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Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
This commit is contained in:
Nikolaj Bjorner 2015-12-09 20:17:00 -08:00
parent fe1039d12f
commit c5a9d81d93
7 changed files with 239 additions and 51 deletions

View file

@ -328,6 +328,9 @@ public:
app * mk_numeral(sexpr const * p, unsigned i) {
return plugin().mk_numeral(p, i);
}
app * mk_int(int i) {
return mk_numeral(rational(i), true);
}
app * mk_le(expr * arg1, expr * arg2) const { return m_manager.mk_app(m_afid, OP_LE, arg1, arg2); }
app * mk_ge(expr * arg1, expr * arg2) const { return m_manager.mk_app(m_afid, OP_GE, arg1, arg2); }
app * mk_lt(expr * arg1, expr * arg2) const { return m_manager.mk_app(m_afid, OP_LT, arg1, arg2); }

View file

@ -55,22 +55,28 @@ br_status seq_rewriter::mk_app_core(func_decl * f, unsigned num_args, expr * con
return mk_seq_concat(args[0], args[1], result);
case OP_SEQ_LENGTH:
SASSERT(num_args == 1);
return mk_str_length(args[0], result);
return mk_seq_length(args[0], result);
case OP_SEQ_EXTRACT:
SASSERT(num_args == 3);
return mk_str_substr(args[0], args[1], args[2], result);
return mk_seq_extract(args[0], args[1], args[2], result);
case OP_SEQ_CONTAINS:
SASSERT(num_args == 2);
return mk_str_strctn(args[0], args[1], result);
return mk_seq_contains(args[0], args[1], result);
case OP_SEQ_AT:
SASSERT(num_args == 2);
return mk_str_at(args[0], args[1], result);
return mk_seq_at(args[0], args[1], result);
case OP_SEQ_PREFIX:
SASSERT(num_args == 2);
return mk_seq_prefix(args[0], args[1], result);
case OP_SEQ_SUFFIX:
SASSERT(num_args == 2);
return mk_seq_suffix(args[0], args[1], result);
case OP_SEQ_INDEX:
SASSERT(num_args == 3);
return mk_seq_index(args[0], args[1], args[2], result);
case OP_SEQ_REPLACE:
SASSERT(num_args == 3);
return mk_seq_replace(args[0], args[1], args[2], result);
case OP_SEQ_TO_RE:
return BR_FAILED;
case OP_SEQ_IN_RE:
@ -78,12 +84,6 @@ br_status seq_rewriter::mk_app_core(func_decl * f, unsigned num_args, expr * con
case OP_STRING_CONST:
return BR_FAILED;
case OP_STRING_STRIDOF:
SASSERT(num_args == 3);
return mk_str_stridof(args[0], args[1], args[2], result);
case OP_STRING_STRREPL:
SASSERT(num_args == 3);
return mk_str_strrepl(args[0], args[1], args[2], result);
case OP_STRING_ITOS:
SASSERT(num_args == 1);
return mk_str_itos(args[0], result);
@ -101,7 +101,8 @@ br_status seq_rewriter::mk_app_core(func_decl * f, unsigned num_args, expr * con
case _OP_STRING_IN_REGEXP:
case _OP_STRING_TO_REGEXP:
case _OP_STRING_SUBSTR:
case _OP_STRING_STRREPL:
case _OP_STRING_STRIDOF:
UNREACHABLE();
}
return BR_FAILED;
@ -143,7 +144,7 @@ br_status seq_rewriter::mk_seq_concat(expr* a, expr* b, expr_ref& result) {
return BR_FAILED;
}
br_status seq_rewriter::mk_str_length(expr* a, expr_ref& result) {
br_status seq_rewriter::mk_seq_length(expr* a, expr_ref& result) {
std::string b;
m_es.reset();
m_util.str.get_concat(a, m_es);
@ -176,7 +177,7 @@ br_status seq_rewriter::mk_str_length(expr* a, expr_ref& result) {
return BR_FAILED;
}
br_status seq_rewriter::mk_str_substr(expr* a, expr* b, expr* c, expr_ref& result) {
br_status seq_rewriter::mk_seq_extract(expr* a, expr* b, expr* c, expr_ref& result) {
std::string s;
rational pos, len;
if (m_util.str.is_string(a, s) && m_autil.is_numeral(b, pos) && m_autil.is_numeral(c, len) &&
@ -188,7 +189,7 @@ br_status seq_rewriter::mk_str_substr(expr* a, expr* b, expr* c, expr_ref& resul
}
return BR_FAILED;
}
br_status seq_rewriter::mk_str_strctn(expr* a, expr* b, expr_ref& result) {
br_status seq_rewriter::mk_seq_contains(expr* a, expr* b, expr_ref& result) {
std::string c, d;
if (m_util.str.is_string(a, c) && m_util.str.is_string(b, d)) {
result = m().mk_bool_val(0 != strstr(d.c_str(), c.c_str()));
@ -212,7 +213,7 @@ br_status seq_rewriter::mk_str_strctn(expr* a, expr* b, expr_ref& result) {
return BR_FAILED;
}
br_status seq_rewriter::mk_str_at(expr* a, expr* b, expr_ref& result) {
br_status seq_rewriter::mk_seq_at(expr* a, expr* b, expr_ref& result) {
std::string c;
rational r;
if (m_util.str.is_string(a, c) && m_autil.is_numeral(b, r) && r.is_unsigned()) {
@ -228,7 +229,7 @@ br_status seq_rewriter::mk_str_at(expr* a, expr* b, expr_ref& result) {
return BR_FAILED;
}
br_status seq_rewriter::mk_str_stridof(expr* a, expr* b, expr* c, expr_ref& result) {
br_status seq_rewriter::mk_seq_index(expr* a, expr* b, expr* c, expr_ref& result) {
std::string s1, s2;
rational r;
bool isc1 = m_util.str.is_string(a, s1);
@ -257,15 +258,17 @@ br_status seq_rewriter::mk_str_stridof(expr* a, expr* b, expr* c, expr_ref& resu
return BR_FAILED;
}
br_status seq_rewriter::mk_str_strrepl(expr* a, expr* b, expr* c, expr_ref& result) {
br_status seq_rewriter::mk_seq_replace(expr* a, expr* b, expr* c, expr_ref& result) {
std::string s1, s2, s3;
if (m_util.str.is_string(a, s1) && m_util.str.is_string(b, s2) &&
m_util.str.is_string(c, s3)) {
std::ostringstream buffer;
bool can_replace = true;
for (size_t i = 0; i < s1.length(); ) {
if (strncmp(s1.c_str() + i, s2.c_str(), s2.length()) == 0) {
if (can_replace && strncmp(s1.c_str() + i, s2.c_str(), s2.length()) == 0) {
buffer << s3;
i += s2.length();
can_replace = false;
}
else {
buffer << s1[i];

View file

@ -35,12 +35,12 @@ class seq_rewriter {
ptr_vector<expr> m_es, m_lhs, m_rhs;
br_status mk_seq_concat(expr* a, expr* b, expr_ref& result);
br_status mk_str_length(expr* a, expr_ref& result);
br_status mk_str_substr(expr* a, expr* b, expr* c, expr_ref& result);
br_status mk_str_strctn(expr* a, expr* b, expr_ref& result);
br_status mk_str_at(expr* a, expr* b, expr_ref& result);
br_status mk_str_stridof(expr* a, expr* b, expr* c, expr_ref& result);
br_status mk_str_strrepl(expr* a, expr* b, expr* c, expr_ref& result);
br_status mk_seq_length(expr* a, expr_ref& result);
br_status mk_seq_extract(expr* a, expr* b, expr* c, expr_ref& result);
br_status mk_seq_contains(expr* a, expr* b, expr_ref& result);
br_status mk_seq_at(expr* a, expr* b, expr_ref& result);
br_status mk_seq_index(expr* a, expr* b, expr* c, expr_ref& result);
br_status mk_seq_replace(expr* a, expr* b, expr* c, expr_ref& result);
br_status mk_seq_prefix(expr* a, expr* b, expr_ref& result);
br_status mk_seq_suffix(expr* a, expr* b, expr_ref& result);
br_status mk_str_itos(expr* a, expr_ref& result);

View file

@ -177,6 +177,7 @@ void seq_decl_plugin::init() {
sort* reAreA[2] = { reA, reA };
sort* AA[2] = { A, A };
sort* seqAint2T[3] = { seqA, intT, intT };
sort* seq2AintT[3] = { seqA, seqA, intT };
sort* str2T[2] = { strT, strT };
sort* str3T[3] = { strT, strT, strT };
sort* strTint2T[3] = { strT, intT, intT };
@ -184,6 +185,7 @@ void seq_decl_plugin::init() {
sort* strTreT[2] = { strT, reT };
sort* str2TintT[3] = { strT, strT, intT };
sort* seqAintT[2] = { seqA, intT };
sort* seq3A[3] = { seqA, seqA, seqA };
m_sigs.resize(LAST_SEQ_OP);
// TBD: have (par ..) construct and load parameterized signature from premable.
m_sigs[OP_SEQ_UNIT] = alloc(psig, m, "seq.unit", 1, 1, &A, seqA);
@ -193,6 +195,8 @@ void seq_decl_plugin::init() {
m_sigs[OP_SEQ_SUFFIX] = alloc(psig, m, "seq.suffixof", 1, 2, seqAseqA, boolT);
m_sigs[OP_SEQ_CONTAINS] = alloc(psig, m, "seq.contains", 1, 2, seqAseqA, boolT);
m_sigs[OP_SEQ_EXTRACT] = alloc(psig, m, "seq.extract", 1, 3, seqAint2T, seqA);
m_sigs[OP_SEQ_REPLACE] = alloc(psig, m, "seq.replace", 1, 3, seq3A, strT);
m_sigs[OP_SEQ_INDEX] = alloc(psig, m, "seq.indexof", 1, 3, seq2AintT, intT);
m_sigs[OP_SEQ_AT] = alloc(psig, m, "seq.at", 1, 2, seqAintT, seqA);
m_sigs[OP_SEQ_LENGTH] = alloc(psig, m, "seq.len", 1, 1, &seqA, intT);
m_sigs[OP_RE_PLUS] = alloc(psig, m, "re.+", 1, 1, &reA, reA);
@ -209,8 +213,8 @@ void seq_decl_plugin::init() {
m_sigs[OP_SEQ_TO_RE] = alloc(psig, m, "seq.to.re", 1, 1, &seqA, reA);
m_sigs[OP_SEQ_IN_RE] = alloc(psig, m, "seq.in.re", 1, 2, seqAreA, boolT);
m_sigs[OP_STRING_CONST] = 0;
m_sigs[OP_STRING_STRIDOF] = alloc(psig, m, "str.indexof", 0, 3, str2TintT, intT);
m_sigs[OP_STRING_STRREPL] = alloc(psig, m, "str.replace", 0, 3, str3T, strT);
m_sigs[_OP_STRING_STRIDOF] = alloc(psig, m, "str.indexof", 0, 3, str2TintT, intT);
m_sigs[_OP_STRING_STRREPL] = alloc(psig, m, "str.replace", 0, 3, str3T, strT);
m_sigs[OP_STRING_ITOS] = alloc(psig, m, "int.to.str", 0, 1, &intT, strT);
m_sigs[OP_STRING_STOI] = alloc(psig, m, "str.to.int", 0, 1, &strT, intT);
m_sigs[OP_REGEXP_LOOP] = alloc(psig, m, "re.loop", 0, 2, strTint2T, reT); // maybe 3 arguments.
@ -347,6 +351,17 @@ func_decl * seq_decl_plugin::mk_func_decl(decl_kind k, unsigned num_parameters,
info.set_left_associative();
return m.mk_func_decl(m_sigs[k]->m_name, rng, rng, rng, info);
}
case OP_SEQ_REPLACE:
return mk_seq_fun(k, arity, domain, range, _OP_STRING_STRREPL);
case _OP_STRING_STRREPL:
return mk_str_fun(k, arity, domain, range, OP_SEQ_REPLACE);
case OP_SEQ_INDEX:
return mk_seq_fun(k, arity, domain, range, _OP_STRING_STRIDOF);
case _OP_STRING_STRIDOF:
return mk_str_fun(k, arity, domain, range, OP_SEQ_INDEX);
case OP_SEQ_PREFIX:
return mk_seq_fun(k, arity, domain, range, _OP_STRING_PREFIX);
case _OP_STRING_PREFIX:
@ -387,8 +402,6 @@ func_decl * seq_decl_plugin::mk_func_decl(decl_kind k, unsigned num_parameters,
case _OP_STRING_SUBSTR:
return mk_str_fun(k, arity, domain, range, OP_SEQ_EXTRACT);
case OP_STRING_STRIDOF:
case OP_STRING_STRREPL:
case OP_STRING_ITOS:
case OP_STRING_STOI:
case OP_REGEXP_LOOP:

View file

@ -39,8 +39,10 @@ enum seq_op_kind {
OP_SEQ_SUFFIX,
OP_SEQ_CONTAINS,
OP_SEQ_EXTRACT,
OP_SEQ_REPLACE,
OP_SEQ_AT,
OP_SEQ_LENGTH,
OP_SEQ_LENGTH,
OP_SEQ_INDEX,
OP_SEQ_TO_RE,
OP_SEQ_IN_RE,
@ -59,12 +61,11 @@ enum seq_op_kind {
// string specific operators.
OP_STRING_CONST,
OP_STRING_STRIDOF, // TBD generalize
OP_STRING_STRREPL, // TBD generalize
OP_STRING_ITOS,
OP_STRING_STOI,
OP_REGEXP_LOOP, // TBD re-loop: integers as parameters or arguments?
// internal only operators. Converted to SEQ variants.
_OP_STRING_STRREPL,
_OP_STRING_CONCAT,
_OP_STRING_LENGTH,
_OP_STRING_STRCTN,
@ -74,6 +75,7 @@ enum seq_op_kind {
_OP_STRING_TO_REGEXP,
_OP_STRING_CHARAT,
_OP_STRING_SUBSTR,
_OP_STRING_STRIDOF,
_OP_SEQ_SKOLEM,
LAST_SEQ_OP
};
@ -175,6 +177,9 @@ public:
app* mk_string(char const* s) { return mk_string(symbol(s)); }
app* mk_string(std::string const& s) { return mk_string(symbol(s.c_str())); }
app* mk_concat(expr* a, expr* b) { expr* es[2] = { a, b }; return m.mk_app(m_fid, OP_SEQ_CONCAT, 2, es); }
app* mk_concat(expr* a, expr* b, expr* c) {
return mk_concat(mk_concat(a, b), c);
}
expr* mk_concat(unsigned n, expr* const* es) { if (n == 1) return es[0]; SASSERT(n > 1); return m.mk_app(m_fid, OP_SEQ_CONCAT, n, es); }
app* mk_length(expr* a) { return m.mk_app(m_fid, OP_SEQ_LENGTH, 1, &a); }
app* mk_substr(expr* a, expr* b, expr* c) { expr* es[3] = { a, b, c }; return m.mk_app(m_fid, OP_SEQ_EXTRACT, 3, es); }
@ -200,8 +205,8 @@ public:
bool is_extract(expr const* n) const { return is_app_of(n, m_fid, OP_SEQ_EXTRACT); }
bool is_contains(expr const* n) const { return is_app_of(n, m_fid, OP_SEQ_CONTAINS); }
bool is_at(expr const* n) const { return is_app_of(n, m_fid, OP_SEQ_AT); }
bool is_stridof(expr const* n) const { return is_app_of(n, m_fid, OP_STRING_STRIDOF); }
bool is_repl(expr const* n) const { return is_app_of(n, m_fid, OP_STRING_STRREPL); }
bool is_index(expr const* n) const { return is_app_of(n, m_fid, OP_SEQ_INDEX); }
bool is_replace(expr const* n) const { return is_app_of(n, m_fid, OP_SEQ_REPLACE); }
bool is_prefix(expr const* n) const { return is_app_of(n, m_fid, OP_SEQ_PREFIX); }
bool is_suffix(expr const* n) const { return is_app_of(n, m_fid, OP_SEQ_SUFFIX); }
bool is_itos(expr const* n) const { return is_app_of(n, m_fid, OP_STRING_ITOS); }
@ -215,8 +220,8 @@ public:
MATCH_TERNARY(is_extract);
MATCH_BINARY(is_contains);
MATCH_BINARY(is_at);
MATCH_BINARY(is_stridof);
MATCH_BINARY(is_repl);
MATCH_BINARY(is_index);
MATCH_TERNARY(is_replace);
MATCH_BINARY(is_prefix);
MATCH_BINARY(is_suffix);
MATCH_UNARY(is_itos);

View file

@ -465,13 +465,6 @@ bool theory_seq::simplify_and_solve_eqs() {
}
final_check_status theory_seq::add_axioms() {
for (unsigned i = 0; i < get_num_vars(); ++i) {
// add axioms for len(x) when x = a ++ b
}
return FC_DONE;
}
bool theory_seq::internalize_atom(app* a, bool) {
return internalize_term(a);
@ -513,8 +506,6 @@ bool theory_seq::internalize_term(app* term) {
!m_util.is_skolem(term)) {
set_incomplete(term);
}
// assert basic axioms
return true;
}
@ -667,6 +658,170 @@ void theory_seq::create_axiom(expr_ref& e) {
m_axioms.push_back(e);
}
/*
encode that s is not a proper prefix of xs
*/
expr_ref theory_seq::tightest_prefix(expr* s, expr* x) {
expr_ref s1 = mk_skolem(symbol("first"), s);
expr_ref c = mk_skolem(symbol("last"), s);
expr_ref fml(m);
fml = m.mk_and(m.mk_eq(s, m_util.str.mk_concat(s1, c)),
m.mk_eq(m_util.str.mk_length(c), m_autil.mk_int(1)),
m.mk_not(m_util.str.mk_contains(s, m_util.str.mk_concat(x, s1))));
return fml;
}
void theory_seq::new_eq_len_concat(enode* n1, enode* n2) {
context& ctx = get_context();
// TBD: walk use list of n1 for concat, walk use-list of n2 for length.
// instantiate length distributes over concatenation axiom.
SASSERT(n1->get_root() != n2->get_root());
if (!m_util.is_seq(n1->get_owner())) {
return;
}
func_decl* f_len = 0;
// TBD: extract length function for sort if it is used.
// TBD: add filter for already processed length equivalence classes
if (!f_len) {
return;
}
enode_vector::const_iterator it = ctx.begin_enodes_of(f_len);
enode_vector::const_iterator end = ctx.end_enodes_of(f_len);
bool has_concat = true;
for (; has_concat && it != end; ++it) {
if ((*it)->get_root() == n1->get_root()) {
enode* start2 = n2;
do {
if (m_util.str.is_concat(n2->get_owner())) {
has_concat = true;
add_len_concat_axiom(n2->get_owner());
}
n2 = n2->get_next();
}
while (n2 != start2);
}
}
}
void theory_seq::add_len_concat_axiom(expr* c) {
// or just internalize lc and have relevancy propagation create axiom?
expr *a, *b;
expr_ref la(m), lb(m), lc(m), fml(m);
VERIFY(m_util.str.is_concat(c, a, b));
la = m_util.str.mk_length(a);
lb = m_util.str.mk_length(b);
lc = m_util.str.mk_length(c);
fml = m.mk_eq(m_autil.mk_add(la, lb), lc);
create_axiom(fml);
}
/*
let i = Index(s, t)
(!contains(s, t) -> i = -1)
(contains(s, t) & s = empty -> i = 0)
(contains(s, t) & s != empty -> t = xsy & tightest_prefix(s, x))
optional lemmas:
(len(s) > len(t) -> i = -1)
(len(s) <= len(t) -> i <= len(t)-len(s))
*/
void theory_seq::add_indexof_axiom(expr* i) {
expr* s, *t;
VERIFY(m_util.str.is_index(i, s, t));
expr_ref cnt(m), fml(m), eq_empty(m);
expr_ref x = mk_skolem(m_contains_left_sym, s, t);
expr_ref y = mk_skolem(m_contains_right_sym, s, t);
eq_empty = m.mk_eq(s, m_util.str.mk_empty(m.get_sort(s)));
cnt = m_util.str.mk_contains(s, t);
fml = m.mk_or(cnt, m.mk_eq(i, m_autil.mk_int(-1)));
create_axiom(fml);
fml = m.mk_or(m.mk_not(cnt), m.mk_not(eq_empty), m.mk_eq(i, m_autil.mk_int(0)));
create_axiom(fml);
fml = m.mk_or(m.mk_not(cnt), eq_empty, m.mk_eq(t, m_util.str.mk_concat(x,s,y)));
create_axiom(fml);
fml = m.mk_or(m.mk_not(cnt), eq_empty, tightest_prefix(s, x));
create_axiom(fml);
}
/*
let r = replace(a, s, t)
(contains(s, a) -> r = xty & a = xsy & tightest_prefix(s,xs)) &
(!contains(s, a) -> r = a)
*/
void theory_seq::add_replace_axiom(expr* r) {
expr* a, *s, *t;
VERIFY(m_util.str.is_replace(r, a, s, t));
expr_ref cnt(m), fml(m);
cnt = m_util.str.mk_contains(s, a);
expr_ref x = mk_skolem(m_contains_left_sym, s, a);
expr_ref y = mk_skolem(m_contains_right_sym, s, a);
fml = m.mk_or(m.mk_not(cnt), m.mk_eq(a, m_util.str.mk_concat(x, s, y)));
create_axiom(fml);
fml = m.mk_or(m.mk_not(cnt), m.mk_eq(r, m_util.str.mk_concat(x, t, y)));
create_axiom(fml);
fml = m.mk_or(m.mk_not(cnt), tightest_prefix(s, x));
create_axiom(fml);
fml = m.mk_or(cnt, m.mk_eq(r, a));
create_axiom(fml);
}
/*
let n = len(x)
len(x) >= 0
len(x) = 0 => x = ""
x = "" => len(x) = 0
len(x) = rewrite(len(x))
*/
void theory_seq::add_len_axiom(expr* n) {
expr* x;
VERIFY(m_util.str.is_length(n, x));
expr_ref fml(m), eq1(m), eq2(m), nr(m);
eq1 = m.mk_eq(m_autil.mk_int(0), n);
eq2 = m.mk_eq(x, m_util.str.mk_empty(m.get_sort(x)));
fml = m_autil.mk_le(m_autil.mk_int(0), n);
create_axiom(fml);
fml = m.mk_or(m.mk_not(eq1), eq2);
create_axiom(fml);
fml = m.mk_or(m.mk_not(eq2), eq1);
create_axiom(fml);
nr = n;
m_rewrite(nr);
if (nr != n) {
fml = m.mk_eq(n, nr);
create_axiom(fml);
}
}
/*
check semantics of extract.
let e = extract(s, i, l)
0 <= i < len(s) -> prefix(xe,s) & len(x) = i
0 <= i < len(s) & l >= len(s) - i -> len(e) = len(s) - i
0 <= i < len(s) & 0 <= l < len(s) - i -> len(e) = l
0 <= i < len(s) & l < 0 -> len(e) = 0
i < 0 -> e = s
i >= len(s) -> e = empty
*/
void theory_seq::add_extract_axiom(expr* e) {
expr* s, *i, *j;
VERIFY(m_util.str.is_extract(e, s, i, j));
expr_ref i_ge_0(m), i_le_j(m), j_lt_s(m);
NOT_IMPLEMENTED_YET();
}
void theory_seq::assert_axiom(expr_ref& e) {
context & ctx = get_context();
if (m.is_true(e)) return;
@ -679,7 +834,7 @@ void theory_seq::assert_axiom(expr_ref& e) {
expr_ref theory_seq::mk_skolem(symbol const& name, expr* e1, expr* e2) {
expr* es[2] = { e1, e2 };
return expr_ref(m_util.mk_skolem(name, 2, es, m.get_sort(e1)), m);
return expr_ref(m_util.mk_skolem(name, e2?2:1, es, m.get_sort(e1)), m);
}
void theory_seq::propagate_eq(bool_var v, expr* e1, expr* e2) {
@ -744,6 +899,9 @@ void theory_seq::new_eq_eh(theory_var v1, theory_var v2) {
m.push_back(m_lhs.back(), n1->get_owner());
m.push_back(m_rhs.back(), n2->get_owner());
m_dam.push_back(m_deps.back(), m_dm.mk_leaf(enode_pair(n1, n2)));
new_eq_len_concat(n1, n2);
new_eq_len_concat(n2, n1);
}
}
@ -807,8 +965,7 @@ void theory_seq::restart_eh() {
void theory_seq::relevant_eh(app* n) {
if (m_util.str.is_length(n)) {
expr_ref e(m);
e = m_autil.mk_le(m_autil.mk_numeral(rational(0), true), n);
create_axiom(e);
add_len_axiom(n);
}
}

View file

@ -166,15 +166,22 @@ namespace smt {
bool is_left_select(expr* a, expr*& b);
bool is_right_select(expr* a, expr*& b);
final_check_status add_axioms();
void assert_axiom(expr_ref& e);
void create_axiom(expr_ref& e);
void add_indexof_axiom(expr* e);
void add_replace_axiom(expr* e);
void add_extract_axiom(expr* e);
void add_len_concat_axiom(expr* c);
void add_len_axiom(expr* n);
void new_eq_len_concat(enode* n1, enode* n2);
expr_ref tightest_prefix(expr* s, expr* x);
expr_ref canonize(expr* e, enode_pair_dependency*& eqs);
expr_ref expand(expr* e, enode_pair_dependency*& eqs);
void add_dependency(enode_pair_dependency*& dep, enode* a, enode* b);
expr_ref mk_skolem(symbol const& s, expr* e1, expr* e2);
expr_ref mk_skolem(symbol const& s, expr* e1, expr* e2 = 0);
void set_incomplete(app* term);