author | wenzelm |
Thu, 22 Oct 1998 12:35:40 +0200 | |
changeset 5728 | 1d1175ef2d56 |
parent 5610 | 377acd99d74c |
child 6391 | 0da748358eff |
permissions | -rw-r--r-- |
5589 | 1 |
(* Title: Provers/Arith/abel_cancel.ML |
2 |
ID: $Id$ |
|
3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
4 |
Copyright 1998 University of Cambridge |
|
5 |
||
6 |
Simplification procedures for abelian groups (e.g. integers, reals) |
|
7 |
||
8 |
- Cancel complementary terms in sums |
|
9 |
- Cancel like terms on opposite sides of relations |
|
10 |
*) |
|
11 |
||
12 |
||
13 |
signature ABEL_CANCEL = |
|
14 |
sig |
|
5610 | 15 |
val ss : simpset (*basic simpset of object-logtic*) |
16 |
val eq_reflection : thm (*object-equality to meta-equality*) |
|
5589 | 17 |
|
5610 | 18 |
val thy : theory (*the theory of the group*) |
19 |
val T : typ (*the type of group elements*) |
|
5589 | 20 |
|
5610 | 21 |
val zero : term |
5589 | 22 |
val add_cancel_21 : thm |
23 |
val add_cancel_end : thm |
|
24 |
val add_left_cancel : thm |
|
5610 | 25 |
val add_assoc : thm |
26 |
val add_commute : thm |
|
27 |
val add_left_commute : thm |
|
28 |
val add_0 : thm |
|
29 |
val add_0_right : thm |
|
5589 | 30 |
|
31 |
val eq_diff_eq : thm |
|
32 |
val eqI_rules : thm list |
|
33 |
val dest_eqI : thm -> term |
|
34 |
||
35 |
val diff_def : thm |
|
36 |
val minus_add_distrib : thm |
|
37 |
val minus_minus : thm |
|
38 |
val minus_0 : thm |
|
39 |
||
40 |
val add_inverses : thm list |
|
41 |
val cancel_simps : thm list |
|
42 |
end; |
|
43 |
||
44 |
||
45 |
functor Abel_Cancel (Data: ABEL_CANCEL) = |
|
46 |
struct |
|
47 |
||
5728
1d1175ef2d56
eliminated 'let ... in structure ...' to make SML/NJ 0.93 happy;
wenzelm
parents:
5610
diff
changeset
|
48 |
open Data; |
1d1175ef2d56
eliminated 'let ... in structure ...' to make SML/NJ 0.93 happy;
wenzelm
parents:
5610
diff
changeset
|
49 |
|
5589 | 50 |
val prepare_ss = Data.ss addsimps [add_assoc, diff_def, |
51 |
minus_add_distrib, minus_minus, |
|
52 |
minus_0, add_0, add_0_right]; |
|
53 |
||
54 |
(*prove while suppressing timing information*) |
|
55 |
fun prove ct = setmp Goals.proof_timing false (prove_goalw_cterm [] ct); |
|
56 |
||
57 |
val plus = Const ("op +", [Data.T,Data.T] ---> Data.T); |
|
58 |
val minus = Const ("uminus", Data.T --> Data.T); |
|
59 |
||
60 |
(*Cancel corresponding terms on the two sides of the equation, NOT on |
|
61 |
the same side!*) |
|
62 |
val cancel_ss = |
|
63 |
Data.ss addsimps [add_cancel_21, add_cancel_end, minus_minus] @ |
|
64 |
Data.cancel_simps; |
|
65 |
||
66 |
val inverse_ss = Data.ss addsimps Data.add_inverses; |
|
67 |
||
68 |
fun mk_sum [] = Data.zero |
|
69 |
| mk_sum tms = foldr1 (fn (x,y) => plus $ x $ y) tms; |
|
70 |
||
71 |
(*We map -t to t and (in other cases) t to -t. No need to check the type of |
|
72 |
uminus, since the simproc is only called on sums of type T.*) |
|
73 |
fun negate (Const("uminus",_) $ t) = t |
|
74 |
| negate t = minus $ t; |
|
75 |
||
76 |
(*Flatten a formula built from +, binary - and unary -. |
|
77 |
No need to check types PROVIDED they are checked upon entry!*) |
|
78 |
fun add_terms neg (Const ("op +", _) $ x $ y, ts) = |
|
79 |
add_terms neg (x, add_terms neg (y, ts)) |
|
80 |
| add_terms neg (Const ("op -", _) $ x $ y, ts) = |
|
81 |
add_terms neg (x, add_terms (not neg) (y, ts)) |
|
82 |
| add_terms neg (Const ("uminus", _) $ x, ts) = |
|
83 |
add_terms (not neg) (x, ts) |
|
84 |
| add_terms neg (x, ts) = |
|
85 |
(if neg then negate x else x) :: ts; |
|
86 |
||
87 |
fun terms fml = add_terms false (fml, []); |
|
88 |
||
89 |
exception Cancel; |
|
90 |
||
91 |
(*Cancels just the first occurrence of u, leaving the rest unchanged*) |
|
92 |
fun cancelled (u, t::ts) = if u aconv t then ts else t :: cancelled(u,ts) |
|
93 |
| cancelled _ = raise Cancel; |
|
94 |
||
95 |
||
96 |
val trace = ref false; |
|
97 |
||
98 |
(*Make a simproc to cancel complementary terms in sums. Examples: |
|
99 |
x-x = 0 x+(y-x) = y -x+(y+(x+z)) = y+z |
|
100 |
It will unfold the definition of diff and associate to the right if |
|
101 |
necessary. Rewriting is faster if the formula is already |
|
102 |
in that form. |
|
103 |
*) |
|
104 |
||
105 |
fun sum_proc sg _ lhs = |
|
106 |
let val _ = if !trace then writeln ("cancel_sums: LHS = " ^ |
|
107 |
string_of_cterm (cterm_of sg lhs)) |
|
108 |
else () |
|
109 |
val (head::tail) = terms lhs |
|
110 |
val head' = negate head |
|
111 |
val rhs = mk_sum (cancelled (head',tail)) |
|
112 |
and chead' = Thm.cterm_of sg head' |
|
113 |
val _ = if !trace then |
|
114 |
writeln ("RHS = " ^ string_of_cterm (Thm.cterm_of sg rhs)) |
|
115 |
else () |
|
5610 | 116 |
val ct = Thm.cterm_of sg (Logic.mk_equals (lhs, rhs)) |
5589 | 117 |
val thm = prove ct |
5610 | 118 |
(fn _ => [rtac eq_reflection 1, |
119 |
simp_tac prepare_ss 1, |
|
5589 | 120 |
IF_UNSOLVED (simp_tac cancel_ss 1), |
121 |
IF_UNSOLVED (simp_tac inverse_ss 1)]) |
|
122 |
handle ERROR => |
|
123 |
error("cancel_sums simproc:\nfailed to prove " ^ |
|
124 |
string_of_cterm ct) |
|
5610 | 125 |
in Some thm end |
5589 | 126 |
handle Cancel => None; |
127 |
||
128 |
||
129 |
val sum_conv = |
|
130 |
Simplifier.mk_simproc "cancel_sums" |
|
131 |
(map (Thm.read_cterm (sign_of Data.thy)) |
|
132 |
[("x + y", Data.T), ("x - y", Data.T)]) |
|
133 |
sum_proc; |
|
134 |
||
135 |
||
136 |
(*A simproc to cancel like terms on the opposite sides of relations: |
|
137 |
(x + y - z < -z + x) = (y < 0) |
|
138 |
Works for (=) and (<=) as well as (<), if the necessary rules are supplied. |
|
139 |
Reduces the problem to subtraction and calls the previous simproc. |
|
140 |
*) |
|
141 |
||
142 |
(*Cancel the FIRST occurrence of a term. If it's repeated, then repeated |
|
143 |
calls to the simproc will be needed.*) |
|
144 |
fun cancel1 ([], u) = raise Match (*impossible: it's a common term*) |
|
145 |
| cancel1 (t::ts, u) = if t aconv u then ts |
|
146 |
else t :: cancel1 (ts,u); |
|
147 |
||
148 |
||
149 |
val sum_cancel_ss = Data.ss addsimprocs [sum_conv] |
|
150 |
addsimps [add_0, add_0_right]; |
|
151 |
||
152 |
val add_ac_ss = Data.ss addsimps [add_assoc,add_commute,add_left_commute]; |
|
153 |
||
154 |
fun rel_proc sg _ (lhs as (rel$lt$rt)) = |
|
155 |
let val _ = if !trace then writeln ("cancel_relations: LHS = " ^ |
|
156 |
string_of_cterm (cterm_of sg lhs)) |
|
157 |
else () |
|
158 |
val ltms = terms lt |
|
159 |
and rtms = terms rt |
|
160 |
val common = (*inter_term miscounts repetitions, so squash them*) |
|
161 |
gen_distinct (op aconv) (inter_term (ltms, rtms)) |
|
162 |
val _ = if null common then raise Cancel (*nothing to do*) |
|
163 |
else () |
|
164 |
||
165 |
fun cancelled tms = mk_sum (foldl cancel1 (tms, common)) |
|
166 |
||
167 |
val lt' = cancelled ltms |
|
168 |
and rt' = cancelled rtms |
|
169 |
||
170 |
val rhs = rel$lt'$rt' |
|
171 |
val _ = if !trace then |
|
172 |
writeln ("RHS = " ^ string_of_cterm (Thm.cterm_of sg rhs)) |
|
173 |
else () |
|
5610 | 174 |
val ct = Thm.cterm_of sg (Logic.mk_equals (lhs,rhs)) |
5589 | 175 |
|
176 |
val thm = prove ct |
|
5610 | 177 |
(fn _ => [rtac eq_reflection 1, |
178 |
resolve_tac eqI_rules 1, |
|
5589 | 179 |
simp_tac prepare_ss 1, |
180 |
simp_tac sum_cancel_ss 1, |
|
181 |
IF_UNSOLVED (simp_tac add_ac_ss 1)]) |
|
182 |
handle ERROR => |
|
183 |
error("cancel_relations simproc:\nfailed to prove " ^ |
|
184 |
string_of_cterm ct) |
|
5610 | 185 |
in Some thm end |
5589 | 186 |
handle Cancel => None; |
187 |
||
188 |
val rel_conv = |
|
189 |
Simplifier.mk_simproc "cancel_relations" |
|
190 |
(map (Thm.cterm_of (sign_of Data.thy) o Data.dest_eqI) eqI_rules) |
|
191 |
rel_proc; |
|
192 |
||
193 |
end; |