4291
|
1 |
(* Title: Provers/Arith/cancel_sums.ML
|
|
2 |
ID: $Id$
|
|
3 |
Author: Markus Wenzel and Stefan Berghofer, TU Muenchen
|
|
4 |
|
|
5 |
Cancel common summands of balanced expressions:
|
|
6 |
|
|
7 |
A + x + B ~~ A' + x + B' == A + B ~~ A' + B'
|
|
8 |
|
|
9 |
where + is AC0 and ~~ an appropriate balancing operation (e.g. =, <=, <, -).
|
|
10 |
*)
|
|
11 |
|
|
12 |
signature CANCEL_SUMS_DATA =
|
|
13 |
sig
|
|
14 |
(*abstract syntax*)
|
|
15 |
val mk_sum: term list -> term
|
|
16 |
val dest_sum: term -> term list
|
|
17 |
val mk_bal: term * term -> term
|
|
18 |
val dest_bal: term -> term * term
|
|
19 |
(*rules*)
|
|
20 |
val prove_conv: tactic -> tactic -> Sign.sg -> term * term -> thm
|
|
21 |
val norm_tac: tactic (*AC0 etc.*)
|
|
22 |
val uncancel_tac: cterm -> tactic (*apply A ~~ B == x + A ~~ x + B*)
|
|
23 |
end;
|
|
24 |
|
|
25 |
signature CANCEL_SUMS =
|
|
26 |
sig
|
|
27 |
val proc: Sign.sg -> thm list -> term -> thm option
|
|
28 |
end;
|
|
29 |
|
|
30 |
|
|
31 |
functor CancelSumsFun(Data: CANCEL_SUMS_DATA): CANCEL_SUMS =
|
|
32 |
struct
|
|
33 |
|
|
34 |
|
|
35 |
(* cancel *)
|
|
36 |
|
|
37 |
fun cons1 x (xs, y, z) = (x :: xs, y, z);
|
|
38 |
fun cons2 y (x, ys, z) = (x, y :: ys, z);
|
|
39 |
fun cons12 x y (xs, ys, z) = (x :: xs, y :: ys, z);
|
|
40 |
|
4346
|
41 |
(*symmetric difference of multisets -- assumed to be sorted wrt. Logic.term_ord*)
|
4291
|
42 |
fun cancel ts [] vs = (ts, [], vs)
|
|
43 |
| cancel [] us vs = ([], us, vs)
|
|
44 |
| cancel (t :: ts) (u :: us) vs =
|
4452
|
45 |
(case Term.term_ord (t, u) of
|
4346
|
46 |
EQUAL => cancel ts us (t :: vs)
|
4291
|
47 |
| LESS => cons1 t (cancel ts (u :: us) vs)
|
|
48 |
| GREATER => cons2 u (cancel (t :: ts) us vs));
|
|
49 |
|
|
50 |
|
|
51 |
(* uncancel *)
|
|
52 |
|
|
53 |
fun uncancel_sums_tac _ [] = all_tac
|
|
54 |
| uncancel_sums_tac sg (t :: ts) =
|
|
55 |
Data.uncancel_tac (Thm.cterm_of sg t) THEN
|
|
56 |
uncancel_sums_tac sg ts;
|
|
57 |
|
|
58 |
|
|
59 |
(* the simplification procedure *)
|
|
60 |
|
|
61 |
fun proc sg _ t =
|
|
62 |
(case try Data.dest_bal t of
|
|
63 |
None => None
|
|
64 |
| Some bal =>
|
|
65 |
let
|
4452
|
66 |
val (ts, us) = pairself (sort Term.term_ord o Data.dest_sum) bal;
|
4291
|
67 |
val (ts', us', vs) = cancel ts us [];
|
|
68 |
in
|
|
69 |
if null vs then None
|
|
70 |
else Some
|
|
71 |
(Data.prove_conv (uncancel_sums_tac sg vs) Data.norm_tac sg
|
|
72 |
(t, Data.mk_bal (Data.mk_sum ts', Data.mk_sum us')))
|
|
73 |
end);
|
|
74 |
|
|
75 |
|
|
76 |
end;
|