(* Title: HOL/OrderedGroup.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
Simplification procedures for abelian groups (e.g. integers, reals)
- Cancel complementary terms in sums
- Cancel like terms on opposite sides of relations
Modification in May 2004 by Steven Obua: polymorphic types work also now
Modification in June 2005 by Tobias Nipkow: cancel_sums works in general now
(using Clemens Ballarin's code for ordered rewriting in abelian groups)
and the whole file is reduced to a fraction of its former size.
*)
signature ABEL_CANCEL =
sig
val cancel_ss : simpset (*abelian group cancel simpset*)
val thy_ref : theory_ref (*signature of the theory of the group*)
val T : typ (*the type of group elements*)
val eq_reflection : thm (*object-equality to meta-equality*)
val eqI_rules : thm list
val dest_eqI : thm -> term
end;
functor Abel_Cancel (Data: ABEL_CANCEL) =
struct
open Data;
fun zero t = Const ("0", t);
fun minus t = Const ("uminus", t --> t);
fun add_terms pos (Const ("op +", _) $ x $ y, ts) =
add_terms pos (x, add_terms pos (y, ts))
| add_terms pos (Const ("op -", _) $ x $ y, ts) =
add_terms pos (x, add_terms (not pos) (y, ts))
| add_terms pos (Const ("uminus", _) $ x, ts) =
add_terms (not pos) (x, ts)
| add_terms pos (x, ts) = (pos,x) :: ts;
fun terms fml = add_terms true (fml, []);
fun zero1 pt (u as (c as Const("op +",_)) $ x $ y) =
(case zero1 pt x of NONE => (case zero1 pt y of NONE => NONE
| SOME z => SOME(c $ x $ z))
| SOME z => SOME(c $ z $ y))
| zero1 (pos,t) (u as (c as Const("op -",_)) $ x $ y) =
(case zero1 (pos,t) x of
NONE => (case zero1 (not pos,t) y of NONE => NONE
| SOME z => SOME(c $ x $ z))
| SOME z => SOME(c $ z $ y))
| zero1 (pos,t) (u as (c as Const("uminus",_)) $ x) =
(case zero1 (not pos,t) x of NONE => NONE
| SOME z => SOME(c $ z))
| zero1 (pos,t) u =
if pos andalso (t aconv u) then SOME(zero(fastype_of t)) else NONE
exception Cancel;
val trace = ref false;
fun find_common _ [] _ = raise Cancel
| find_common opp ((p,l)::ls) rs =
let val pr = if opp then not p else p
in if exists (fn (q,r) => pr = q andalso l aconv r) rs then (p,l)
else find_common opp ls rs
end
(* turns t1(t) OP t2(t) into t1(0) OP t2(0) where OP can be +, -, =, etc.
If OP = +, it must be t2(-t) rather than t2(t)
*)
fun cancel sg t =
let val _ = if !trace
then tracing ("Abel cancel: " ^ string_of_cterm (cterm_of sg t))
else ()
val c $ lhs $ rhs = t
val opp = case c of Const("op +",_) => true | _ => false;
val (pos,l) = find_common opp (terms lhs) (terms rhs)
val posr = if opp then not pos else pos
val t' = c $ (the(zero1 (pos,l) lhs)) $ (the(zero1 (posr,l) rhs))
val _ = if !trace
then tracing ("cancelled: " ^ string_of_cterm (cterm_of sg t'))
else ()
in t' end;
(* A simproc to cancel complementary terms in arbitrary sums. *)
fun sum_proc sg ss t =
let val t' = cancel sg t
val thm = Tactic.prove sg [] [] (Logic.mk_equals (t, t'))
(fn _ => simp_tac (Simplifier.inherit_context ss cancel_ss) 1)
in SOME thm end
handle Cancel => NONE;
val sum_conv =
Simplifier.mk_simproc "cancel_sums"
(map (Drule.cterm_fun Logic.varify o Thm.read_cterm (Theory.deref thy_ref))
[("x + y", Data.T), ("x - y", Data.T)]) (* FIXME depends on concrete syntax !???!!??! *)
sum_proc;
(*A simproc to cancel like terms on the opposite sides of relations:
(x + y - z < -z + x) = (y < 0)
Works for (=) and (<=) as well as (<), if the necessary rules are supplied.
Reduces the problem to subtraction.
*)
fun rel_proc sg ss t =
let val t' = cancel sg t
val thm = Tactic.prove sg [] [] (Logic.mk_equals (t, t'))
(fn _ => rtac eq_reflection 1 THEN
resolve_tac eqI_rules 1 THEN
simp_tac (Simplifier.inherit_context ss cancel_ss) 1)
in SOME thm end
handle Cancel => NONE;
val rel_conv =
Simplifier.simproc_i (Theory.deref thy_ref) "cancel_relations"
(map Data.dest_eqI eqI_rules) rel_proc;
end;