# File ‹Tools/nat_arith.ML›

```(* Author: Markus Wenzel, Stefan Berghofer, and Tobias Nipkow
Author: Brian Huffman

Basic arithmetic for natural numbers.
*)

signature NAT_ARITH =
sig
val cancel_diff_conv: Proof.context -> conv
val cancel_eq_conv: Proof.context -> conv
val cancel_le_conv: Proof.context -> conv
val cancel_less_conv: Proof.context -> conv
end;

structure Nat_Arith: NAT_ARITH =
struct

fun move_to_front ctxt path = Conv.every_conv
[Conv.rewr_conv (Library.foldl (op RS) (@{thm nat_arith.rule0}, path)),
Conv.arg_conv (Raw_Simplifier.rewrite ctxt false norm_rules)]

fun add_atoms path (Const (\<^const_name>‹Groups.plus›, _) \$ x \$ y) =
| add_atoms path (Const (\<^const_name>‹Nat.Suc›, _) \$ x) =
| add_atoms _ (Const (\<^const_name>‹Groups.zero›, _)) = I
| add_atoms path x = cons (x, path)

fun atoms t = add_atoms [] t []

exception Cancel

fun find_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)
| LESS => find xs' ys
| GREATER => find xs ys')
| find _ _ = raise Cancel
fun ord' ((x, _), (y, _)) = ord (x, y)
in
find (sort ord' xs) (sort ord' ys)
end

fun cancel_conv rule ctxt ct =
let
val ((_, lhs), rhs) = (apfst dest_comb o dest_comb) (Thm.term_of ct)
val (lpath, rpath) = find_common Term_Ord.term_ord (atoms lhs) (atoms rhs)
val lconv = move_to_front ctxt lpath
val rconv = move_to_front ctxt rpath
val conv1 = Conv.combination_conv (Conv.arg_conv lconv) rconv
val conv = conv1 then_conv Conv.rewr_conv rule
in conv ct end
handle Cancel => raise CTERM ("no_conversion", [])

val cancel_diff_conv = cancel_conv (mk_meta_eq @{thm add_diff_cancel_left [where ?'a = nat]})
val cancel_eq_conv = cancel_conv (mk_meta_eq @{thm add_left_cancel [where ?'a = nat]})
val cancel_le_conv = cancel_conv (mk_meta_eq @{thm add_le_cancel_left [where ?'a = nat]})
val cancel_less_conv = cancel_conv (mk_meta_eq @{thm add_less_cancel_left [where ?'a = nat]})

end;
```