/usr/share/hol88-2.02.19940316/Library/arith/sup-inf.ml is in hol88-library-source 2.02.19940316-31.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 | %****************************************************************************%
% FILE : sup-inf.ml %
% DESCRIPTION : SUP-INF method for deciding a subset of Presburger %
% arithmetic (R.E.Shostak, JACM Vol.24 No.4 Pages 529-543) %
% %
% READS FILES : <none> %
% WRITES FILES : <none> %
% %
% AUTHOR : R.J.Boulton %
% DATE : 4th March 1991 %
% %
% LAST MODIFIED : R.J.Boulton %
% DATE : 2nd July 1992 %
%****************************************************************************%
%============================================================================%
% SUP-INF algorithm %
%============================================================================%
%----------------------------------------------------------------------------%
% Datatype for representing the bounds of a normalised expression %
%----------------------------------------------------------------------------%
rectype bound = Bound of rat # (string # rat) list
| Max_bound of bound list
| Min_bound of bound list
| Pos_inf
| Neg_inf;;
%----------------------------------------------------------------------------%
% Datatype for representing the bounds of an non-normalised expression %
%----------------------------------------------------------------------------%
rectype internal_bound = Ibound of bound
| Mult_ibound of rat # internal_bound
| Plus_ibound of internal_bound # internal_bound
| Max_ibound of internal_bound list
| Min_ibound of internal_bound list;;
%----------------------------------------------------------------------------%
% solve_ineqs : %
% (int # (string # int) list) list -> %
% string -> %
% ((rat # (string # rat) list) list # (rat # (string # rat) list) list) %
%----------------------------------------------------------------------------%
letrec solve_ineqs ineqs var =
if (null ineqs)
then ([],[])
else let (const,bind) = hd ineqs
and (restl,restr) = solve_ineqs (tl ineqs) var
in (let i = snd (assoc var bind)
in let const' = Rat (const,(-i))
and bind' = map (I # (\n. Rat (n,(-i))))
(filter (\(name,_) . not (name = var)) bind)
in if (i < 0)
then (((const',bind').restl),restr)
else (restl,((const',bind').restr)))
? (restl,restr);;
%----------------------------------------------------------------------------%
% UPPER : (int # (string # int) list) list -> string -> bound %
%----------------------------------------------------------------------------%
let UPPER s x =
let uppers = fst (solve_ineqs s x)
in if (null uppers)
then Pos_inf
else if (null (tl uppers))
then Bound (hd uppers)
else Min_bound (map Bound uppers);;
%----------------------------------------------------------------------------%
% LOWER : (int # (string # int) list) list -> string -> bound %
%----------------------------------------------------------------------------%
let LOWER s x =
let lowers = snd (solve_ineqs s x)
in if (null lowers)
then Neg_inf
else if (null (tl lowers))
then Bound (hd lowers)
else Max_bound (map Bound lowers);;
%----------------------------------------------------------------------------%
% SIMP_mult : rat -> bound -> bound %
%----------------------------------------------------------------------------%
letrec SIMP_mult r b =
case b
of (Bound (const,bind)) .
(Bound (rat_mult r const,map (I # (rat_mult r)) bind))
| (Max_bound bl) .
(if ((Numerator r) < 0)
then (Min_bound (map (SIMP_mult r) bl))
else (Max_bound (map (SIMP_mult r) bl)))
| (Min_bound bl) .
(if ((Numerator r) < 0)
then (Max_bound (map (SIMP_mult r) bl))
else (Min_bound (map (SIMP_mult r) bl)))
| Pos_inf . (if ((Numerator r) < 0) then Neg_inf else Pos_inf)
| Neg_inf . (if ((Numerator r) < 0) then Pos_inf else Neg_inf);;
%----------------------------------------------------------------------------%
% sum_bindings : %
% (string # rat) list -> (string # rat) list -> (string # rat) list %
%----------------------------------------------------------------------------%
letrec sum_bindings bind1 bind2 =
if (null bind1) then bind2
if (null bind2) then bind1
else (let (name1,coeff1) = hd bind1
and (name2,coeff2) = hd bind2
in if (name1 = name2) then
(let coeff = rat_plus coeff1 coeff2
and bind = sum_bindings (tl bind1) (tl bind2)
in if ((Numerator coeff) = 0)
then bind
else (name1,coeff).bind)
if (string_less name1 name2) then
(name1,coeff1).(sum_bindings (tl bind1) bind2)
else (name2,coeff2).(sum_bindings bind1 (tl bind2)));;
%----------------------------------------------------------------------------%
% SIMP_plus : bound -> bound -> bound %
%----------------------------------------------------------------------------%
letrec SIMP_plus b1 b2 =
(case (b1,b2)
of (Bound (const1,bind1),Bound (const2,bind2)) .
(Bound (rat_plus const1 const2,sum_bindings bind1 bind2))
| (Bound _,Max_bound bl) . (Max_bound (map (SIMP_plus b1) bl))
| (Bound _,Min_bound bl) . (Min_bound (map (SIMP_plus b1) bl))
| (Bound _,Pos_inf) . Pos_inf
| (Bound _,Neg_inf) . Neg_inf
| (Max_bound bl,_) . (Max_bound (map (\b. SIMP_plus b b2) bl))
| (Min_bound bl,_) . (Min_bound (map (\b. SIMP_plus b b2) bl))
| (Pos_inf,Pos_inf) . Pos_inf
| (Pos_inf,Neg_inf) . fail
| (Pos_inf,_) . (SIMP_plus b2 b1)
| (Neg_inf,Neg_inf) . Neg_inf
| (Neg_inf,Pos_inf) . fail
| (Neg_inf,_) . (SIMP_plus b2 b1)
) ? failwith `SIMP_plus`;;
%----------------------------------------------------------------------------%
% SIMP : internal_bound -> bound %
%----------------------------------------------------------------------------%
letrec SIMP ib =
case ib
of (Ibound b) . b
| (Mult_ibound (r,ib')) . (SIMP_mult r (SIMP ib'))
| (Plus_ibound (ib1,ib2)) . (SIMP_plus (SIMP ib1) (SIMP ib2))
| (Max_ibound ibl) . (Max_bound (map SIMP ibl))
| (Min_ibound ibl) . (Min_bound (map SIMP ibl));;
%----------------------------------------------------------------------------%
% SUPP : (string # bound) -> bound %
% INFF : (string # bound) -> bound %
%----------------------------------------------------------------------------%
letrec SUPP (x,y) =
case y
of (Bound (_,[])) . y
| Pos_inf . y
| Neg_inf . y
| (Min_bound bl) . (Min_bound (map (\y. SUPP (x,y)) bl))
| (Bound (const,bind)) .
(let b = snd (assoc x bind) ? rat_zero
and bind' = filter (\p. not (fst p = x)) bind
in if ((null bind') & (const = rat_zero) & (b = rat_one))
then Pos_inf
else let b' = rat_minus rat_one b
in if (Numerator b' < 0) then Pos_inf
if (Numerator b' > 0) then
(Bound (rat_div const b',
map (I # (\r. rat_div r b')) bind'))
else if (not (null bind')) then Pos_inf
if (Numerator const < 0) then Neg_inf
else Pos_inf)
| (_) . failwith `SUPP`;;
letrec INFF (x,y) =
case y
of (Bound (_,[])) . y
| Pos_inf . y
| Neg_inf . y
| (Max_bound bl) . (Max_bound (map (\y. INFF (x,y)) bl))
| (Bound (const,bind)) .
(let b = snd (assoc x bind) ? rat_zero
and bind' = filter (\p. not (fst p = x)) bind
in if ((null bind') & (const = rat_zero) & (b = rat_one))
then Neg_inf
else let b' = rat_minus rat_one b
in if (Numerator b' < 0) then Neg_inf
if (Numerator b' > 0) then
(Bound (rat_div const b',
map (I # (\r. rat_div r b')) bind'))
else if (not (null bind')) then Neg_inf
if (Numerator const > 0) then Pos_inf
else Neg_inf)
| (_) . failwith `INFF`;;
%----------------------------------------------------------------------------%
% occurs_in_bound : string -> bound -> bool %
%----------------------------------------------------------------------------%
letrec occurs_in_bound v b =
case b
of (Bound (_,bind)) . (mem v (map fst bind))
| (Max_bound bl) .
(itlist (\x y. x or y) (map (occurs_in_bound v) bl) false)
| (Min_bound bl) .
(itlist (\x y. x or y) (map (occurs_in_bound v) bl) false)
| (_) . false;;
%----------------------------------------------------------------------------%
% occurs_in_ibound : string -> internal_bound -> bool %
%----------------------------------------------------------------------------%
letrec occurs_in_ibound v ib =
case ib
of (Ibound b) . (occurs_in_bound v b)
| (Mult_ibound (_,ib')) . (occurs_in_ibound v ib')
| (Plus_ibound (ib1,ib2)) .
((occurs_in_ibound v ib1) or (occurs_in_ibound v ib2))
| (Max_ibound ibl) .
(itlist (\x y. x or y) (map (occurs_in_ibound v) ibl) false)
| (Min_ibound ibl) .
(itlist (\x y. x or y) (map (occurs_in_ibound v) ibl) false);;
%----------------------------------------------------------------------------%
% SUP : (int # (string # int) list) list -> %
% (bound # (string list)) -> %
% internal_bound %
% INF : (int # (string # int) list) list -> %
% (bound # (string list)) -> %
% internal_bound %
%----------------------------------------------------------------------------%
letrec SUP s (J,H) =
case J
of (Bound (_,[])) . (Ibound J)
| Pos_inf . (Ibound J)
| Neg_inf . (Ibound J)
| (Min_bound bl) . (Min_ibound (map (\j. SUP s (j,H)) bl))
| (Bound (const,bind)) .
(let (rv.bind') = bind
in let (v,r) = rv
in if ((const = rat_zero) & (null bind'))
then (if (r = rat_one) then
(if (mem v H)
then Ibound J
else let Q = UPPER s v
in let Z = SUP s (Q,union H [v])
in Ibound (SUPP (v,SIMP Z)))
if (Numerator r < 0)
then (Mult_ibound
(r,INF s (Bound (rat_zero,[v,rat_one]),H)))
else (Mult_ibound
(r,SUP s (Bound (rat_zero,[v,rat_one]),H)))
)
else let B' = SUP s (Bound (const,bind'),union H [v])
and rvb = Bound (rat_zero,[rv])
in if (occurs_in_ibound v B')
then let J' = SIMP (Plus_ibound (Ibound rvb,B'))
in SUP s (J',H)
else Plus_ibound (SUP s (rvb,H),B'))
| (_) . failwith `SUP`
and INF s (J,H) =
case J
of (Bound (_,[])) . (Ibound J)
| Pos_inf . (Ibound J)
| Neg_inf . (Ibound J)
| (Max_bound bl) . (Max_ibound (map (\j. INF s (j,H)) bl))
| (Bound (const,bind)) .
(let (rv.bind') = bind
in let (v,r) = rv
in if ((const = rat_zero) & (null bind'))
then (if (r = rat_one) then
(if (mem v H)
then Ibound J
else let Q = LOWER s v
in let Z = INF s (Q,union H [v])
in Ibound (INFF (v,SIMP Z)))
if (Numerator r < 0)
then (Mult_ibound
(r,SUP s (Bound (rat_zero,[v,rat_one]),H)))
else (Mult_ibound
(r,INF s (Bound (rat_zero,[v,rat_one]),H)))
)
else let B' = INF s (Bound (const,bind'),union H [v])
and rvb = Bound (rat_zero,[rv])
in if (occurs_in_ibound v B')
then let J' = SIMP (Plus_ibound (Ibound rvb,B'))
in INF s (J',H)
else Plus_ibound (INF s (rvb,H),B'))
| (_) . failwith `INF`;;
%----------------------------------------------------------------------------%
% eval_max_bound : bound list -> bound %
%----------------------------------------------------------------------------%
letrec eval_max_bound bl =
if (null bl) then failwith `eval_max_bound`
if (null (tl bl)) then (hd bl)
else let b = hd bl
and max = eval_max_bound (tl bl)
in case (b,max)
of (Pos_inf,_) . Pos_inf
| (_,Pos_inf) . Pos_inf
| (Neg_inf,_) . max
| (_,Neg_inf) . b
| (Bound (r1,[]),Bound (r2,[])) .
(if (Numerator (rat_minus r1 r2) < 0) then max else b)
| (_) . failwith `eval_max_bound`;;
%----------------------------------------------------------------------------%
% eval_min_bound : bound list -> bound %
%----------------------------------------------------------------------------%
letrec eval_min_bound bl =
if (null bl) then failwith `eval_min_bound`
if (null (tl bl)) then (hd bl)
else let b = hd bl
and min = eval_min_bound (tl bl)
in case (b,min)
of (Pos_inf,_) . min
| (_,Pos_inf) . b
| (_,Neg_inf) . Neg_inf
| (Neg_inf,_) . Neg_inf
| (Bound (r1,[]),Bound (r2,[])) .
(if (Numerator (rat_minus r1 r2) < 0) then b else min)
| (_) . failwith `eval_min_bound`;;
%----------------------------------------------------------------------------%
% eval_bound : bound -> bound %
%----------------------------------------------------------------------------%
letrec eval_bound b =
case b
of (Bound (_,[])) . b
| (Max_bound bl) . (eval_max_bound (map eval_bound bl))
| (Min_bound bl) . (eval_min_bound (map eval_bound bl))
| Pos_inf . b
| Neg_inf . b;;
%----------------------------------------------------------------------------%
% SUP_INF : %
% (int # (string # int) list) list -> (string # bound # bound) list %
%----------------------------------------------------------------------------%
let SUP_INF set =
let vars_of_coeffs coeffsl = setify (flat (map ((map fst) o snd) coeffsl))
in
let vars = vars_of_coeffs set
and make_bound v = Bound (rat_zero,[v,rat_one])
and eval = eval_bound o SIMP
in map (\v. let b = make_bound v
in (v,eval (INF set (b,[])),eval (SUP set (b,[])))) vars;;
|