path: root/backends/lean/Base/Primitives/Scalar.lean

import Lean
import Lean.Meta.Tactic.Simp
import Mathlib.Tactic.Linarith
import Base.Primitives.Base
import Base.Diverge.Base
import Base.Progress.Base
import Base.Arith.Int

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    : Int   := -128
def I8.max    : Int   := 127
def I16.min   : Int  := -32768
def I16.max   : Int  := 32767
def I32.min   : Int  := -2147483648
def I32.max   : Int  := 2147483647
def I64.min   : Int  := -9223372036854775808
def I64.max   : Int  := 9223372036854775807
def I128.min  : Int := -170141183460469231731687303715884105728
def I128.max  : Int := 170141183460469231731687303715884105727
@[simp]
def U8.min    : Int   := 0
def U8.max    : Int   := 255
@[simp]
def U16.min   : Int  := 0
def U16.max   : Int  := 65535
@[simp]
def U32.min   : Int  := 0
def U32.max   : Int  := 4294967295
@[simp]
def U64.min   : Int  := 0
def U64.max   : Int  := 18446744073709551615
@[simp]
def U128.min  : Int := 0
def U128.max  : Int := 340282366920938463463374607431768211455
@[simp]
def Usize.min : Int := 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 ScalarTy.isSigned (ty : ScalarTy) : Bool :=
  match ty with
  | Isize
  | I8
  | I16
  | I32
  | I64
  | I128 => true
  | Usize
  | U8
  | U16
  | U32
  | U64
  | U128 => false


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)

-- 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|)

@[simp]
def scalar_rem_nonneg {x y : Int} (hx : 0 ≤ x) : scalar_rem x y = x % y := by
  intros
  simp [*, scalar_rem]

-- 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|

@[simp]
def scalar_div_nonneg {x y : Int} (hx : 0 ≤ x) (hy : 0 ≤ y) : scalar_div x y = x / y := by
  intros
  simp [*, scalar_div]

-- 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.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
  if y.val != 0 then Scalar.tryMk ty (scalar_div x.val y.val) else fail divisionByZero

def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
  if y.val != 0 then Scalar.tryMk ty (scalar_rem 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

-- Generic theorem - shouldn't be used much
@[cpspec]
theorem Scalar.add_spec {ty} {x y : Scalar ty}
  (hmin : Scalar.min ty ≤ x.val + y.val)
  (hmax : x.val + y.val ≤ Scalar.max ty) :
  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
  simp [HAdd.hAdd, add, Add.add]
  simp [tryMk]
  split
  . simp [pure]
    rfl
  . tauto

theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
  (hmax : x.val + y.val ≤ Scalar.max ty) :
  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
  have hmin : Scalar.min ty ≤ x.val + y.val := by
    have hx := x.hmin
    have hy := y.hmin
    cases ty <;> simp [min] at * <;> linarith
  apply add_spec <;> assumption

/- Fine-grained theorems -/
@[cepspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) :
  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) :
  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) :
  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) :
  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) :
  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) :
  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]

-- Generic theorem - shouldn't be used much
@[cpspec]
theorem Scalar.sub_spec {ty} {x y : Scalar ty}
  (hmin : Scalar.min ty ≤ x.val - y.val)
  (hmax : x.val - y.val ≤ Scalar.max ty) :
  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
  simp [HSub.hSub, sub, Sub.sub]
  simp [tryMk]
  split
  . simp [pure]
    rfl
  . tauto

theorem Scalar.sub_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
  (hmin : Scalar.min ty ≤ x.val - y.val) :
  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
  have : x.val - y.val ≤ Scalar.max ty := by
    have hx := x.hmin
    have hxm := x.hmax
    have hy := y.hmin
    cases ty <;> simp [min, max] at * <;> linarith
  intros
  apply sub_spec <;> assumption

/- Fine-grained theorems -/
@[cepspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) :
  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]

@[cepspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) :
  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]

@[cepspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) :
  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]

@[cepspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) :
  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]

@[cepspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) :
  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]

@[cepspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) :
  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]

-- Generic theorem - shouldn't be used much
theorem Scalar.mul_spec {ty} {x y : Scalar ty}
  (hmin : Scalar.min ty ≤ x.val * y.val)
  (hmax : x.val * y.val ≤ Scalar.max ty) :
  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
  simp [HMul.hMul, mul, Mul.mul]
  simp [tryMk]
  split
  . simp [pure]
    rfl
  . tauto

theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
  (hmax : x.val * y.val ≤ Scalar.max ty) :
  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
  have : Scalar.min ty ≤ x.val * y.val := by
    have hx := x.hmin
    have hy := y.hmin
    cases ty <;> simp at * <;> apply mul_nonneg hx hy
  apply mul_spec <;> assumption

/- Fine-grained theorems -/
@[cepspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) :
  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) :
  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) :
  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) :
  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) :
  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]

@[cepspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) :
  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]

-- Generic theorem - shouldn't be used much
@[cpspec]
theorem Scalar.div_spec {ty} {x y : Scalar ty}
  (hnz : y.val ≠ 0)
  (hmin : Scalar.min ty ≤ scalar_div x.val y.val)
  (hmax : scalar_div x.val y.val ≤ Scalar.max ty) :
  ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := by
  simp [HDiv.hDiv, div, Div.div]
  simp [tryMk, *]
  simp [pure]
  rfl

theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
  (hnz : y.val ≠ 0) :
  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
  have h : Scalar.min ty = 0 := by cases ty <;> simp at *
  have hx := x.hmin
  have hy := y.hmin
  simp [h] at hx hy
  have hmin : 0 ≤ x.val / y.val := Int.ediv_nonneg hx hy
  have hmax : x.val / y.val ≤ Scalar.max ty := by
    have := Int.ediv_le_self y.val hx
    have := x.hmax
    linarith
  have hs := @div_spec ty x y hnz
  simp [*] at hs
  apply hs

/- Fine-grained theorems -/
@[cepspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
  apply Scalar.div_unsigned_spec <;> simp [*]

@[cepspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]

@[cepspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]

@[cepspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]

@[cepspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]

@[cepspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]

-- Generic theorem - shouldn't be used much
@[cpspec]
theorem Scalar.rem_spec {ty} {x y : Scalar ty}
  (hnz : y.val ≠ 0)
  (hmin : Scalar.min ty ≤ scalar_rem x.val y.val)
  (hmax : scalar_rem x.val y.val ≤ Scalar.max ty) :
  ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := by
  simp [HMod.hMod, rem]
  simp [tryMk, *]
  simp [pure]
  rfl

theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
  (hnz : y.val ≠ 0) :
  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
  have h : Scalar.min ty = 0 := by cases ty <;> simp at *
  have hx := x.hmin
  have hy := y.hmin
  simp [h] at hx hy
  have hmin : 0 ≤ x.val % y.val := Int.emod_nonneg x.val hnz
  have hmax : x.val % y.val ≤ Scalar.max ty := by
    have h : 0 < y.val := by int_tac
    have h := Int.emod_lt_of_pos x.val h
    have := y.hmax
    linarith
  have hs := @rem_spec ty x y hnz
  simp [*] at hs
  simp [*]

@[cepspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
  apply Scalar.rem_unsigned_spec <;> simp [*]

@[cepspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]

@[cepspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]

@[cepspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]

@[cepspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]

@[cepspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]

-- 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?
@[reducible] def Isize.ofInt := @Scalar.ofInt .Isize
@[reducible] def I8.ofInt    := @Scalar.ofInt .I8
@[reducible] def I16.ofInt   := @Scalar.ofInt .I16
@[reducible] def I32.ofInt   := @Scalar.ofInt .I32
@[reducible] def I64.ofInt   := @Scalar.ofInt .I64
@[reducible] def I128.ofInt  := @Scalar.ofInt .I128
@[reducible] def Usize.ofInt := @Scalar.ofInt .Usize
@[reducible] def U8.ofInt    := @Scalar.ofInt .U8
@[reducible] def U16.ofInt   := @Scalar.ofInt .U16
@[reducible] def U32.ofInt   := @Scalar.ofInt .U32
@[reducible] def U64.ofInt   := @Scalar.ofInt .U64
@[reducible] def U128.ofInt  := @Scalar.ofInt .U128

@[simp] theorem Scalar.ofInt_val_eq {ty} (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty) : (Scalar.ofInt x h).val = x := by
  simp [Scalar.ofInt, Scalar.ofIntCore]

-- 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)

instance (ty : ScalarTy) : CoeOut (Scalar ty) Int where
  coe := λ v => v.val

@[simp] theorem Scalar.neq_to_neq_val {ty} : ∀ {i j : Scalar ty}, (¬ i = j) ↔ ¬ i.val = j.val := by
  intro i j; cases i; cases j; simp

-- -- 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