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|
signature MATRIX =
sig
type 'entry matrix
val make: int * int * (int * int -> 'entry) -> 'entry matrix
val height: 'entry matrix -> int
val width: 'entry matrix -> int
val fetch: 'entry matrix * int * int -> 'entry
val fetchRow: 'entry matrix * int -> int -> 'entry
val fetchCol: 'entry matrix * int -> int -> 'entry
val store: 'entry matrix * int * int * 'entry -> unit
val storeRow: 'entry matrix * int -> int * 'entry -> unit
val storeCol: 'entry matrix * int -> int * 'entry -> unit
val rowSwap: 'entry matrix * int * int -> unit
val colSwap: 'entry matrix * int * int -> unit
val rowOp: 'entry matrix * int * int * ('entry * 'entry -> 'entry) -> unit
val colOp: 'entry matrix * int * int * ('entry * 'entry -> 'entry) -> unit
val copy: 'entry matrix -> 'entry matrix
val map: 'entry1 matrix * ('entry1 -> 'entry2) -> 'entry2 matrix
val toString: 'entry matrix * ('entry -> string) -> string
end
structure Matrix:> MATRIX =
struct
type 'entry matrix = int * int * 'entry array
exception sizeError
exception index
exception foldError
fun make (height: int, width: int, generator: int * int -> 'entry)
: 'entry matrix =
if height < 0 orelse width < 0
then raise sizeError
else (height,
width,
Array.tabulate (height*width,
fn z => generator (z div width,
z mod width)))
fun height (height, _, _) = height
fun width (width, _, _) = width
fun fetch ((height, width, mat), row, col) =
if 0 <= row
andalso row < height
andalso 0 <= col
andalso col < width
then Array.sub (mat, col + width*row)
else raise index
fun fetchRow ((height, width, mat), row) =
if 0 <= row andalso row < height
then let val offset = width * row
in fn col =>
if 0 <= col andalso col < width
then Array.sub (mat, col + offset)
else raise index
end
else raise index
fun fetchCol ((height, width, mat), col) =
if 0 <= col andalso col < width
then fn row =>
if 0 <= row andalso row < height
then Array.sub (mat, col + width*row)
else raise index
else raise index
fun store ((height, width, mat), row, col, entry) =
if 0 <= row
andalso row < height
andalso 0 <= col
andalso col < width
then Array.update (mat, col + width*row, entry)
else raise index
fun storeRow ((height, width, mat), row) =
if 0 <= row andalso row < height
then let val offset = width * row
in fn (col, entry) =>
if 0 <= col andalso col < width
then Array.update (mat, col + offset, entry)
else raise index
end
else raise index
fun storeCol ((height, width, mat), col) =
if 0 <= col andalso col < width
then fn (row, entry) =>
if 0 <= row andalso row < height
then Array.update (mat, col + width*row, entry)
else raise index
else raise index
fun swapLoop (from1: int -> 'entry,
to1: int * 'entry -> unit,
from2: int -> 'entry,
to2: int * 'entry -> unit,
limit: int): unit =
let fun loop (i: int): unit =
if i = limit
then ()
else let val tmp = from1 i
in to1 (i, from2 i);
to2 (i, tmp);
loop (i + 1)
end
in loop 0
end
fun rowSwap (mat as (height, width, _), row1, row2): unit =
if 0 <= row1 andalso row1 < height
andalso 0 <= row2 andalso row2 < height
then if row1 = row2
then ()
else swapLoop (fetchRow (mat, row1),
storeRow (mat, row1),
fetchRow (mat, row2),
storeRow (mat, row2),
width)
else raise index
fun colSwap (mat as (height, width, _), col1, col2): unit =
if 0 <= col1 andalso col1 < width
andalso 0 <= col2 andalso col2 < width
then if col1 = col2
then ()
else swapLoop (fetchCol (mat, col1),
storeCol (mat, col1),
fetchCol (mat, col2),
storeCol (mat, col2),
height)
else raise index
fun opLoop (from1: int -> 'entry,
from2: int -> 'entry,
to2: int * 'entry -> unit,
limit: int,
f: 'entry * 'entry -> 'entry): unit =
let fun loop (i: int): unit =
if i = limit
then ()
else (
to2 (i,
f (from1 i, from2 i));
loop (i + 1))
in loop 0
end
fun rowOp (mat as (height, width, _),
row1,
row2,
f: 'entry * 'entry -> 'entry): unit =
if 0 <= row1 andalso row1 < height
andalso 0 <= row2 andalso row2 < height
andalso row1 <> row2
then opLoop (fetchRow (mat, row1),
fetchRow (mat, row2),
storeRow (mat, row2),
width,
f)
else raise index
fun colOp (mat as (height, width, _),
col1,
col2,
f: 'entry * 'entry -> 'entry): unit =
if 0 <= col1 andalso col1 < width
andalso 0 <= col2 andalso col2 < width
andalso col1 <> col2
then opLoop (fetchCol (mat, col1),
fetchCol (mat, col2),
storeCol (mat, col2),
height,
f)
else raise index
fun copy ((height, width, mat)) =
(height,
width,
Array.tabulate (Array.length mat,
fn i => Array.sub (mat, i)))
fun map ((height, width, mat: 'entry1 Array.array),
f: 'entry1 -> 'entry2)
: 'entry2 matrix =
(height,
width,
Array.tabulate (Array.length mat,
fn i => f (Array.sub (mat, i))))
(* Natural fold a range of integers in reverse. *)
fun naturalFold (limit: int,
state: 'state,
folder: int * 'state -> 'state): 'state =
let fun loop (i: int, state: 'state) =
if i = 0
then state
else loop (i - 1, folder (i - 1, state))
in if limit < 0
then raise foldError
else loop (limit, state)
end
local val blank8 = Byte.charToByte #" "
fun makeBlanks size =
let val blanks = Word8Vector.tabulate (size,
fn _ => blank8)
in Byte.bytesToString blanks
end
in fun toString (mat: 'entry matrix, f: 'entry -> string): string =
let val mat as (height, width, _) = map (mat, f)
fun maxSize from (i, width) = Int.max (String.size (from i),
width)
fun colWidth col = naturalFold (height,
0,
maxSize (fetchCol (mat,
col)))
val widths = Vector.tabulate (width, colWidth)
fun doRow (row: int, ac: string list): string list =
let val from = fetchRow (mat, row)
fun loop (col: int, ac: string list) =
let val next = from col
val ac = next::ac
val s = String.size next
val pad = Vector.sub (widths, col) - s
val ac = if pad <= 0
then ac
else (makeBlanks pad)::ac
in if col = 0
then ac
else loop (col - 1,
" "::ac)
end
val ac = "\n"::ac
in if width = 0
then ac
else loop (width - 1, ac)
end
val pieces = naturalFold (height,
[],
doRow)
in String.concat pieces
end
end
end
val zero = IntInf.fromInt 0
fun smaller (a: IntInf.int, b: IntInf.int): bool =
(not (a = zero))
andalso (b = zero orelse IntInf.< (IntInf.abs a , IntInf.abs b))
fun smithNormalForm (mat: IntInf.int Matrix.matrix): IntInf.int Matrix.matrix =
let val height = Matrix.height mat
val width = Matrix.width mat
val mat = Matrix.copy mat
val range = Int.min (width, height)
fun dd pos =
let val matCol = Matrix.fetchCol (mat, pos)
val matRow = Matrix.fetchRow (mat, pos)
val _ = print ("dd: pos = " ^ (Int.toString pos) ^ "\n")
fun swapRowLoop (best, bestRow, bestCol, row) =
if row >= height
then (Matrix.rowSwap (mat, pos, bestRow);
Matrix.colSwap (mat, pos, bestCol))
else let val matRow = Matrix.fetchRow (mat, row)
fun swapColLoop (best, bestRow, bestCol, col) =
if col >= width
then swapRowLoop (best, bestRow, bestCol, row + 1)
else let val next = matRow col
in if smaller (next, best)
then swapColLoop (next, row, col, col + 1)
else swapColLoop (best, bestRow, bestCol, col + 1)
end
in swapColLoop (best, bestRow, bestCol, pos)
end
fun rowLoop row =
if row < height
then if (matCol row) = zero
then rowLoop (row + 1)
else (Matrix.rowOp (mat,
pos,
row,
let val x = IntInf.~ (IntInf.quot(matCol row, matCol pos))
in fn (lhs, rhs) => IntInf.+ (IntInf.* (lhs, x), rhs)
end);
if (matCol row) = zero
then rowLoop (row + 1)
else hitPosAgain ())
else let fun colLoop col =
if col < width
then if (matRow col) = zero
then colLoop (col + 1)
else (Matrix.colOp (mat,
pos,
col,
let val x = IntInf.~ (IntInf.quot (matRow col, matRow pos))
in fn (lhs, rhs) => IntInf.+ (IntInf.* (lhs, x), rhs)
end);
if (matRow col) = zero
then colLoop (col + 1)
else hitPosAgain ())
else ()
in colLoop (pos + 1)
end
and hitPosAgain () = (swapRowLoop (zero, pos, pos, pos);
rowLoop (pos + 1))
in hitPosAgain ()
end
fun loop pos =
if pos = range
then mat
else (dd pos;
loop (pos + 1))
in loop 0
end
val table = [[ 8, ~3, 1, 3, 6, 9, ~2, 4, ~9, ~9, 2, 3, 8, ~1, 3, ~5, 4, ~3, ~5, ~6, 8, 1, 4, ~5, 7, ~4, ~4, ~7, 7, 1, 4, ~3, 8, 4, ~4, ~8, 5, ~9, 3, ~4, 1, 9, ~8, ~6, ~2, 8, ~9, ~5, ~3, ~3],
[ 0, 8, ~6, ~2, ~3, 4, 5, ~2, 7, ~7, ~6, ~7, ~3, ~4, 9, 7, ~3, 3, 0, 3, 3, ~8, ~8, 2, 3, 8, 3, ~2, ~4, 3, ~6, ~6, ~2, 6, 5, ~1, ~3, 1, 8, ~8, 2, 1, ~7, ~7, ~7, ~3, ~6, 6, ~4, ~9],
[ 0, ~5, 8, ~9, 2, 4, 2, 7, ~4, 9, ~3, 6, ~2, 3, ~3, 0, ~9, 5, 8, ~1, 2, ~8, 3, 4, ~6, 5, ~6, ~5, ~8, 0, ~5, 3, ~2, ~5, 8, 7, ~1, 1, ~1, 7, 6, 3, 6, 5, 6, 8, 7, 9, 7, ~3],
[ 5, 4, 7, 2, 3, ~9, 7, ~7, 3, ~8, 7, 5, 5, ~2, ~6, ~3, 6, 5, 3, ~1, ~1, 4, 5, ~5, 5, 9, 9, 3, 8, ~3, ~1, 9, ~9, 6, ~7, 7, 4, 6, ~8, ~9, 0, ~3, ~2, ~7, 1, ~2, ~6, 7, 7, 7],
[ 2, 9, 9, 3, ~4, 0, 9, 2, 5, 3, ~5, ~3, ~1, 1, 8, ~6, 2, ~4, ~8, ~7, ~8, 4, 5, 8, ~1, ~1, 7, 2, 5, 5, ~4, ~7, ~3, ~7, 6, ~4, ~5, ~8, ~5, ~9, ~8, 5, ~5, ~5, 0, 8, 8, 6, 4, ~1],
[ 5, 5, 1, ~7, 3, ~5, 4, 9, 3, 4, 4, ~5, 7, ~1, 7, 4, ~7, 7, ~7, ~2, 9, ~9, 0, ~4, ~4, 0, 2, 6, 3, ~1, 6, 6, 8, ~6, ~4, ~9, 3, ~2, ~5, 5, ~3, 2, ~1, ~6, 9, 3, ~3, ~8, ~9, 7],
[ 7, 1, 2, 7, 6, 5, ~6, ~3, ~4, ~8, 0, 9, 6, 1, 2, ~5, 4, 4, 4, ~6, ~7, ~9, ~6, 2, ~4, 5, ~2, 1, 0, 1, ~8, 7, ~7, ~5, 4, 1, ~5, 4, ~4, ~2, ~3, 1, 1, 3, 4, ~4, ~5, 9, 8, ~2],
[ 6, 2, ~1, ~8, 4, ~7, 7, ~3, ~2, ~5, 3, 0, 3, ~9, 3, 3, 9, ~1, 4, 8, ~9, 6, ~5, 9, 5, ~1, ~1, ~9, 7, ~2, 3, 9, 8, 9, 2, 7, 7, 6, ~1, ~1, ~2, ~2, ~7, 3, ~6, 0, ~9, 4, 3, 7],
[ 0, ~6, ~3, ~7, ~1, 5, ~2, 8, ~5, ~3, ~8, 7, ~2, ~2, 0, ~8, 4, 8, 9, ~5, ~4, ~8, ~1, 7, 1, 1, 6, ~9, ~4, 0, 8, 4, 3, ~7, 6, 0, 1, 8, 6, ~1, ~1, ~7, 9, ~9, ~5, ~2, ~2, ~1, 1, 0],
[~4, 9, 6, ~3, ~2, ~6, ~3, 4, 8, ~8, 1, ~5, 9, 7, 9, 7, ~9, ~6, 6, 1, ~3, 3, ~3, ~7, 1, 7, ~7, 0, ~2, 7, ~4, ~6, 0, 1, ~3, ~5, ~9, ~7, 8, 4, 9, ~8, ~8, ~7, ~6, 7, 6, ~3, ~8, 5],
[ 6, 7, ~5, ~9, 6, 1, 8, 4, ~2, 7, ~7, ~1, ~9, 1, ~6, ~5, 4, 9, 6, 0, ~8, ~3, 1, ~3, 8, ~3, 2, 9, ~3, ~9, ~1, ~3, 4, 3, 2, ~9, ~5, ~3, 8, ~4, 8, 5, ~4, 7, 6, ~8, 7, 6, ~5, 5],
[ 1, 7, ~8, ~9, ~7, ~3, 8, 9, ~7, ~1, ~7, 4, 0, 0, 1, ~5, 9, ~8, ~1, ~2, 3, 5, 9, ~9, 5, 4, ~9, 1, ~4, ~2, 3, ~4, 8, ~6, ~4, ~8, ~5, ~5, 4, ~2, ~4, ~1, ~9, ~5, 2, ~9, 2, ~9, ~2, ~3],
[~5, ~4, ~4, 9, 2, 7, ~2, 6, 7, 2, ~9, 4, 2, 7, 8, ~9, 2, 5, 3, 9, 6, 3, 0, ~7, ~6, ~7, 6, ~2, 9, ~3, ~6, 9, ~9, 2, 2, ~6, ~1, 4, ~3, 3, 0, 6, ~3, 4, 9, 9, ~6, 5, 5, ~5],
[ 5, ~7, 8, ~4, 8, 8, ~4, ~9, 6, 0, ~3, 6, 0, 8, 8, ~6, ~2, 5, 4, ~1, ~8, 1, ~3, ~1, 2, 3, ~9, ~9, ~5, 1, 8, ~5, ~3, 0, ~4, ~9, 0, ~6, 3, ~1, ~7, 0, 8, 9, ~6, ~1, ~9, 1, ~6, 2],
[ 7, ~5, ~1, 5, ~2, 7, 0, ~7, ~1, 8, 8, ~3, 9, ~5, 7, ~8, ~8, ~4, 3, 2, ~1, 8, ~2, 1, 2, 5, 0, ~6, 7, 3, 3, 7, ~5, 5, ~1, 1, 0, ~8, 1, 0, 0, ~4, 6, 9, ~5, ~6, 3, ~5, 8, 5],
[~4, ~2, 3, ~3, ~1, 2, ~2, ~1, ~9, ~5, 1, 0, 0, 2, 9, ~3, ~9, 2, 9, 3, 8, ~3, 4, 8, 8, 3, ~3, ~1, ~4, 4, ~6, ~9, 5, ~2, 1, 3, ~7, ~5, ~6, ~5, ~8, 4, ~8, ~3, 5, 0, 7, ~9, 6, 2],
[ 5, 1, 4, ~3, ~1, ~9, 5, ~8, ~8, 6, 1, 1, ~2, 7, 5, 6, ~4, 2, ~7, 0, ~7, ~3, ~5, 9, 3, 4, ~6, 8, ~4, 3, 6, 0, 2, 3, ~6, 3, 9, 4, 1, ~4, 6, ~5, ~7, 0, ~1, ~8, ~3, ~9, 9, 7],
[ 2, ~6, ~1, 8, 4, ~3, ~1, ~6, ~2, ~8, ~2, ~1, ~1, ~5, ~9, ~8, 9, ~9, 5, 1, 9, ~1, ~6, 9, ~7, 2, 8, ~7, 4, ~9, 7, 6, ~2, 1, ~2, ~7, 8, 0, 5, 0, ~5, ~7, ~6, 0, 4, 0, 3, ~8, 5, 4],
[~2, 9, ~9, ~6, 1, ~8, 8, 4, ~6, 8, 1, ~3, ~7, 8, ~5, 2, ~8, 1, 3, ~2, 6, 6, 6, 1, 0, 0, ~7, 7, ~3, ~3, 0, ~4, 3, ~7, ~6, 7, 5, 9, ~5, 7, ~8, 2, 3, ~8, ~7, 6, ~5, ~5, ~8, ~9],
[~7, ~4, 4, 1, ~1, ~3, ~8, 3, 7, 9, 8, 3, 0, 4, 4, ~1, ~5, 4, 2, 2, 0, 6, ~6, 2, ~9, 8, ~9, 3, ~2, 2, 6, 6, 1, 7, 1, 0, ~8, 2, 3, ~3, 8, 9, 5, 5, ~6, 4, ~7, ~4, ~2, ~3],
[~5, 8, 6, 1, ~6, ~6, 6, 1, 1, ~3, ~9, ~6, 2, ~7, 2, ~1, 6, ~6, 0, 2, ~7, 8, ~8, 4, 9, ~3, 9, ~7, ~9, ~6, ~4, ~4, ~5, 8, 2, ~5, ~4, ~3, 5, 2, 1, ~3, ~3, ~7, ~9, 3, 7, ~7, 3, ~8],
[~4, ~7, ~2, 2, ~4, ~2, 6, ~3, ~1, ~4, 0, ~5, 9, 7, ~6, ~9, 7, ~9, ~6, 2, ~3, 1, 5, ~9, 4, ~5, 4, ~9, 1, ~2, ~2, 4, 0, 4, ~8, ~8, 3, ~1, ~5, ~4, ~9, ~7, 7, 6, 3, ~9, 6, 4, ~4, ~7],
[~9, 6, 6, ~5, ~1, ~7, 4, ~9, 4, ~1, 6, ~4, 7, 2, 8, 7, 3, 1, ~7, 7, 7, 9, 8, ~9, 7, 2, 1, 2, ~8, 4, 5, 6, 7, 2, ~7, 6, 8, 4, ~9, 7, ~5, 6, 9, ~1, 9, 2, 0, 9, 3, 6],
[ 4, ~3, 8, 0, ~2, ~2, 2, ~3, 8, 3, 1, ~8, ~5, ~2, 5, 6, 8, 0, ~3, 4, ~2, 4, ~9, ~5, 7, 6, ~4, ~7, 2, 4, ~3, ~8, ~9, 9, 8, ~9, 3, ~7, 4, ~7, ~5, 4, 9, 3, ~6, ~3, ~7, 4, 2, ~2],
[~8, ~8, 6, ~2, ~6, 8, ~3, 3, ~1, ~7, 1, 9, 1, 7, ~6, 8, ~2, ~9, ~1, 3, ~4, 7, 8, ~1, 9, ~9, 6, ~3, 5, 0, 2, 5, ~1, ~6, ~6, 1, 8, 6, ~3, ~9, ~1, 9, ~2, 9, ~8, ~7, ~3, 6, ~3, ~3],
[ 5, ~2, 3, 0, ~9, ~8, ~6, 1, 8, 0, 1, 2, ~8, ~2, 0, ~9, ~8, 0, 5, ~3, ~4, 5, 6, ~2, ~5, 0, ~9, 9, ~9, ~5, 9, 9, ~5, ~2, 4, 3, 8, ~8, ~7, 5, ~3, ~2, 2, 3, 9, 7, ~1, 0, 4, ~1],
[~4, 5, ~5, 7, 8, 9, 7, ~3, 1, 9, ~7, ~1, 8, ~5, ~1, 2, ~8, 1, 0, 9, ~8, ~1, 6, ~1, 9, ~8, 7, 4, ~8, 7, 0, ~6, 2, 3, 7, 4, ~3, ~5, 9, ~3, 0, 6, ~9, 2, 4, ~8, 6, ~7, 9, 1],
[ 7, 0, ~9, 6, 8, 2, 2, 5, ~6, ~6, 9, ~5, 9, 2, 2, ~8, 0, ~6, ~9, ~6, ~4, ~9, 8, ~2, 9, 7, ~5, ~1, 7, 2, ~7, 7, ~1, ~3, 6, 6, 1, ~4, 0, ~1, ~6, ~5, 6, ~7, ~3, ~2, 8, 2, ~9, 8],
[ 8, ~7, ~9, ~6, 9, ~7, ~7, 6, ~8, 9, 5, ~4, 1, ~7, ~8, ~6, ~3, 8, ~8, 1, ~8, 6, 9, ~3, ~7, 7, 1, 6, 1, 0, 8, ~5, ~8, 8, ~9, 0, 4, 4, 3, ~4, 6, ~3, ~9, 0, 4, ~4, ~5, ~9, ~5, ~8],
[~3, ~2, 8, 1, ~1, ~1, ~4, 3, 7, ~2, ~9, 9, ~8, ~9, 6, ~4, 7, ~1, ~5, ~3, ~9, 0, ~3, 0, 7, 9, 1, ~2, 7, ~9, ~6, 3, 3, ~4, ~7, ~3, ~4, ~8, ~2, ~3, ~9, ~2, ~6, 3, ~6, ~4, 7, ~5, ~8, ~1],
[~9, ~9, ~2, ~9, ~9, 9, 6, 6, 7, 5, ~1, ~2, 1, 5, 2, ~3, ~4, 1, ~6, 0, ~3, ~9, ~1, 7, 0, ~9, 5, ~2, ~2, 5, 3, 4, ~1, 6, ~6, 3, ~6, 7, ~1, 5, ~8, ~4, ~2, ~2, ~6, ~5, ~6, 3, ~1, 4],
[ 7, 7, 8, 7, 6, 1, ~2, 5, ~6, 9, 4, 8, 5, 0, ~4, ~2, ~2, ~5, ~2, ~6, 9, ~8, ~2, ~5, ~9, 3, ~6, ~3, ~4, ~5, ~2, 6, 1, 6, ~5, 0, ~3, ~2, 4, ~6, 1, 6, ~1, 3, ~9, 2, ~3, 1, 5, ~6],
[ 6, 4, ~7, 3, ~7, 9, 1, ~7, ~8, 0, ~6, 8, 4, 1, 9, 6, 8, 3, 0, 9, 0, 4, 9, ~7, ~7, 1, 5, 1, ~5, 6, 9, 2, 4, 1, ~9, 8, 4, 5, 8, 3, 2, ~9, ~6, ~9, 9, ~9, 7, ~6, ~4, 3],
[~3, ~9, ~4, 2, 3, 9, ~9, 8, ~9, 9, ~4, ~9, ~5, 5, 0, 7, 3, ~5, ~8, 2, ~3, 0, ~9, ~3, 1, 9, 4, 5, ~1, 8, 0, ~4, ~2, 9, ~4, ~1, 3, 5, 9, ~1, 1, 4, ~8, ~2, ~3, 5, 1, 5, ~6, 7],
[ 9, ~3, 2, ~9, 3, 4, 0, 7, ~5, 9, 0, ~6, 7, ~2, 3, ~7, 2, ~5, ~2, 6, 3, ~9, ~5, ~9, 5, 2, ~5, ~3, 8, ~5, 6, 2, 9, ~7, ~7, ~7, ~6, 9, ~3, 6, 0, 6, ~6, ~9, 4, ~3, ~9, 0, ~4, ~9],
[~4, ~8, 8, ~7, 7, 0, ~6, ~6, 8, ~9, ~4, 5, ~3, ~1, 7, ~5, ~6, ~1, 8, 6, ~2, 1, ~1, 5, ~9, 1, ~1, ~7, ~6, ~6, ~6, ~4, 6, 3, ~5, ~5, ~6, 2, 3, ~6, ~8, ~3, 8, ~2, ~5, ~4, ~3, 1, 4, ~4],
[ 4, ~6, 2, 6, 2, ~8, 8, 5, 8, ~2, 0, ~6, ~1, ~6, ~2, 2, 6, ~9, ~7, ~6, ~4, ~4, ~7, ~2, 8, 6, 3, ~7, ~6, 8, 2, 3, 4, 5, 3, 4, ~6, 8, 8, ~1, 4, ~5, 6, 2, 8, ~3, ~9, ~2, 6, 7],
[ 3, ~4, 0, ~3, ~5, 0, ~2, ~6, ~2, 8, 5, ~9, ~4, ~8, ~6, 0, 8, 9, 1, ~2, 8, 2, ~2, 8, 9, 3, 3, 5, ~9, ~3, ~2, 7, 2, 9, 0, 4, 8, ~9, 0, ~6, 9, ~9, 9, ~4, 8, ~8, ~8, 2, ~3, 2],
[~1, 3, ~9, ~8, ~7, 6, ~6, 3, 0, 5, ~5, 1, 2, ~2, ~3, 7, 7, 3, ~4, ~2, ~9, ~5, ~1, 9, 6, 8, 2, 8, 7, ~3, 4, 6, 6, 0, ~2, 2, ~7, ~7, 6, ~3, 8, 2, 1, 0, 8, ~1, 3, 9, 8, 6],
[ 1, ~2, ~3, 6, 5, 5, ~6, ~4, ~5, 1, 1, 6, ~7, ~4, ~3, 4, 4, ~8, ~9, 7, ~2, ~3, ~7, ~2, 1, 2, 0, 8, ~6, ~5, ~5, 7, 8, 5, ~2, 3, 9, 0, 5, 1, 3, ~4, ~6, 1, 4, ~9, ~2, 5, 4, 3],
[ 3, 3, 9, ~2, 6, 9, 4, 9, 4, ~8, 5, ~1, 3, ~2, 1, ~7, ~3, 2, 2, 0, ~3, 3, 8, 2, 0, ~5, 7, 1, 4, ~8, 8, ~9, ~1, 1, ~9, ~4, 5, 2, 2, 8, 6, 1, 6, ~2, 2, 7, 1, ~6, ~1, ~1],
[ 4, ~2, 4, ~1, ~5, ~1, 5, ~2, 3, ~4, ~5, 0, 2, ~4, 6, 4, ~3, 2, 2, 5, ~6, ~7, ~9, ~1, ~9, ~9, 6, 0, 6, 5, 9, ~1, 3, ~3, ~8, 8, ~8, 8, 4, 5, ~1, ~5, 1, 0, 3, ~2, 5, 6, 6, 5],
[~4, 9, 6, 8, ~9, 5, 5, ~3, ~7, 7, 6, 8, ~8, 0, 4, ~1, 9, 5, ~7, 0, ~1, ~2, 3, 6, 0, 4, ~3, 1, 4, 6, 4, 0, 5, ~1, 7, ~7, ~6, ~8, ~3, ~6, 7, ~1, ~3, ~2, ~3, ~5, 3, 1, ~8, ~9],
[~6, 4, ~5, 9, 9, ~7, ~1, ~8, ~4, 2, ~6, 0, ~6, ~6, 7, 6, 0, 1, 7, ~7, 0, ~4, ~6, ~8, ~9, 5, ~6, ~9, 2, ~7, ~2, ~6, 9, 4, ~5, 0, 4, ~4, ~5, 6, 9, 1, ~6, ~5, 3, ~1, 7, ~7, ~6, 7],
[~8, 7, 7, ~6, 7, ~4, 8, 0, ~9, ~8, ~3, 7, ~3, 3, 8, ~7, ~2, ~7, 5, 5, ~5, 4, 6, 2, 4, 1, 4, ~9, ~3, 8, 8, ~9, ~4, ~2, 1, ~3, 1, 3, 9, ~5, ~8, ~2, 7, 8, 9, 2, 0, 1, ~9, 6],
[~7, 1, ~9, 5, ~5, ~5, 7, 6, ~5, ~9, ~6, ~8, ~6, 9, 7, 9, 0, ~5, 7, 7, ~6, 4, 5, ~9, ~1, ~2, ~7, 3, ~5, ~2, ~5, 5, ~3, ~4, ~2, ~8, 2, ~8, 0, ~8, 0, ~8, 9, 8, ~5, ~5, 1, 3, 5, ~4],
[~8, ~8, 0, ~5, ~8, ~6, 3, ~6, ~4, 6, 1, ~5, ~6, ~8, ~4, ~6, ~2, ~6, 6, ~4, 8, 8, 4, ~5, ~1, 0, 9, ~8, ~3, ~1, ~8, 7, ~3, 0, ~7, 1, ~7, ~1, ~7, 3, ~7, 3, ~4, ~8, 8, ~7, ~9, ~8, 3, 2],
[ 3, 6, 8, ~9, 7, 1, ~9, 9, 3, 8, 6, 4, ~2, 1, ~8, 4, ~7, ~4, ~3, 3, ~5, ~6, ~7, ~2, 0, ~4, 5, 2, 5, 6, 3, ~8, 2, ~5, ~7, 6, 8, ~2, ~5, ~4, 9, 9, 2, ~2, ~2, 7, 4, 4, ~2, 3],
[ 6, 6, ~5, ~2, ~8, ~2, ~9, 0, 2, 4, ~6, ~9, 9, 0, ~8, ~3, ~1, ~2, ~1, 6, 8, 2, ~9, 5, ~2, 1, 7, ~6, 5, 1, ~1, 4, ~4, ~7, ~6, ~3, ~8, 2, 2, 5, 5, ~6, 5, 3, 3, 7, 4, 7, ~3, ~9],
[~9, 6, ~4, 1, 3, ~8, ~8, ~8, ~1, 5, 1, 1, ~1, 6, 5, 1, ~1, 5, ~8, 8, ~7, ~5, ~1, ~1, 6, ~8, ~3, ~1, ~2, ~6, ~5, ~5, ~6, 0, 2, 2, 7, ~1, ~5, ~7, ~1, ~3, 7, 6, 0, 2, 4, ~5, 0, ~4]]
fun f (x, y) = List.nth (List.nth (table, x), y)
fun show m = print (Matrix.toString (m, IntInf.toString))
val _ =
let val dim = 32
val big = Matrix.map (Matrix.make (dim, dim, f), IntInf.fromInt)
in show big;
print "\n";
show (smithNormalForm big)
end
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