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seq

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Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>
This commit is contained in:
Nikolaj Bjorner 2015-12-29 10:13:19 -08:00
parent e2fab0a555
commit bd9b5b5735
5 changed files with 133 additions and 54 deletions
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4
src/ast/rewriter/seq_rewriter.cpp
View file

@ -39,7 +39,10 @@ re2automaton::re2automaton(ast_manager& m): m(m), u(m) {}
eautomaton* re2automaton::operator()(expr* e) { eautomaton* re2automaton::operator()(expr* e) {
eautomaton* r = re2aut(e); eautomaton* r = re2aut(e);
if (r) { if (r) {
//display_expr1 disp(m);
//r->display(std::cout, disp);
r->compress(); r->compress();
//r->display(std::cout, disp);
} }
return r; return r;
} }
@ -61,6 +64,7 @@ eautomaton* re2automaton::re2aut(expr* e) {
else if (u.re.is_star(e, e1) && (a = re2aut(e1))) { else if (u.re.is_star(e, e1) && (a = re2aut(e1))) {
a->add_final_to_init_moves(); a->add_final_to_init_moves();
a->add_init_to_final_states(); a->add_init_to_final_states();
return a.detach(); return a.detach();
} }
else if (u.re.is_plus(e, e1) && (a = re2aut(e1))) { else if (u.re.is_plus(e, e1) && (a = re2aut(e1))) {

26
src/math/automata/automaton.h
View file

@ -209,24 +209,16 @@ public:
moves mvs; moves mvs;
unsigned_vector final; unsigned_vector final;
unsigned init = 0; unsigned init = 0;
if (a.has_single_final_sink() && b.initial_state_is_source() && b.init() == 0) { unsigned offset1 = 1;
unsigned offset2 = a.num_states(); unsigned offset2 = a.num_states() + offset1;
init = a.init(); mvs.push_back(move(m, 0, a.init() + offset1));
append_moves(0, a, mvs); append_moves(offset1, a, mvs);
append_moves(offset2, b, mvs); for (unsigned i = 0; i < a.m_final_states.size(); ++i) {
append_final(offset2, b, final); mvs.push_back(move(m, a.m_final_states[i] + offset1, b.init() + offset2));
}
else {
unsigned offset1 = 1;
unsigned offset2 = a.num_states() + offset1;
mvs.push_back(move(m, 0, a.init() + offset1));
append_moves(offset1, a, mvs);
for (unsigned i = 0; i < a.m_final_states.size(); ++i) {
mvs.push_back(move(m, a.m_final_states[i], b.init() + offset2));
}
append_moves(offset2, b, mvs);
append_final(offset2, b, final);
} }
append_moves(offset2, b, mvs);
append_final(offset2, b, final);
return alloc(automaton, m, init, final, mvs); return alloc(automaton, m, init, final, mvs);
} }

1
src/smt/smt_model_generator.cpp
View file

@ -361,6 +361,7 @@ namespace smt {
} }
else { else {
enode * child = d.get_enode(); enode * child = d.get_enode();
TRACE("mg_top_sort", tout << "#" << n->get_owner_id() << " (" << mk_pp(n->get_owner(), m_manager) << "): " << mk_pp(child->get_owner(), m_manager) << " " << mk_pp(child->get_root()->get_owner(), m_manager) << "\n";);
child = child->get_root(); child = child->get_root();
app * val = 0; app * val = 0;
m_root2value.find(child, val); m_root2value.find(child, val);

139
src/smt/theory_seq.cpp
View file

@ -77,6 +77,14 @@ expr* theory_seq::solution_map::find(expr* e, enode_pair_dependency*& d) {
return result; return result;
} }
expr* theory_seq::solution_map::find(expr* e) {
std::pair<expr*, enode_pair_dependency*> value;
while (m_map.find(e, value)) {
e = value.first;
}
return e;
}
void theory_seq::solution_map::pop_scope(unsigned num_scopes) { void theory_seq::solution_map::pop_scope(unsigned num_scopes) {
if (num_scopes == 0) return; if (num_scopes == 0) return;
m_cache.reset(); m_cache.reset();
@ -185,26 +193,32 @@ final_check_status theory_seq::final_check_eh() {
context & ctx = get_context(); context & ctx = get_context();
TRACE("seq", display(tout);); TRACE("seq", display(tout););
if (!check_ineqs()) { if (!check_ineqs()) {
++m_stats.m_check_ineqs;
TRACE("seq", tout << ">>check_ineqs\n";); TRACE("seq", tout << ">>check_ineqs\n";);
return FC_CONTINUE; return FC_CONTINUE;
} }
if (simplify_and_solve_eqs()) { if (simplify_and_solve_eqs()) {
++m_stats.m_solve_eqs;
TRACE("seq", tout << ">>solve_eqs\n";); TRACE("seq", tout << ">>solve_eqs\n";);
return FC_CONTINUE; return FC_CONTINUE;
} }
if (solve_nqs()) { if (solve_nqs()) {
++m_stats.m_solve_nqs;
TRACE("seq", tout << ">>solve_nqs\n";); TRACE("seq", tout << ">>solve_nqs\n";);
return FC_CONTINUE; return FC_CONTINUE;
} }
if (branch_variable()) { if (branch_variable()) {
++m_stats.m_branch_variable;
TRACE("seq", tout << ">>branch_variable\n";); TRACE("seq", tout << ">>branch_variable\n";);
return FC_CONTINUE; return FC_CONTINUE;
} }
if (!check_length_coherence()) { if (!check_length_coherence()) {
++m_stats.m_check_length_coherence;
TRACE("seq", tout << ">>check_length_coherence\n";); TRACE("seq", tout << ">>check_length_coherence\n";);
return FC_CONTINUE; return FC_CONTINUE;
} }
if (propagate_automata()) { if (propagate_automata()) {
++m_stats.m_propagate_automata;
TRACE("seq", tout << ">>propagate_automata\n";); TRACE("seq", tout << ">>propagate_automata\n";);
return FC_CONTINUE; return FC_CONTINUE;
} }
@ -312,23 +326,27 @@ bool theory_seq::assume_equality(expr* l, expr* r) {
} }
bool theory_seq::propagate_length_coherence(expr* e) { bool theory_seq::propagate_length_coherence(expr* e) {
expr_ref head(m), tail(m), emp(m); expr_ref head(m), tail(m);
rational lo, hi; rational lo, hi;
if (!is_var(e) || !m_rep.is_root(e)) {
return false;
}
if (!lower_bound(e, lo) || !lo.is_pos() || lo >= rational(2048)) {
return false;
}
TRACE("seq", tout << "Unsolved " << mk_pp(e, m); TRACE("seq", tout << "Unsolved " << mk_pp(e, m);
if (!lower_bound(e, lo)) lo = -rational::one(); if (!lower_bound(e, lo)) lo = -rational::one();
if (!upper_bound(e, hi)) hi = -rational::one(); if (!upper_bound(e, hi)) hi = -rational::one();
tout << " lo: " << lo << " hi: " << hi << "\n"; tout << " lo: " << lo << " hi: " << hi << "\n";
); );
if (!lower_bound(e, lo) || !lo.is_pos() || lo >= rational(2048)) {
return false;
}
literal low(mk_literal(m_autil.mk_ge(m_util.str.mk_length(e), m_autil.mk_numeral(lo, true)))); literal low(mk_literal(m_autil.mk_ge(m_util.str.mk_length(e), m_autil.mk_numeral(lo, true))));
expr_ref seq(e, m); expr_ref seq(e, m);
expr_ref_vector elems(m); expr_ref_vector elems(m);
unsigned _lo = lo.get_unsigned(); unsigned _lo = lo.get_unsigned();
for (unsigned j = 0; j < _lo; ++j) { for (unsigned j = 0; j < _lo; ++j) {
mk_decompose(seq, emp, head, tail); mk_decompose(seq, head, tail);
elems.push_back(head); elems.push_back(head);
seq = tail; seq = tail;
} }
@ -342,6 +360,10 @@ bool theory_seq::propagate_length_coherence(expr* e) {
expr_ref high2(m_autil.mk_le(m_util.str.mk_length(seq), m_autil.mk_numeral(hi-lo, true)), m); expr_ref high2(m_autil.mk_le(m_util.str.mk_length(seq), m_autil.mk_numeral(hi-lo, true)), m);
add_axiom(~mk_literal(high1), mk_literal(high2)); add_axiom(~mk_literal(high1), mk_literal(high2));
} }
//m_replay_length_coherence.push_back(e);
//m_replay_length_coherence_qhead = m_replay_length_coherence.size();
return true; return true;
} }
@ -357,14 +379,12 @@ bool theory_seq::check_length_coherence() {
expr_ref head(m), tail(m); expr_ref head(m), tail(m);
if (propagate_length_coherence(e)) { if (propagate_length_coherence(e)) {
//m_replay_length_coherence.push_back(e);
//m_replay_length_coherence_qhead = m_replay_length_coherence.size();
} }
else if (!assume_equality(e, emp)) { else if (!assume_equality(e, emp)) {
mk_decompose(e, emp, head, tail); mk_decompose(e, head, tail);
// e = emp \/ e = unit(head.elem(e))*tail(e) // e = emp \/ e = unit(head.elem(e))*tail(e)
expr_ref conc(m_util.str.mk_concat(head, tail), m); expr_ref conc(m_util.str.mk_concat(head, tail), m);
add_axiom(mk_eq(e, emp, false), mk_eq(e, conc, false)); add_axiom(mk_eq_empty(e), mk_eq(e, conc, false));
assume_equality(tail, emp); assume_equality(tail, emp);
} }
return false; return false;
@ -379,10 +399,9 @@ expr_ref theory_seq::mk_nth(expr* s, expr* idx) {
return mk_skolem(m_nth, s, idx, 0, char_sort); return mk_skolem(m_nth, s, idx, 0, char_sort);
} }
void theory_seq::mk_decompose(expr* e, expr_ref& emp, expr_ref& head, expr_ref& tail) { void theory_seq::mk_decompose(expr* e, expr_ref& head, expr_ref& tail) {
expr* e1, *e2; expr* e1, *e2;
zstring s; zstring s;
emp = m_util.str.mk_empty(m.get_sort(e));
if (m_util.str.is_empty(e)) { if (m_util.str.is_empty(e)) {
head = m_util.str.mk_unit(mk_nth(e, m_autil.mk_int(0))); head = m_util.str.mk_unit(mk_nth(e, m_autil.mk_int(0)));
tail = e; tail = e;
@ -393,7 +412,7 @@ void theory_seq::mk_decompose(expr* e, expr_ref& emp, expr_ref& head, expr_ref&
} }
else if (m_util.str.is_unit(e)) { else if (m_util.str.is_unit(e)) {
head = e; head = e;
tail = emp; tail = m_util.str.mk_empty(m.get_sort(e));
} }
else if (m_util.str.is_concat(e, e1, e2) && m_util.str.is_unit(e1)) { else if (m_util.str.is_concat(e, e1, e2) && m_util.str.is_unit(e1)) {
head = e1; head = e1;
@ -585,7 +604,7 @@ bool theory_seq::is_var(expr* a) {
} }
bool theory_seq::is_head_elem(expr* e) const { bool theory_seq::is_nth(expr* e) const {
return is_skolem(m_nth, e); return is_skolem(m_nth, e);
} }
@ -884,6 +903,13 @@ void theory_seq::display_deps(std::ostream& out, enode_pair_dependency* dep) con
void theory_seq::collect_statistics(::statistics & st) const { void theory_seq::collect_statistics(::statistics & st) const {
st.update("seq num splits", m_stats.m_num_splits); st.update("seq num splits", m_stats.m_num_splits);
st.update("seq num reductions", m_stats.m_num_reductions); st.update("seq num reductions", m_stats.m_num_reductions);
st.update("e", m_stats.m_propagate_automata);
st.update("seq length coherence", m_stats.m_check_length_coherence);
st.update("seq branch", m_stats.m_branch_variable);
st.update("seq solve !=", m_stats.m_solve_nqs);
st.update("seq solve =", m_stats.m_solve_eqs);
st.update("seq check negations", m_stats.m_check_ineqs);
} }
void theory_seq::init_model(model_generator & mg) { void theory_seq::init_model(model_generator & mg) {
@ -910,7 +936,6 @@ public:
if (values.empty()) { if (values.empty()) {
return th.mk_value(n); return th.mk_value(n);
} }
SASSERT(values.size() == n->get_num_args());
return th.mk_value(mg.get_manager().mk_app(n->get_decl(), values.size(), values.c_ptr())); return th.mk_value(mg.get_manager().mk_app(n->get_decl(), values.size(), values.c_ptr()));
} }
}; };
@ -918,13 +943,35 @@ public:
model_value_proc * theory_seq::mk_value(enode * n, model_generator & mg) { model_value_proc * theory_seq::mk_value(enode * n, model_generator & mg) {
context& ctx = get_context(); context& ctx = get_context();
enode_pair_dependency* dep = 0; expr_ref e(m);
expr* e = m_rep.find(n->get_owner(), dep); expr* e1;
expr* e1, *e2; ptr_vector<expr> concats;
get_concat(n->get_owner(), concats);
switch (concats.size()) {
case 0:
e = m_util.str.mk_empty(m.get_sort(n->get_owner()));
break;
case 1:
e = concats[0];
SASSERT(!m_util.str.is_concat(e));
break;
default:
e = m_rep.find(n->get_owner());
SASSERT(m_util.str.is_concat(e));
break;
}
seq_value_proc* sv = alloc(seq_value_proc, *this, to_app(e)); seq_value_proc* sv = alloc(seq_value_proc, *this, to_app(e));
if (m_util.str.is_concat(e, e1, e2)) { TRACE("seq", tout << mk_pp(n->get_owner(), m) << " ";
sv->add_dependency(ctx.get_enode(e1)); for (unsigned i = 0; i < concats.size(); ++i) {
sv->add_dependency(ctx.get_enode(e2)); tout << mk_pp(concats[i], m) << " ";
}
tout << "\n";
);
if (concats.size() > 1) {
for (unsigned i = 0; i < concats.size(); ++i) {
sv->add_dependency(ctx.get_enode(concats[i]));
}
} }
else if (m_util.str.is_unit(e, e1)) { else if (m_util.str.is_unit(e, e1)) {
sv->add_dependency(ctx.get_enode(e1)); sv->add_dependency(ctx.get_enode(e1));
@ -932,6 +979,22 @@ model_value_proc * theory_seq::mk_value(enode * n, model_generator & mg) {
return sv; return sv;
} }
void theory_seq::get_concat(expr* e, ptr_vector<expr>& concats) {
expr* e1, *e2;
while (true) {
e = m_rep.find(e);
if (m_util.str.is_concat(e, e1, e2)) {
get_concat(e1, concats);
e = e2;
continue;
}
if (!m_util.str.is_empty(e)) {
concats.push_back(e);
}
return;
}
}
app* theory_seq::mk_value(app* e) { app* theory_seq::mk_value(app* e) {
expr* e1; expr* e1;
expr_ref result(e, m); expr_ref result(e, m);
@ -964,7 +1027,7 @@ app* theory_seq::mk_value(app* e) {
result = e; result = e;
} }
} }
else if (is_head_elem(e)) { else if (is_nth(e)) {
enode* n = get_context().get_enode(e)->get_root(); enode* n = get_context().get_enode(e)->get_root();
enode* n0 = n; enode* n0 = n;
bool found_value = false; bool found_value = false;
@ -1088,6 +1151,9 @@ void theory_seq::deque_axiom(expr* n) {
if (m_util.str.is_length(n)) { if (m_util.str.is_length(n)) {
add_length_axiom(n); add_length_axiom(n);
} }
else if (m_util.str.is_empty(n) && !has_length(n)) {
enforce_length(get_context().get_enode(n));
}
else if (m_util.str.is_index(n)) { else if (m_util.str.is_index(n)) {
add_indexof_axiom(n); add_indexof_axiom(n);
} }
@ -1120,8 +1186,7 @@ void theory_seq::tightest_prefix(expr* s, expr* x, literal lit1, literal lit2) {
expr_ref s1c(m_util.str.mk_concat(s1, c), m); expr_ref s1c(m_util.str.mk_concat(s1, c), m);
expr_ref lc(m_util.str.mk_length(c), m); expr_ref lc(m_util.str.mk_length(c), m);
expr_ref one(m_autil.mk_int(1), m); expr_ref one(m_autil.mk_int(1), m);
expr_ref emp(m_util.str.mk_empty(m.get_sort(s)), m); literal s_eq_emp = mk_eq_empty(s);
literal s_eq_emp = mk_eq(s, emp, false);
add_axiom(lit1, lit2, s_eq_emp, mk_eq(s, s1c, false)); add_axiom(lit1, lit2, s_eq_emp, mk_eq(s, s1c, false));
add_axiom(lit1, lit2, s_eq_emp, mk_eq(lc, one, false)); add_axiom(lit1, lit2, s_eq_emp, mk_eq(lc, one, false));
add_axiom(lit1, lit2, s_eq_emp, ~mk_literal(m_util.str.mk_contains(s, m_util.str.mk_concat(x, s1)))); add_axiom(lit1, lit2, s_eq_emp, ~mk_literal(m_util.str.mk_contains(s, m_util.str.mk_concat(x, s1))));
@ -1160,7 +1225,6 @@ void theory_seq::add_indexof_axiom(expr* i) {
expr_ref emp(m), minus_one(m), zero(m), xsy(m); expr_ref emp(m), minus_one(m), zero(m), xsy(m);
minus_one = m_autil.mk_int(-1); minus_one = m_autil.mk_int(-1);
zero = m_autil.mk_int(0); zero = m_autil.mk_int(0);
emp = m_util.str.mk_empty(m.get_sort(s));
literal offset_ne_zero = null_literal; literal offset_ne_zero = null_literal;
bool is_num = m_autil.is_numeral(offset, r); bool is_num = m_autil.is_numeral(offset, r);
if (is_num && r.is_zero()) { if (is_num && r.is_zero()) {
@ -1174,7 +1238,7 @@ void theory_seq::add_indexof_axiom(expr* i) {
expr_ref y = mk_skolem(m_contains_right, t, s); expr_ref y = mk_skolem(m_contains_right, t, s);
xsy = m_util.str.mk_concat(x,s,y); xsy = m_util.str.mk_concat(x,s,y);
literal cnt = mk_literal(m_util.str.mk_contains(t, s)); literal cnt = mk_literal(m_util.str.mk_contains(t, s));
literal eq_empty = mk_eq(s, emp, false); literal eq_empty = mk_eq_empty(s);
add_axiom(offset_ne_zero, cnt, mk_eq(i, minus_one, false)); add_axiom(offset_ne_zero, cnt, mk_eq(i, minus_one, false));
add_axiom(offset_ne_zero, ~eq_empty, mk_eq(i, zero, false)); add_axiom(offset_ne_zero, ~eq_empty, mk_eq(i, zero, false));
add_axiom(offset_ne_zero, ~cnt, eq_empty, mk_eq(t, xsy, false)); add_axiom(offset_ne_zero, ~cnt, eq_empty, mk_eq(t, xsy, false));
@ -1263,12 +1327,12 @@ void theory_seq::add_length_axiom(expr* n) {
} }
else { else {
expr_ref zero(m_autil.mk_int(0), m); expr_ref zero(m_autil.mk_int(0), m);
expr_ref emp(m_util.str.mk_empty(m.get_sort(x)), m);
literal eq1(mk_eq(zero, n, false));
literal eq2(mk_eq(x, emp, false));
add_axiom(mk_literal(m_autil.mk_ge(n, zero))); add_axiom(mk_literal(m_autil.mk_ge(n, zero)));
add_axiom(~eq1, eq2);
add_axiom(~eq2, eq1); //expr_ref emp(m_util.str.mk_empty(m.get_sort(x)), m);
//literal eq1(mk_eq(zero, n, false));
//literal eq2(mk_eq(x, emp, false));
//add_axiom(~eq1, eq2);
} }
} }
@ -1510,6 +1574,17 @@ literal theory_seq::mk_literal(expr* _e) {
return ctx.get_literal(e); return ctx.get_literal(e);
} }
literal theory_seq::mk_eq_empty(expr* _e) {
expr_ref e(_e, m);
context& ctx = get_context();
SASSERT(m_util.is_seq(e));
expr_ref emp(m);
emp = m_util.str.mk_empty(m.get_sort(e));
literal lit = mk_eq(e, emp, false);
ctx.force_phase(lit);
return lit;
}
void theory_seq::add_axiom(literal l1, literal l2, literal l3, literal l4) { void theory_seq::add_axiom(literal l1, literal l2, literal l3, literal l4) {
context& ctx = get_context(); context& ctx = get_context();
literal_vector lits; literal_vector lits;
@ -1681,8 +1756,8 @@ void theory_seq::relevant_eh(app* n) {
m_util.str.is_replace(n) || m_util.str.is_replace(n) ||
m_util.str.is_extract(n) || m_util.str.is_extract(n) ||
m_util.str.is_at(n) || m_util.str.is_at(n) ||
m_util.str.is_string(n) || m_util.str.is_empty(n) ||
is_step(n)) { m_util.str.is_string(n)) {
enque_axiom(n); enque_axiom(n);
} }

17
src/smt/theory_seq.h
View file

@ -72,6 +72,7 @@ namespace smt {
void add_cache(expr* v, expr_dep& r) { m_cache.insert(v, r); } void add_cache(expr* v, expr_dep& r) { m_cache.insert(v, r); }
bool find_cache(expr* v, expr_dep& r) { return m_cache.find(v, r); } bool find_cache(expr* v, expr_dep& r) { return m_cache.find(v, r); }
expr* find(expr* e, enode_pair_dependency*& d); expr* find(expr* e, enode_pair_dependency*& d);
expr* find(expr* e);
bool is_root(expr* e) const; bool is_root(expr* e) const;
void cache(expr* e, expr* r, enode_pair_dependency* d); void cache(expr* e, expr* r, enode_pair_dependency* d);
void reset_cache() { m_cache.reset(); } void reset_cache() { m_cache.reset(); }
@ -246,6 +247,12 @@ namespace smt {
void reset() { memset(this, 0, sizeof(stats)); } void reset() { memset(this, 0, sizeof(stats)); }
unsigned m_num_splits; unsigned m_num_splits;
unsigned m_num_reductions; unsigned m_num_reductions;
unsigned m_propagate_automata;
unsigned m_check_length_coherence;
unsigned m_branch_variable;
unsigned m_solve_nqs;
unsigned m_solve_eqs;
unsigned m_check_ineqs;
}; };
ast_manager& m; ast_manager& m;
enode_pair_dependency_manager m_dm; enode_pair_dependency_manager m_dm;
@ -335,13 +342,13 @@ namespace smt {
bool occurs(expr* a, expr* b); bool occurs(expr* a, expr* b);
bool is_var(expr* b); bool is_var(expr* b);
bool add_solution(expr* l, expr* r, enode_pair_dependency* dep); bool add_solution(expr* l, expr* r, enode_pair_dependency* dep);
bool is_left_select(expr* a, expr*& b); bool is_nth(expr* a) const;
bool is_right_select(expr* a, expr*& b); expr_ref mk_nth(expr* s, expr* idx);
bool is_head_elem(expr* a) const;
expr_ref canonize(expr* e, enode_pair_dependency*& eqs); expr_ref canonize(expr* e, enode_pair_dependency*& eqs);
expr_ref expand(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); void add_dependency(enode_pair_dependency*& dep, enode* a, enode* b);
void get_concat(expr* e, ptr_vector<expr>& concats);
// terms whose meaning are encoded using axioms. // terms whose meaning are encoded using axioms.
void enque_axiom(expr* e); void enque_axiom(expr* e);
@ -360,6 +367,7 @@ namespace smt {
void add_at_axiom(expr* n); void add_at_axiom(expr* n);
void add_in_re_axiom(expr* n); void add_in_re_axiom(expr* n);
literal mk_literal(expr* n); literal mk_literal(expr* n);
literal mk_eq_empty(expr* n);
void tightest_prefix(expr* s, expr* x, literal lit, literal lit2 = null_literal); void tightest_prefix(expr* s, expr* x, literal lit, literal lit2 = null_literal);
expr_ref mk_sub(expr* a, expr* b); expr_ref mk_sub(expr* a, expr* b);
enode* ensure_enode(expr* a); enode* ensure_enode(expr* a);
@ -369,9 +377,8 @@ namespace smt {
bool upper_bound(expr* s, rational& hi); bool upper_bound(expr* s, rational& hi);
bool get_length(expr* s, rational& val); bool get_length(expr* s, rational& val);
void mk_decompose(expr* e, expr_ref& emp, expr_ref& head, expr_ref& tail); void mk_decompose(expr* e, expr_ref& head, expr_ref& tail);
expr_ref mk_skolem(symbol const& s, expr* e1, expr* e2 = 0, expr* e3 = 0, sort* range = 0); expr_ref mk_skolem(symbol const& s, expr* e1, expr* e2 = 0, expr* e3 = 0, sort* range = 0);
expr_ref mk_nth(expr* s, expr* idx);
bool is_skolem(symbol const& s, expr* e) const; bool is_skolem(symbol const& s, expr* e) const;
void set_incomplete(app* term); void set_incomplete(app* term);