berghofe@13876
|
1 |
(* Title: HOL/Integ/cooper_dec.ML
|
berghofe@13876
|
2 |
ID: $Id$
|
berghofe@13876
|
3 |
Author: Amine Chaieb and Tobias Nipkow, TU Muenchen
|
berghofe@13876
|
4 |
|
berghofe@13876
|
5 |
File containing the implementation of Cooper Algorithm
|
berghofe@13876
|
6 |
decision procedure (intensively inspired from J.Harrison)
|
berghofe@13876
|
7 |
*)
|
chaieb@14920
|
8 |
|
chaieb@14920
|
9 |
|
berghofe@13876
|
10 |
signature COOPER_DEC =
|
berghofe@13876
|
11 |
sig
|
berghofe@13876
|
12 |
exception COOPER
|
chaieb@14941
|
13 |
exception COOPER_ORACLE of term
|
berghofe@13876
|
14 |
val is_arith_rel : term -> bool
|
berghofe@13876
|
15 |
val mk_numeral : int -> term
|
berghofe@13876
|
16 |
val dest_numeral : term -> int
|
chaieb@15164
|
17 |
val is_numeral : term -> bool
|
berghofe@13876
|
18 |
val zero : term
|
berghofe@13876
|
19 |
val one : term
|
berghofe@13876
|
20 |
val linear_cmul : int -> term -> term
|
berghofe@13876
|
21 |
val linear_add : string list -> term -> term -> term
|
berghofe@13876
|
22 |
val linear_sub : string list -> term -> term -> term
|
berghofe@13876
|
23 |
val linear_neg : term -> term
|
berghofe@13876
|
24 |
val lint : string list -> term -> term
|
berghofe@13876
|
25 |
val linform : string list -> term -> term
|
berghofe@13876
|
26 |
val formlcm : term -> term -> int
|
berghofe@13876
|
27 |
val adjustcoeff : term -> int -> term -> term
|
berghofe@13876
|
28 |
val unitycoeff : term -> term -> term
|
berghofe@13876
|
29 |
val divlcm : term -> term -> int
|
berghofe@13876
|
30 |
val bset : term -> term -> term list
|
berghofe@13876
|
31 |
val aset : term -> term -> term list
|
berghofe@13876
|
32 |
val linrep : string list -> term -> term -> term -> term
|
berghofe@13876
|
33 |
val list_disj : term list -> term
|
chaieb@14758
|
34 |
val list_conj : term list -> term
|
berghofe@13876
|
35 |
val simpl : term -> term
|
berghofe@13876
|
36 |
val fv : term -> string list
|
berghofe@13876
|
37 |
val negate : term -> term
|
berghofe@13876
|
38 |
val operations : (string * (int * int -> bool)) list
|
chaieb@14758
|
39 |
val conjuncts : term -> term list
|
chaieb@14758
|
40 |
val disjuncts : term -> term list
|
chaieb@14758
|
41 |
val has_bound : term -> bool
|
chaieb@14758
|
42 |
val minusinf : term -> term -> term
|
chaieb@14758
|
43 |
val plusinf : term -> term -> term
|
chaieb@14877
|
44 |
val onatoms : (term -> term) -> term -> term
|
chaieb@14877
|
45 |
val evalc : term -> term
|
chaieb@15107
|
46 |
val cooper_w : string list -> term -> (term option * term)
|
chaieb@14920
|
47 |
val integer_qelim : Term.term -> Term.term
|
chaieb@14941
|
48 |
val mk_presburger_oracle : (Sign.sg * exn) -> term
|
berghofe@13876
|
49 |
end;
|
berghofe@13876
|
50 |
|
berghofe@13876
|
51 |
structure CooperDec : COOPER_DEC =
|
berghofe@13876
|
52 |
struct
|
berghofe@13876
|
53 |
|
berghofe@13876
|
54 |
(* ========================================================================= *)
|
berghofe@13876
|
55 |
(* Cooper's algorithm for Presburger arithmetic. *)
|
berghofe@13876
|
56 |
(* ========================================================================= *)
|
berghofe@13876
|
57 |
exception COOPER;
|
berghofe@13876
|
58 |
|
chaieb@14941
|
59 |
(* Exception for the oracle *)
|
chaieb@14941
|
60 |
exception COOPER_ORACLE of term;
|
chaieb@14941
|
61 |
|
chaieb@14941
|
62 |
|
berghofe@13876
|
63 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
64 |
(* Lift operations up to numerals. *)
|
berghofe@13876
|
65 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
66 |
|
berghofe@13876
|
67 |
(*Assumption : The construction of atomar formulas in linearl arithmetic is based on
|
berghofe@13876
|
68 |
relation operations of Type : [int,int]---> bool *)
|
berghofe@13876
|
69 |
|
berghofe@13876
|
70 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
71 |
|
berghofe@13876
|
72 |
(*Function is_arith_rel returns true if and only if the term is an atomar presburger
|
berghofe@13876
|
73 |
formula *)
|
berghofe@13876
|
74 |
fun is_arith_rel tm = case tm of
|
berghofe@13876
|
75 |
Const(p,Type ("fun",[Type ("Numeral.bin", []),Type ("fun",[Type ("Numeral.bin",
|
berghofe@13876
|
76 |
[]),Type ("bool",[])] )])) $ _ $_ => true
|
berghofe@13876
|
77 |
|Const(p,Type ("fun",[Type ("IntDef.int", []),Type ("fun",[Type ("IntDef.int",
|
berghofe@13876
|
78 |
[]),Type ("bool",[])] )])) $ _ $_ => true
|
berghofe@13876
|
79 |
|_ => false;
|
berghofe@13876
|
80 |
|
berghofe@13876
|
81 |
(*Function is_arith_rel returns true if and only if the term is an operation of the
|
berghofe@13876
|
82 |
form [int,int]---> int*)
|
berghofe@13876
|
83 |
|
berghofe@13876
|
84 |
(*Transform a natural number to a term*)
|
berghofe@13876
|
85 |
|
berghofe@13876
|
86 |
fun mk_numeral 0 = Const("0",HOLogic.intT)
|
berghofe@13876
|
87 |
|mk_numeral 1 = Const("1",HOLogic.intT)
|
berghofe@13876
|
88 |
|mk_numeral n = (HOLogic.number_of_const HOLogic.intT) $ (HOLogic.mk_bin n);
|
berghofe@13876
|
89 |
|
berghofe@13876
|
90 |
(*Transform an Term to an natural number*)
|
berghofe@13876
|
91 |
|
berghofe@13876
|
92 |
fun dest_numeral (Const("0",Type ("IntDef.int", []))) = 0
|
berghofe@13876
|
93 |
|dest_numeral (Const("1",Type ("IntDef.int", []))) = 1
|
berghofe@13876
|
94 |
|dest_numeral (Const ("Numeral.number_of",_) $ n)= HOLogic.dest_binum n;
|
berghofe@13876
|
95 |
(*Some terms often used for pattern matching*)
|
berghofe@13876
|
96 |
|
berghofe@13876
|
97 |
val zero = mk_numeral 0;
|
berghofe@13876
|
98 |
val one = mk_numeral 1;
|
berghofe@13876
|
99 |
|
berghofe@13876
|
100 |
(*Tests if a Term is representing a number*)
|
berghofe@13876
|
101 |
|
berghofe@13876
|
102 |
fun is_numeral t = (t = zero) orelse (t = one) orelse (can dest_numeral t);
|
berghofe@13876
|
103 |
|
berghofe@13876
|
104 |
(*maps a unary natural function on a term containing an natural number*)
|
berghofe@13876
|
105 |
|
berghofe@13876
|
106 |
fun numeral1 f n = mk_numeral (f(dest_numeral n));
|
berghofe@13876
|
107 |
|
berghofe@13876
|
108 |
(*maps a binary natural function on 2 term containing natural numbers*)
|
berghofe@13876
|
109 |
|
berghofe@13876
|
110 |
fun numeral2 f m n = mk_numeral(f(dest_numeral m) (dest_numeral n));
|
berghofe@13876
|
111 |
|
berghofe@13876
|
112 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
113 |
(* Operations on canonical linear terms c1 * x1 + ... + cn * xn + k *)
|
berghofe@13876
|
114 |
(* *)
|
berghofe@13876
|
115 |
(* Note that we're quite strict: the ci must be present even if 1 *)
|
berghofe@13876
|
116 |
(* (but if 0 we expect the monomial to be omitted) and k must be there *)
|
berghofe@13876
|
117 |
(* even if it's zero. Thus, it's a constant iff not an addition term. *)
|
berghofe@13876
|
118 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
119 |
|
berghofe@13876
|
120 |
|
berghofe@13876
|
121 |
fun linear_cmul n tm = if n = 0 then zero else let fun times n k = n*k in
|
berghofe@13876
|
122 |
( case tm of
|
berghofe@13876
|
123 |
(Const("op +",T) $ (Const ("op *",T1 ) $c1 $ x1) $ rest) =>
|
berghofe@13876
|
124 |
Const("op +",T) $ ((Const("op *",T1) $ (numeral1 (times n) c1) $ x1)) $ (linear_cmul n rest)
|
berghofe@13876
|
125 |
|_ => numeral1 (times n) tm)
|
berghofe@13876
|
126 |
end ;
|
berghofe@13876
|
127 |
|
berghofe@13876
|
128 |
|
berghofe@13876
|
129 |
|
berghofe@13876
|
130 |
|
berghofe@13876
|
131 |
(* Whether the first of two items comes earlier in the list *)
|
berghofe@13876
|
132 |
fun earlier [] x y = false
|
berghofe@13876
|
133 |
|earlier (h::t) x y =if h = y then false
|
berghofe@13876
|
134 |
else if h = x then true
|
berghofe@13876
|
135 |
else earlier t x y ;
|
berghofe@13876
|
136 |
|
berghofe@13876
|
137 |
fun earlierv vars (Bound i) (Bound j) = i < j
|
berghofe@13876
|
138 |
|earlierv vars (Bound _) _ = true
|
berghofe@13876
|
139 |
|earlierv vars _ (Bound _) = false
|
berghofe@13876
|
140 |
|earlierv vars (Free (x,_)) (Free (y,_)) = earlier vars x y;
|
berghofe@13876
|
141 |
|
berghofe@13876
|
142 |
|
berghofe@13876
|
143 |
fun linear_add vars tm1 tm2 =
|
berghofe@13876
|
144 |
let fun addwith x y = x + y in
|
berghofe@13876
|
145 |
(case (tm1,tm2) of
|
berghofe@13876
|
146 |
((Const ("op +",T1) $ ( Const("op *",T2) $ c1 $ x1) $ rest1),(Const
|
berghofe@13876
|
147 |
("op +",T3)$( Const("op *",T4) $ c2 $ x2) $ rest2)) =>
|
berghofe@13876
|
148 |
if x1 = x2 then
|
berghofe@13876
|
149 |
let val c = (numeral2 (addwith) c1 c2)
|
berghofe@13876
|
150 |
in
|
berghofe@13876
|
151 |
if c = zero then (linear_add vars rest1 rest2)
|
berghofe@13876
|
152 |
else (Const("op +",T1) $ (Const("op *",T2) $ c $ x1) $ (linear_add vars rest1 rest2))
|
berghofe@13876
|
153 |
end
|
berghofe@13876
|
154 |
else
|
berghofe@13876
|
155 |
if earlierv vars x1 x2 then (Const("op +",T1) $
|
berghofe@13876
|
156 |
(Const("op *",T2)$ c1 $ x1) $ (linear_add vars rest1 tm2))
|
berghofe@13876
|
157 |
else (Const("op +",T1) $ (Const("op *",T2) $ c2 $ x2) $ (linear_add vars tm1 rest2))
|
berghofe@13876
|
158 |
|((Const("op +",T1) $ (Const("op *",T2) $ c1 $ x1) $ rest1) ,_) =>
|
berghofe@13876
|
159 |
(Const("op +",T1)$ (Const("op *",T2) $ c1 $ x1) $ (linear_add vars
|
berghofe@13876
|
160 |
rest1 tm2))
|
berghofe@13876
|
161 |
|(_, (Const("op +",T1) $(Const("op *",T2) $ c2 $ x2) $ rest2)) =>
|
berghofe@13876
|
162 |
(Const("op +",T1) $ (Const("op *",T2) $ c2 $ x2) $ (linear_add vars tm1
|
berghofe@13876
|
163 |
rest2))
|
berghofe@13876
|
164 |
| (_,_) => numeral2 (addwith) tm1 tm2)
|
berghofe@13876
|
165 |
|
berghofe@13876
|
166 |
end;
|
berghofe@13876
|
167 |
|
berghofe@13876
|
168 |
(*To obtain the unary - applyed on a formula*)
|
berghofe@13876
|
169 |
|
berghofe@13876
|
170 |
fun linear_neg tm = linear_cmul (0 - 1) tm;
|
berghofe@13876
|
171 |
|
berghofe@13876
|
172 |
(*Substraction of two terms *)
|
berghofe@13876
|
173 |
|
berghofe@13876
|
174 |
fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2);
|
berghofe@13876
|
175 |
|
berghofe@13876
|
176 |
|
berghofe@13876
|
177 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
178 |
(* Linearize a term. *)
|
berghofe@13876
|
179 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
180 |
|
berghofe@13876
|
181 |
(* linearises a term from the point of view of Variable Free (x,T).
|
berghofe@13876
|
182 |
After this fuction the all expressions containig ths variable will have the form
|
berghofe@13876
|
183 |
c*Free(x,T) + t where c is a constant ant t is a Term which is not containing
|
berghofe@13876
|
184 |
Free(x,T)*)
|
berghofe@13876
|
185 |
|
berghofe@13876
|
186 |
fun lint vars tm = if is_numeral tm then tm else case tm of
|
berghofe@13876
|
187 |
(Free (x,T)) => (HOLogic.mk_binop "op +" ((HOLogic.mk_binop "op *" ((mk_numeral 1),Free (x,T))), zero))
|
berghofe@13876
|
188 |
|(Bound i) => (Const("op +",HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $
|
berghofe@13876
|
189 |
(Const("op *",HOLogic.intT -->HOLogic.intT -->HOLogic.intT) $ (mk_numeral 1) $ (Bound i)) $ zero)
|
berghofe@13876
|
190 |
|(Const("uminus",_) $ t ) => (linear_neg (lint vars t))
|
berghofe@13876
|
191 |
|(Const("op +",_) $ s $ t) => (linear_add vars (lint vars s) (lint vars t))
|
berghofe@13876
|
192 |
|(Const("op -",_) $ s $ t) => (linear_sub vars (lint vars s) (lint vars t))
|
berghofe@13876
|
193 |
|(Const ("op *",_) $ s $ t) =>
|
berghofe@13876
|
194 |
let val s' = lint vars s
|
berghofe@13876
|
195 |
val t' = lint vars t
|
berghofe@13876
|
196 |
in
|
berghofe@13876
|
197 |
if is_numeral s' then (linear_cmul (dest_numeral s') t')
|
berghofe@13876
|
198 |
else if is_numeral t' then (linear_cmul (dest_numeral t') s')
|
berghofe@13876
|
199 |
|
berghofe@13876
|
200 |
else (warning "lint: apparent nonlinearity"; raise COOPER)
|
berghofe@13876
|
201 |
end
|
chaieb@15107
|
202 |
|_ => error ("COOPER:lint: unknown term \n");
|
berghofe@13876
|
203 |
|
berghofe@13876
|
204 |
|
berghofe@13876
|
205 |
|
berghofe@13876
|
206 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
207 |
(* Linearize the atoms in a formula, and eliminate non-strict inequalities. *)
|
berghofe@13876
|
208 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
209 |
|
berghofe@13876
|
210 |
fun mkatom vars p t = Const(p,HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ zero $ (lint vars t);
|
berghofe@13876
|
211 |
|
chaieb@15164
|
212 |
fun linform vars (Const ("Divides.op dvd",_) $ c $ t) =
|
chaieb@15164
|
213 |
if is_numeral c then
|
berghofe@13876
|
214 |
let val c' = (mk_numeral(abs(dest_numeral c)))
|
berghofe@13876
|
215 |
in (HOLogic.mk_binrel "Divides.op dvd" (c,lint vars t))
|
berghofe@13876
|
216 |
end
|
chaieb@15164
|
217 |
else (warning "Nonlinear term --- Non numeral leftside at dvd"
|
chaieb@15164
|
218 |
;raise COOPER)
|
berghofe@13876
|
219 |
|linform vars (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ s $ t ) = (mkatom vars "op =" (Const ("op -",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s) )
|
berghofe@13876
|
220 |
|linform vars (Const("op <",_)$ s $t ) = (mkatom vars "op <" (Const ("op -",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ t $ s))
|
berghofe@13876
|
221 |
|linform vars (Const("op >",_) $ s $ t ) = (mkatom vars "op <" (Const ("op -",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ s $ t))
|
berghofe@13876
|
222 |
|linform vars (Const("op <=",_)$ s $ t ) =
|
berghofe@13876
|
223 |
(mkatom vars "op <" (Const ("op -",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $ (Const("op +",HOLogic.intT --> HOLogic.intT --> HOLogic.intT) $t $(mk_numeral 1)) $ s))
|
berghofe@13876
|
224 |
|linform vars (Const("op >=",_)$ s $ t ) =
|
berghofe@13876
|
225 |
(mkatom vars "op <" (Const ("op -",HOLogic.intT --> HOLogic.intT -->
|
berghofe@13876
|
226 |
HOLogic.intT) $ (Const("op +",HOLogic.intT --> HOLogic.intT -->
|
berghofe@13876
|
227 |
HOLogic.intT) $s $(mk_numeral 1)) $ t))
|
berghofe@13876
|
228 |
|
berghofe@13876
|
229 |
|linform vars fm = fm;
|
berghofe@13876
|
230 |
|
berghofe@13876
|
231 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
232 |
(* Post-NNF transformation eliminating negated inequalities. *)
|
berghofe@13876
|
233 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
234 |
|
berghofe@13876
|
235 |
fun posineq fm = case fm of
|
berghofe@13876
|
236 |
(Const ("Not",_)$(Const("op <",_)$ c $ t)) =>
|
berghofe@13876
|
237 |
(HOLogic.mk_binrel "op <" (zero , (linear_sub [] (mk_numeral 1) (linear_add [] c t ) )))
|
berghofe@13876
|
238 |
| ( Const ("op &",_) $ p $ q) => HOLogic.mk_conj (posineq p,posineq q)
|
berghofe@13876
|
239 |
| ( Const ("op |",_) $ p $ q ) => HOLogic.mk_disj (posineq p,posineq q)
|
berghofe@13876
|
240 |
| _ => fm;
|
berghofe@13876
|
241 |
|
berghofe@13876
|
242 |
|
berghofe@13876
|
243 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
244 |
(* Find the LCM of the coefficients of x. *)
|
berghofe@13876
|
245 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
246 |
(*gcd calculates gcd (a,b) and helps lcm_num calculating lcm (a,b)*)
|
berghofe@13876
|
247 |
|
berghofe@13876
|
248 |
fun gcd a b = if a=0 then b else gcd (b mod a) a ;
|
berghofe@13876
|
249 |
fun lcm_num a b = (abs a*b) div (gcd (abs a) (abs b));
|
berghofe@13876
|
250 |
|
berghofe@13876
|
251 |
fun formlcm x fm = case fm of
|
berghofe@13876
|
252 |
(Const (p,_)$ _ $(Const ("op +", _)$(Const ("op *",_)$ c $ y ) $z ) ) => if
|
berghofe@13876
|
253 |
(is_arith_rel fm) andalso (x = y) then abs(dest_numeral c) else 1
|
berghofe@13876
|
254 |
| ( Const ("Not", _) $p) => formlcm x p
|
berghofe@13876
|
255 |
| ( Const ("op &",_) $ p $ q) => lcm_num (formlcm x p) (formlcm x q)
|
berghofe@13876
|
256 |
| ( Const ("op |",_) $ p $ q )=> lcm_num (formlcm x p) (formlcm x q)
|
berghofe@13876
|
257 |
| _ => 1;
|
berghofe@13876
|
258 |
|
berghofe@13876
|
259 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
260 |
(* Adjust all coefficients of x in formula; fold in reduction to +/- 1. *)
|
berghofe@13876
|
261 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
262 |
|
berghofe@13876
|
263 |
fun adjustcoeff x l fm =
|
berghofe@13876
|
264 |
case fm of
|
berghofe@13876
|
265 |
(Const(p,_) $d $( Const ("op +", _)$(Const ("op *",_) $
|
berghofe@13876
|
266 |
c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then
|
berghofe@13876
|
267 |
let val m = l div (dest_numeral c)
|
berghofe@13876
|
268 |
val n = (if p = "op <" then abs(m) else m)
|
berghofe@13876
|
269 |
val xtm = HOLogic.mk_binop "op *" ((mk_numeral (m div n)), x)
|
berghofe@13876
|
270 |
in
|
berghofe@13876
|
271 |
(HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "op +" ( xtm ,( linear_cmul n z) ))))
|
berghofe@13876
|
272 |
end
|
berghofe@13876
|
273 |
else fm
|
berghofe@13876
|
274 |
|( Const ("Not", _) $ p) => HOLogic.Not $ (adjustcoeff x l p)
|
berghofe@13876
|
275 |
|( Const ("op &",_) $ p $ q) => HOLogic.conj$(adjustcoeff x l p) $(adjustcoeff x l q)
|
berghofe@13876
|
276 |
|( Const ("op |",_) $ p $ q) => HOLogic.disj $(adjustcoeff x l p)$ (adjustcoeff x l q)
|
berghofe@13876
|
277 |
|_ => fm;
|
berghofe@13876
|
278 |
|
berghofe@13876
|
279 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
280 |
(* Hence make coefficient of x one in existential formula. *)
|
berghofe@13876
|
281 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
282 |
|
berghofe@13876
|
283 |
fun unitycoeff x fm =
|
berghofe@13876
|
284 |
let val l = formlcm x fm
|
berghofe@13876
|
285 |
val fm' = adjustcoeff x l fm in
|
berghofe@13876
|
286 |
if l = 1 then fm' else
|
berghofe@13876
|
287 |
let val xp = (HOLogic.mk_binop "op +"
|
berghofe@13876
|
288 |
((HOLogic.mk_binop "op *" ((mk_numeral 1), x )), zero)) in
|
berghofe@13876
|
289 |
HOLogic.conj $(HOLogic.mk_binrel "Divides.op dvd" ((mk_numeral l) , xp )) $ (adjustcoeff x l fm)
|
berghofe@13876
|
290 |
end
|
berghofe@13876
|
291 |
end;
|
berghofe@13876
|
292 |
|
berghofe@13876
|
293 |
(* adjustcoeffeq l fm adjusts the coeffitients c_i of x overall in fm to l*)
|
berghofe@13876
|
294 |
(* Here l must be a multiple of all c_i otherwise the obtained formula is not equivalent*)
|
berghofe@13876
|
295 |
(*
|
berghofe@13876
|
296 |
fun adjustcoeffeq x l fm =
|
berghofe@13876
|
297 |
case fm of
|
berghofe@13876
|
298 |
(Const(p,_) $d $( Const ("op +", _)$(Const ("op *",_) $
|
berghofe@13876
|
299 |
c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then
|
berghofe@13876
|
300 |
let val m = l div (dest_numeral c)
|
berghofe@13876
|
301 |
val n = (if p = "op <" then abs(m) else m)
|
berghofe@13876
|
302 |
val xtm = (HOLogic.mk_binop "op *" ((mk_numeral ((m div n)*l) ), x))
|
berghofe@13876
|
303 |
in (HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "op +" ( xtm ,( linear_cmul n z) ))))
|
berghofe@13876
|
304 |
end
|
berghofe@13876
|
305 |
else fm
|
berghofe@13876
|
306 |
|( Const ("Not", _) $ p) => HOLogic.Not $ (adjustcoeffeq x l p)
|
berghofe@13876
|
307 |
|( Const ("op &",_) $ p $ q) => HOLogic.conj$(adjustcoeffeq x l p) $(adjustcoeffeq x l q)
|
berghofe@13876
|
308 |
|( Const ("op |",_) $ p $ q) => HOLogic.disj $(adjustcoeffeq x l p)$ (adjustcoeffeq x l q)
|
berghofe@13876
|
309 |
|_ => fm;
|
berghofe@13876
|
310 |
|
berghofe@13876
|
311 |
|
berghofe@13876
|
312 |
*)
|
berghofe@13876
|
313 |
|
berghofe@13876
|
314 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
315 |
(* The "minus infinity" version. *)
|
berghofe@13876
|
316 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
317 |
|
berghofe@13876
|
318 |
fun minusinf x fm = case fm of
|
berghofe@13876
|
319 |
(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ (c1 ) $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
|
berghofe@13876
|
320 |
if (is_arith_rel fm) andalso (x=y) andalso (c2 = one) andalso (c1 =zero) then HOLogic.false_const
|
berghofe@13876
|
321 |
else fm
|
berghofe@13876
|
322 |
|
berghofe@13876
|
323 |
|(Const("op <",_) $ c $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z
|
berghofe@13876
|
324 |
)) =>
|
berghofe@13876
|
325 |
if (x =y) andalso (pm1 = one) andalso (c = zero) then HOLogic.false_const else HOLogic.true_const
|
berghofe@13876
|
326 |
|
berghofe@13876
|
327 |
|(Const ("Not", _) $ p) => HOLogic.Not $ (minusinf x p)
|
berghofe@13876
|
328 |
|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (minusinf x p) $ (minusinf x q)
|
berghofe@13876
|
329 |
|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (minusinf x p) $ (minusinf x q)
|
berghofe@13876
|
330 |
|_ => fm;
|
berghofe@13876
|
331 |
|
berghofe@13876
|
332 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
333 |
(* The "Plus infinity" version. *)
|
berghofe@13876
|
334 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
335 |
|
berghofe@13876
|
336 |
fun plusinf x fm = case fm of
|
berghofe@13876
|
337 |
(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ (c1 ) $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
|
berghofe@13876
|
338 |
if (is_arith_rel fm) andalso (x=y) andalso (c2 = one) andalso (c1 =zero) then HOLogic.false_const
|
berghofe@13876
|
339 |
else fm
|
berghofe@13876
|
340 |
|
berghofe@13876
|
341 |
|(Const("op <",_) $ c $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z
|
berghofe@13876
|
342 |
)) =>
|
berghofe@13876
|
343 |
if (x =y) andalso (pm1 = one) andalso (c = zero) then HOLogic.true_const else HOLogic.false_const
|
berghofe@13876
|
344 |
|
berghofe@13876
|
345 |
|(Const ("Not", _) $ p) => HOLogic.Not $ (plusinf x p)
|
berghofe@13876
|
346 |
|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (plusinf x p) $ (plusinf x q)
|
berghofe@13876
|
347 |
|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (plusinf x p) $ (plusinf x q)
|
berghofe@13876
|
348 |
|_ => fm;
|
berghofe@13876
|
349 |
|
berghofe@13876
|
350 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
351 |
(* The LCM of all the divisors that involve x. *)
|
berghofe@13876
|
352 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
353 |
|
berghofe@13876
|
354 |
fun divlcm x (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z ) ) =
|
berghofe@13876
|
355 |
if x = y then abs(dest_numeral d) else 1
|
berghofe@13876
|
356 |
|divlcm x ( Const ("Not", _) $ p) = divlcm x p
|
berghofe@13876
|
357 |
|divlcm x ( Const ("op &",_) $ p $ q) = lcm_num (divlcm x p) (divlcm x q)
|
berghofe@13876
|
358 |
|divlcm x ( Const ("op |",_) $ p $ q ) = lcm_num (divlcm x p) (divlcm x q)
|
berghofe@13876
|
359 |
|divlcm x _ = 1;
|
berghofe@13876
|
360 |
|
berghofe@13876
|
361 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
362 |
(* Construct the B-set. *)
|
berghofe@13876
|
363 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
364 |
|
berghofe@13876
|
365 |
fun bset x fm = case fm of
|
berghofe@13876
|
366 |
(Const ("Not", _) $ p) => if (is_arith_rel p) then
|
berghofe@13876
|
367 |
(case p of
|
berghofe@13876
|
368 |
(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $c2 $y) $a ) )
|
berghofe@13876
|
369 |
=> if (is_arith_rel p) andalso (x= y) andalso (c2 = one) andalso (c1 = zero)
|
berghofe@13876
|
370 |
then [linear_neg a]
|
berghofe@13876
|
371 |
else bset x p
|
berghofe@13876
|
372 |
|_ =>[])
|
berghofe@13876
|
373 |
|
berghofe@13876
|
374 |
else bset x p
|
berghofe@13876
|
375 |
|(Const ("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +",_) $ (Const ("op *",_) $c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_neg(linear_add [] a (mk_numeral 1))] else []
|
berghofe@13876
|
376 |
|(Const ("op <",_) $ c1$ (Const ("op +",_) $(Const ("op *",_)$ c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_neg a] else []
|
berghofe@13876
|
377 |
|(Const ("op &",_) $ p $ q) => (bset x p) union (bset x q)
|
berghofe@13876
|
378 |
|(Const ("op |",_) $ p $ q) => (bset x p) union (bset x q)
|
berghofe@13876
|
379 |
|_ => [];
|
berghofe@13876
|
380 |
|
berghofe@13876
|
381 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
382 |
(* Construct the A-set. *)
|
berghofe@13876
|
383 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
384 |
|
berghofe@13876
|
385 |
fun aset x fm = case fm of
|
berghofe@13876
|
386 |
(Const ("Not", _) $ p) => if (is_arith_rel p) then
|
berghofe@13876
|
387 |
(case p of
|
berghofe@13876
|
388 |
(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $c2 $y) $a ) )
|
berghofe@13876
|
389 |
=> if (x= y) andalso (c2 = one) andalso (c1 = zero)
|
berghofe@13876
|
390 |
then [linear_neg a]
|
berghofe@13876
|
391 |
else []
|
berghofe@13876
|
392 |
|_ =>[])
|
berghofe@13876
|
393 |
|
berghofe@13876
|
394 |
else aset x p
|
berghofe@13876
|
395 |
|(Const ("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +",_) $ (Const ("op *",_) $c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = one) then [linear_sub [] (mk_numeral 1) a] else []
|
berghofe@13876
|
396 |
|(Const ("op <",_) $ c1$ (Const ("op +",_) $(Const ("op *",_)$ c2 $ x) $ a)) => if (c1 =zero) andalso (c2 = (mk_numeral (~1))) then [a] else []
|
berghofe@13876
|
397 |
|(Const ("op &",_) $ p $ q) => (aset x p) union (aset x q)
|
berghofe@13876
|
398 |
|(Const ("op |",_) $ p $ q) => (aset x p) union (aset x q)
|
berghofe@13876
|
399 |
|_ => [];
|
berghofe@13876
|
400 |
|
berghofe@13876
|
401 |
|
berghofe@13876
|
402 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
403 |
(* Replace top variable with another linear form, retaining canonicality. *)
|
berghofe@13876
|
404 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
405 |
|
berghofe@13876
|
406 |
fun linrep vars x t fm = case fm of
|
berghofe@13876
|
407 |
((Const(p,_)$ d $ (Const("op +",_)$(Const("op *",_)$ c $ y) $ z))) =>
|
berghofe@13876
|
408 |
if (x = y) andalso (is_arith_rel fm)
|
berghofe@13876
|
409 |
then
|
berghofe@13876
|
410 |
let val ct = linear_cmul (dest_numeral c) t
|
berghofe@13876
|
411 |
in (HOLogic.mk_binrel p (d, linear_add vars ct z))
|
berghofe@13876
|
412 |
end
|
berghofe@13876
|
413 |
else fm
|
berghofe@13876
|
414 |
|(Const ("Not", _) $ p) => HOLogic.Not $ (linrep vars x t p)
|
berghofe@13876
|
415 |
|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (linrep vars x t p) $ (linrep vars x t q)
|
berghofe@13876
|
416 |
|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (linrep vars x t p) $ (linrep vars x t q)
|
berghofe@13876
|
417 |
|_ => fm;
|
berghofe@13876
|
418 |
|
berghofe@13876
|
419 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
420 |
(* Evaluation of constant expressions. *)
|
berghofe@13876
|
421 |
(* ------------------------------------------------------------------------- *)
|
chaieb@15107
|
422 |
|
chaieb@15107
|
423 |
(* An other implementation of divides, that covers more cases*)
|
chaieb@15107
|
424 |
|
chaieb@15107
|
425 |
exception DVD_UNKNOWN
|
chaieb@15107
|
426 |
|
chaieb@15107
|
427 |
fun dvd_op (d, t) =
|
chaieb@15107
|
428 |
if not(is_numeral d) then raise DVD_UNKNOWN
|
chaieb@15107
|
429 |
else let
|
chaieb@15107
|
430 |
val dn = dest_numeral d
|
chaieb@15107
|
431 |
fun coeffs_of x = case x of
|
chaieb@15107
|
432 |
Const(p,_) $ tl $ tr =>
|
chaieb@15107
|
433 |
if p = "op +" then (coeffs_of tl) union (coeffs_of tr)
|
chaieb@15107
|
434 |
else if p = "op *"
|
chaieb@15107
|
435 |
then if (is_numeral tr)
|
chaieb@15107
|
436 |
then [(dest_numeral tr) * (dest_numeral tl)]
|
chaieb@15107
|
437 |
else [dest_numeral tl]
|
chaieb@15107
|
438 |
else []
|
chaieb@15107
|
439 |
|_ => if (is_numeral t) then [dest_numeral t] else []
|
chaieb@15107
|
440 |
val ts = coeffs_of t
|
chaieb@15107
|
441 |
in case ts of
|
chaieb@15107
|
442 |
[] => raise DVD_UNKNOWN
|
chaieb@15107
|
443 |
|_ => foldr (fn(k,r) => r andalso (k mod dn = 0)) (ts,true)
|
chaieb@15107
|
444 |
end;
|
chaieb@15107
|
445 |
|
chaieb@15107
|
446 |
|
berghofe@13876
|
447 |
val operations =
|
berghofe@13876
|
448 |
[("op =",op=), ("op <",op<), ("op >",op>), ("op <=",op<=) , ("op >=",op>=),
|
berghofe@13876
|
449 |
("Divides.op dvd",fn (x,y) =>((y mod x) = 0))];
|
berghofe@13876
|
450 |
|
berghofe@13876
|
451 |
fun applyoperation (Some f) (a,b) = f (a, b)
|
berghofe@13876
|
452 |
|applyoperation _ (_, _) = false;
|
berghofe@13876
|
453 |
|
berghofe@13876
|
454 |
(*Evaluation of constant atomic formulas*)
|
chaieb@15107
|
455 |
(*FIXME : This is an optimation but still incorrect !! *)
|
chaieb@15107
|
456 |
(*
|
berghofe@13876
|
457 |
fun evalc_atom at = case at of
|
chaieb@15107
|
458 |
(Const (p,_) $ s $ t) =>
|
chaieb@15107
|
459 |
(if p="Divides.op dvd" then
|
chaieb@15107
|
460 |
((if dvd_op(s,t) then HOLogic.true_const
|
chaieb@15107
|
461 |
else HOLogic.false_const)
|
chaieb@15107
|
462 |
handle _ => at)
|
chaieb@15107
|
463 |
else
|
chaieb@15107
|
464 |
case assoc (operations,p) of
|
chaieb@15107
|
465 |
Some f => ((if (f ((dest_numeral s),(dest_numeral t))) then HOLogic.true_const else HOLogic.false_const)
|
chaieb@15107
|
466 |
handle _ => at)
|
chaieb@15107
|
467 |
| _ => at)
|
chaieb@15107
|
468 |
|Const("Not",_)$(Const (p,_) $ s $ t) =>(
|
chaieb@15107
|
469 |
case assoc (operations,p) of
|
chaieb@15107
|
470 |
Some f => ((if (f ((dest_numeral s),(dest_numeral t))) then
|
chaieb@15107
|
471 |
HOLogic.false_const else HOLogic.true_const)
|
chaieb@15107
|
472 |
handle _ => at)
|
chaieb@15107
|
473 |
| _ => at)
|
chaieb@15107
|
474 |
| _ => at;
|
chaieb@15107
|
475 |
|
chaieb@15107
|
476 |
*)
|
chaieb@15107
|
477 |
|
chaieb@15107
|
478 |
fun evalc_atom at = case at of
|
chaieb@15107
|
479 |
(Const (p,_) $ s $ t) =>
|
chaieb@15107
|
480 |
( case assoc (operations,p) of
|
chaieb@15107
|
481 |
Some f => ((if (f ((dest_numeral s),(dest_numeral t))) then HOLogic.true_const else HOLogic.false_const)
|
chaieb@15107
|
482 |
handle _ => at)
|
chaieb@15107
|
483 |
| _ => at)
|
chaieb@15107
|
484 |
|Const("Not",_)$(Const (p,_) $ s $ t) =>(
|
chaieb@15107
|
485 |
case assoc (operations,p) of
|
chaieb@15107
|
486 |
Some f => ((if (f ((dest_numeral s),(dest_numeral t))) then
|
chaieb@15107
|
487 |
HOLogic.false_const else HOLogic.true_const)
|
chaieb@15107
|
488 |
handle _ => at)
|
chaieb@15107
|
489 |
| _ => at)
|
chaieb@15107
|
490 |
| _ => at;
|
chaieb@15107
|
491 |
|
chaieb@15107
|
492 |
(*Function onatoms apllys function f on the atomic formulas involved in a.*)
|
berghofe@13876
|
493 |
|
berghofe@13876
|
494 |
fun onatoms f a = if (is_arith_rel a) then f a else case a of
|
berghofe@13876
|
495 |
|
berghofe@13876
|
496 |
(Const ("Not",_) $ p) => if is_arith_rel p then HOLogic.Not $ (f p)
|
berghofe@13876
|
497 |
|
berghofe@13876
|
498 |
else HOLogic.Not $ (onatoms f p)
|
berghofe@13876
|
499 |
|(Const ("op &",_) $ p $ q) => HOLogic.conj $ (onatoms f p) $ (onatoms f q)
|
berghofe@13876
|
500 |
|(Const ("op |",_) $ p $ q) => HOLogic.disj $ (onatoms f p) $ (onatoms f q)
|
berghofe@13876
|
501 |
|(Const ("op -->",_) $ p $ q) => HOLogic.imp $ (onatoms f p) $ (onatoms f q)
|
berghofe@13876
|
502 |
|((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q) => (Const ("op =", [HOLogic.boolT, HOLogic.boolT] ---> HOLogic.boolT)) $ (onatoms f p) $ (onatoms f q)
|
berghofe@13876
|
503 |
|(Const("All",_) $ Abs(x,T,p)) => Const("All", [HOLogic.intT -->
|
berghofe@13876
|
504 |
HOLogic.boolT] ---> HOLogic.boolT)$ Abs (x ,T, (onatoms f p))
|
berghofe@13876
|
505 |
|(Const("Ex",_) $ Abs(x,T,p)) => Const("Ex", [HOLogic.intT --> HOLogic.boolT]---> HOLogic.boolT) $ Abs( x ,T, (onatoms f p))
|
berghofe@13876
|
506 |
|_ => a;
|
berghofe@13876
|
507 |
|
berghofe@13876
|
508 |
val evalc = onatoms evalc_atom;
|
berghofe@13876
|
509 |
|
berghofe@13876
|
510 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
511 |
(* Hence overall quantifier elimination. *)
|
berghofe@13876
|
512 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
513 |
|
berghofe@13876
|
514 |
(*Applyes a function iteratively on the list*)
|
berghofe@13876
|
515 |
|
berghofe@13876
|
516 |
fun end_itlist f [] = error "end_itlist"
|
berghofe@13876
|
517 |
|end_itlist f [x] = x
|
berghofe@13876
|
518 |
|end_itlist f (h::t) = f h (end_itlist f t);
|
berghofe@13876
|
519 |
|
berghofe@13876
|
520 |
|
berghofe@13876
|
521 |
(*list_disj[conj] makes a disj[conj] of a given list. used with conjucts or disjuncts
|
berghofe@13876
|
522 |
it liearises iterated conj[disj]unctions. *)
|
berghofe@13876
|
523 |
|
berghofe@13876
|
524 |
fun disj_help p q = HOLogic.disj $ p $ q ;
|
berghofe@13876
|
525 |
|
berghofe@13876
|
526 |
fun list_disj l =
|
berghofe@13876
|
527 |
if l = [] then HOLogic.false_const else end_itlist disj_help l;
|
berghofe@13876
|
528 |
|
berghofe@13876
|
529 |
fun conj_help p q = HOLogic.conj $ p $ q ;
|
berghofe@13876
|
530 |
|
berghofe@13876
|
531 |
fun list_conj l =
|
berghofe@13876
|
532 |
if l = [] then HOLogic.true_const else end_itlist conj_help l;
|
berghofe@13876
|
533 |
|
berghofe@13876
|
534 |
(*Simplification of Formulas *)
|
berghofe@13876
|
535 |
|
berghofe@13876
|
536 |
(*Function q_bnd_chk checks if a quantified Formula makes sens : Means if in
|
berghofe@13876
|
537 |
the body of the existential quantifier there are bound variables to the
|
berghofe@13876
|
538 |
existential quantifier.*)
|
berghofe@13876
|
539 |
|
berghofe@13876
|
540 |
fun has_bound fm =let fun has_boundh fm i = case fm of
|
berghofe@13876
|
541 |
Bound n => (i = n)
|
berghofe@13876
|
542 |
|Abs (_,_,p) => has_boundh p (i+1)
|
berghofe@13876
|
543 |
|t1 $ t2 => (has_boundh t1 i) orelse (has_boundh t2 i)
|
berghofe@13876
|
544 |
|_ =>false
|
berghofe@13876
|
545 |
|
berghofe@13876
|
546 |
in case fm of
|
berghofe@13876
|
547 |
Bound _ => true
|
berghofe@13876
|
548 |
|Abs (_,_,p) => has_boundh p 0
|
berghofe@13876
|
549 |
|t1 $ t2 => (has_bound t1 ) orelse (has_bound t2 )
|
berghofe@13876
|
550 |
|_ =>false
|
berghofe@13876
|
551 |
end;
|
berghofe@13876
|
552 |
|
berghofe@13876
|
553 |
(*has_sub_abs checks if in a given Formula there are subformulas which are quantifed
|
berghofe@13876
|
554 |
too. Is no used no more.*)
|
berghofe@13876
|
555 |
|
berghofe@13876
|
556 |
fun has_sub_abs fm = case fm of
|
berghofe@13876
|
557 |
Abs (_,_,_) => true
|
berghofe@13876
|
558 |
|t1 $ t2 => (has_bound t1 ) orelse (has_bound t2 )
|
berghofe@13876
|
559 |
|_ =>false ;
|
berghofe@13876
|
560 |
|
berghofe@13876
|
561 |
(*update_bounds called with i=0 udates the numeration of bounded variables because the
|
berghofe@13876
|
562 |
formula will not be quantified any more.*)
|
berghofe@13876
|
563 |
|
berghofe@13876
|
564 |
fun update_bounds fm i = case fm of
|
berghofe@13876
|
565 |
Bound n => if n >= i then Bound (n-1) else fm
|
berghofe@13876
|
566 |
|Abs (x,T,p) => Abs(x,T,(update_bounds p (i+1)))
|
berghofe@13876
|
567 |
|t1 $ t2 => (update_bounds t1 i) $ (update_bounds t2 i)
|
berghofe@13876
|
568 |
|_ => fm ;
|
berghofe@13876
|
569 |
|
berghofe@13876
|
570 |
(*psimpl : Simplification of propositions (general purpose)*)
|
berghofe@13876
|
571 |
fun psimpl1 fm = case fm of
|
berghofe@13876
|
572 |
Const("Not",_) $ Const ("False",_) => HOLogic.true_const
|
berghofe@13876
|
573 |
| Const("Not",_) $ Const ("True",_) => HOLogic.false_const
|
berghofe@13876
|
574 |
| Const("op &",_) $ Const ("False",_) $ q => HOLogic.false_const
|
berghofe@13876
|
575 |
| Const("op &",_) $ p $ Const ("False",_) => HOLogic.false_const
|
berghofe@13876
|
576 |
| Const("op &",_) $ Const ("True",_) $ q => q
|
berghofe@13876
|
577 |
| Const("op &",_) $ p $ Const ("True",_) => p
|
berghofe@13876
|
578 |
| Const("op |",_) $ Const ("False",_) $ q => q
|
berghofe@13876
|
579 |
| Const("op |",_) $ p $ Const ("False",_) => p
|
berghofe@13876
|
580 |
| Const("op |",_) $ Const ("True",_) $ q => HOLogic.true_const
|
berghofe@13876
|
581 |
| Const("op |",_) $ p $ Const ("True",_) => HOLogic.true_const
|
berghofe@13876
|
582 |
| Const("op -->",_) $ Const ("False",_) $ q => HOLogic.true_const
|
berghofe@13876
|
583 |
| Const("op -->",_) $ Const ("True",_) $ q => q
|
berghofe@13876
|
584 |
| Const("op -->",_) $ p $ Const ("True",_) => HOLogic.true_const
|
berghofe@13876
|
585 |
| Const("op -->",_) $ p $ Const ("False",_) => HOLogic.Not $ p
|
berghofe@13876
|
586 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ Const ("True",_) $ q => q
|
berghofe@13876
|
587 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ Const ("True",_) => p
|
berghofe@13876
|
588 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ Const ("False",_) $ q => HOLogic.Not $ q
|
berghofe@13876
|
589 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ Const ("False",_) => HOLogic.Not $ p
|
berghofe@13876
|
590 |
| _ => fm;
|
berghofe@13876
|
591 |
|
berghofe@13876
|
592 |
fun psimpl fm = case fm of
|
berghofe@13876
|
593 |
Const ("Not",_) $ p => psimpl1 (HOLogic.Not $ (psimpl p))
|
berghofe@13876
|
594 |
| Const("op &",_) $ p $ q => psimpl1 (HOLogic.mk_conj (psimpl p,psimpl q))
|
berghofe@13876
|
595 |
| Const("op |",_) $ p $ q => psimpl1 (HOLogic.mk_disj (psimpl p,psimpl q))
|
berghofe@13876
|
596 |
| Const("op -->",_) $ p $ q => psimpl1 (HOLogic.mk_imp(psimpl p,psimpl q))
|
berghofe@13876
|
597 |
| Const("op =", Type ("fun",[Type ("bool", []),_])) $ p $ q => psimpl1 (HOLogic.mk_eq(psimpl p,psimpl q))
|
berghofe@13876
|
598 |
| _ => fm;
|
berghofe@13876
|
599 |
|
berghofe@13876
|
600 |
|
berghofe@13876
|
601 |
(*simpl : Simplification of Terms involving quantifiers too.
|
berghofe@13876
|
602 |
This function is able to drop out some quantified expressions where there are no
|
berghofe@13876
|
603 |
bound varaibles.*)
|
berghofe@13876
|
604 |
|
berghofe@13876
|
605 |
fun simpl1 fm =
|
berghofe@13876
|
606 |
case fm of
|
berghofe@13876
|
607 |
Const("All",_) $Abs(x,_,p) => if (has_bound fm ) then fm
|
berghofe@13876
|
608 |
else (update_bounds p 0)
|
berghofe@13876
|
609 |
| Const("Ex",_) $ Abs (x,_,p) => if has_bound fm then fm
|
berghofe@13876
|
610 |
else (update_bounds p 0)
|
berghofe@13876
|
611 |
| _ => psimpl1 fm;
|
berghofe@13876
|
612 |
|
berghofe@13876
|
613 |
fun simpl fm = case fm of
|
berghofe@13876
|
614 |
Const ("Not",_) $ p => simpl1 (HOLogic.Not $(simpl p))
|
berghofe@13876
|
615 |
| Const ("op &",_) $ p $ q => simpl1 (HOLogic.mk_conj (simpl p ,simpl q))
|
berghofe@13876
|
616 |
| Const ("op |",_) $ p $ q => simpl1 (HOLogic.mk_disj (simpl p ,simpl q ))
|
berghofe@13876
|
617 |
| Const ("op -->",_) $ p $ q => simpl1 (HOLogic.mk_imp(simpl p ,simpl q ))
|
berghofe@13876
|
618 |
| Const("op =", Type ("fun",[Type ("bool", []),_]))$ p $ q => simpl1
|
berghofe@13876
|
619 |
(HOLogic.mk_eq(simpl p ,simpl q ))
|
chaieb@14920
|
620 |
(* | Const ("All",Ta) $ Abs(Vn,VT,p) => simpl1(Const("All",Ta) $
|
berghofe@13876
|
621 |
Abs(Vn,VT,simpl p ))
|
berghofe@13876
|
622 |
| Const ("Ex",Ta) $ Abs(Vn,VT,p) => simpl1(Const("Ex",Ta) $
|
berghofe@13876
|
623 |
Abs(Vn,VT,simpl p ))
|
chaieb@14920
|
624 |
*)
|
berghofe@13876
|
625 |
| _ => fm;
|
berghofe@13876
|
626 |
|
berghofe@13876
|
627 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
628 |
|
berghofe@13876
|
629 |
(* Puts fm into NNF*)
|
berghofe@13876
|
630 |
|
berghofe@13876
|
631 |
fun nnf fm = if (is_arith_rel fm) then fm
|
berghofe@13876
|
632 |
else (case fm of
|
berghofe@13876
|
633 |
( Const ("op &",_) $ p $ q) => HOLogic.conj $ (nnf p) $(nnf q)
|
berghofe@13876
|
634 |
| (Const("op |",_) $ p $q) => HOLogic.disj $ (nnf p)$(nnf q)
|
berghofe@13876
|
635 |
| (Const ("op -->",_) $ p $ q) => HOLogic.disj $ (nnf (HOLogic.Not $ p)) $ (nnf q)
|
berghofe@13876
|
636 |
| ((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q) =>(HOLogic.disj $ (HOLogic.conj $ (nnf p) $ (nnf q)) $ (HOLogic.conj $ (nnf (HOLogic.Not $ p) ) $ (nnf(HOLogic.Not $ q))))
|
berghofe@13876
|
637 |
| (Const ("Not",_)) $ ((Const ("Not",_)) $ p) => (nnf p)
|
berghofe@13876
|
638 |
| (Const ("Not",_)) $ (( Const ("op &",_)) $ p $ q) =>HOLogic.disj $ (nnf(HOLogic.Not $ p)) $ (nnf(HOLogic.Not $q))
|
berghofe@13876
|
639 |
| (Const ("Not",_)) $ (( Const ("op |",_)) $ p $ q) =>HOLogic.conj $ (nnf(HOLogic.Not $ p)) $ (nnf(HOLogic.Not $ q))
|
berghofe@13876
|
640 |
| (Const ("Not",_)) $ (( Const ("op -->",_)) $ p $ q ) =>HOLogic.conj $ (nnf p) $(nnf(HOLogic.Not $ q))
|
berghofe@13876
|
641 |
| (Const ("Not",_)) $ ((Const ("op =", Type ("fun",[Type ("bool", []),_]))) $ p $ q ) =>(HOLogic.disj $ (HOLogic.conj $(nnf p) $ (nnf(HOLogic.Not $ q))) $ (HOLogic.conj $(nnf(HOLogic.Not $ p)) $ (nnf q)))
|
berghofe@13876
|
642 |
| _ => fm);
|
berghofe@13876
|
643 |
|
berghofe@13876
|
644 |
|
berghofe@13876
|
645 |
(* Function remred to remove redundancy in a list while keeping the order of appearance of the
|
berghofe@13876
|
646 |
elements. but VERY INEFFICIENT!! *)
|
berghofe@13876
|
647 |
|
berghofe@13876
|
648 |
fun remred1 el [] = []
|
berghofe@13876
|
649 |
|remred1 el (h::t) = if el=h then (remred1 el t) else h::(remred1 el t);
|
berghofe@13876
|
650 |
|
berghofe@13876
|
651 |
fun remred [] = []
|
berghofe@13876
|
652 |
|remred (x::l) = x::(remred1 x (remred l));
|
berghofe@13876
|
653 |
|
berghofe@13876
|
654 |
(*Makes sure that all free Variables are of the type integer but this function is only
|
berghofe@13876
|
655 |
used temporarily, this job must be done by the parser later on.*)
|
berghofe@13876
|
656 |
|
berghofe@13876
|
657 |
fun mk_uni_vars T (node $ rest) = (case node of
|
berghofe@13876
|
658 |
Free (name,_) => Free (name,T) $ (mk_uni_vars T rest)
|
berghofe@13876
|
659 |
|_=> (mk_uni_vars T node) $ (mk_uni_vars T rest ) )
|
berghofe@13876
|
660 |
|mk_uni_vars T (Free (v,_)) = Free (v,T)
|
berghofe@13876
|
661 |
|mk_uni_vars T tm = tm;
|
berghofe@13876
|
662 |
|
berghofe@13876
|
663 |
fun mk_uni_int T (Const ("0",T2)) = if T = T2 then (mk_numeral 0) else (Const ("0",T2))
|
berghofe@13876
|
664 |
|mk_uni_int T (Const ("1",T2)) = if T = T2 then (mk_numeral 1) else (Const ("1",T2))
|
berghofe@13876
|
665 |
|mk_uni_int T (node $ rest) = (mk_uni_int T node) $ (mk_uni_int T rest )
|
berghofe@13876
|
666 |
|mk_uni_int T (Abs(AV,AT,p)) = Abs(AV,AT,mk_uni_int T p)
|
berghofe@13876
|
667 |
|mk_uni_int T tm = tm;
|
berghofe@13876
|
668 |
|
berghofe@13876
|
669 |
|
berghofe@13876
|
670 |
(* Minusinfinity Version*)
|
berghofe@13876
|
671 |
fun coopermi vars1 fm =
|
berghofe@13876
|
672 |
case fm of
|
berghofe@13876
|
673 |
Const ("Ex",_) $ Abs(x0,T,p0) => let
|
berghofe@13876
|
674 |
val (xn,p1) = variant_abs (x0,T,p0)
|
berghofe@13876
|
675 |
val x = Free (xn,T)
|
berghofe@13876
|
676 |
val vars = (xn::vars1)
|
berghofe@13876
|
677 |
val p = unitycoeff x (posineq (simpl p1))
|
berghofe@13876
|
678 |
val p_inf = simpl (minusinf x p)
|
berghofe@13876
|
679 |
val bset = bset x p
|
berghofe@13876
|
680 |
val js = 1 upto divlcm x p
|
berghofe@13876
|
681 |
fun p_element j b = linrep vars x (linear_add vars b (mk_numeral j)) p
|
berghofe@13876
|
682 |
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) bset)
|
berghofe@13876
|
683 |
in (list_disj (map stage js))
|
berghofe@13876
|
684 |
end
|
berghofe@13876
|
685 |
| _ => error "cooper: not an existential formula";
|
berghofe@13876
|
686 |
|
berghofe@13876
|
687 |
|
berghofe@13876
|
688 |
|
berghofe@13876
|
689 |
(* The plusinfinity version of cooper*)
|
berghofe@13876
|
690 |
fun cooperpi vars1 fm =
|
berghofe@13876
|
691 |
case fm of
|
berghofe@13876
|
692 |
Const ("Ex",_) $ Abs(x0,T,p0) => let
|
berghofe@13876
|
693 |
val (xn,p1) = variant_abs (x0,T,p0)
|
berghofe@13876
|
694 |
val x = Free (xn,T)
|
berghofe@13876
|
695 |
val vars = (xn::vars1)
|
berghofe@13876
|
696 |
val p = unitycoeff x (posineq (simpl p1))
|
berghofe@13876
|
697 |
val p_inf = simpl (plusinf x p)
|
berghofe@13876
|
698 |
val aset = aset x p
|
berghofe@13876
|
699 |
val js = 1 upto divlcm x p
|
berghofe@13876
|
700 |
fun p_element j a = linrep vars x (linear_sub vars a (mk_numeral j)) p
|
berghofe@13876
|
701 |
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) aset)
|
berghofe@13876
|
702 |
in (list_disj (map stage js))
|
berghofe@13876
|
703 |
end
|
berghofe@13876
|
704 |
| _ => error "cooper: not an existential formula";
|
berghofe@13876
|
705 |
|
berghofe@13876
|
706 |
|
chaieb@15107
|
707 |
(* Try to find a withness for the formula *)
|
chaieb@15107
|
708 |
|
chaieb@15107
|
709 |
fun inf_w mi d vars x p =
|
chaieb@15107
|
710 |
let val f = if mi then minusinf else plusinf in
|
chaieb@15107
|
711 |
case (simpl (minusinf x p)) of
|
chaieb@15107
|
712 |
Const("True",_) => (Some (mk_numeral 1), HOLogic.true_const)
|
chaieb@15107
|
713 |
|Const("False",_) => (None,HOLogic.false_const)
|
chaieb@15107
|
714 |
|F =>
|
chaieb@15107
|
715 |
let
|
chaieb@15107
|
716 |
fun h n =
|
chaieb@15107
|
717 |
case ((simpl o evalc) (linrep vars x (mk_numeral n) F)) of
|
chaieb@15107
|
718 |
Const("True",_) => (Some (mk_numeral n),HOLogic.true_const)
|
chaieb@15107
|
719 |
|F' => if n=1 then (None,F')
|
chaieb@15107
|
720 |
else let val (rw,rf) = h (n-1) in
|
chaieb@15107
|
721 |
(rw,HOLogic.mk_disj(F',rf))
|
chaieb@15107
|
722 |
end
|
chaieb@15107
|
723 |
|
chaieb@15107
|
724 |
in (h d)
|
chaieb@15107
|
725 |
end
|
chaieb@15107
|
726 |
end;
|
chaieb@15107
|
727 |
|
chaieb@15107
|
728 |
fun set_w d b st vars x p = let
|
chaieb@15107
|
729 |
fun h ns = case ns of
|
chaieb@15107
|
730 |
[] => (None,HOLogic.false_const)
|
chaieb@15107
|
731 |
|n::nl => ( case ((simpl o evalc) (linrep vars x n p)) of
|
chaieb@15107
|
732 |
Const("True",_) => (Some n,HOLogic.true_const)
|
chaieb@15107
|
733 |
|F' => let val (rw,rf) = h nl
|
chaieb@15107
|
734 |
in (rw,HOLogic.mk_disj(F',rf))
|
chaieb@15107
|
735 |
end)
|
chaieb@15107
|
736 |
val f = if b then linear_add else linear_sub
|
chaieb@15107
|
737 |
val p_elements = foldr (fn (i,l) => l union (map (fn e => f [] e (mk_numeral i)) st)) (1 upto d,[])
|
chaieb@15107
|
738 |
in h p_elements
|
chaieb@15107
|
739 |
end;
|
chaieb@15107
|
740 |
|
chaieb@15107
|
741 |
fun withness d b st vars x p = case (inf_w b d vars x p) of
|
chaieb@15107
|
742 |
(Some n,_) => (Some n,HOLogic.true_const)
|
chaieb@15107
|
743 |
|(None,Pinf) => (case (set_w d b st vars x p) of
|
chaieb@15107
|
744 |
(Some n,_) => (Some n,HOLogic.true_const)
|
chaieb@15107
|
745 |
|(_,Pst) => (None,HOLogic.mk_disj(Pinf,Pst)));
|
chaieb@15107
|
746 |
|
chaieb@15107
|
747 |
|
chaieb@15107
|
748 |
|
berghofe@13876
|
749 |
|
berghofe@13876
|
750 |
(*Cooper main procedure*)
|
berghofe@13876
|
751 |
|
berghofe@13876
|
752 |
fun cooper vars1 fm =
|
berghofe@13876
|
753 |
case fm of
|
berghofe@13876
|
754 |
Const ("Ex",_) $ Abs(x0,T,p0) => let
|
berghofe@13876
|
755 |
val (xn,p1) = variant_abs (x0,T,p0)
|
berghofe@13876
|
756 |
val x = Free (xn,T)
|
berghofe@13876
|
757 |
val vars = (xn::vars1)
|
chaieb@14920
|
758 |
(* val p = unitycoeff x (posineq (simpl p1)) *)
|
chaieb@14920
|
759 |
val p = unitycoeff x p1
|
berghofe@13876
|
760 |
val ast = aset x p
|
berghofe@13876
|
761 |
val bst = bset x p
|
berghofe@13876
|
762 |
val js = 1 upto divlcm x p
|
berghofe@13876
|
763 |
val (p_inf,f,S ) =
|
berghofe@13876
|
764 |
if (length bst) < (length ast)
|
berghofe@13876
|
765 |
then (minusinf x p,linear_add,bst)
|
berghofe@13876
|
766 |
else (plusinf x p, linear_sub,ast)
|
berghofe@13876
|
767 |
fun p_element j a = linrep vars x (f vars a (mk_numeral j)) p
|
berghofe@13876
|
768 |
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) S)
|
berghofe@13876
|
769 |
in (list_disj (map stage js))
|
berghofe@13876
|
770 |
end
|
berghofe@13876
|
771 |
| _ => error "cooper: not an existential formula";
|
berghofe@13876
|
772 |
|
berghofe@13876
|
773 |
|
chaieb@15107
|
774 |
(* A Version of cooper that returns a withness *)
|
chaieb@15107
|
775 |
fun cooper_w vars1 fm =
|
chaieb@15107
|
776 |
case fm of
|
chaieb@15107
|
777 |
Const ("Ex",_) $ Abs(x0,T,p0) => let
|
chaieb@15107
|
778 |
val (xn,p1) = variant_abs (x0,T,p0)
|
chaieb@15107
|
779 |
val x = Free (xn,T)
|
chaieb@15107
|
780 |
val vars = (xn::vars1)
|
chaieb@15107
|
781 |
(* val p = unitycoeff x (posineq (simpl p1)) *)
|
chaieb@15107
|
782 |
val p = unitycoeff x p1
|
chaieb@15107
|
783 |
val ast = aset x p
|
chaieb@15107
|
784 |
val bst = bset x p
|
chaieb@15107
|
785 |
val d = divlcm x p
|
chaieb@15107
|
786 |
val (p_inf,S ) =
|
chaieb@15107
|
787 |
if (length bst) <= (length ast)
|
chaieb@15107
|
788 |
then (true,bst)
|
chaieb@15107
|
789 |
else (false,ast)
|
chaieb@15107
|
790 |
in withness d p_inf S vars x p
|
chaieb@15107
|
791 |
(* fun p_element j a = linrep vars x (f vars a (mk_numeral j)) p
|
chaieb@15107
|
792 |
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) S)
|
chaieb@15107
|
793 |
in (list_disj (map stage js))
|
chaieb@15107
|
794 |
*)
|
chaieb@15107
|
795 |
end
|
chaieb@15107
|
796 |
| _ => error "cooper: not an existential formula";
|
berghofe@13876
|
797 |
|
berghofe@13876
|
798 |
|
berghofe@13876
|
799 |
(*Function itlist applys a double parametred function f : 'a->'b->b iteratively to a List l : 'a
|
berghofe@13876
|
800 |
list With End condition b. ict calculates f(e1,f(f(e2,f(e3,...(...f(en,b))..)))))
|
berghofe@13876
|
801 |
assuming l = [e1,e2,...,en]*)
|
berghofe@13876
|
802 |
|
berghofe@13876
|
803 |
fun itlist f l b = case l of
|
berghofe@13876
|
804 |
[] => b
|
berghofe@13876
|
805 |
| (h::t) => f h (itlist f t b);
|
berghofe@13876
|
806 |
|
berghofe@13876
|
807 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
808 |
(* Free variables in terms and formulas. *)
|
berghofe@13876
|
809 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
810 |
|
berghofe@13876
|
811 |
fun fvt tml = case tml of
|
berghofe@13876
|
812 |
[] => []
|
berghofe@13876
|
813 |
| Free(x,_)::r => x::(fvt r)
|
berghofe@13876
|
814 |
|
berghofe@13876
|
815 |
fun fv fm = fvt (term_frees fm);
|
berghofe@13876
|
816 |
|
berghofe@13876
|
817 |
|
berghofe@13876
|
818 |
(* ========================================================================= *)
|
berghofe@13876
|
819 |
(* Quantifier elimination. *)
|
berghofe@13876
|
820 |
(* ========================================================================= *)
|
berghofe@13876
|
821 |
(*conj[/disj]uncts lists iterated conj[disj]unctions*)
|
berghofe@13876
|
822 |
|
berghofe@13876
|
823 |
fun disjuncts fm = case fm of
|
berghofe@13876
|
824 |
Const ("op |",_) $ p $ q => (disjuncts p) @ (disjuncts q)
|
berghofe@13876
|
825 |
| _ => [fm];
|
berghofe@13876
|
826 |
|
berghofe@13876
|
827 |
fun conjuncts fm = case fm of
|
berghofe@13876
|
828 |
Const ("op &",_) $p $ q => (conjuncts p) @ (conjuncts q)
|
berghofe@13876
|
829 |
| _ => [fm];
|
berghofe@13876
|
830 |
|
berghofe@13876
|
831 |
|
berghofe@13876
|
832 |
|
berghofe@13876
|
833 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
834 |
(* Lift procedure given literal modifier, formula normalizer & basic quelim. *)
|
chaieb@14920
|
835 |
(* ------------------------------------------------------------------------- *)
|
chaieb@14920
|
836 |
(*
|
chaieb@14920
|
837 |
fun lift_qelim afn nfn qfn isat =
|
chaieb@14920
|
838 |
let
|
chaieb@14920
|
839 |
fun qelift vars fm = if (isat fm) then afn vars fm
|
chaieb@14920
|
840 |
else
|
chaieb@14920
|
841 |
case fm of
|
chaieb@14920
|
842 |
Const ("Not",_) $ p => HOLogic.Not $ (qelift vars p)
|
chaieb@14920
|
843 |
| Const ("op &",_) $ p $q => HOLogic.conj $ (qelift vars p) $ (qelift vars q)
|
chaieb@14920
|
844 |
| Const ("op |",_) $ p $ q => HOLogic.disj $ (qelift vars p) $ (qelift vars q)
|
chaieb@14920
|
845 |
| Const ("op -->",_) $ p $ q => HOLogic.imp $ (qelift vars p) $ (qelift vars q)
|
chaieb@14920
|
846 |
| Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q => HOLogic.mk_eq ((qelift vars p),(qelift vars q))
|
chaieb@14920
|
847 |
| Const ("All",QT) $ Abs(x,T,p) => HOLogic.Not $(qelift vars (Const ("Ex",QT) $ Abs(x,T,(HOLogic.Not $ p))))
|
chaieb@14920
|
848 |
| (e as Const ("Ex",_)) $ Abs (x,T,p) => qfn vars (e$Abs (x,T,(nfn(qelift (x::vars) p))))
|
chaieb@14920
|
849 |
| _ => fm
|
chaieb@14920
|
850 |
|
chaieb@14920
|
851 |
in (fn fm => qelift (fv fm) fm)
|
chaieb@14920
|
852 |
end;
|
chaieb@14920
|
853 |
*)
|
chaieb@14920
|
854 |
|
berghofe@13876
|
855 |
|
berghofe@13876
|
856 |
fun lift_qelim afn nfn qfn isat =
|
berghofe@13876
|
857 |
let fun qelim x vars p =
|
berghofe@13876
|
858 |
let val cjs = conjuncts p
|
berghofe@13876
|
859 |
val (ycjs,ncjs) = partition (has_bound) cjs in
|
berghofe@13876
|
860 |
(if ycjs = [] then p else
|
berghofe@13876
|
861 |
let val q = (qfn vars ((HOLogic.exists_const HOLogic.intT
|
berghofe@13876
|
862 |
) $ Abs(x,HOLogic.intT,(list_conj ycjs)))) in
|
berghofe@13876
|
863 |
(itlist conj_help ncjs q)
|
berghofe@13876
|
864 |
end)
|
berghofe@13876
|
865 |
end
|
berghofe@13876
|
866 |
|
berghofe@13876
|
867 |
fun qelift vars fm = if (isat fm) then afn vars fm
|
berghofe@13876
|
868 |
else
|
berghofe@13876
|
869 |
case fm of
|
berghofe@13876
|
870 |
Const ("Not",_) $ p => HOLogic.Not $ (qelift vars p)
|
berghofe@13876
|
871 |
| Const ("op &",_) $ p $q => HOLogic.conj $ (qelift vars p) $ (qelift vars q)
|
berghofe@13876
|
872 |
| Const ("op |",_) $ p $ q => HOLogic.disj $ (qelift vars p) $ (qelift vars q)
|
berghofe@13876
|
873 |
| Const ("op -->",_) $ p $ q => HOLogic.imp $ (qelift vars p) $ (qelift vars q)
|
berghofe@13876
|
874 |
| Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q => HOLogic.mk_eq ((qelift vars p),(qelift vars q))
|
berghofe@13876
|
875 |
| Const ("All",QT) $ Abs(x,T,p) => HOLogic.Not $(qelift vars (Const ("Ex",QT) $ Abs(x,T,(HOLogic.Not $ p))))
|
berghofe@13876
|
876 |
| Const ("Ex",_) $ Abs (x,T,p) => let val djs = disjuncts(nfn(qelift (x::vars) p)) in
|
berghofe@13876
|
877 |
list_disj(map (qelim x vars) djs) end
|
berghofe@13876
|
878 |
| _ => fm
|
berghofe@13876
|
879 |
|
berghofe@13876
|
880 |
in (fn fm => simpl(qelift (fv fm) fm))
|
berghofe@13876
|
881 |
end;
|
chaieb@14920
|
882 |
|
berghofe@13876
|
883 |
|
berghofe@13876
|
884 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
885 |
(* Cleverer (proposisional) NNF with conditional and literal modification. *)
|
berghofe@13876
|
886 |
(* ------------------------------------------------------------------------- *)
|
berghofe@13876
|
887 |
|
berghofe@13876
|
888 |
(*Function Negate used by cnnf, negates a formula p*)
|
berghofe@13876
|
889 |
|
berghofe@13876
|
890 |
fun negate (Const ("Not",_) $ p) = p
|
berghofe@13876
|
891 |
|negate p = (HOLogic.Not $ p);
|
berghofe@13876
|
892 |
|
berghofe@13876
|
893 |
fun cnnf lfn =
|
berghofe@13876
|
894 |
let fun cnnfh fm = case fm of
|
berghofe@13876
|
895 |
(Const ("op &",_) $ p $ q) => HOLogic.mk_conj(cnnfh p,cnnfh q)
|
berghofe@13876
|
896 |
| (Const ("op |",_) $ p $ q) => HOLogic.mk_disj(cnnfh p,cnnfh q)
|
berghofe@13876
|
897 |
| (Const ("op -->",_) $ p $q) => HOLogic.mk_disj(cnnfh(HOLogic.Not $ p),cnnfh q)
|
berghofe@13876
|
898 |
| (Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q) => HOLogic.mk_disj(
|
berghofe@13876
|
899 |
HOLogic.mk_conj(cnnfh p,cnnfh q),
|
berghofe@13876
|
900 |
HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $q)))
|
berghofe@13876
|
901 |
|
berghofe@13876
|
902 |
| (Const ("Not",_) $ (Const("Not",_) $ p)) => cnnfh p
|
berghofe@13876
|
903 |
| (Const ("Not",_) $ (Const ("op &",_) $ p $ q)) => HOLogic.mk_disj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))
|
berghofe@13876
|
904 |
| (Const ("Not",_) $(Const ("op |",_) $ (Const ("op &",_) $ p $ q) $
|
berghofe@13876
|
905 |
(Const ("op &",_) $ p1 $ r))) => if p1 = negate p then
|
berghofe@13876
|
906 |
HOLogic.mk_disj(
|
berghofe@13876
|
907 |
cnnfh (HOLogic.mk_conj(p,cnnfh(HOLogic.Not $ q))),
|
berghofe@13876
|
908 |
cnnfh (HOLogic.mk_conj(p1,cnnfh(HOLogic.Not $ r))))
|
berghofe@13876
|
909 |
else HOLogic.mk_conj(
|
berghofe@13876
|
910 |
cnnfh (HOLogic.mk_disj(cnnfh (HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))),
|
berghofe@13876
|
911 |
cnnfh (HOLogic.mk_disj(cnnfh (HOLogic.Not $ p1),cnnfh(HOLogic.Not $ r)))
|
berghofe@13876
|
912 |
)
|
berghofe@13876
|
913 |
| (Const ("Not",_) $ (Const ("op |",_) $ p $ q)) => HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh(HOLogic.Not $ q))
|
berghofe@13876
|
914 |
| (Const ("Not",_) $ (Const ("op -->",_) $ p $q)) => HOLogic.mk_conj(cnnfh p,cnnfh(HOLogic.Not $ q))
|
berghofe@13876
|
915 |
| (Const ("Not",_) $ (Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q)) => HOLogic.mk_disj(HOLogic.mk_conj(cnnfh p,cnnfh(HOLogic.Not $ q)),HOLogic.mk_conj(cnnfh(HOLogic.Not $ p),cnnfh q))
|
berghofe@13876
|
916 |
| _ => lfn fm
|
chaieb@14920
|
917 |
in cnnfh
|
chaieb@14920
|
918 |
end;
|
berghofe@13876
|
919 |
|
berghofe@13876
|
920 |
(*End- function the quantifierelimination an decion procedure of presburger formulas.*)
|
chaieb@14920
|
921 |
|
chaieb@14920
|
922 |
(*
|
berghofe@13876
|
923 |
val integer_qelim = simpl o evalc o (lift_qelim linform (simpl o (cnnf posineq o evalc)) cooper is_arith_rel) ;
|
chaieb@14920
|
924 |
*)
|
chaieb@14920
|
925 |
val integer_qelim = simpl o evalc o (lift_qelim linform (cnnf posineq o evalc) cooper is_arith_rel) ;
|
chaieb@14920
|
926 |
|
chaieb@14941
|
927 |
fun mk_presburger_oracle (sg,COOPER_ORACLE t) =
|
chaieb@14941
|
928 |
if (!quick_and_dirty)
|
chaieb@14941
|
929 |
then HOLogic.mk_Trueprop (HOLogic.mk_eq(t,integer_qelim t))
|
chaieb@15107
|
930 |
else raise COOPER_ORACLE t
|
chaieb@15107
|
931 |
|mk_presburger_oracle (sg,_) = error "Oops";
|
berghofe@13876
|
932 |
end;
|
chaieb@14920
|
933 |
|
chaieb@14920
|
934 |
|