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