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)