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z3-z3-4.12.6.src.math.dd.dd_pdd.h Maven / Gradle / Ivy
/*++
Copyright (c) 2019 Microsoft Corporation
Module Name:
dd_pdd
Abstract:
Poly DD package
It is a mild variant of ZDDs.
In PDDs arithmetic is either standard or using mod 2^n.
Non-leaf nodes are of the form x*hi + lo
where
- maxdegree(x, lo) = 0, meaning x does not appear in lo
Leaf nodes are of the form (0*idx + 0), where idx is an index into m_values.
- reduce(a, b) reduces a using the polynomial equality b = 0. That is, given b = lt(b) + tail(b),
the occurrences in a where lm(b) divides a monomial a_i, replace a_i in a by -tail(b)*a_i/lt(b)
- try_spoly(a, b, c) returns true if lt(a) and lt(b) have a non-trivial overlap. c is the resolvent (S polynomial).
- reduce(v, a, b) reduces 'a' using b = 0 with respect to eliminating v.
Given b = v^l*b1 + b2, where l is the leading coefficient of v in b
Given a = v^m*a1 + b2, where m >= l is the leading coefficient of v in b.
reduce(a1, b1)*v^{m - l} + reduce(v, a2, b);
Author:
Nikolaj Bjorner (nbjorner) 2019-12-17
--*/
#pragma once
#include "util/vector.h"
#include "util/map.h"
#include "util/small_object_allocator.h"
#include "util/rational.h"
namespace dd {
class test;
class pdd;
class pdd_manager;
class pdd_iterator;
class pdd_linear_iterator;
class pdd_manager {
public:
enum semantics { free_e, mod2_e, zero_one_vars_e, mod2N_e };
private:
friend test;
friend pdd;
friend pdd_iterator;
friend pdd_linear_iterator;
typedef unsigned PDD;
typedef vector> monomials_t;
static constexpr PDD null_pdd = UINT_MAX;
static constexpr PDD zero_pdd = 0;
static constexpr PDD one_pdd = 1;
enum pdd_op {
pdd_add_op = 2,
pdd_sub_op = 3,
pdd_minus_op = 4,
pdd_mul_op = 5,
pdd_reduce_op = 6,
pdd_subst_val_op = 7,
pdd_subst_add_op = 8,
pdd_div_const_op = 9,
pdd_no_op = 10
};
struct node {
node(unsigned level, PDD lo, PDD hi):
m_refcount(0),
m_level(level),
m_lo(lo),
m_hi(hi),
m_index(0)
{}
node(unsigned value):
m_refcount(0),
m_level(0),
m_lo(value),
m_hi(0),
m_index(0)
{}
node(): m_refcount(0), m_level(0), m_lo(0), m_hi(0), m_index(0) {}
unsigned m_refcount : 10;
unsigned m_level : 22;
PDD m_lo;
PDD m_hi;
unsigned m_index;
unsigned hash() const { return mk_mix(m_level, m_lo, m_hi); }
bool is_val() const { return m_hi == 0 && (m_lo != 0 || m_index == 0); }
bool is_internal() const { return m_hi == 0 && m_lo == 0 && m_index != 0; }
void set_internal() { m_lo = 0; m_hi = 0; }
};
struct hash_node {
unsigned operator()(node const& n) const { return n.hash(); }
};
struct eq_node {
bool operator()(node const& a, node const& b) const {
return a.m_lo == b.m_lo && a.m_hi == b.m_hi && a.m_level == b.m_level;
}
};
typedef hashtable node_table;
struct const_info {
unsigned m_value_index;
unsigned m_node_index;
};
typedef map mpq_table;
struct op_entry {
op_entry(PDD l, PDD r, PDD op):
m_pdd1(l),
m_pdd2(r),
m_op(op),
m_result(0)
{}
PDD m_pdd1;
PDD m_pdd2;
PDD m_op;
PDD m_result;
unsigned hash() const { return mk_mix(m_pdd1, m_pdd2, m_op); }
};
struct hash_entry {
unsigned operator()(op_entry* e) const { return e->hash(); }
};
struct eq_entry {
bool operator()(op_entry * a, op_entry * b) const {
return a->m_pdd1 == b->m_pdd2 && a->m_pdd2 == b->m_pdd2 && a->m_op == b->m_op;
}
};
typedef ptr_hashtable op_table;
struct factor_entry {
factor_entry(PDD p, unsigned v, unsigned degree):
m_p(p),
m_v(v),
m_degree(degree),
m_lc(UINT_MAX),
m_rest(UINT_MAX)
{}
factor_entry(): m_p(0), m_v(0), m_degree(0), m_lc(UINT_MAX), m_rest(UINT_MAX) {}
PDD m_p; // input
unsigned m_v; // input
unsigned m_degree; // input
PDD m_lc; // output
PDD m_rest; // output
bool is_valid() { return m_lc != UINT_MAX; }
unsigned hash() const { return mk_mix(m_p, m_v, m_degree); }
};
struct hash_factor_entry {
unsigned operator()(factor_entry const& e) const { return e.hash(); }
};
struct eq_factor_entry {
bool operator()(factor_entry const& a, factor_entry const& b) const {
return a.m_p == b.m_p && a.m_v == b.m_v && a.m_degree == b.m_degree;
}
};
typedef hashtable factor_table;
svector m_nodes;
vector m_values;
op_table m_op_cache;
factor_table m_factor_cache;
node_table m_node_table;
mpq_table m_mpq_table;
svector m_pdd_stack;
op_entry* m_spare_entry;
svector m_var2pdd;
unsigned_vector m_var2level, m_level2var;
unsigned_vector m_free_nodes;
small_object_allocator m_alloc;
mutable svector m_mark;
mutable unsigned m_mark_level;
mutable svector m_todo;
mutable unsigned_vector m_degree;
mutable svector m_tree_size;
bool m_disable_gc;
bool m_is_new_node;
unsigned m_max_num_nodes;
semantics m_semantics;
unsigned_vector m_free_vars;
unsigned_vector m_free_values;
rational m_freeze_value;
rational m_mod2N;
unsigned m_power_of_2 = 0;
rational m_max_value;
void reset_op_cache();
void init_nodes(unsigned_vector const& l2v);
void init_vars(unsigned_vector const& l2v);
PDD make_node(unsigned level, PDD l, PDD r);
PDD insert_node(node const& n);
bool is_new_node() const { return m_is_new_node; }
PDD apply(PDD arg1, PDD arg2, pdd_op op);
PDD apply_rec(PDD arg1, PDD arg2, pdd_op op);
PDD minus_rec(PDD p);
PDD div_rec(PDD p, rational const& c, PDD c_pdd);
PDD pow(PDD p, unsigned j);
PDD pow_rec(PDD p, unsigned j);
PDD reduce_on_match(PDD a, PDD b);
bool lm_occurs(PDD p, PDD q) const;
PDD lt_quotient(PDD p, PDD q);
PDD lt_quotient_hi(PDD p, PDD q);
PDD imk_val(rational const& r);
void init_value(const_info& info, rational const& r);
void init_value(rational const& v, unsigned r);
void push(PDD b);
void pop(unsigned num_scopes);
PDD read(unsigned index);
op_entry* pop_entry(PDD l, PDD r, PDD op);
void push_entry(op_entry* e);
bool check_result(op_entry*& e1, op_entry const* e2, PDD a, PDD b, PDD c);
void alloc_free_nodes(unsigned n);
void init_mark();
void set_mark(unsigned i) { m_mark[i] = m_mark_level; }
bool is_marked(unsigned i) { return m_mark[i] == m_mark_level; }
static const unsigned max_rc = (1 << 10) - 1;
inline bool is_zero(PDD p) const { return p == zero_pdd; }
inline bool is_one(PDD p) const { return p == one_pdd; }
inline bool is_val(PDD p) const { return m_nodes[p].is_val(); }
inline bool is_internal(PDD p) const { return m_nodes[p].is_internal(); }
inline bool is_var(PDD p) const { return !is_val(p) && is_zero(lo(p)) && is_one(hi(p)); }
inline bool is_max(PDD p) const { SASSERT(m_semantics == mod2_e || m_semantics == mod2N_e); return is_val(p) && val(p) == max_value(); }
bool is_never_zero(PDD p);
unsigned min_parity(PDD p);
inline unsigned level(PDD p) const { return m_nodes[p].m_level; }
inline unsigned var(PDD p) const { return m_level2var[level(p)]; }
inline PDD lo(PDD p) const { return m_nodes[p].m_lo; }
inline PDD hi(PDD p) const { return m_nodes[p].m_hi; }
inline rational const& val(PDD p) const { SASSERT(is_val(p)); return m_values[lo(p)]; }
inline rational get_signed_val(PDD p) const { SASSERT(m_semantics == mod2_e || m_semantics == mod2N_e); rational const& a = val(p); return a.get_bit(power_of_2() - 1) ? a - two_to_N() : a; }
inline void inc_ref(PDD p) { if (m_nodes[p].m_refcount != max_rc) m_nodes[p].m_refcount++; SASSERT(!m_free_nodes.contains(p)); }
inline void dec_ref(PDD p) { if (m_nodes[p].m_refcount != max_rc) m_nodes[p].m_refcount--; SASSERT(!m_free_nodes.contains(p)); }
inline PDD level2pdd(unsigned l) const { return m_var2pdd[m_level2var[l]]; }
unsigned dag_size(pdd const& p);
mutable svector m_dmark;
mutable unsigned m_dmark_level;
void init_dmark();
void set_dmark(unsigned i) const { m_dmark[i] = m_dmark_level; }
bool is_dmarked(unsigned i) const { return m_dmark[i] == m_dmark_level; }
unsigned degree(pdd const& p) const;
unsigned degree(PDD p) const;
unsigned degree(PDD p, unsigned v);
unsigned max_pow2_divisor(PDD p);
unsigned max_pow2_divisor(pdd const& p);
template pdd map_coefficients(pdd const& p, Fn f);
void factor(pdd const& p, unsigned v, unsigned degree, pdd& lc, pdd& rest);
bool factor(pdd const& p, unsigned v, unsigned degree, pdd& lc);
pdd reduce(unsigned v, pdd const& a, unsigned m, pdd const& b1, pdd const& b2);
bool var_is_leaf(PDD p, unsigned v);
bool is_reachable(PDD p);
void compute_reachable(bool_vector& reachable);
void try_gc();
void reserve_var(unsigned v);
bool well_formed();
bool well_formed(node const& n);
unsigned_vector m_p, m_q;
rational m_pc, m_qc;
pdd spoly(pdd const& a, pdd const& b, unsigned_vector const& p, unsigned_vector const& q, rational const& pc, rational const& qc);
bool common_factors(pdd const& a, pdd const& b, unsigned_vector& p, unsigned_vector& q, rational& pc, rational& qc);
PDD first_leading(PDD p) const;
PDD next_leading(PDD p) const;
monomials_t to_monomials(pdd const& p);
struct scoped_push {
pdd_manager& m;
unsigned m_size;
scoped_push(pdd_manager& m) :m(m), m_size(m.m_pdd_stack.size()) {}
~scoped_push() { m.m_pdd_stack.shrink(m_size); }
};
public:
struct mem_out {};
pdd_manager(unsigned num_vars, semantics s = free_e, unsigned power_of_2 = 0);
~pdd_manager();
pdd_manager(pdd_manager const&) = delete;
pdd_manager(pdd_manager&&) = delete;
pdd_manager& operator=(pdd_manager const&) = delete;
pdd_manager& operator=(pdd_manager&&) = delete;
semantics get_semantics() const { return m_semantics; }
void reset(unsigned_vector const& level2var);
void set_max_num_nodes(unsigned n);
unsigned_vector const& get_level2var() const { return m_level2var; }
unsigned num_nodes() const { return m_nodes.size() - m_free_nodes.size(); }
pdd mk_var(unsigned i);
pdd mk_val(rational const& r);
pdd mk_val(unsigned r);
pdd zero();
pdd one();
pdd minus(pdd const& a);
pdd add(pdd const& a, pdd const& b);
pdd add(rational const& a, pdd const& b);
pdd sub(pdd const& a, pdd const& b);
pdd mul(pdd const& a, pdd const& b);
pdd mul(rational const& c, pdd const& b);
pdd div(pdd const& a, rational const& c);
bool try_div(pdd const& a, rational const& c, pdd& out_result);
pdd mk_and(pdd const& p, pdd const& q);
pdd mk_or(pdd const& p, pdd const& q);
pdd mk_xor(pdd const& p, pdd const& q);
pdd mk_xor(pdd const& p, unsigned x);
pdd mk_not(pdd const& p);
pdd reduce(pdd const& a, pdd const& b);
pdd subst_val0(pdd const& a, vector> const& s);
pdd subst_val(pdd const& a, unsigned v, rational const& val);
pdd subst_val(pdd const& a, pdd const& s);
pdd subst_add(pdd const& s, unsigned v, rational const& val);
bool subst_get(pdd const& s, unsigned v, rational& out_val);
bool resolve(unsigned v, pdd const& p, pdd const& q, pdd& r);
pdd reduce(unsigned v, pdd const& a, pdd const& b);
void quot_rem(pdd const& a, pdd const& b, pdd& q, pdd& r);
pdd pow(pdd const& p, unsigned j);
bool is_linear(PDD p) { return degree(p) == 1; }
bool is_linear(pdd const& p);
bool is_binary(PDD p);
bool is_binary(pdd const& p);
bool is_monomial(PDD p);
bool is_univariate(PDD p);
bool is_univariate_in(PDD p, unsigned v);
void get_univariate_coefficients(PDD p, vector& coeff);
rational const& offset(PDD p) const;
// create an spoly r if leading monomials of a and b overlap
bool try_spoly(pdd const& a, pdd const& b, pdd& r);
// simple lexicographic comparison
bool lex_lt(pdd const& a, pdd const& b);
// more elaborate comparison based on leading monomials
bool lm_lt(pdd const& a, pdd const& b);
bool different_leading_term(pdd const& a, pdd const& b);
double tree_size(pdd const& p);
unsigned num_vars() const { return m_var2pdd.size(); }
unsigned power_of_2() const { return m_power_of_2; }
rational const& max_value() const { return m_max_value; }
rational const& two_to_N() const { return m_mod2N; }
rational normalize(rational const& n) const { return mod(-n, m_mod2N) < n ? -mod(-n, m_mod2N) : n; }
unsigned_vector const& free_vars(pdd const& p);
std::ostream& display(std::ostream& out);
std::ostream& display(std::ostream& out, pdd const& b);
void gc();
};
class pdd {
friend test;
friend class pdd_manager;
friend class pdd_iterator;
friend class pdd_linear_iterator;
unsigned root;
pdd_manager* m;
pdd(unsigned root, pdd_manager& pm): root(root), m(&pm) { m->inc_ref(root); }
pdd(unsigned root, pdd_manager* pm): root(root), m(pm) { m->inc_ref(root); }
public:
pdd(pdd_manager& m): pdd(0, m) { SASSERT(is_zero()); }
pdd(pdd const& other): pdd(other.root, other.m) { m->inc_ref(root); }
pdd(pdd && other) noexcept : pdd(0, other.m) { std::swap(root, other.root); }
pdd& operator=(pdd const& other);
pdd& operator=(unsigned k);
pdd& operator=(rational const& k);
// TODO: pdd& operator=(pdd&& other); (just swap like move constructor?)
~pdd() { m->dec_ref(root); }
void reset(pdd_manager& new_m);
pdd lo() const { return pdd(m->lo(root), m); }
pdd hi() const { return pdd(m->hi(root), m); }
unsigned index() const { return root; }
unsigned var() const { return m->var(root); }
rational const& val() const { return m->val(root); }
rational get_signed_val() const { return m->get_signed_val(root); }
rational const& leading_coefficient() const;
rational const& offset() const { return m->offset(root); }
bool is_val() const { return m->is_val(root); }
bool is_one() const { return m->is_one(root); }
bool is_zero() const { return m->is_zero(root); }
bool is_linear() const { return m->is_linear(root); }
bool is_var() const { return m->is_var(root); }
bool is_max() const { return m->is_max(root); }
/** Polynomial is of the form a * x + b for some numerals a, b. */
bool is_unilinear() const { return !is_val() && lo().is_val() && hi().is_val(); }
/** Polynomial is of the form a * x for some numeral a. */
bool is_unary() const { return !is_val() && lo().is_zero() && hi().is_val(); }
bool is_offset() const { return !is_val() && lo().is_val() && hi().is_one(); }
bool is_binary() const { return m->is_binary(root); }
bool is_monomial() const { return m->is_monomial(root); }
bool is_univariate() const { return m->is_univariate(root); }
bool is_univariate_in(unsigned v) const { return m->is_univariate_in(root, v); }
void get_univariate_coefficients(vector& coeff) const { m->get_univariate_coefficients(root, coeff); }
vector get_univariate_coefficients() const { vector coeff; m->get_univariate_coefficients(root, coeff); return coeff; }
bool is_never_zero() const { return m->is_never_zero(root); }
unsigned min_parity() const { return m->min_parity(root); }
bool var_is_leaf(unsigned v) const { return m->var_is_leaf(root, v); }
pdd operator-() const { return m->minus(*this); }
pdd operator+(pdd const& other) const { VERIFY_EQ(m, other.m); return m->add(*this, other); }
pdd operator-(pdd const& other) const { VERIFY_EQ(m, other.m); return m->sub(*this, other); }
pdd operator*(pdd const& other) const { VERIFY_EQ(m, other.m); return m->mul(*this, other); }
pdd operator&(pdd const& other) const { VERIFY_EQ(m, other.m); return m->mk_and(*this, other); }
pdd operator|(pdd const& other) const { VERIFY_EQ(m, other.m); return m->mk_or(*this, other); }
pdd operator^(pdd const& other) const { VERIFY_EQ(m, other.m); return m->mk_xor(*this, other); }
pdd operator^(unsigned other) const { return m->mk_xor(*this, m->mk_val(other)); }
pdd operator*(rational const& other) const { return m->mul(other, *this); }
pdd operator+(rational const& other) const { return m->add(other, *this); }
pdd operator~() const { return m->mk_not(*this); }
pdd shl(unsigned n) const;
pdd rev_sub(rational const& r) const { return m->sub(m->mk_val(r), *this); }
pdd div(rational const& other) const { return m->div(*this, other); }
bool try_div(rational const& other, pdd& out_result) const { VERIFY_EQ(m, out_result.m); return m->try_div(*this, other, out_result); }
pdd pow(unsigned j) const { return m->pow(*this, j); }
pdd reduce(pdd const& other) const { VERIFY_EQ(m, other.m); return m->reduce(*this, other); }
bool different_leading_term(pdd const& other) const { VERIFY_EQ(m, other.m); return m->different_leading_term(*this, other); }
void factor(unsigned v, unsigned degree, pdd& lc, pdd& rest) const { VERIFY_EQ(m, lc.m); VERIFY_EQ(m, rest.m); m->factor(*this, v, degree, lc, rest); }
bool factor(unsigned v, unsigned degree, pdd& lc) const { VERIFY_EQ(m, lc.m); return m->factor(*this, v, degree, lc); }
bool resolve(unsigned v, pdd const& other, pdd& result) { VERIFY_EQ(m, other.m); VERIFY_EQ(m, result.m); return m->resolve(v, *this, other, result); }
pdd reduce(unsigned v, pdd const& other) const { VERIFY_EQ(m, other.m); return m->reduce(v, *this, other); }
/**
* \brief factor out variables
*/
std::pair var_factors() const;
pdd subst_val0(vector> const& s) const { return m->subst_val0(*this, s); }
pdd subst_val(pdd const& s) const { VERIFY_EQ(m, s.m); return m->subst_val(*this, s); }
pdd subst_val(unsigned v, rational const& val) const { return m->subst_val(*this, v, val); }
pdd subst_add(unsigned var, rational const& val) const { return m->subst_add(*this, var, val); }
bool subst_get(unsigned var, rational& out_val) const { return m->subst_get(*this, var, out_val); }
/**
* \brief substitute variable v by r.
*/
pdd subst_pdd(unsigned v, pdd const& r) const;
std::ostream& display(std::ostream& out) const { return m->display(out, *this); }
bool operator==(pdd const& other) const { return root == other.root && m == other.m; }
bool operator!=(pdd const& other) const { return !operator==(other); }
unsigned hash() const { return root; }
unsigned power_of_2() const { return m->power_of_2(); }
unsigned dag_size() const { return m->dag_size(*this); }
double tree_size() const { return m->tree_size(*this); }
unsigned degree() const { return m->degree(*this); }
unsigned degree(unsigned v) const { return m->degree(root, v); }
unsigned max_pow2_divisor() const { return m->max_pow2_divisor(root); }
unsigned_vector const& free_vars() const { return m->free_vars(*this); }
void swap(pdd& other) noexcept { VERIFY_EQ(m, other.m); std::swap(root, other.root); }
pdd_iterator begin() const;
pdd_iterator end() const;
class pdd_linear_monomials {
friend class pdd;
pdd const& m_pdd;
pdd_linear_monomials(pdd const& p): m_pdd(p) {}
public:
pdd_linear_iterator begin() const;
pdd_linear_iterator end() const;
};
pdd_linear_monomials linear_monomials() const { return pdd_linear_monomials(*this); }
pdd_manager& manager() const { return *m; }
};
inline pdd operator*(rational const& r, pdd const& b) { return b * r; }
inline pdd operator*(int x, pdd const& b) { return b * rational(x); }
inline pdd operator*(pdd const& b, int x) { return b * rational(x); }
inline pdd operator+(rational const& r, pdd const& b) { return b + r; }
inline pdd operator+(int x, pdd const& b) { return b + rational(x); }
inline pdd operator+(pdd const& b, int x) { return b + rational(x); }
inline pdd operator^(unsigned x, pdd const& b) { return b ^ x; }
inline pdd operator^(bool x, pdd const& b) { return b ^ x; }
inline pdd operator-(rational const& r, pdd const& b) { return b.rev_sub(r); }
inline pdd operator-(int x, pdd const& b) { return rational(x) - b; }
inline pdd operator-(pdd const& b, int x) { return b + (-rational(x)); }
inline pdd operator-(pdd const& b, rational const& r) { return b + (-r); }
inline pdd& operator&=(pdd & p, pdd const& q) { p = p & q; return p; }
inline pdd& operator^=(pdd & p, pdd const& q) { p = p ^ q; return p; }
inline pdd& operator|=(pdd & p, pdd const& q) { p = p | q; return p; }
inline pdd& operator*=(pdd & p, pdd const& q) { p = p * q; return p; }
inline pdd& operator-=(pdd & p, pdd const& q) { p = p - q; return p; }
inline pdd& operator+=(pdd & p, pdd const& q) { p = p + q; return p; }
inline pdd& operator*=(pdd & p, rational const& q) { p = p * q; return p; }
inline pdd& operator-=(pdd & p, rational const& q) { p = p - q; return p; }
inline pdd& operator+=(pdd & p, rational const& q) { p = p + q; return p; }
inline void swap(pdd& p, pdd& q) noexcept { p.swap(q); }
std::ostream& operator<<(std::ostream& out, pdd const& b);
struct pdd_monomial {
rational coeff;
unsigned_vector vars;
};
std::ostream& operator<<(std::ostream& out, pdd_monomial const& m);
class pdd_iterator {
friend class pdd;
pdd m_pdd;
svector> m_nodes;
pdd_monomial m_mono;
pdd_iterator(pdd const& p, bool at_start): m_pdd(p) { if (at_start) first(); }
void first();
void next();
public:
pdd_monomial const& operator*() const { return m_mono; }
pdd_monomial const* operator->() const { return &m_mono; }
pdd_iterator& operator++() { next(); return *this; }
pdd_iterator operator++(int) { auto tmp = *this; next(); return tmp; }
bool operator==(pdd_iterator const& other) const { return m_nodes == other.m_nodes; }
bool operator!=(pdd_iterator const& other) const { return !operator==(other); }
};
class pdd_linear_iterator {
friend class pdd::pdd_linear_monomials;
pdd m_pdd;
std::pair m_mono;
pdd_manager::PDD m_next = pdd_manager::null_pdd;
pdd_linear_iterator(pdd const& p, bool at_start): m_pdd(p) { if (at_start) first(); }
void first();
void next();
public:
using value_type = std::pair; // coefficient and variable
using reference = value_type const&;
using pointer = value_type const*;
reference operator*() const { return m_mono; }
pointer operator->() const { return &m_mono; }
pdd_linear_iterator& operator++() { next(); return *this; }
pdd_linear_iterator operator++(int) { auto tmp = *this; next(); return tmp; }
bool operator==(pdd_linear_iterator const& other) const { return m_next == other.m_next; }
bool operator!=(pdd_linear_iterator const& other) const { return m_next != other.m_next; }
};
class val_pp {
pdd_manager const& m;
rational const& val;
bool require_parens;
char const* lparen() const { return require_parens ? "(" : ""; }
char const* rparen() const { return require_parens ? ")" : ""; }
public:
val_pp(pdd_manager const& m, rational const& val, bool require_parens = false): m(m), val(val), require_parens(require_parens) {}
std::ostream& display(std::ostream& out) const;
};
inline std::ostream& operator<<(std::ostream& out, val_pp const& v) { return v.display(out); }
}
