(* 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
val cancel_add_conv: conv
end
structure Group_Cancel: GROUP_CANCEL =
struct
val add1 = @{lemma "(A::'a::comm_monoid_add) == k + a ==> A + b == k + (a + b)"
by (simp only: add_ac)}
val add2 = @{lemma "(B::'a::comm_monoid_add) == k + b ==> a + B == k + (a + b)"
by (simp only: add_ac)}
val sub1 = @{lemma "(A::'a::ab_group_add) == k + a ==> A - b == k + (a - b)"
by (simp only: add_diff_eq)}
val sub2 = @{lemma "(B::'a::ab_group_add) == k + b ==> a - B == - k + (a - b)"
by (simp only: minus_add diff_conv_add_uminus add_ac)}
val neg1 = @{lemma "(A::'a::ab_group_add) == k + a ==> - A == - k + - a"
by (simp only: minus_add_distrib)}
val rule0 = @{lemma "(a::'a::comm_monoid_add) == a + 0"
by (simp only: add_0_right)}
val minus_minus = mk_meta_eq @{thm minus_minus}
fun move_to_front path = Conv.every_conv
[Conv.rewr_conv (Library.foldl (op RS) (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 (add1::path) x #> add_atoms pos (add2::path) y
| add_atoms pos path (Const (@{const_name Groups.minus}, _) $ x $ y) =
add_atoms pos (sub1::path) x #> add_atoms (not pos) (sub2::path) y
| add_atoms pos path (Const (@{const_name Groups.uminus}, _) $ x) =
add_atoms (not pos) (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})
val diff_minus_eq_add = mk_meta_eq @{thm diff_minus_eq_add}
val add_eq_diff_minus = Thm.symmetric diff_minus_eq_add
val cancel_add_conv = Conv.every_conv
[Conv.rewr_conv add_eq_diff_minus,
cancel_diff_conv,
Conv.rewr_conv diff_minus_eq_add]
end