1 (* Title: HOL/Tools/lin_arith.ML
3 Author: Tjark Weber and Tobias Nipkow
5 HOL setup for linear arithmetic (see Provers/Arith/fast_lin_arith.ML).
8 signature BASIC_LIN_ARITH =
11 val mk_arith_tactic: string -> (Proof.context -> int -> tactic) -> arith_tactic
12 val eq_arith_tactic: arith_tactic * arith_tactic -> bool
13 val arith_split_add: attribute
14 val arith_discrete: string -> Context.generic -> Context.generic
15 val arith_inj_const: string * typ -> Context.generic -> Context.generic
16 val arith_tactic_add: arith_tactic -> Context.generic -> Context.generic
17 val fast_arith_split_limit: int Config.T
18 val fast_arith_neq_limit: int Config.T
19 val lin_arith_pre_tac: Proof.context -> int -> tactic
20 val fast_arith_tac: Proof.context -> int -> tactic
21 val fast_ex_arith_tac: Proof.context -> bool -> int -> tactic
22 val trace_arith: bool ref
23 val lin_arith_simproc: simpset -> term -> thm option
24 val fast_nat_arith_simproc: simproc
25 val simple_arith_tac: Proof.context -> int -> tactic
26 val arith_tac: Proof.context -> int -> tactic
27 val silent_arith_tac: Proof.context -> int -> tactic
32 include BASIC_LIN_ARITH
34 ({add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
35 lessD: thm list, neqE: thm list, simpset: Simplifier.simpset} ->
36 {add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
37 lessD: thm list, neqE: thm list, simpset: Simplifier.simpset}) ->
38 Context.generic -> Context.generic
39 val setup: Context.generic -> Context.generic
42 structure LinArith: LIN_ARITH =
45 (* Parameters data for general linear arithmetic functor *)
47 structure LA_Logic: LIN_ARITH_LOGIC =
54 val not_lessD = @{thm linorder_not_less} RS iffD1;
55 val not_leD = @{thm linorder_not_le} RS iffD1;
58 fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
60 val mk_Trueprop = HOLogic.mk_Trueprop;
62 fun atomize thm = case Thm.prop_of thm of
63 Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) =>
64 atomize(thm RS conjunct1) @ atomize(thm RS conjunct2)
67 fun neg_prop ((TP as Const("Trueprop",_)) $ (Const("Not",_) $ t)) = TP $ t
68 | neg_prop ((TP as Const("Trueprop",_)) $ t) = TP $ (HOLogic.Not $t)
69 | neg_prop t = raise TERM ("neg_prop", [t]);
72 let val _ $ t = Thm.prop_of thm
73 in t = Const("False",HOLogic.boolT) end;
75 fun is_nat(t) = fastype_of1 t = HOLogic.natT;
78 let val ct = cterm_of sg t and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
79 in instantiate ([],[(cn,ct)]) le0 end;
84 (* arith context data *)
86 datatype arith_tactic =
87 ArithTactic of {name: string, tactic: Proof.context -> int -> tactic, id: stamp};
89 fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()};
91 fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2);
93 structure ArithContextData = GenericDataFun
95 type T = {splits: thm list,
96 inj_consts: (string * typ) list,
97 discrete: string list,
98 tactics: arith_tactic list};
99 val empty = {splits = [], inj_consts = [], discrete = [], tactics = []};
101 fun merge _ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1},
102 {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) =
103 {splits = Library.merge Thm.eq_thm_prop (splits1, splits2),
104 inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
105 discrete = Library.merge (op =) (discrete1, discrete2),
106 tactics = Library.merge eq_arith_tactic (tactics1, tactics2)};
109 val get_arith_data = ArithContextData.get o Context.Proof;
111 val arith_split_add = Thm.declaration_attribute (fn thm =>
112 ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
113 {splits = update Thm.eq_thm_prop thm splits,
114 inj_consts = inj_consts, discrete = discrete, tactics = tactics}));
116 fun arith_discrete d = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
117 {splits = splits, inj_consts = inj_consts,
118 discrete = update (op =) d discrete, tactics = tactics});
120 fun arith_inj_const c = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
121 {splits = splits, inj_consts = update (op =) c inj_consts,
122 discrete = discrete, tactics= tactics});
124 fun arith_tactic_add tac = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
125 {splits = splits, inj_consts = inj_consts, discrete = discrete,
126 tactics = update eq_arith_tactic tac tactics});
129 val (fast_arith_split_limit, setup1) = Attrib.config_int "fast_arith_split_limit" 9;
130 val (fast_arith_neq_limit, setup2) = Attrib.config_int "fast_arith_neq_limit" 9;
131 val setup_options = setup1 #> setup2;
134 structure LA_Data_Ref =
137 val fast_arith_neq_limit = fast_arith_neq_limit;
140 (* Decomposition of terms *)
142 (*internal representation of linear (in-)equations*)
144 ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);
146 fun nT (Type ("fun", [N, _])) = (N = HOLogic.natT)
149 fun add_atom (t : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) :
150 (term * Rat.rat) list * Rat.rat =
151 case AList.lookup (op =) p t of
152 NONE => ((t, m) :: p, i)
153 | SOME n => (AList.update (op =) (t, Rat.add n m) p, i);
155 (* decompose nested multiplications, bracketing them to the right and combining
156 all their coefficients
158 inj_consts: list of constants to be ignored when encountered
159 (e.g. arithmetic type conversions that preserve value)
161 m: multiplicity associated with the entire product
163 returns either (SOME term, associated multiplicity) or (NONE, constant)
165 fun demult (inj_consts : (string * typ) list) : term * Rat.rat -> term option * Rat.rat =
167 fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) =
168 (case s of Const (@{const_name HOL.times}, _) $ s1 $ s2 =>
169 (* bracketing to the right: '(s1 * s2) * t' becomes 's1 * (s2 * t)' *)
170 demult (mC $ s1 $ (mC $ s2 $ t), m)
172 (* product 's * t', where either factor can be 'NONE' *)
173 (case demult (s, m) of
175 (case demult (t, m') of
176 (SOME t', m'') => (SOME (mC $ s' $ t'), m'')
177 | (NONE, m'') => (SOME s', m''))
178 | (NONE, m') => demult (t, m')))
179 | demult ((mC as Const (@{const_name HOL.divide}, _)) $ s $ t, m) =
180 (* FIXME: Shouldn't we simplify nested quotients, e.g. '(s/t)/u' could
181 become 's/(t*u)', and '(s*t)/u' could become 's*(t/u)' ? Note that
182 if we choose to do so here, the simpset used by arith must be able to
183 perform the same simplifications. *)
184 (* FIXME: Currently we treat the numerator as atomic unless the
185 denominator can be reduced to a numeric constant. It might be better
186 to demult the numerator in any case, and invent a new term of the form
187 '1 / t' if the numerator can be reduced, but the denominator cannot. *)
188 (* FIXME: Currently we even treat the whole fraction as atomic unless the
189 denominator can be reduced to a numeric constant. It might be better
190 to use the partially reduced denominator (i.e. 's / (2* t)' could be
191 demult'ed to 's / t' with multiplicity .5). This would require a
192 very simple change only below, but it breaks existing proofs. *)
193 (* quotient 's / t', where the denominator t can be NONE *)
194 (* Note: will raise Rat.DIVZERO iff m' is Rat.zero *)
195 (case demult (t, Rat.one) of
196 (SOME _, _) => (SOME (mC $ s $ t), m)
197 | (NONE, m') => apsnd (Rat.mult (Rat.inv m')) (demult (s, m)))
198 (* terms that evaluate to numeric constants *)
199 | demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m)
200 | demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero)
201 | demult (Const (@{const_name HOL.one}, _), m) = (NONE, m)
202 (*Warning: in rare cases number_of encloses a non-numeral,
203 in which case dest_numeral raises TERM; hence all the handles below.
204 Same for Suc-terms that turn out not to be numerals -
205 although the simplifier should eliminate those anyway ...*)
206 | demult (t as Const ("Numeral.number_class.number_of", _) $ n, m) =
207 ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
208 handle TERM _ => (SOME t, m))
209 | demult (t as Const (@{const_name Suc}, _) $ _, m) =
210 ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat t)))
211 handle TERM _ => (SOME t, m))
212 (* injection constants are ignored *)
213 | demult (t as Const f $ x, m) =
214 if member (op =) inj_consts f then demult (x, m) else (SOME t, m)
215 (* everything else is considered atomic *)
216 | demult (atom, m) = (SOME atom, m)
219 fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) :
220 ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option =
222 (* Turns a term 'all' and associated multiplicity 'm' into a list 'p' of
223 summands and associated multiplicities, plus a constant 'i' (with implicit
225 fun poly (Const (@{const_name HOL.plus}, _) $ s $ t,
226 m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) = poly (s, m, poly (t, m, pi))
227 | poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) =
228 if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi))
229 | poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) =
230 if nT T then add_atom all m pi else poly (t, Rat.neg m, pi)
231 | poly (Const (@{const_name HOL.zero}, _), _, pi) =
233 | poly (Const (@{const_name HOL.one}, _), m, (p, i)) =
235 | poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =
236 poly (t, m, (p, Rat.add i m))
237 | poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) =
238 (case demult inj_consts (all, m) of
239 (NONE, m') => (p, Rat.add i m')
240 | (SOME u, m') => add_atom u m' pi)
241 | poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) =
242 (case demult inj_consts (all, m) of
243 (NONE, m') => (p, Rat.add i m')
244 | (SOME u, m') => add_atom u m' pi)
245 | poly (all as Const ("Numeral.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) =
246 (let val k = HOLogic.dest_numeral t
247 val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k
248 in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end
249 handle TERM _ => add_atom all m pi)
250 | poly (all as Const f $ x, m, pi) =
251 if f mem inj_consts then poly (x, m, pi) else add_atom all m pi
252 | poly (all, m, pi) =
254 val (p, i) = poly (lhs, Rat.one, ([], Rat.zero))
255 val (q, j) = poly (rhs, Rat.one, ([], Rat.zero))
258 @{const_name HOL.less} => SOME (p, i, "<", q, j)
259 | @{const_name HOL.less_eq} => SOME (p, i, "<=", q, j)
260 | "op =" => SOME (p, i, "=", q, j)
262 end handle Rat.DIVZERO => NONE;
264 fun of_lin_arith_sort thy U =
265 Sign.of_sort thy (U, ["Ring_and_Field.ordered_idom"]);
267 fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool =
268 if of_lin_arith_sort sg U then
269 (true, D mem discrete)
270 else (* special cases *)
271 if D mem discrete then (true, true) else (false, false)
272 | allows_lin_arith sg discrete U =
273 (of_lin_arith_sort sg U, false);
275 fun decomp_typecheck (thy, discrete, inj_consts) (T : typ, xxx) : decompT option =
277 Type ("fun", [U, _]) =>
278 (case allows_lin_arith thy discrete U of
280 (case decomp0 inj_consts xxx of
282 | SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d))
287 fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d)
288 | negate NONE = NONE;
290 fun decomp_negation data
291 ((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decompT option =
292 decomp_typecheck data (T, (rel, lhs, rhs))
293 | decomp_negation data ((Const ("Trueprop", _)) $
294 (Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) =
295 negate (decomp_typecheck data (T, (rel, lhs, rhs)))
296 | decomp_negation data _ =
299 fun decomp ctxt : term -> decompT option =
301 val thy = ProofContext.theory_of ctxt
302 val {discrete, inj_consts, ...} = get_arith_data ctxt
303 in decomp_negation (thy, discrete, inj_consts) end;
305 fun domain_is_nat (_ $ (Const (_, T) $ _ $ _)) = nT T
306 | domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T
307 | domain_is_nat _ = false;
309 fun number_of (n, T) = HOLogic.mk_number T n;
311 (*---------------------------------------------------------------------------*)
312 (* the following code performs splitting of certain constants (e.g. min, *)
313 (* max) in a linear arithmetic problem; similar to what split_tac later does *)
314 (* to the proof state *)
315 (*---------------------------------------------------------------------------*)
317 (* checks if splitting with 'thm' is implemented *)
319 fun is_split_thm (thm : thm) : bool =
320 case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => (
321 (* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)
323 Const (a, _) => member (op =) [@{const_name Orderings.max},
324 @{const_name Orderings.min},
325 @{const_name HOL.abs},
326 @{const_name HOL.minus},
328 "Divides.div_class.mod",
329 "Divides.div_class.div"] a
330 | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
331 Display.string_of_thm thm);
333 | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
334 Display.string_of_thm thm);
337 (* substitute new for occurrences of old in a term, incrementing bound *)
338 (* variables as needed when substituting inside an abstraction *)
340 fun subst_term ([] : (term * term) list) (t : term) = t
341 | subst_term pairs t =
342 (case AList.lookup (op aconv) pairs t of
346 (case t of Abs (a, T, body) =>
347 let val pairs' = map (pairself (incr_boundvars 1)) pairs
348 in Abs (a, T, subst_term pairs' body) end
350 subst_term pairs t1 $ subst_term pairs t2
353 (* approximates the effect of one application of split_tac (followed by NNF *)
354 (* normalization) on the subgoal represented by '(Ts, terms)'; returns a *)
355 (* list of new subgoals (each again represented by a typ list for bound *)
356 (* variables and a term list for premises), or NONE if split_tac would fail *)
359 (* FIXME: currently only the effect of certain split theorems is reproduced *)
360 (* (which is why we need 'is_split_thm'). A more canonical *)
361 (* implementation should analyze the right-hand side of the split *)
362 (* theorem that can be applied, and modify the subgoal accordingly. *)
363 (* Or even better, the splitter should be extended to provide *)
364 (* splitting on terms as well as splitting on theorems (where the *)
365 (* former can have a faster implementation as it does not need to be *)
366 (* proof-producing). *)
368 fun split_once_items ctxt (Ts : typ list, terms : term list) :
369 (typ list * term list) list option =
371 val thy = ProofContext.theory_of ctxt
372 (* takes a list [t1, ..., tn] to the term *)
373 (* tn' --> ... --> t1' --> False , *)
374 (* where ti' = HOLogic.dest_Trueprop ti *)
375 fun REPEAT_DETERM_etac_rev_mp terms' =
376 fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop terms') HOLogic.false_const
377 val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
378 val cmap = Splitter.cmap_of_split_thms split_thms
379 val splits = Splitter.split_posns cmap thy Ts (REPEAT_DETERM_etac_rev_mp terms)
380 val split_limit = Config.get ctxt fast_arith_split_limit
382 if length splits > split_limit then
383 (tracing ("fast_arith_split_limit exceeded (current value is " ^
384 string_of_int split_limit ^ ")"); NONE)
387 (* split_tac would fail: no possible split *)
389 | ((_, _, _, split_type, split_term) :: _) => (
390 (* ignore all but the first possible split *)
391 case strip_comb split_term of
392 (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
393 (Const (@{const_name Orderings.max}, _), [t1, t2]) =>
395 val rev_terms = rev terms
396 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
397 val terms2 = map (subst_term [(split_term, t2)]) rev_terms
398 val t1_leq_t2 = Const (@{const_name HOL.less_eq},
399 split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
400 val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
401 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
402 val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
403 val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
405 SOME [(Ts, subgoal1), (Ts, subgoal2)]
407 (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
408 | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
410 val rev_terms = rev terms
411 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
412 val terms2 = map (subst_term [(split_term, t2)]) rev_terms
413 val t1_leq_t2 = Const (@{const_name HOL.less_eq},
414 split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
415 val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
416 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
417 val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
418 val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
420 SOME [(Ts, subgoal1), (Ts, subgoal2)]
422 (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
423 | (Const (@{const_name HOL.abs}, _), [t1]) =>
425 val rev_terms = rev terms
426 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
427 val terms2 = map (subst_term [(split_term, Const (@{const_name HOL.uminus},
428 split_type --> split_type) $ t1)]) rev_terms
429 val zero = Const (@{const_name HOL.zero}, split_type)
430 val zero_leq_t1 = Const (@{const_name HOL.less_eq},
431 split_type --> split_type --> HOLogic.boolT) $ zero $ t1
432 val t1_lt_zero = Const (@{const_name HOL.less},
433 split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
434 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
435 val subgoal1 = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
436 val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
438 SOME [(Ts, subgoal1), (Ts, subgoal2)]
440 (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
441 | (Const (@{const_name HOL.minus}, _), [t1, t2]) =>
443 (* "d" in the above theorem becomes a new bound variable after NNF *)
444 (* transformation, therefore some adjustment of indices is necessary *)
445 val rev_terms = rev terms
446 val zero = Const (@{const_name HOL.zero}, split_type)
448 val terms1 = map (subst_term [(split_term, zero)]) rev_terms
449 val terms2 = map (subst_term [(incr_boundvars 1 split_term, d)])
450 (map (incr_boundvars 1) rev_terms)
451 val t1' = incr_boundvars 1 t1
452 val t2' = incr_boundvars 1 t2
453 val t1_lt_t2 = Const (@{const_name HOL.less},
454 split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
455 val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
456 (Const (@{const_name HOL.plus},
457 split_type --> split_type --> split_type) $ t2' $ d)
458 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
459 val subgoal1 = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
460 val subgoal2 = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
462 SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
464 (* ?P (nat ?i) = ((ALL n. ?i = int n --> ?P n) & (?i < 0 --> ?P 0)) *)
465 | (Const ("IntDef.nat", _), [t1]) =>
467 val rev_terms = rev terms
468 val zero_int = Const (@{const_name HOL.zero}, HOLogic.intT)
469 val zero_nat = Const (@{const_name HOL.zero}, HOLogic.natT)
471 val terms1 = map (subst_term [(incr_boundvars 1 split_term, n)])
472 (map (incr_boundvars 1) rev_terms)
473 val terms2 = map (subst_term [(split_term, zero_nat)]) rev_terms
474 val t1' = incr_boundvars 1 t1
475 val t1_eq_int_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
476 (Const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) $ n)
477 val t1_lt_zero = Const (@{const_name HOL.less},
478 HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
479 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
480 val subgoal1 = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false]
481 val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
483 SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
485 (* "?P ((?n::nat) mod (number_of ?k)) =
486 ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
487 (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
488 | (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
490 val rev_terms = rev terms
491 val zero = Const (@{const_name HOL.zero}, split_type)
494 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
495 val terms2 = map (subst_term [(incr_boundvars 2 split_term, j)])
496 (map (incr_boundvars 2) rev_terms)
497 val t1' = incr_boundvars 2 t1
498 val t2' = incr_boundvars 2 t2
499 val t2_eq_zero = Const ("op =",
500 split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
501 val t2_neq_zero = HOLogic.mk_not (Const ("op =",
502 split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
503 val j_lt_t2 = Const (@{const_name HOL.less},
504 split_type --> split_type--> HOLogic.boolT) $ j $ t2'
505 val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
506 (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
507 (Const (@{const_name HOL.times},
508 split_type --> split_type --> split_type) $ t2' $ i) $ j)
509 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
510 val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
511 val subgoal2 = (map HOLogic.mk_Trueprop
512 [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
513 @ terms2 @ [not_false]
515 SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
517 (* "?P ((?n::nat) div (number_of ?k)) =
518 ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
519 (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
520 | (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
522 val rev_terms = rev terms
523 val zero = Const (@{const_name HOL.zero}, split_type)
526 val terms1 = map (subst_term [(split_term, zero)]) rev_terms
527 val terms2 = map (subst_term [(incr_boundvars 2 split_term, i)])
528 (map (incr_boundvars 2) rev_terms)
529 val t1' = incr_boundvars 2 t1
530 val t2' = incr_boundvars 2 t2
531 val t2_eq_zero = Const ("op =",
532 split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
533 val t2_neq_zero = HOLogic.mk_not (Const ("op =",
534 split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
535 val j_lt_t2 = Const (@{const_name HOL.less},
536 split_type --> split_type--> HOLogic.boolT) $ j $ t2'
537 val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
538 (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
539 (Const (@{const_name HOL.times},
540 split_type --> split_type --> split_type) $ t2' $ i) $ j)
541 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
542 val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
543 val subgoal2 = (map HOLogic.mk_Trueprop
544 [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
545 @ terms2 @ [not_false]
547 SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
549 (* "?P ((?n::int) mod (number_of ?k)) =
550 ((iszero (number_of ?k) --> ?P ?n) &
551 (neg (number_of (uminus ?k)) -->
552 (ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
553 (neg (number_of ?k) -->
554 (ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
555 | (Const ("Divides.div_class.mod",
556 Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) =>
558 val rev_terms = rev terms
559 val zero = Const (@{const_name HOL.zero}, split_type)
562 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
563 val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, j)])
564 (map (incr_boundvars 2) rev_terms)
565 val t1' = incr_boundvars 2 t1
566 val (t2' as (_ $ k')) = incr_boundvars 2 t2
567 val iszero_t2 = Const ("IntDef.iszero", split_type --> HOLogic.boolT) $ t2
568 val neg_minus_k = Const ("IntDef.neg", split_type --> HOLogic.boolT) $
570 (Const (@{const_name HOL.uminus},
571 HOLogic.intT --> HOLogic.intT) $ k'))
572 val zero_leq_j = Const (@{const_name HOL.less_eq},
573 split_type --> split_type --> HOLogic.boolT) $ zero $ j
574 val j_lt_t2 = Const (@{const_name HOL.less},
575 split_type --> split_type--> HOLogic.boolT) $ j $ t2'
576 val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
577 (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
578 (Const (@{const_name HOL.times},
579 split_type --> split_type --> split_type) $ t2' $ i) $ j)
580 val neg_t2 = Const ("IntDef.neg", split_type --> HOLogic.boolT) $ t2'
581 val t2_lt_j = Const (@{const_name HOL.less},
582 split_type --> split_type--> HOLogic.boolT) $ t2' $ j
583 val j_leq_zero = Const (@{const_name HOL.less_eq},
584 split_type --> split_type --> HOLogic.boolT) $ j $ zero
585 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
586 val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
587 val subgoal2 = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j])
589 :: (if tl terms2_3 = [] then [not_false] else [])
590 @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
591 @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
592 val subgoal3 = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j])
594 :: (if tl terms2_3 = [] then [not_false] else [])
595 @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
596 @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
597 val Ts' = split_type :: split_type :: Ts
599 SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
601 (* "?P ((?n::int) div (number_of ?k)) =
602 ((iszero (number_of ?k) --> ?P 0) &
603 (neg (number_of (uminus ?k)) -->
604 (ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) --> ?P i)) &
605 (neg (number_of ?k) -->
606 (ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) --> ?P i))) *)
607 | (Const ("Divides.div_class.div",
608 Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) =>
610 val rev_terms = rev terms
611 val zero = Const (@{const_name HOL.zero}, split_type)
614 val terms1 = map (subst_term [(split_term, zero)]) rev_terms
615 val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, i)])
616 (map (incr_boundvars 2) rev_terms)
617 val t1' = incr_boundvars 2 t1
618 val (t2' as (_ $ k')) = incr_boundvars 2 t2
619 val iszero_t2 = Const ("IntDef.iszero", split_type --> HOLogic.boolT) $ t2
620 val neg_minus_k = Const ("IntDef.neg", split_type --> HOLogic.boolT) $
622 (Const (@{const_name HOL.uminus},
623 HOLogic.intT --> HOLogic.intT) $ k'))
624 val zero_leq_j = Const (@{const_name HOL.less_eq},
625 split_type --> split_type --> HOLogic.boolT) $ zero $ j
626 val j_lt_t2 = Const (@{const_name HOL.less},
627 split_type --> split_type--> HOLogic.boolT) $ j $ t2'
628 val t1_eq_t2_times_i_plus_j = Const ("op =",
629 split_type --> split_type --> HOLogic.boolT) $ t1' $
630 (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
631 (Const (@{const_name HOL.times},
632 split_type --> split_type --> split_type) $ t2' $ i) $ j)
633 val neg_t2 = Const ("IntDef.neg", split_type --> HOLogic.boolT) $ t2'
634 val t2_lt_j = Const (@{const_name HOL.less},
635 split_type --> split_type--> HOLogic.boolT) $ t2' $ j
636 val j_leq_zero = Const (@{const_name HOL.less_eq},
637 split_type --> split_type --> HOLogic.boolT) $ j $ zero
638 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
639 val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
640 val subgoal2 = (HOLogic.mk_Trueprop neg_minus_k)
643 :: (map HOLogic.mk_Trueprop
644 [zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j])
645 val subgoal3 = (HOLogic.mk_Trueprop neg_t2)
648 :: (map HOLogic.mk_Trueprop
649 [t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j])
650 val Ts' = split_type :: split_type :: Ts
652 SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
654 (* this will only happen if a split theorem can be applied for which no *)
655 (* code exists above -- in which case either the split theorem should be *)
656 (* implemented above, or 'is_split_thm' should be modified to filter it *)
659 warning ("Lin. Arith.: split rule for " ^ Syntax.string_of_term ctxt t ^
660 " (with " ^ string_of_int (length ts) ^
661 " argument(s)) not implemented; proof reconstruction is likely to fail");
667 (* remove terms that do not satisfy 'p'; change the order of the remaining *)
668 (* terms in the same way as filter_prems_tac does *)
670 fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
672 fun filter_prems (t, (left, right)) =
673 if p t then (left, right @ [t]) else (left @ right, [])
674 val (left, right) = foldl filter_prems ([], []) terms
679 (* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a *)
680 (* subgoal that has 'terms' as premises *)
682 fun negated_term_occurs_positively (terms : term list) : bool =
684 (fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t)
688 fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =
690 (* repeatedly split (including newly emerging subgoals) until no further *)
691 (* splitting is possible *)
692 fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list)
693 | split_loop (subgoal::subgoals) = (
694 case split_once_items ctxt subgoal of
695 SOME new_subgoals => split_loop (new_subgoals @ subgoals)
696 | NONE => subgoal :: split_loop subgoals
698 fun is_relevant t = isSome (decomp ctxt t)
699 (* filter_prems_tac is_relevant: *)
700 val relevant_terms = filter_prems_tac_items is_relevant terms
701 (* split_tac, NNF normalization: *)
702 val split_goals = split_loop [(Ts, relevant_terms)]
703 (* necessary because split_once_tac may normalize terms: *)
704 val beta_eta_norm = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals
705 (* TRY (etac notE) THEN eq_assume_tac: *)
706 val result = List.filter (not o negated_term_occurs_positively o snd) beta_eta_norm
711 (* takes the i-th subgoal [| A1; ...; An |] ==> B to *)
712 (* An --> ... --> A1 --> B, performs splitting with the given 'split_thms' *)
713 (* (resulting in a different subgoal P), takes P to ~P ==> False, *)
714 (* performs NNF-normalization of ~P, and eliminates conjunctions, *)
715 (* disjunctions and existential quantifiers from the premises, possibly (in *)
716 (* the case of disjunctions) resulting in several new subgoals, each of the *)
717 (* general form [| Q1; ...; Qm |] ==> False. Fails if more than *)
718 (* !fast_arith_split_limit splits are possible. *)
722 empty_ss setmkeqTrue mk_eq_True
723 setmksimps (mksimps mksimps_pairs)
724 addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj,
725 not_all, not_ex, not_not]
726 fun prem_nnf_tac i st =
727 full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st
730 fun split_once_tac ctxt split_thms =
732 val thy = ProofContext.theory_of ctxt
733 val cond_split_tac = SUBGOAL (fn (subgoal, i) =>
735 val Ts = rev (map snd (Logic.strip_params subgoal))
736 val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)
737 val cmap = Splitter.cmap_of_split_thms split_thms
738 val splits = Splitter.split_posns cmap thy Ts concl
739 val split_limit = Config.get ctxt fast_arith_split_limit
741 if length splits > split_limit then no_tac
742 else split_tac split_thms i
746 REPEAT_DETERM o etac rev_mp,
750 TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))
756 (* remove irrelevant premises, then split the i-th subgoal (and all new *)
757 (* subgoals) by using 'split_once_tac' repeatedly. Beta-eta-normalize new *)
758 (* subgoals and finally attempt to solve them by finding an immediate *)
759 (* contradiction (i.e. a term and its negation) in their premises. *)
763 val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
764 fun is_relevant t = isSome (decomp ctxt t)
767 TRY (filter_prems_tac is_relevant i)
769 (TRY o REPEAT_ALL_NEW (split_once_tac ctxt split_thms))
771 (CONVERSION Drule.beta_eta_conversion
773 (TRY o (etac notE THEN' eq_assume_tac)))
778 end; (* LA_Data_Ref *)
781 val lin_arith_pre_tac = LA_Data_Ref.pre_tac;
783 structure Fast_Arith =
784 Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref);
786 val map_data = Fast_Arith.map_data;
788 fun fast_arith_tac ctxt = Fast_Arith.lin_arith_tac ctxt false;
789 val fast_ex_arith_tac = Fast_Arith.lin_arith_tac;
790 val trace_arith = Fast_Arith.trace;
792 (* reduce contradictory <= to False.
793 Most of the work is done by the cancel tactics. *)
795 val init_arith_data =
796 Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
797 {add_mono_thms = add_mono_thms @
798 @{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field},
799 mult_mono_thms = mult_mono_thms,
801 lessD = lessD @ [thm "Suc_leI"],
802 neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}],
803 simpset = HOL_basic_ss
805 [@{thm "monoid_add_class.zero_plus.add_0_left"},
806 @{thm "monoid_add_class.zero_plus.add_0_right"},
807 @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
808 @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
809 @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
810 @{thm "not_one_less_zero"}]
811 addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
812 (*abel_cancel helps it work in abstract algebraic domains*)
813 addsimprocs nat_cancel_sums_add}) #>
814 arith_discrete "nat";
816 val lin_arith_simproc = Fast_Arith.lin_arith_simproc;
818 val fast_nat_arith_simproc =
819 Simplifier.simproc (the_context ()) "fast_nat_arith"
820 ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K Fast_Arith.lin_arith_simproc);
822 (* Because of fast_nat_arith_simproc, the arithmetic solver is really only
823 useful to detect inconsistencies among the premises for subgoals which are
824 *not* themselves (in)equalities, because the latter activate
825 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
826 solver all the time rather than add the additional check. *)
829 (* arith proof method *)
833 fun raw_arith_tac ctxt ex =
834 (* FIXME: K true should be replaced by a sensible test (perhaps "isSome o
835 decomp sg"? -- but note that the test is applied to terms already before
836 they are split/normalized) to speed things up in case there are lots of
837 irrelevant terms involved; elimination of min/max can be optimized:
838 (max m n + k <= r) = (m+k <= r & n+k <= r)
839 (l <= min m n + k) = (l <= m+k & l <= n+k)
842 (* Splitting is also done inside fast_arith_tac, but not completely -- *)
843 (* split_tac may use split theorems that have not been implemented in *)
844 (* fast_arith_tac (cf. pre_decomp and split_once_items above), and *)
845 (* fast_arith_split_limit may trigger. *)
846 (* Therefore splitting outside of fast_arith_tac may allow us to prove *)
847 (* some goals that fast_arith_tac alone would fail on. *)
848 (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
849 (fast_ex_arith_tac ctxt ex);
851 fun more_arith_tacs ctxt =
852 let val tactics = #tactics (get_arith_data ctxt)
853 in FIRST' (map (fn ArithTactic {tactic, ...} => tactic ctxt) tactics) end;
857 fun simple_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
858 ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true];
860 fun arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
861 ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true,
862 more_arith_tacs ctxt];
864 fun silent_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
865 ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt false,
866 more_arith_tacs ctxt];
868 fun arith_method src =
869 Method.syntax Args.bang_facts src
870 #> (fn (prems, ctxt) => Method.METHOD (fn facts =>
871 HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac ctxt)));
880 Simplifier.map_ss (fn ss => ss addsimprocs [fast_nat_arith_simproc]
881 addSolver (mk_solver' "lin_arith" Fast_Arith.cut_lin_arith_tac)) #>
885 [("arith", arith_method, "decide linear arithmethic")] #>
886 Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add,
887 "declaration of split rules for arithmetic procedure")]) I;
891 structure BasicLinArith: BASIC_LIN_ARITH = LinArith;