path: root/tests/lean/misc/paper/Base/Primitives.lean

import Lean
import Lean.Meta.Tactic.Simp
import Init.Data.List.Basic
import Mathlib.Tactic.RunCmd

-------------
-- PRELUDE --
-------------

-- Results & monadic combinators

-- TODO: use syntactic conventions and capitalize error, result, etc.

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 -/

-- TODO: is there automated syntax for these discriminators?
def is_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) (_: is_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 : (o : result α) → result { x : α // o = ret x }
  | .ret x => .ret ⟨x, rfl⟩
  | .fail e   => .fail e

macro "let" h:ident " : " e:term " <-- " f:term : doElem =>
  `(doElem| let ⟨$e, $h⟩ ← result.attach $f)

-- Silly example of the kind of reasoning that this notation enables
#eval do
  let h: y <-- .ret (0: Nat)
  let _: y = 0 := by cases h; decide
  let r: { x: Nat // x = 0 } := ⟨ y, by assumption ⟩
  .ret r

----------------------
-- MACHINE INTEGERS --
----------------------

-- NOTE: we reuse the USize type from prelude.lean, because at least we know
-- it's defined in an idiomatic style that is going to make proofs easy (and
-- indeed, several proofs here are much shortened compared to Aymeric's earlier
-- attempt.) This is not stricto sensu the *correct* thing to do, because one
-- can query at run-time the value of USize, which we do *not* want to do (we
-- don't know what target we'll run on!), but when the day comes, we'll just
-- define our own USize.
-- ANOTHER NOTE: there is USize.sub but it has wraparound semantics, which is
-- not something we want to define (I think), so we use our own monadic sub (but
-- is it in line with the Rust behavior?)

-- TODO: I am somewhat under the impression that subtraction is defined as a
-- total function over nats...? the hypothesis in the if condition is not used
-- in the then-branch which confuses me quite a bit

-- TODO: add a refinement for the result (just like vec_push_back below) that
-- explains that the toNat of the result (in the case of success) is the sub of
-- the toNat of the arguments (i.e. intrinsic specification)
-- ... do we want intrinsic specifications for the builtins? that might require
-- some careful type annotations in the monadic notation for clients, but may
-- give us more "for free"

-- 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." Try to settle this with a Lean expert on what is the most
-- productive way to go about this?

-- One needs to perform a little bit of reasoning in order to successfully
-- inject constants into USize, so we provide a general-purpose macro

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)

-- 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

def USize.checked_add (n: USize) (m: USize): result USize :=
  if h: n.val.val + m.val.val <= 4294967295 then
    .ret ⟨ n.val.val + m.val.val, by
      have h': 4294967295 < USize.size := by intlit
      apply Nat.lt_of_le_of_lt h h'
    ⟩
  else 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.val * m.val.val <= 4294967295 then
    .ret ⟨ n.val.val * m.val.val, by
      have h': 4294967295 < USize.size := by intlit
      apply Nat.lt_of_le_of_lt h h'
    ⟩
  else 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

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
  ))

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


-- 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

-------------
-- 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

--------------------
-- ASSERT COMMAND --
--------------------

open Lean Elab Command Term Meta

syntax (name := assert) "#assert" term: command

@[command_elab assert]
def assertImpl : CommandElab := fun (_stx: Syntax) => do
  logInfo "Reducing and asserting: "
  logInfo _stx[1]
  runTermElabM (fun _ => do
    let e ← Term.elabTerm _stx[1] none
    logInfo (Expr.dbgToString e)
    -- How to evaluate the term and compare the result to true?
    pure ())
  -- logInfo (Expr.dbgToString (``true))
  -- throwError "TODO: assert"

#eval 2 == 2
#assert (2 == 2)