(* 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
val add1 = @{lemma "(A::'a::comm_monoid_add) == k + a ==> A + b == k + (a + b)"
by (simp only: ac_simps)}
val add2 = @{lemma "(B::'a::comm_monoid_add) == k + b ==> a + B == k + (a + b)"
by (simp only: ac_simps)}
val suc1 = @{lemma "A == k + a ==> Suc A == k + Suc a"
by (simp only: add_Suc_right)}
val rule0 = @{lemma "(a::'a::comm_monoid_add) == a + 0"
by (simp only: add_0_right)}
val norm_rules = map mk_meta_eq @{thms add_0_left add_0_right}
fun move_to_front ctxt path = Conv.every_conv
[Conv.rewr_conv (Library.foldl (op RS) (rule0, path)),
Conv.arg_conv (Raw_Simplifier.rewrite ctxt false norm_rules)]
fun add_atoms path (Const (\<^const_name>\<open>Groups.plus\<close>, _) $ x $ y) =
add_atoms (add1::path) x #> add_atoms (add2::path) y
| add_atoms path (Const (\<^const_name>\<open>Nat.Suc\<close>, _) $ x) =
add_atoms (suc1::path) x
| add_atoms _ (Const (\<^const_name>\<open>Groups.zero\<close>, _)) = 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;