src/HOL/SMT.thy
author boehmes
Tue Oct 26 11:45:12 2010 +0200 (2010-10-26)
changeset 40162 7f58a9a843c2
parent 39483 9f0e5684f04b
child 40274 6486c610a549
permissions -rw-r--r--
joined setup of SMT solvers in one place; turned Z3-specific options into SMT options (renamed configuration options from z3_* to smt_*); more detailed SMT exception; improved SMT filter interface
boehmes@36898
     1
(*  Title:      HOL/SMT.thy
boehmes@36898
     2
    Author:     Sascha Boehme, TU Muenchen
boehmes@36898
     3
*)
boehmes@36898
     4
boehmes@36898
     5
header {* Bindings to Satisfiability Modulo Theories (SMT) solvers *}
boehmes@36898
     6
boehmes@36898
     7
theory SMT
boehmes@36898
     8
imports List
boehmes@36898
     9
uses
boehmes@39483
    10
  "Tools/Datatype/datatype_selectors.ML"
boehmes@36898
    11
  ("Tools/SMT/smt_monomorph.ML")
boehmes@36898
    12
  ("Tools/SMT/smt_normalize.ML")
boehmes@36898
    13
  ("Tools/SMT/smt_translate.ML")
boehmes@36898
    14
  ("Tools/SMT/smt_solver.ML")
boehmes@36898
    15
  ("Tools/SMT/smtlib_interface.ML")
boehmes@36898
    16
  ("Tools/SMT/z3_proof_parser.ML")
boehmes@36898
    17
  ("Tools/SMT/z3_proof_tools.ML")
boehmes@36898
    18
  ("Tools/SMT/z3_proof_literals.ML")
boehmes@36898
    19
  ("Tools/SMT/z3_proof_reconstruction.ML")
boehmes@36898
    20
  ("Tools/SMT/z3_model.ML")
boehmes@36898
    21
  ("Tools/SMT/z3_interface.ML")
boehmes@40162
    22
  ("Tools/SMT/smt_setup_solvers.ML")
boehmes@36898
    23
begin
boehmes@36898
    24
boehmes@36898
    25
boehmes@36898
    26
huffman@36902
    27
subsection {* Triggers for quantifier instantiation *}
boehmes@36898
    28
boehmes@36898
    29
text {*
boehmes@36898
    30
Some SMT solvers support triggers for quantifier instantiation.
boehmes@36898
    31
Each trigger consists of one ore more patterns.  A pattern may either
boehmes@37124
    32
be a list of positive subterms (each being tagged by "pat"), or a
boehmes@37124
    33
list of negative subterms (each being tagged by "nopat").
boehmes@37124
    34
boehmes@37124
    35
When an SMT solver finds a term matching a positive pattern (a
boehmes@37124
    36
pattern with positive subterms only), it instantiates the
boehmes@37124
    37
corresponding quantifier accordingly.  Negative patterns inhibit
boehmes@37124
    38
quantifier instantiations.  Each pattern should mention all preceding
boehmes@37124
    39
bound variables.
boehmes@36898
    40
*}
boehmes@36898
    41
boehmes@36898
    42
datatype pattern = Pattern
boehmes@36898
    43
boehmes@37124
    44
definition pat :: "'a \<Rightarrow> pattern" where "pat _ = Pattern"
boehmes@37124
    45
definition nopat :: "'a \<Rightarrow> pattern" where "nopat _ = Pattern"
boehmes@36898
    46
boehmes@37124
    47
definition trigger :: "pattern list list \<Rightarrow> bool \<Rightarrow> bool"
boehmes@36898
    48
where "trigger _ P = P"
boehmes@36898
    49
boehmes@36898
    50
boehmes@36898
    51
huffman@36902
    52
subsection {* Higher-order encoding *}
boehmes@36898
    53
boehmes@36898
    54
text {*
boehmes@36898
    55
Application is made explicit for constants occurring with varying
boehmes@36898
    56
numbers of arguments.  This is achieved by the introduction of the
boehmes@36898
    57
following constant.
boehmes@36898
    58
*}
boehmes@36898
    59
boehmes@37153
    60
definition fun_app where "fun_app f x = f x"
boehmes@36898
    61
boehmes@36898
    62
text {*
boehmes@36898
    63
Some solvers support a theory of arrays which can be used to encode
boehmes@36898
    64
higher-order functions.  The following set of lemmas specifies the
boehmes@36898
    65
properties of such (extensional) arrays.
boehmes@36898
    66
*}
boehmes@36898
    67
boehmes@36898
    68
lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other
boehmes@37157
    69
  fun_upd_upd fun_app_def
boehmes@36898
    70
boehmes@36898
    71
boehmes@36898
    72
huffman@36902
    73
subsection {* First-order logic *}
boehmes@36898
    74
boehmes@36898
    75
text {*
boehmes@36898
    76
Some SMT solvers require a strict separation between formulas and
boehmes@36898
    77
terms.  When translating higher-order into first-order problems,
boehmes@36898
    78
all uninterpreted constants (those not builtin in the target solver)
boehmes@36898
    79
are treated as function symbols in the first-order sense.  Their
boehmes@36898
    80
occurrences as head symbols in atoms (i.e., as predicate symbols) is
boehmes@36898
    81
turned into terms by equating such atoms with @{term True} using the
boehmes@36898
    82
following term-level equation symbol.
boehmes@36898
    83
*}
boehmes@36898
    84
boehmes@37124
    85
definition term_eq :: "bool \<Rightarrow> bool \<Rightarrow> bool" where "term_eq x y = (x = y)"
boehmes@36898
    86
boehmes@36898
    87
boehmes@36898
    88
boehmes@37151
    89
subsection {* Integer division and modulo for Z3 *}
boehmes@37151
    90
boehmes@37151
    91
definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
boehmes@37151
    92
  "z3div k l = (if 0 \<le> l then k div l else -(k div (-l)))"
boehmes@37151
    93
boehmes@37151
    94
definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
boehmes@37151
    95
  "z3mod k l = (if 0 \<le> l then k mod l else k mod (-l))"
boehmes@37151
    96
boehmes@37151
    97
lemma div_by_z3div: "k div l = (
boehmes@37151
    98
     if k = 0 \<or> l = 0 then 0
boehmes@37151
    99
     else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3div k l
boehmes@37151
   100
     else z3div (-k) (-l))"
boehmes@37151
   101
  by (auto simp add: z3div_def)
boehmes@37151
   102
boehmes@37151
   103
lemma mod_by_z3mod: "k mod l = (
boehmes@37151
   104
     if l = 0 then k
boehmes@37151
   105
     else if k = 0 then 0
boehmes@37151
   106
     else if (0 < k \<and> 0 < l) \<or> (k < 0 \<and> 0 < l) then z3mod k l
boehmes@37151
   107
     else - z3mod (-k) (-l))"
boehmes@37151
   108
  by (auto simp add: z3mod_def)
boehmes@37151
   109
boehmes@37151
   110
boehmes@37151
   111
huffman@36902
   112
subsection {* Setup *}
boehmes@36898
   113
boehmes@36898
   114
use "Tools/SMT/smt_monomorph.ML"
boehmes@36898
   115
use "Tools/SMT/smt_normalize.ML"
boehmes@36898
   116
use "Tools/SMT/smt_translate.ML"
boehmes@36898
   117
use "Tools/SMT/smt_solver.ML"
boehmes@36898
   118
use "Tools/SMT/smtlib_interface.ML"
boehmes@36898
   119
use "Tools/SMT/z3_interface.ML"
boehmes@36898
   120
use "Tools/SMT/z3_proof_parser.ML"
boehmes@36898
   121
use "Tools/SMT/z3_proof_tools.ML"
boehmes@36898
   122
use "Tools/SMT/z3_proof_literals.ML"
boehmes@36898
   123
use "Tools/SMT/z3_proof_reconstruction.ML"
boehmes@36898
   124
use "Tools/SMT/z3_model.ML"
boehmes@40162
   125
use "Tools/SMT/smt_setup_solvers.ML"
boehmes@36898
   126
boehmes@36898
   127
setup {*
boehmes@36898
   128
  SMT_Solver.setup #>
boehmes@36898
   129
  Z3_Proof_Reconstruction.setup #>
boehmes@40162
   130
  SMT_Setup_Solvers.setup
boehmes@36898
   131
*}
boehmes@36898
   132
boehmes@36898
   133
boehmes@36898
   134
huffman@36902
   135
subsection {* Configuration *}
boehmes@36898
   136
boehmes@36898
   137
text {*
boehmes@36899
   138
The current configuration can be printed by the command
boehmes@36899
   139
@{text smt_status}, which shows the values of most options.
boehmes@36898
   140
*}
boehmes@36898
   141
boehmes@36898
   142
boehmes@36898
   143
boehmes@36898
   144
subsection {* General configuration options *}
boehmes@36898
   145
boehmes@36898
   146
text {*
boehmes@36898
   147
The option @{text smt_solver} can be used to change the target SMT
boehmes@36898
   148
solver.  The possible values are @{text cvc3}, @{text yices}, and
boehmes@36898
   149
@{text z3}.  It is advisable to locally install the selected solver,
boehmes@36898
   150
although this is not necessary for @{text cvc3} and @{text z3}, which
boehmes@36898
   151
can also be used over an Internet-based service.
boehmes@36898
   152
boehmes@36898
   153
When using local SMT solvers, the path to their binaries should be
boehmes@36898
   154
declared by setting the following environment variables:
boehmes@36898
   155
@{text CVC3_SOLVER}, @{text YICES_SOLVER}, and @{text Z3_SOLVER}.
boehmes@36898
   156
*}
boehmes@36898
   157
boehmes@36898
   158
declare [[ smt_solver = z3 ]]
boehmes@36898
   159
boehmes@36898
   160
text {*
boehmes@36898
   161
Since SMT solvers are potentially non-terminating, there is a timeout
boehmes@36898
   162
(given in seconds) to restrict their runtime.  A value greater than
boehmes@36898
   163
120 (seconds) is in most cases not advisable.
boehmes@36898
   164
*}
boehmes@36898
   165
boehmes@36898
   166
declare [[ smt_timeout = 20 ]]
boehmes@36898
   167
boehmes@40162
   168
text {*
boehmes@40162
   169
In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
boehmes@40162
   170
solvers are fully trusted without additional checks.  The following
boehmes@40162
   171
option can cause the SMT solver to run in proof-producing mode, giving
boehmes@40162
   172
a checkable certificate.  This is currently only implemented for Z3.
boehmes@40162
   173
*}
boehmes@40162
   174
boehmes@40162
   175
declare [[ smt_oracle = false ]]
boehmes@40162
   176
boehmes@40162
   177
text {*
boehmes@40162
   178
Each SMT solver provides several commandline options to tweak its
boehmes@40162
   179
behaviour.  They can be passed to the solver by setting the following
boehmes@40162
   180
options.
boehmes@40162
   181
*}
boehmes@40162
   182
boehmes@40162
   183
declare [[ cvc3_options = "", yices_options = "", z3_options = "" ]]
boehmes@40162
   184
boehmes@40162
   185
text {*
boehmes@40162
   186
Enable the following option to use built-in support for datatypes and
boehmes@40162
   187
records.  Currently, this is only implemented for Z3 running in oracle
boehmes@40162
   188
mode.
boehmes@40162
   189
*}
boehmes@40162
   190
boehmes@40162
   191
declare [[ smt_datatypes = false ]]
boehmes@40162
   192
boehmes@36898
   193
boehmes@36898
   194
boehmes@36898
   195
subsection {* Certificates *}
boehmes@36898
   196
boehmes@36898
   197
text {*
boehmes@36898
   198
By setting the option @{text smt_certificates} to the name of a file,
boehmes@36898
   199
all following applications of an SMT solver a cached in that file.
boehmes@36898
   200
Any further application of the same SMT solver (using the very same
boehmes@36898
   201
configuration) re-uses the cached certificate instead of invoking the
boehmes@36898
   202
solver.  An empty string disables caching certificates.
boehmes@36898
   203
boehmes@36898
   204
The filename should be given as an explicit path.  It is good
boehmes@36898
   205
practice to use the name of the current theory (with ending
boehmes@36898
   206
@{text ".certs"} instead of @{text ".thy"}) as the certificates file.
boehmes@36898
   207
*}
boehmes@36898
   208
boehmes@36898
   209
declare [[ smt_certificates = "" ]]
boehmes@36898
   210
boehmes@36898
   211
text {*
boehmes@36898
   212
The option @{text smt_fixed} controls whether only stored
boehmes@36898
   213
certificates are should be used or invocation of an SMT solver is
boehmes@36898
   214
allowed.  When set to @{text true}, no SMT solver will ever be
boehmes@36898
   215
invoked and only the existing certificates found in the configured
boehmes@36898
   216
cache are used;  when set to @{text false} and there is no cached
boehmes@36898
   217
certificate for some proposition, then the configured SMT solver is
boehmes@36898
   218
invoked.
boehmes@36898
   219
*}
boehmes@36898
   220
boehmes@36898
   221
declare [[ smt_fixed = false ]]
boehmes@36898
   222
boehmes@36898
   223
boehmes@36898
   224
boehmes@36898
   225
subsection {* Tracing *}
boehmes@36898
   226
boehmes@36898
   227
text {*
boehmes@36898
   228
For tracing the generated problem file given to the SMT solver as
boehmes@36898
   229
well as the returned result of the solver, the option
boehmes@36898
   230
@{text smt_trace} should be set to @{text true}.
boehmes@36898
   231
*}
boehmes@36898
   232
boehmes@36898
   233
declare [[ smt_trace = false ]]
boehmes@36898
   234
boehmes@36898
   235
text {*
boehmes@40162
   236
From the set of assumptions given to the SMT solver, those assumptions
boehmes@40162
   237
used in the proof are traced when the following option is set to
boehmes@40162
   238
@{term true}.  This only works for Z3 when it runs in non-oracle mode
boehmes@40162
   239
(see options @{text smt_solver} and @{text smt_oracle} above).
boehmes@36898
   240
*}
boehmes@36898
   241
boehmes@40162
   242
declare [[ smt_trace_used_facts = false ]]
boehmes@39298
   243
boehmes@36898
   244
boehmes@36898
   245
huffman@36902
   246
subsection {* Schematic rules for Z3 proof reconstruction *}
boehmes@36898
   247
boehmes@36898
   248
text {*
boehmes@36898
   249
Several prof rules of Z3 are not very well documented.  There are two
boehmes@36898
   250
lemma groups which can turn failing Z3 proof reconstruction attempts
boehmes@36898
   251
into succeeding ones: the facts in @{text z3_rule} are tried prior to
boehmes@36898
   252
any implemented reconstruction procedure for all uncertain Z3 proof
boehmes@36898
   253
rules;  the facts in @{text z3_simp} are only fed to invocations of
boehmes@36898
   254
the simplifier when reconstructing theory-specific proof steps.
boehmes@36898
   255
*}
boehmes@36898
   256
boehmes@36898
   257
lemmas [z3_rule] =
boehmes@36898
   258
  refl eq_commute conj_commute disj_commute simp_thms nnf_simps
boehmes@36898
   259
  ring_distribs field_simps times_divide_eq_right times_divide_eq_left
boehmes@36898
   260
  if_True if_False not_not
boehmes@36898
   261
boehmes@36898
   262
lemma [z3_rule]:
boehmes@36898
   263
  "(P \<longrightarrow> Q) = (Q \<or> \<not>P)"
boehmes@36898
   264
  "(\<not>P \<longrightarrow> Q) = (P \<or> Q)"
boehmes@36898
   265
  "(\<not>P \<longrightarrow> Q) = (Q \<or> P)"
boehmes@36898
   266
  by auto
boehmes@36898
   267
boehmes@36898
   268
lemma [z3_rule]:
boehmes@36898
   269
  "((P = Q) \<longrightarrow> R) = (R | (Q = (\<not>P)))"
boehmes@36898
   270
  by auto
boehmes@36898
   271
boehmes@36898
   272
lemma [z3_rule]:
boehmes@36898
   273
  "((\<not>P) = P) = False"
boehmes@36898
   274
  "(P = (\<not>P)) = False"
boehmes@36898
   275
  "(P \<noteq> Q) = (Q = (\<not>P))"
boehmes@36898
   276
  "(P = Q) = ((\<not>P \<or> Q) \<and> (P \<or> \<not>Q))"
boehmes@36898
   277
  "(P \<noteq> Q) = ((\<not>P \<or> \<not>Q) \<and> (P \<or> Q))"
boehmes@36898
   278
  by auto
boehmes@36898
   279
boehmes@36898
   280
lemma [z3_rule]:
boehmes@36898
   281
  "(if P then P else \<not>P) = True"
boehmes@36898
   282
  "(if \<not>P then \<not>P else P) = True"
boehmes@36898
   283
  "(if P then True else False) = P"
boehmes@36898
   284
  "(if P then False else True) = (\<not>P)"
boehmes@36898
   285
  "(if \<not>P then x else y) = (if P then y else x)"
boehmes@36898
   286
  by auto
boehmes@36898
   287
boehmes@36898
   288
lemma [z3_rule]:
boehmes@36898
   289
  "P = Q \<or> P \<or> Q"
boehmes@36898
   290
  "P = Q \<or> \<not>P \<or> \<not>Q"
boehmes@36898
   291
  "(\<not>P) = Q \<or> \<not>P \<or> Q"
boehmes@36898
   292
  "(\<not>P) = Q \<or> P \<or> \<not>Q"
boehmes@36898
   293
  "P = (\<not>Q) \<or> \<not>P \<or> Q"
boehmes@36898
   294
  "P = (\<not>Q) \<or> P \<or> \<not>Q"
boehmes@36898
   295
  "P \<noteq> Q \<or> P \<or> \<not>Q"
boehmes@36898
   296
  "P \<noteq> Q \<or> \<not>P \<or> Q"
boehmes@36898
   297
  "P \<noteq> (\<not>Q) \<or> P \<or> Q"
boehmes@36898
   298
  "(\<not>P) \<noteq> Q \<or> P \<or> Q"
boehmes@36898
   299
  "P \<or> Q \<or> P \<noteq> (\<not>Q)"
boehmes@36898
   300
  "P \<or> Q \<or> (\<not>P) \<noteq> Q"
boehmes@36898
   301
  "P \<or> \<not>Q \<or> P \<noteq> Q"
boehmes@36898
   302
  "\<not>P \<or> Q \<or> P \<noteq> Q"
boehmes@36898
   303
  by auto
boehmes@36898
   304
boehmes@36898
   305
lemma [z3_rule]:
boehmes@36898
   306
  "0 + (x::int) = x"
boehmes@36898
   307
  "x + 0 = x"
boehmes@36898
   308
  "0 * x = 0"
boehmes@36898
   309
  "1 * x = x"
boehmes@36898
   310
  "x + y = y + x"
boehmes@36898
   311
  by auto
boehmes@36898
   312
boehmes@37124
   313
boehmes@37124
   314
boehmes@37124
   315
hide_type (open) pattern
boehmes@37153
   316
hide_const Pattern term_eq
boehmes@37153
   317
hide_const (open) trigger pat nopat fun_app z3div z3mod
boehmes@37124
   318
boehmes@39483
   319
boehmes@39483
   320
boehmes@39483
   321
subsection {* Selectors for datatypes *}
boehmes@39483
   322
boehmes@39483
   323
setup {* Datatype_Selectors.setup *}
boehmes@39483
   324
boehmes@39483
   325
declare [[ selector Pair 1 = fst, selector Pair 2 = snd ]]
boehmes@39483
   326
declare [[ selector Cons 1 = hd, selector Cons 2 = tl ]]
boehmes@39483
   327
boehmes@36898
   328
end