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 warning_count: int ref
40 val setup: Context.generic -> Context.generic
43 structure LinArith: LIN_ARITH =
46 (* Parameters data for general linear arithmetic functor *)
48 structure LA_Logic: LIN_ARITH_LOGIC =
55 val not_lessD = @{thm linorder_not_less} RS iffD1;
56 val not_leD = @{thm linorder_not_le} RS iffD1;
59 fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
61 val mk_Trueprop = HOLogic.mk_Trueprop;
63 fun atomize thm = case Thm.prop_of thm of
64 Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) =>
65 atomize(thm RS conjunct1) @ atomize(thm RS conjunct2)
68 fun neg_prop ((TP as Const("Trueprop",_)) $ (Const("Not",_) $ t)) = TP $ t
69 | neg_prop ((TP as Const("Trueprop",_)) $ t) = TP $ (HOLogic.Not $t)
70 | neg_prop t = raise TERM ("neg_prop", [t]);
73 let val _ $ t = Thm.prop_of thm
74 in t = Const("False",HOLogic.boolT) end;
76 fun is_nat(t) = fastype_of1 t = HOLogic.natT;
79 let val ct = cterm_of sg t and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
80 in instantiate ([],[(cn,ct)]) le0 end;
85 (* arith context data *)
87 datatype arith_tactic =
88 ArithTactic of {name: string, tactic: Proof.context -> int -> tactic, id: stamp};
90 fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()};
92 fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2);
94 structure ArithContextData = GenericDataFun
96 type T = {splits: thm list,
97 inj_consts: (string * typ) list,
98 discrete: string list,
99 tactics: arith_tactic list};
100 val empty = {splits = [], inj_consts = [], discrete = [], tactics = []};
102 fun merge _ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1},
103 {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) =
104 {splits = Library.merge Thm.eq_thm_prop (splits1, splits2),
105 inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
106 discrete = Library.merge (op =) (discrete1, discrete2),
107 tactics = Library.merge eq_arith_tactic (tactics1, tactics2)};
110 val get_arith_data = ArithContextData.get o Context.Proof;
112 val arith_split_add = Thm.declaration_attribute (fn thm =>
113 ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
114 {splits = update Thm.eq_thm_prop thm splits,
115 inj_consts = inj_consts, discrete = discrete, tactics = tactics}));
117 fun arith_discrete d = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
118 {splits = splits, inj_consts = inj_consts,
119 discrete = update (op =) d discrete, tactics = tactics});
121 fun arith_inj_const c = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
122 {splits = splits, inj_consts = update (op =) c inj_consts,
123 discrete = discrete, tactics= tactics});
125 fun arith_tactic_add tac = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
126 {splits = splits, inj_consts = inj_consts, discrete = discrete,
127 tactics = update eq_arith_tactic tac tactics});
130 val (fast_arith_split_limit, setup1) = Attrib.config_int "fast_arith_split_limit" 9;
131 val (fast_arith_neq_limit, setup2) = Attrib.config_int "fast_arith_neq_limit" 9;
132 val setup_options = setup1 #> setup2;
135 structure LA_Data_Ref =
138 val fast_arith_neq_limit = fast_arith_neq_limit;
141 (* Decomposition of terms *)
143 (*internal representation of linear (in-)equations*)
145 ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);
147 fun nT (Type ("fun", [N, _])) = (N = HOLogic.natT)
150 fun add_atom (t : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) :
151 (term * Rat.rat) list * Rat.rat =
152 case AList.lookup (op =) p t of
153 NONE => ((t, m) :: p, i)
154 | SOME n => (AList.update (op =) (t, Rat.add n m) p, i);
156 (* decompose nested multiplications, bracketing them to the right and combining
157 all their coefficients
159 inj_consts: list of constants to be ignored when encountered
160 (e.g. arithmetic type conversions that preserve value)
162 m: multiplicity associated with the entire product
164 returns either (SOME term, associated multiplicity) or (NONE, constant)
166 fun demult (inj_consts : (string * typ) list) : term * Rat.rat -> term option * Rat.rat =
168 fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) =
169 (case s of Const (@{const_name HOL.times}, _) $ s1 $ s2 =>
170 (* bracketing to the right: '(s1 * s2) * t' becomes 's1 * (s2 * t)' *)
171 demult (mC $ s1 $ (mC $ s2 $ t), m)
173 (* product 's * t', where either factor can be 'NONE' *)
174 (case demult (s, m) of
176 (case demult (t, m') of
177 (SOME t', m'') => (SOME (mC $ s' $ t'), m'')
178 | (NONE, m'') => (SOME s', m''))
179 | (NONE, m') => demult (t, m')))
180 | demult ((mC as Const (@{const_name HOL.divide}, _)) $ s $ t, m) =
181 (* FIXME: Shouldn't we simplify nested quotients, e.g. '(s/t)/u' could
182 become 's/(t*u)', and '(s*t)/u' could become 's*(t/u)' ? Note that
183 if we choose to do so here, the simpset used by arith must be able to
184 perform the same simplifications. *)
185 (* FIXME: Currently we treat the numerator as atomic unless the
186 denominator can be reduced to a numeric constant. It might be better
187 to demult the numerator in any case, and invent a new term of the form
188 '1 / t' if the numerator can be reduced, but the denominator cannot. *)
189 (* FIXME: Currently we even treat the whole fraction as atomic unless the
190 denominator can be reduced to a numeric constant. It might be better
191 to use the partially reduced denominator (i.e. 's / (2*t)' could be
192 demult'ed to 's / t' with multiplicity .5). This would require a
193 very simple change only below, but it breaks existing proofs. *)
194 (* quotient 's / t', where the denominator t can be NONE *)
195 (* Note: will raise Rat.DIVZERO iff m' is Rat.zero *)
196 (case demult (t, Rat.one) of
197 (SOME _, _) => (SOME (mC $ s $ t), m)
198 | (NONE, m') => apsnd (Rat.mult (Rat.inv m')) (demult (s, m)))
199 (* terms that evaluate to numeric constants *)
200 | demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m)
201 | demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero)
202 | demult (Const (@{const_name HOL.one}, _), m) = (NONE, m)
203 (*Warning: in rare cases number_of encloses a non-numeral,
204 in which case dest_numeral raises TERM; hence all the handles below.
205 Same for Suc-terms that turn out not to be numerals -
206 although the simplifier should eliminate those anyway ...*)
207 | demult (t as Const ("Int.number_class.number_of", _) $ n, m) =
208 ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
209 handle TERM _ => (SOME t, m))
210 | demult (t as Const (@{const_name Suc}, _) $ _, m) =
211 ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat t)))
212 handle TERM _ => (SOME t, m))
213 (* injection constants are ignored *)
214 | demult (t as Const f $ x, m) =
215 if member (op =) inj_consts f then demult (x, m) else (SOME t, m)
216 (* everything else is considered atomic *)
217 | demult (atom, m) = (SOME atom, m)
220 fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) :
221 ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option =
223 (* Turns a term 'all' and associated multiplicity 'm' into a list 'p' of
224 summands and associated multiplicities, plus a constant 'i' (with implicit
226 fun poly (Const (@{const_name HOL.plus}, _) $ s $ t,
227 m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) = poly (s, m, poly (t, m, pi))
228 | poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) =
229 if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi))
230 | poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) =
231 if nT T then add_atom all m pi else poly (t, Rat.neg m, pi)
232 | poly (Const (@{const_name HOL.zero}, _), _, pi) =
234 | poly (Const (@{const_name HOL.one}, _), m, (p, i)) =
236 | poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =
237 poly (t, m, (p, Rat.add i m))
238 | poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) =
239 (case demult inj_consts (all, m) of
240 (NONE, m') => (p, Rat.add i m')
241 | (SOME u, m') => add_atom u m' pi)
242 | poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) =
243 (case demult inj_consts (all, m) of
244 (NONE, m') => (p, Rat.add i m')
245 | (SOME u, m') => add_atom u m' pi)
246 | poly (all as Const ("Int.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) =
247 (let val k = HOLogic.dest_numeral t
248 val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k
249 in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end
250 handle TERM _ => add_atom all m pi)
251 | poly (all as Const f $ x, m, pi) =
252 if f mem inj_consts then poly (x, m, pi) else add_atom all m pi
253 | poly (all, m, pi) =
255 val (p, i) = poly (lhs, Rat.one, ([], Rat.zero))
256 val (q, j) = poly (rhs, Rat.one, ([], Rat.zero))
259 @{const_name HOL.less} => SOME (p, i, "<", q, j)
260 | @{const_name HOL.less_eq} => SOME (p, i, "<=", q, j)
261 | "op =" => SOME (p, i, "=", q, j)
263 end handle Rat.DIVZERO => NONE;
265 fun of_lin_arith_sort thy U =
266 Sign.of_sort thy (U, ["Ring_and_Field.ordered_idom"]);
268 fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool =
269 if of_lin_arith_sort sg U then
270 (true, D mem discrete)
271 else (* special cases *)
272 if D mem discrete then (true, true) else (false, false)
273 | allows_lin_arith sg discrete U =
274 (of_lin_arith_sort sg U, false);
276 fun decomp_typecheck (thy, discrete, inj_consts) (T : typ, xxx) : decomp option =
278 Type ("fun", [U, _]) =>
279 (case allows_lin_arith thy discrete U of
281 (case decomp0 inj_consts xxx of
283 | SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d))
288 fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d)
289 | negate NONE = NONE;
291 fun decomp_negation data
292 ((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decomp option =
293 decomp_typecheck data (T, (rel, lhs, rhs))
294 | decomp_negation data ((Const ("Trueprop", _)) $
295 (Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) =
296 negate (decomp_typecheck data (T, (rel, lhs, rhs)))
297 | decomp_negation data _ =
300 fun decomp ctxt : term -> decomp option =
302 val thy = ProofContext.theory_of ctxt
303 val {discrete, inj_consts, ...} = get_arith_data ctxt
304 in decomp_negation (thy, discrete, inj_consts) end;
306 fun domain_is_nat (_ $ (Const (_, T) $ _ $ _)) = nT T
307 | domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T
308 | domain_is_nat _ = false;
310 fun number_of (n, T) = HOLogic.mk_number T n;
312 (*---------------------------------------------------------------------------*)
313 (* the following code performs splitting of certain constants (e.g. min, *)
314 (* max) in a linear arithmetic problem; similar to what split_tac later does *)
315 (* to the proof state *)
316 (*---------------------------------------------------------------------------*)
318 (* checks if splitting with 'thm' is implemented *)
320 fun is_split_thm (thm : thm) : bool =
321 case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => (
322 (* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)
324 Const (a, _) => member (op =) [@{const_name Orderings.max},
325 @{const_name Orderings.min},
326 @{const_name HOL.abs},
327 @{const_name HOL.minus},
329 "Divides.div_class.mod",
330 "Divides.div_class.div"] a
331 | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
332 Display.string_of_thm thm);
334 | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
335 Display.string_of_thm thm);
338 (* substitute new for occurrences of old in a term, incrementing bound *)
339 (* variables as needed when substituting inside an abstraction *)
341 fun subst_term ([] : (term * term) list) (t : term) = t
342 | subst_term pairs t =
343 (case AList.lookup (op aconv) pairs t of
347 (case t of Abs (a, T, body) =>
348 let val pairs' = map (pairself (incr_boundvars 1)) pairs
349 in Abs (a, T, subst_term pairs' body) end
351 subst_term pairs t1 $ subst_term pairs t2
354 (* approximates the effect of one application of split_tac (followed by NNF *)
355 (* normalization) on the subgoal represented by '(Ts, terms)'; returns a *)
356 (* list of new subgoals (each again represented by a typ list for bound *)
357 (* variables and a term list for premises), or NONE if split_tac would fail *)
360 (* FIXME: currently only the effect of certain split theorems is reproduced *)
361 (* (which is why we need 'is_split_thm'). A more canonical *)
362 (* implementation should analyze the right-hand side of the split *)
363 (* theorem that can be applied, and modify the subgoal accordingly. *)
364 (* Or even better, the splitter should be extended to provide *)
365 (* splitting on terms as well as splitting on theorems (where the *)
366 (* former can have a faster implementation as it does not need to be *)
367 (* proof-producing). *)
369 fun split_once_items ctxt (Ts : typ list, terms : term list) :
370 (typ list * term list) list option =
372 val thy = ProofContext.theory_of ctxt
373 (* takes a list [t1, ..., tn] to the term *)
374 (* tn' --> ... --> t1' --> False , *)
375 (* where ti' = HOLogic.dest_Trueprop ti *)
376 fun REPEAT_DETERM_etac_rev_mp terms' =
377 fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop terms') HOLogic.false_const
378 val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
379 val cmap = Splitter.cmap_of_split_thms split_thms
380 val splits = Splitter.split_posns cmap thy Ts (REPEAT_DETERM_etac_rev_mp terms)
381 val split_limit = Config.get ctxt fast_arith_split_limit
383 if length splits > split_limit then
384 (tracing ("fast_arith_split_limit exceeded (current value is " ^
385 string_of_int split_limit ^ ")"); NONE)
388 (* split_tac would fail: no possible split *)
390 | ((_, _, _, split_type, split_term) :: _) => (
391 (* ignore all but the first possible split *)
392 case strip_comb split_term of
393 (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
394 (Const (@{const_name Orderings.max}, _), [t1, t2]) =>
396 val rev_terms = rev terms
397 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
398 val terms2 = map (subst_term [(split_term, t2)]) rev_terms
399 val t1_leq_t2 = Const (@{const_name HOL.less_eq},
400 split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
401 val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
402 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
403 val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
404 val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
406 SOME [(Ts, subgoal1), (Ts, subgoal2)]
408 (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
409 | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
411 val rev_terms = rev terms
412 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
413 val terms2 = map (subst_term [(split_term, t2)]) rev_terms
414 val t1_leq_t2 = Const (@{const_name HOL.less_eq},
415 split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
416 val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
417 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
418 val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
419 val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
421 SOME [(Ts, subgoal1), (Ts, subgoal2)]
423 (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
424 | (Const (@{const_name HOL.abs}, _), [t1]) =>
426 val rev_terms = rev terms
427 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
428 val terms2 = map (subst_term [(split_term, Const (@{const_name HOL.uminus},
429 split_type --> split_type) $ t1)]) rev_terms
430 val zero = Const (@{const_name HOL.zero}, split_type)
431 val zero_leq_t1 = Const (@{const_name HOL.less_eq},
432 split_type --> split_type --> HOLogic.boolT) $ zero $ t1
433 val t1_lt_zero = Const (@{const_name HOL.less},
434 split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
435 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
436 val subgoal1 = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
437 val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
439 SOME [(Ts, subgoal1), (Ts, subgoal2)]
441 (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
442 | (Const (@{const_name HOL.minus}, _), [t1, t2]) =>
444 (* "d" in the above theorem becomes a new bound variable after NNF *)
445 (* transformation, therefore some adjustment of indices is necessary *)
446 val rev_terms = rev terms
447 val zero = Const (@{const_name HOL.zero}, split_type)
449 val terms1 = map (subst_term [(split_term, zero)]) rev_terms
450 val terms2 = map (subst_term [(incr_boundvars 1 split_term, d)])
451 (map (incr_boundvars 1) rev_terms)
452 val t1' = incr_boundvars 1 t1
453 val t2' = incr_boundvars 1 t2
454 val t1_lt_t2 = Const (@{const_name HOL.less},
455 split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
456 val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
457 (Const (@{const_name HOL.plus},
458 split_type --> split_type --> split_type) $ t2' $ d)
459 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
460 val subgoal1 = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
461 val subgoal2 = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
463 SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
465 (* ?P (nat ?i) = ((ALL n. ?i = int n --> ?P n) & (?i < 0 --> ?P 0)) *)
466 | (Const ("Int.nat", _), [t1]) =>
468 val rev_terms = rev terms
469 val zero_int = Const (@{const_name HOL.zero}, HOLogic.intT)
470 val zero_nat = Const (@{const_name HOL.zero}, HOLogic.natT)
472 val terms1 = map (subst_term [(incr_boundvars 1 split_term, n)])
473 (map (incr_boundvars 1) rev_terms)
474 val terms2 = map (subst_term [(split_term, zero_nat)]) rev_terms
475 val t1' = incr_boundvars 1 t1
476 val t1_eq_int_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
477 (Const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) $ n)
478 val t1_lt_zero = Const (@{const_name HOL.less},
479 HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
480 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
481 val subgoal1 = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false]
482 val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
484 SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
486 (* "?P ((?n::nat) mod (number_of ?k)) =
487 ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
488 (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
489 | (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
491 val rev_terms = rev terms
492 val zero = Const (@{const_name HOL.zero}, split_type)
495 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
496 val terms2 = map (subst_term [(incr_boundvars 2 split_term, j)])
497 (map (incr_boundvars 2) rev_terms)
498 val t1' = incr_boundvars 2 t1
499 val t2' = incr_boundvars 2 t2
500 val t2_eq_zero = Const ("op =",
501 split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
502 val t2_neq_zero = HOLogic.mk_not (Const ("op =",
503 split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
504 val j_lt_t2 = Const (@{const_name HOL.less},
505 split_type --> split_type--> HOLogic.boolT) $ j $ t2'
506 val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
507 (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
508 (Const (@{const_name HOL.times},
509 split_type --> split_type --> split_type) $ t2' $ i) $ j)
510 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
511 val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
512 val subgoal2 = (map HOLogic.mk_Trueprop
513 [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
514 @ terms2 @ [not_false]
516 SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
518 (* "?P ((?n::nat) div (number_of ?k)) =
519 ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
520 (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
521 | (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
523 val rev_terms = rev terms
524 val zero = Const (@{const_name HOL.zero}, split_type)
527 val terms1 = map (subst_term [(split_term, zero)]) rev_terms
528 val terms2 = map (subst_term [(incr_boundvars 2 split_term, i)])
529 (map (incr_boundvars 2) rev_terms)
530 val t1' = incr_boundvars 2 t1
531 val t2' = incr_boundvars 2 t2
532 val t2_eq_zero = Const ("op =",
533 split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
534 val t2_neq_zero = HOLogic.mk_not (Const ("op =",
535 split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
536 val j_lt_t2 = Const (@{const_name HOL.less},
537 split_type --> split_type--> HOLogic.boolT) $ j $ t2'
538 val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
539 (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
540 (Const (@{const_name HOL.times},
541 split_type --> split_type --> split_type) $ t2' $ i) $ j)
542 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
543 val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
544 val subgoal2 = (map HOLogic.mk_Trueprop
545 [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
546 @ terms2 @ [not_false]
548 SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
550 (* "?P ((?n::int) mod (number_of ?k)) =
551 ((iszero (number_of ?k) --> ?P ?n) &
552 (neg (number_of (uminus ?k)) -->
553 (ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
554 (neg (number_of ?k) -->
555 (ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
556 | (Const ("Divides.div_class.mod",
557 Type ("fun", [Type ("Int.int", []), _])), [t1, t2 as (number_of $ k)]) =>
559 val rev_terms = rev terms
560 val zero = Const (@{const_name HOL.zero}, split_type)
563 val terms1 = map (subst_term [(split_term, t1)]) rev_terms
564 val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, j)])
565 (map (incr_boundvars 2) rev_terms)
566 val t1' = incr_boundvars 2 t1
567 val (t2' as (_ $ k')) = incr_boundvars 2 t2
568 val iszero_t2 = Const ("Int.iszero", split_type --> HOLogic.boolT) $ t2
569 val neg_minus_k = Const ("Int.neg", split_type --> HOLogic.boolT) $
571 (Const (@{const_name HOL.uminus},
572 HOLogic.intT --> HOLogic.intT) $ k'))
573 val zero_leq_j = Const (@{const_name HOL.less_eq},
574 split_type --> split_type --> HOLogic.boolT) $ zero $ j
575 val j_lt_t2 = Const (@{const_name HOL.less},
576 split_type --> split_type--> HOLogic.boolT) $ j $ t2'
577 val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
578 (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
579 (Const (@{const_name HOL.times},
580 split_type --> split_type --> split_type) $ t2' $ i) $ j)
581 val neg_t2 = Const ("Int.neg", split_type --> HOLogic.boolT) $ t2'
582 val t2_lt_j = Const (@{const_name HOL.less},
583 split_type --> split_type--> HOLogic.boolT) $ t2' $ j
584 val j_leq_zero = Const (@{const_name HOL.less_eq},
585 split_type --> split_type --> HOLogic.boolT) $ j $ zero
586 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
587 val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
588 val subgoal2 = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j])
590 :: (if tl terms2_3 = [] then [not_false] else [])
591 @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
592 @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
593 val subgoal3 = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j])
595 :: (if tl terms2_3 = [] then [not_false] else [])
596 @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
597 @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
598 val Ts' = split_type :: split_type :: Ts
600 SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
602 (* "?P ((?n::int) div (number_of ?k)) =
603 ((iszero (number_of ?k) --> ?P 0) &
604 (neg (number_of (uminus ?k)) -->
605 (ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) --> ?P i)) &
606 (neg (number_of ?k) -->
607 (ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) --> ?P i))) *)
608 | (Const ("Divides.div_class.div",
609 Type ("fun", [Type ("Int.int", []), _])), [t1, t2 as (number_of $ k)]) =>
611 val rev_terms = rev terms
612 val zero = Const (@{const_name HOL.zero}, split_type)
615 val terms1 = map (subst_term [(split_term, zero)]) rev_terms
616 val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, i)])
617 (map (incr_boundvars 2) rev_terms)
618 val t1' = incr_boundvars 2 t1
619 val (t2' as (_ $ k')) = incr_boundvars 2 t2
620 val iszero_t2 = Const ("Int.iszero", split_type --> HOLogic.boolT) $ t2
621 val neg_minus_k = Const ("Int.neg", split_type --> HOLogic.boolT) $
623 (Const (@{const_name HOL.uminus},
624 HOLogic.intT --> HOLogic.intT) $ k'))
625 val zero_leq_j = Const (@{const_name HOL.less_eq},
626 split_type --> split_type --> HOLogic.boolT) $ zero $ j
627 val j_lt_t2 = Const (@{const_name HOL.less},
628 split_type --> split_type--> HOLogic.boolT) $ j $ t2'
629 val t1_eq_t2_times_i_plus_j = Const ("op =",
630 split_type --> split_type --> HOLogic.boolT) $ t1' $
631 (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
632 (Const (@{const_name HOL.times},
633 split_type --> split_type --> split_type) $ t2' $ i) $ j)
634 val neg_t2 = Const ("Int.neg", split_type --> HOLogic.boolT) $ t2'
635 val t2_lt_j = Const (@{const_name HOL.less},
636 split_type --> split_type--> HOLogic.boolT) $ t2' $ j
637 val j_leq_zero = Const (@{const_name HOL.less_eq},
638 split_type --> split_type --> HOLogic.boolT) $ j $ zero
639 val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
640 val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
641 val subgoal2 = (HOLogic.mk_Trueprop neg_minus_k)
644 :: (map HOLogic.mk_Trueprop
645 [zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j])
646 val subgoal3 = (HOLogic.mk_Trueprop neg_t2)
649 :: (map HOLogic.mk_Trueprop
650 [t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j])
651 val Ts' = split_type :: split_type :: Ts
653 SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
655 (* this will only happen if a split theorem can be applied for which no *)
656 (* code exists above -- in which case either the split theorem should be *)
657 (* implemented above, or 'is_split_thm' should be modified to filter it *)
660 warning ("Lin. Arith.: split rule for " ^ Syntax.string_of_term ctxt t ^
661 " (with " ^ string_of_int (length ts) ^
662 " argument(s)) not implemented; proof reconstruction is likely to fail");
668 (* remove terms that do not satisfy 'p'; change the order of the remaining *)
669 (* terms in the same way as filter_prems_tac does *)
671 fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
673 fun filter_prems (t, (left, right)) =
674 if p t then (left, right @ [t]) else (left @ right, [])
675 val (left, right) = foldl filter_prems ([], []) terms
680 (* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a *)
681 (* subgoal that has 'terms' as premises *)
683 fun negated_term_occurs_positively (terms : term list) : bool =
685 (fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t)
689 fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =
691 (* repeatedly split (including newly emerging subgoals) until no further *)
692 (* splitting is possible *)
693 fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list)
694 | split_loop (subgoal::subgoals) = (
695 case split_once_items ctxt subgoal of
696 SOME new_subgoals => split_loop (new_subgoals @ subgoals)
697 | NONE => subgoal :: split_loop subgoals
699 fun is_relevant t = isSome (decomp ctxt t)
700 (* filter_prems_tac is_relevant: *)
701 val relevant_terms = filter_prems_tac_items is_relevant terms
702 (* split_tac, NNF normalization: *)
703 val split_goals = split_loop [(Ts, relevant_terms)]
704 (* necessary because split_once_tac may normalize terms: *)
705 val beta_eta_norm = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals
706 (* TRY (etac notE) THEN eq_assume_tac: *)
707 val result = List.filter (not o negated_term_occurs_positively o snd) beta_eta_norm
712 (* takes the i-th subgoal [| A1; ...; An |] ==> B to *)
713 (* An --> ... --> A1 --> B, performs splitting with the given 'split_thms' *)
714 (* (resulting in a different subgoal P), takes P to ~P ==> False, *)
715 (* performs NNF-normalization of ~P, and eliminates conjunctions, *)
716 (* disjunctions and existential quantifiers from the premises, possibly (in *)
717 (* the case of disjunctions) resulting in several new subgoals, each of the *)
718 (* general form [| Q1; ...; Qm |] ==> False. Fails if more than *)
719 (* !fast_arith_split_limit splits are possible. *)
723 empty_ss setmkeqTrue mk_eq_True
724 setmksimps (mksimps mksimps_pairs)
725 addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj,
726 not_all, not_ex, not_not]
727 fun prem_nnf_tac i st =
728 full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st
731 fun split_once_tac ctxt split_thms =
733 val thy = ProofContext.theory_of ctxt
734 val cond_split_tac = SUBGOAL (fn (subgoal, i) =>
736 val Ts = rev (map snd (Logic.strip_params subgoal))
737 val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)
738 val cmap = Splitter.cmap_of_split_thms split_thms
739 val splits = Splitter.split_posns cmap thy Ts concl
740 val split_limit = Config.get ctxt fast_arith_split_limit
742 if length splits > split_limit then no_tac
743 else split_tac split_thms i
747 REPEAT_DETERM o etac rev_mp,
751 TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))
757 (* remove irrelevant premises, then split the i-th subgoal (and all new *)
758 (* subgoals) by using 'split_once_tac' repeatedly. Beta-eta-normalize new *)
759 (* subgoals and finally attempt to solve them by finding an immediate *)
760 (* contradiction (i.e. a term and its negation) in their premises. *)
764 val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
765 fun is_relevant t = isSome (decomp ctxt t)
768 TRY (filter_prems_tac is_relevant i)
770 (TRY o REPEAT_ALL_NEW (split_once_tac ctxt split_thms))
772 (CONVERSION Drule.beta_eta_conversion
774 (TRY o (etac notE THEN' eq_assume_tac)))
779 end; (* LA_Data_Ref *)
782 val lin_arith_pre_tac = LA_Data_Ref.pre_tac;
784 structure Fast_Arith = 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;
791 val warning_count = Fast_Arith.warning_count;
793 (* reduce contradictory <= to False.
794 Most of the work is done by the cancel tactics. *)
796 val init_arith_data =
797 Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
798 {add_mono_thms = add_mono_thms @
799 @{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field},
800 mult_mono_thms = mult_mono_thms,
802 lessD = lessD @ [thm "Suc_leI"],
803 neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}],
804 simpset = HOL_basic_ss
806 [@{thm "monoid_add_class.add_0_left"},
807 @{thm "monoid_add_class.add_0_right"},
808 @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
809 @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
810 @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
811 @{thm "not_one_less_zero"}]
812 addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
813 (*abel_cancel helps it work in abstract algebraic domains*)
814 addsimprocs ArithData.nat_cancel_sums_add}) #>
815 arith_discrete "nat";
817 val lin_arith_simproc = Fast_Arith.lin_arith_simproc;
819 val fast_nat_arith_simproc =
820 Simplifier.simproc (the_context ()) "fast_nat_arith"
821 ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K Fast_Arith.lin_arith_simproc);
823 (* Because of fast_nat_arith_simproc, the arithmetic solver is really only
824 useful to detect inconsistencies among the premises for subgoals which are
825 *not* themselves (in)equalities, because the latter activate
826 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
827 solver all the time rather than add the additional check. *)
830 (* generic refutation procedure *)
835 tests if a term is at all relevant to the refutation proof;
836 if not, then it can be discarded. Can improve performance,
837 esp. if disjunctions can be discarded (no case distinction needed!).
839 prep_tac: int -> tactic
840 A preparation tactic to be applied to the goal once all relevant premises
841 have been moved to the conclusion.
843 ref_tac: int -> tactic
844 the actual refutation tactic. Should be able to deal with goals
845 [| A1; ...; An |] ==> False
846 where the Ai are atomic, i.e. no top-level &, | or EX
851 empty_ss setmkeqTrue mk_eq_True
852 setmksimps (mksimps mksimps_pairs)
853 addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},
854 @{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm not_not}];
855 fun prem_nnf_tac i st =
856 full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
858 fun refute_tac test prep_tac ref_tac =
859 let val refute_prems_tac =
861 (eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE
862 filter_prems_tac test 1 ORELSE
863 etac @{thm disjE} 1) THEN
864 (DETERM (etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE
866 in EVERY'[TRY o filter_prems_tac test,
867 REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac,
868 SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
873 (* arith proof method *)
877 fun raw_arith_tac ctxt ex =
878 (* FIXME: K true should be replaced by a sensible test (perhaps "isSome o
879 decomp sg"? -- but note that the test is applied to terms already before
880 they are split/normalized) to speed things up in case there are lots of
881 irrelevant terms involved; elimination of min/max can be optimized:
882 (max m n + k <= r) = (m+k <= r & n+k <= r)
883 (l <= min m n + k) = (l <= m+k & l <= n+k)
886 (* Splitting is also done inside fast_arith_tac, but not completely -- *)
887 (* split_tac may use split theorems that have not been implemented in *)
888 (* fast_arith_tac (cf. pre_decomp and split_once_items above), and *)
889 (* fast_arith_split_limit may trigger. *)
890 (* Therefore splitting outside of fast_arith_tac may allow us to prove *)
891 (* some goals that fast_arith_tac alone would fail on. *)
892 (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
893 (fast_ex_arith_tac ctxt ex);
895 fun more_arith_tacs ctxt =
896 let val tactics = #tactics (get_arith_data ctxt)
897 in FIRST' (map (fn ArithTactic {tactic, ...} => tactic ctxt) tactics) end;
901 fun simple_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
902 ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true];
904 fun arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
905 ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true,
906 more_arith_tacs ctxt];
908 fun silent_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
909 ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt false,
910 more_arith_tacs ctxt];
912 fun arith_method src =
913 Method.syntax Args.bang_facts src
914 #> (fn (prems, ctxt) => Method.METHOD (fn facts =>
915 HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac ctxt)));
924 Simplifier.map_ss (fn ss => ss addsimprocs [fast_nat_arith_simproc]
925 addSolver (mk_solver' "lin_arith" Fast_Arith.cut_lin_arith_tac)) #>
929 [("arith", arith_method, "decide linear arithmetic")] #>
930 Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add,
931 "declaration of split rules for arithmetic procedure")]) I;
935 structure BasicLinArith: BASIC_LIN_ARITH = LinArith;