# File ‹Tools/group_cancel.ML›

```(*  Title:      HOL/Tools/group_cancel.ML
Author:     Brian Huffman, TU Munich

Simplification procedures for abelian groups:
- Cancel complementary terms in sums.
- Cancel like terms on opposite sides of relations.
*)

signature GROUP_CANCEL =
sig
val cancel_diff_conv: conv
val cancel_eq_conv: conv
val cancel_le_conv: conv
val cancel_less_conv: conv
end

structure Group_Cancel: GROUP_CANCEL =
struct

val minus_minus = mk_meta_eq @{thm minus_minus}

fun move_to_front path = Conv.every_conv
[Conv.rewr_conv (Library.foldl (op RS) (@{thm group_cancel.rule0}, path)),
Conv.arg1_conv (Conv.repeat_conv (Conv.rewr_conv minus_minus))]

fun add_atoms pos path (Const (\<^const_name>‹Groups.plus›, _) \$ x \$ y) =
| add_atoms pos path (Const (\<^const_name>‹Groups.minus›, _) \$ x \$ y) =
add_atoms pos (@{thm group_cancel.sub1}::path) x #>
add_atoms (not pos) (@{thm group_cancel.sub2}::path) y
| add_atoms pos path (Const (\<^const_name>‹Groups.uminus›, _) \$ x) =
add_atoms (not pos) (@{thm group_cancel.neg1}::path) x
| add_atoms _ _ (Const (\<^const_name>‹Groups.zero›, _)) = I
| add_atoms pos path x = cons ((pos, x), path)

fun atoms t = add_atoms true [] t []

val coeff_ord = prod_ord bool_ord Term_Ord.term_ord

fun find_all_common ord xs ys =
let
fun find (xs as (x, px)::xs') (ys as (y, py)::ys') =
(case ord (x, y) of
EQUAL => (px, py) :: find xs' ys'
| LESS => find xs' ys
| GREATER => find xs ys')
| find _ _ = []
fun ord' ((x, _), (y, _)) = ord (x, y)
in
find (sort ord' xs) (sort ord' ys)
end

fun cancel_conv rule ct =
let
fun cancel1_conv (lpath, rpath) =
let
val lconv = move_to_front lpath
val rconv = move_to_front rpath
val conv1 = Conv.combination_conv (Conv.arg_conv lconv) rconv
in
conv1 then_conv Conv.rewr_conv rule
end
val ((_, lhs), rhs) = (apfst dest_comb o dest_comb) (Thm.term_of ct)
val common = find_all_common coeff_ord (atoms lhs) (atoms rhs)
val conv =
if null common then Conv.no_conv
else Conv.every_conv (map cancel1_conv common)
in conv ct end

val cancel_diff_conv = cancel_conv (mk_meta_eq @{thm add_diff_cancel_left})
val cancel_eq_conv = cancel_conv (mk_meta_eq @{thm add_left_cancel})
val cancel_le_conv = cancel_conv (mk_meta_eq @{thm add_le_cancel_left})
val cancel_less_conv = cancel_conv (mk_meta_eq @{thm add_less_cancel_left})