasoning already!
+def USize.checked_sub (n: USize) (m: USize): Result USize :=
+ -- 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 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
+
+-------------
+-- 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
+ } ⟩
+
+#check vec_new
+
+def vec_len (α : Type u) (v : Vec α) : USize :=
+ let ⟨ v, l ⟩ := v
+ USize.ofNatCore (List.length v) l
+
+#eval vec_len Nat (vec_new Nat)
+
+def vec_push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
+
+-- 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
+ ⟩
+ else
+ fail maximumSizeExceeded
+
+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) :=
+ if i.val < List.length v.val then
+ .ret ⟨ List.set v.val i.val x, by
+ have h: List.length v.val < USize.size := v.property
+ 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
+ .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 :=
+ 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
+ .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) :=
+ if i.val < List.length v.val then
+ .ret ⟨ List.set v.val i.val x, by
+ have h: List.length v.val < USize.size := v.property
+ rewrite [ List.length_set v.val i.val x ]
+ assumption
+ ⟩
+ else
+ .fail arrayOutOfBounds
+
+----------
+-- MISC --
+----------
+
+def mem_replace_fwd (a : Type) (x : a) (_ : a) : a :=
+ x
+
+def mem_replace_back (a : Type) (_ : a) (y : a) : a :=
+ y
+
+/-- Aeneas-translated function -- useful to reduce non-recursive definitions.
+ Use with `simp [ aeneas ]` -/
+register_simp_attr aeneas
+
+--------------------
+-- ASSERT COMMAND --
+--------------------
+
+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)