src/HOL/SMT.thy
 author blanchet Mon Oct 06 19:19:16 2014 +0200 (2014-10-06) changeset 58598 d9892c88cb56 parent 58481 62bc7c79212b child 58776 95e58e04e534 permissions -rw-r--r--
strengthened 'moura' method
1 (*  Title:      HOL/SMT.thy
2     Author:     Sascha Boehme, TU Muenchen
3 *)
5 header {* Bindings to Satisfiability Modulo Theories (SMT) solvers based on SMT-LIB 2 *}
7 theory SMT
8 imports Divides
9 keywords "smt_status" :: diag
10 begin
12 subsection {* A skolemization tactic and proof method *}
14 lemma choices:
15   "\<And>Q. \<forall>x. \<exists>y ya. Q x y ya \<Longrightarrow> \<exists>f fa. \<forall>x. Q x (f x) (fa x)"
16   "\<And>Q. \<forall>x. \<exists>y ya yb. Q x y ya yb \<Longrightarrow> \<exists>f fa fb. \<forall>x. Q x (f x) (fa x) (fb x)"
17   "\<And>Q. \<forall>x. \<exists>y ya yb yc. Q x y ya yb yc \<Longrightarrow> \<exists>f fa fb fc. \<forall>x. Q x (f x) (fa x) (fb x) (fc x)"
18   "\<And>Q. \<forall>x. \<exists>y ya yb yc yd. Q x y ya yb yc yd \<Longrightarrow>
19      \<exists>f fa fb fc fd. \<forall>x. Q x (f x) (fa x) (fb x) (fc x) (fd x)"
20   "\<And>Q. \<forall>x. \<exists>y ya yb yc yd ye. Q x y ya yb yc yd ye \<Longrightarrow>
21      \<exists>f fa fb fc fd fe. \<forall>x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x)"
22   "\<And>Q. \<forall>x. \<exists>y ya yb yc yd ye yf. Q x y ya yb yc yd ye yf \<Longrightarrow>
23      \<exists>f fa fb fc fd fe ff. \<forall>x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x)"
24   "\<And>Q. \<forall>x. \<exists>y ya yb yc yd ye yf yg. Q x y ya yb yc yd ye yf yg \<Longrightarrow>
25      \<exists>f fa fb fc fd fe ff fg. \<forall>x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x) (fg x)"
26   by metis+
28 lemma bchoices:
29   "\<And>Q. \<forall>x \<in> S. \<exists>y ya. Q x y ya \<Longrightarrow> \<exists>f fa. \<forall>x \<in> S. Q x (f x) (fa x)"
30   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb. Q x y ya yb \<Longrightarrow> \<exists>f fa fb. \<forall>x \<in> S. Q x (f x) (fa x) (fb x)"
31   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc. Q x y ya yb yc \<Longrightarrow> \<exists>f fa fb fc. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x)"
32   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc yd. Q x y ya yb yc yd \<Longrightarrow>
33     \<exists>f fa fb fc fd. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x) (fd x)"
34   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc yd ye. Q x y ya yb yc yd ye \<Longrightarrow>
35     \<exists>f fa fb fc fd fe. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x)"
36   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc yd ye yf. Q x y ya yb yc yd ye yf \<Longrightarrow>
37     \<exists>f fa fb fc fd fe ff. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x)"
38   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc yd ye yf yg. Q x y ya yb yc yd ye yf yg \<Longrightarrow>
39     \<exists>f fa fb fc fd fe ff fg. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x) (fg x)"
40   by metis+
42 ML {*
43 fun moura_tac ctxt =
44   Atomize_Elim.atomize_elim_tac ctxt THEN'
45   SELECT_GOAL (Clasimp.auto_tac (ctxt addSIs @{thms choice choices bchoice bchoices}) THEN
46     ALLGOALS (Metis_Tactic.metis_tac (take 1 ATP_Proof_Reconstruct.partial_type_encs)
47         ATP_Proof_Reconstruct.default_metis_lam_trans ctxt [] ORELSE'
48       blast_tac ctxt))
49 *}
51 method_setup moura = {*
52  Scan.succeed (SIMPLE_METHOD' o moura_tac)
53 *} "solve skolemization goals, especially those arising from Z3 proofs"
55 hide_fact (open) choices bchoices
58 subsection {* Triggers for quantifier instantiation *}
60 text {*
61 Some SMT solvers support patterns as a quantifier instantiation
62 heuristics. Patterns may either be positive terms (tagged by "pat")
63 triggering quantifier instantiations -- when the solver finds a
64 term matching a positive pattern, it instantiates the corresponding
65 quantifier accordingly -- or negative terms (tagged by "nopat")
66 inhibiting quantifier instantiations. A list of patterns
67 of the same kind is called a multipattern, and all patterns in a
68 multipattern are considered conjunctively for quantifier instantiation.
69 A list of multipatterns is called a trigger, and their multipatterns
70 act disjunctively during quantifier instantiation. Each multipattern
71 should mention at least all quantified variables of the preceding
72 quantifier block.
73 *}
75 typedecl 'a symb_list
77 consts
78   Symb_Nil :: "'a symb_list"
79   Symb_Cons :: "'a \<Rightarrow> 'a symb_list \<Rightarrow> 'a symb_list"
81 typedecl pattern
83 consts
84   pat :: "'a \<Rightarrow> pattern"
85   nopat :: "'a \<Rightarrow> pattern"
87 definition trigger :: "pattern symb_list symb_list \<Rightarrow> bool \<Rightarrow> bool" where
88   "trigger _ P = P"
91 subsection {* Higher-order encoding *}
93 text {*
94 Application is made explicit for constants occurring with varying
95 numbers of arguments. This is achieved by the introduction of the
96 following constant.
97 *}
99 definition fun_app :: "'a \<Rightarrow> 'a" where "fun_app f = f"
101 text {*
102 Some solvers support a theory of arrays which can be used to encode
103 higher-order functions. The following set of lemmas specifies the
104 properties of such (extensional) arrays.
105 *}
107 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other  fun_upd_upd fun_app_def
110 subsection {* Normalization *}
112 lemma case_bool_if[abs_def]: "case_bool x y P = (if P then x else y)"
113   by simp
115 lemmas Ex1_def_raw = Ex1_def[abs_def]
116 lemmas Ball_def_raw = Ball_def[abs_def]
117 lemmas Bex_def_raw = Bex_def[abs_def]
118 lemmas abs_if_raw = abs_if[abs_def]
119 lemmas min_def_raw = min_def[abs_def]
120 lemmas max_def_raw = max_def[abs_def]
123 subsection {* Integer division and modulo for Z3 *}
125 text {*
126 The following Z3-inspired definitions are overspecified for the case where @{text "l = 0"}. This
127 Schönheitsfehler is corrected in the @{text div_as_z3div} and @{text mod_as_z3mod} theorems.
128 *}
130 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
131   "z3div k l = (if l \<ge> 0 then k div l else - (k div - l))"
133 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
134   "z3mod k l = k mod (if l \<ge> 0 then l else - l)"
136 lemma div_as_z3div:
137   "\<forall>k l. k div l = (if l = 0 then 0 else if l > 0 then z3div k l else z3div (- k) (- l))"
138   by (simp add: z3div_def)
140 lemma mod_as_z3mod:
141   "\<forall>k l. k mod l = (if l = 0 then k else if l > 0 then z3mod k l else - z3mod (- k) (- l))"
142   by (simp add: z3mod_def)
145 subsection {* Setup *}
147 ML_file "Tools/SMT/smt_util.ML"
148 ML_file "Tools/SMT/smt_failure.ML"
149 ML_file "Tools/SMT/smt_config.ML"
150 ML_file "Tools/SMT/smt_builtin.ML"
151 ML_file "Tools/SMT/smt_datatypes.ML"
152 ML_file "Tools/SMT/smt_normalize.ML"
153 ML_file "Tools/SMT/smt_translate.ML"
154 ML_file "Tools/SMT/smtlib.ML"
155 ML_file "Tools/SMT/smtlib_interface.ML"
156 ML_file "Tools/SMT/smtlib_proof.ML"
157 ML_file "Tools/SMT/smtlib_isar.ML"
158 ML_file "Tools/SMT/z3_proof.ML"
159 ML_file "Tools/SMT/z3_isar.ML"
160 ML_file "Tools/SMT/smt_solver.ML"
161 ML_file "Tools/SMT/cvc4_interface.ML"
162 ML_file "Tools/SMT/verit_proof.ML"
163 ML_file "Tools/SMT/verit_isar.ML"
164 ML_file "Tools/SMT/verit_proof_parse.ML"
165 ML_file "Tools/SMT/z3_interface.ML"
166 ML_file "Tools/SMT/z3_replay_util.ML"
167 ML_file "Tools/SMT/z3_replay_literals.ML"
168 ML_file "Tools/SMT/z3_replay_rules.ML"
169 ML_file "Tools/SMT/z3_replay_methods.ML"
170 ML_file "Tools/SMT/z3_replay.ML"
171 ML_file "Tools/SMT/smt_systems.ML"
173 method_setup smt = {*
174   Scan.optional Attrib.thms [] >>
175     (fn thms => fn ctxt =>
176       METHOD (fn facts => HEADGOAL (SMT_Solver.smt_tac ctxt (thms @ facts))))
177 *} "apply an SMT solver to the current goal"
180 subsection {* Configuration *}
182 text {*
183 The current configuration can be printed by the command
184 @{text smt_status}, which shows the values of most options.
185 *}
188 subsection {* General configuration options *}
190 text {*
191 The option @{text smt_solver} can be used to change the target SMT
192 solver. The possible values can be obtained from the @{text smt_status}
193 command.
195 Due to licensing restrictions, Z3 is not enabled by default. Z3 is free
196 for non-commercial applications and can be enabled by setting Isabelle
197 system option @{text z3_non_commercial} to @{text yes}.
198 *}
200 declare [[smt_solver = z3]]
202 text {*
203 Since SMT solvers are potentially nonterminating, there is a timeout
204 (given in seconds) to restrict their runtime.
205 *}
207 declare [[smt_timeout = 20]]
209 text {*
210 SMT solvers apply randomized heuristics. In case a problem is not
211 solvable by an SMT solver, changing the following option might help.
212 *}
214 declare [[smt_random_seed = 1]]
216 text {*
217 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
218 solvers are fully trusted without additional checks. The following
219 option can cause the SMT solver to run in proof-producing mode, giving
220 a checkable certificate. This is currently only implemented for Z3.
221 *}
223 declare [[smt_oracle = false]]
225 text {*
226 Each SMT solver provides several commandline options to tweak its
227 behaviour. They can be passed to the solver by setting the following
228 options.
229 *}
231 declare [[cvc3_options = ""]]
232 declare [[cvc4_options = "--full-saturate-quant --quant-cf"]]
233 declare [[veriT_options = ""]]
234 declare [[z3_options = ""]]
236 text {*
237 The SMT method provides an inference mechanism to detect simple triggers
238 in quantified formulas, which might increase the number of problems
239 solvable by SMT solvers (note: triggers guide quantifier instantiations
240 in the SMT solver). To turn it on, set the following option.
241 *}
243 declare [[smt_infer_triggers = false]]
245 text {*
246 Enable the following option to use built-in support for datatypes,
247 codatatypes, and records in CVC4. Currently, this is implemented only
248 in oracle mode.
249 *}
251 declare [[cvc4_extensions = false]]
253 text {*
254 Enable the following option to use built-in support for div/mod, datatypes,
255 and records in Z3. Currently, this is implemented only in oracle mode.
256 *}
258 declare [[z3_extensions = false]]
261 subsection {* Certificates *}
263 text {*
264 By setting the option @{text smt_certificates} to the name of a file,
265 all following applications of an SMT solver a cached in that file.
266 Any further application of the same SMT solver (using the very same
267 configuration) re-uses the cached certificate instead of invoking the
268 solver. An empty string disables caching certificates.
270 The filename should be given as an explicit path. It is good
271 practice to use the name of the current theory (with ending
272 @{text ".certs"} instead of @{text ".thy"}) as the certificates file.
273 Certificate files should be used at most once in a certain theory context,
274 to avoid race conditions with other concurrent accesses.
275 *}
277 declare [[smt_certificates = ""]]
279 text {*
280 The option @{text smt_read_only_certificates} controls whether only
281 stored certificates are should be used or invocation of an SMT solver
282 is allowed. When set to @{text true}, no SMT solver will ever be
283 invoked and only the existing certificates found in the configured
284 cache are used;  when set to @{text false} and there is no cached
285 certificate for some proposition, then the configured SMT solver is
286 invoked.
287 *}
289 declare [[smt_read_only_certificates = false]]
292 subsection {* Tracing *}
294 text {*
295 The SMT method, when applied, traces important information. To
296 make it entirely silent, set the following option to @{text false}.
297 *}
299 declare [[smt_verbose = true]]
301 text {*
302 For tracing the generated problem file given to the SMT solver as
303 well as the returned result of the solver, the option
304 @{text smt_trace} should be set to @{text true}.
305 *}
307 declare [[smt_trace = false]]
310 subsection {* Schematic rules for Z3 proof reconstruction *}
312 text {*
313 Several prof rules of Z3 are not very well documented. There are two
314 lemma groups which can turn failing Z3 proof reconstruction attempts
315 into succeeding ones: the facts in @{text z3_rule} are tried prior to
316 any implemented reconstruction procedure for all uncertain Z3 proof
317 rules;  the facts in @{text z3_simp} are only fed to invocations of
318 the simplifier when reconstructing theory-specific proof steps.
319 *}
321 lemmas [z3_rule] =
322   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
323   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
324   if_True if_False not_not
326 lemma [z3_rule]:
327   "(P \<and> Q) = (\<not> (\<not> P \<or> \<not> Q))"
328   "(P \<and> Q) = (\<not> (\<not> Q \<or> \<not> P))"
329   "(\<not> P \<and> Q) = (\<not> (P \<or> \<not> Q))"
330   "(\<not> P \<and> Q) = (\<not> (\<not> Q \<or> P))"
331   "(P \<and> \<not> Q) = (\<not> (\<not> P \<or> Q))"
332   "(P \<and> \<not> Q) = (\<not> (Q \<or> \<not> P))"
333   "(\<not> P \<and> \<not> Q) = (\<not> (P \<or> Q))"
334   "(\<not> P \<and> \<not> Q) = (\<not> (Q \<or> P))"
335   by auto
337 lemma [z3_rule]:
338   "(P \<longrightarrow> Q) = (Q \<or> \<not> P)"
339   "(\<not> P \<longrightarrow> Q) = (P \<or> Q)"
340   "(\<not> P \<longrightarrow> Q) = (Q \<or> P)"
341   "(True \<longrightarrow> P) = P"
342   "(P \<longrightarrow> True) = True"
343   "(False \<longrightarrow> P) = True"
344   "(P \<longrightarrow> P) = True"
345   by auto
347 lemma [z3_rule]:
348   "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not> P)))"
349   by auto
351 lemma [z3_rule]:
352   "(\<not> True) = False"
353   "(\<not> False) = True"
354   "(x = x) = True"
355   "(P = True) = P"
356   "(True = P) = P"
357   "(P = False) = (\<not> P)"
358   "(False = P) = (\<not> P)"
359   "((\<not> P) = P) = False"
360   "(P = (\<not> P)) = False"
361   "((\<not> P) = (\<not> Q)) = (P = Q)"
362   "\<not> (P = (\<not> Q)) = (P = Q)"
363   "\<not> ((\<not> P) = Q) = (P = Q)"
364   "(P \<noteq> Q) = (Q = (\<not> P))"
365   "(P = Q) = ((\<not> P \<or> Q) \<and> (P \<or> \<not> Q))"
366   "(P \<noteq> Q) = ((\<not> P \<or> \<not> Q) \<and> (P \<or> Q))"
367   by auto
369 lemma [z3_rule]:
370   "(if P then P else \<not> P) = True"
371   "(if \<not> P then \<not> P else P) = True"
372   "(if P then True else False) = P"
373   "(if P then False else True) = (\<not> P)"
374   "(if P then Q else True) = ((\<not> P) \<or> Q)"
375   "(if P then Q else True) = (Q \<or> (\<not> P))"
376   "(if P then Q else \<not> Q) = (P = Q)"
377   "(if P then Q else \<not> Q) = (Q = P)"
378   "(if P then \<not> Q else Q) = (P = (\<not> Q))"
379   "(if P then \<not> Q else Q) = ((\<not> Q) = P)"
380   "(if \<not> P then x else y) = (if P then y else x)"
381   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not> Q) then y else x)"
382   "(if P then (if Q then x else y) else x) = (if (\<not> Q) \<and> P then y else x)"
383   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
384   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
385   "(if P then x else if P then y else z) = (if P then x else z)"
386   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
387   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
388   "(if P then x = y else x = z) = (x = (if P then y else z))"
389   "(if P then x = y else y = z) = (y = (if P then x else z))"
390   "(if P then x = y else z = y) = (y = (if P then x else z))"
391   by auto
393 lemma [z3_rule]:
394   "0 + (x::int) = x"
395   "x + 0 = x"
396   "x + x = 2 * x"
397   "0 * x = 0"
398   "1 * x = x"
399   "x + y = y + x"
400   by (auto simp add: mult_2)
402 lemma [z3_rule]:  (* for def-axiom *)
403   "P = Q \<or> P \<or> Q"
404   "P = Q \<or> \<not> P \<or> \<not> Q"
405   "(\<not> P) = Q \<or> \<not> P \<or> Q"
406   "(\<not> P) = Q \<or> P \<or> \<not> Q"
407   "P = (\<not> Q) \<or> \<not> P \<or> Q"
408   "P = (\<not> Q) \<or> P \<or> \<not> Q"
409   "P \<noteq> Q \<or> P \<or> \<not> Q"
410   "P \<noteq> Q \<or> \<not> P \<or> Q"
411   "P \<noteq> (\<not> Q) \<or> P \<or> Q"
412   "(\<not> P) \<noteq> Q \<or> P \<or> Q"
413   "P \<or> Q \<or> P \<noteq> (\<not> Q)"
414   "P \<or> Q \<or> (\<not> P) \<noteq> Q"
415   "P \<or> \<not> Q \<or> P \<noteq> Q"
416   "\<not> P \<or> Q \<or> P \<noteq> Q"
417   "P \<or> y = (if P then x else y)"
418   "P \<or> (if P then x else y) = y"
419   "\<not> P \<or> x = (if P then x else y)"
420   "\<not> P \<or> (if P then x else y) = x"
421   "P \<or> R \<or> \<not> (if P then Q else R)"
422   "\<not> P \<or> Q \<or> \<not> (if P then Q else R)"
423   "\<not> (if P then Q else R) \<or> \<not> P \<or> Q"
424   "\<not> (if P then Q else R) \<or> P \<or> R"
425   "(if P then Q else R) \<or> \<not> P \<or> \<not> Q"
426   "(if P then Q else R) \<or> P \<or> \<not> R"
427   "(if P then \<not> Q else R) \<or> \<not> P \<or> Q"
428   "(if P then Q else \<not> R) \<or> P \<or> R"
429   by auto
431 hide_type (open) symb_list pattern
432 hide_const (open) Symb_Nil Symb_Cons trigger pat nopat fun_app z3div z3mod
434 end