z3/src/ast/rewriter/arith_rewriter.cpp

/*++
Copyright (c) 2011 Microsoft Corporation

Module Name:

    arith_rewriter.cpp

Abstract:

    Basic rewriting rules for arithmetic

Author:

    Leonardo (leonardo) 2011-04-10

Notes:

--*/

#include "params/arith_rewriter_params.hpp"
#include "ast/rewriter/arith_rewriter.h"
#include "ast/rewriter/poly_rewriter_def.h"
#include "math/polynomial/algebraic_numbers.h"
#include "ast/ast_pp.h"

seq_util& arith_rewriter_core::seq() {
    if (!m_seq) 
        m_seq = alloc(seq_util, m);    
    return *m_seq;
}

void arith_rewriter::updt_local_params(params_ref const & _p) {
    arith_rewriter_params p(_p);
    m_arith_lhs       = p.arith_lhs();
    m_arith_ineq_lhs  = p.arith_ineq_lhs();
    m_gcd_rounding    = p.gcd_rounding();
    m_elim_to_real    = p.elim_to_real();
    m_push_to_real    = p.push_to_real();
    m_anum_simp       = p.algebraic_number_evaluator();
    m_max_degree      = p.max_degree();
    m_expand_power    = p.expand_power();
    m_mul2power       = p.mul_to_power();
    m_elim_rem        = p.elim_rem();
    m_expand_tan      = p.expand_tan();
    m_eq2ineq         = p.eq2ineq();
    set_sort_sums(p.sort_sums());
}

void arith_rewriter::updt_params(params_ref const & p) {
    poly_rewriter<arith_rewriter_core>::updt_params(p);
    updt_local_params(p);
}

void arith_rewriter::get_param_descrs(param_descrs & r) {
    poly_rewriter<arith_rewriter_core>::get_param_descrs(r);
    arith_rewriter_params::collect_param_descrs(r);
}

br_status arith_rewriter::mk_app_core(func_decl * f, unsigned num_args, expr * const * args, expr_ref & result) {
    br_status st = BR_FAILED;
    SASSERT(f->get_family_id() == get_fid());
    switch (f->get_decl_kind()) {
    case OP_NUM: st = BR_FAILED; break;
    case OP_IRRATIONAL_ALGEBRAIC_NUM: st = BR_FAILED; break;
    case OP_LE:  SASSERT(num_args == 2); st = mk_le_core(args[0], args[1], result); break;
    case OP_GE:  SASSERT(num_args == 2); st = mk_ge_core(args[0], args[1], result); break;
    case OP_LT:  SASSERT(num_args == 2); st = mk_lt_core(args[0], args[1], result); break;
    case OP_GT:  SASSERT(num_args == 2); st = mk_gt_core(args[0], args[1], result); break;
    case OP_ADD: st = mk_add_core(num_args, args, result); break;
    case OP_MUL: st = mk_mul_core(num_args, args, result); break;
    case OP_SUB: st = mk_sub(num_args, args, result); break;
    case OP_DIV: if (num_args == 1) { result = args[0]; st = BR_DONE; break; }
        SASSERT(num_args == 2); st = mk_div_core(args[0], args[1], result); break;
    case OP_IDIV: if (num_args == 1) { result = args[0]; st = BR_DONE; break; }
        SASSERT(num_args == 2); st = mk_idiv_core(args[0], args[1], result); break;
    case OP_IDIVIDES: SASSERT(num_args == 1); st = mk_idivides(f->get_parameter(0).get_int(), args[0], result); break;
    case OP_MOD: SASSERT(num_args == 2); st = mk_mod_core(args[0], args[1], result); break;
    case OP_REM: SASSERT(num_args == 2); st = mk_rem_core(args[0], args[1], result); break;
    case OP_UMINUS: SASSERT(num_args == 1);  st = mk_uminus(args[0], result); break;
    case OP_TO_REAL: SASSERT(num_args == 1); st = mk_to_real_core(args[0], result); break;
    case OP_TO_INT: SASSERT(num_args == 1);  st = mk_to_int_core(args[0], result); break;
    case OP_IS_INT: SASSERT(num_args == 1);  st = mk_is_int(args[0], result); break;
    case OP_POWER:  SASSERT(num_args == 2);  st = mk_power_core(args[0], args[1], result); break;
    case OP_ABS:    SASSERT(num_args == 1);  st = mk_abs_core(args[0], result); break;
    case OP_SIN: SASSERT(num_args == 1); st = mk_sin_core(args[0], result); break;
    case OP_COS: SASSERT(num_args == 1); st = mk_cos_core(args[0], result); break;
    case OP_TAN: SASSERT(num_args == 1); st = mk_tan_core(args[0], result); break;
    case OP_ASIN: SASSERT(num_args == 1); st = mk_asin_core(args[0], result); break;
    case OP_ACOS: SASSERT(num_args == 1); st = mk_acos_core(args[0], result); break;
    case OP_ATAN: SASSERT(num_args == 1); st = mk_atan_core(args[0], result); break;
    case OP_SINH: SASSERT(num_args == 1); st = mk_sinh_core(args[0], result); break;
    case OP_COSH: SASSERT(num_args == 1); st = mk_cosh_core(args[0], result); break;
    case OP_TANH: SASSERT(num_args == 1); st = mk_tanh_core(args[0], result); break;
    case OP_ARITH_BAND: SASSERT(num_args == 2);  st = mk_band_core(f->get_parameter(0).get_int(), args[0], args[1], result); break;
    case OP_ARITH_SHL: SASSERT(num_args == 2);  st = mk_shl_core(f->get_parameter(0).get_int(), args[0], args[1], result); break;
    case OP_ARITH_ASHR: SASSERT(num_args == 2);  st = mk_ashr_core(f->get_parameter(0).get_int(), args[0], args[1], result); break;
    case OP_ARITH_LSHR: SASSERT(num_args == 2);  st = mk_lshr_core(f->get_parameter(0).get_int(), args[0], args[1], result); break;
    default: st = BR_FAILED; break;
    }
    CTRACE(arith_rewriter, st != BR_FAILED, tout << st << ": " << mk_pp(f, m);
           for (unsigned i = 0; i < num_args; ++i) tout << mk_pp(args[i], m) << " ";
           tout << "\n==>\n" << mk_pp(result.get(), m) << "\n";
           if (is_app(result)) tout << "args: " << to_app(result)->get_num_args() << "\n";
           );
    return st;
}

void arith_rewriter::get_coeffs_gcd(expr * t, numeral & g, bool & first, unsigned & num_consts) {
    unsigned sz;
    expr * const * ms = get_monomials(t, sz);
    SASSERT(sz >= 1);
    numeral a;
    for (unsigned i = 0; i < sz; i++) {
        expr * arg = ms[i];
        if (is_numeral(arg, a)) {
            if (!a.is_zero())
                num_consts++;
            continue;
        }
        if (first) {
            get_power_product(arg, g);
            SASSERT(g.is_int());
            first = false;
        }
        else {
            get_power_product(arg, a);
            SASSERT(a.is_int());
            g = gcd(abs(a), g);
        }
        if (g.is_one())
            return;
    }
}

bool arith_rewriter::div_polynomial(expr * t, numeral const & g, const_treatment ct, expr_ref & result) {
    SASSERT(m_util.is_int(t));
    SASSERT(!g.is_one());
    unsigned sz;
    expr * const * ms = get_monomials(t, sz);
    expr_ref_buffer new_args(m);
    numeral a;
    for (unsigned i = 0; i < sz; i++) {
        expr * arg = ms[i];
        if (is_numeral(arg, a)) {
            a /= g;
            if (!a.is_int()) {
                switch (ct) {
                case CT_FLOOR:
                    a = floor(a);
                    break;
                case CT_CEIL:
                    a = ceil(a);
                    break;
                case CT_FALSE:
                    return false;
                }
            }
            if (!a.is_zero())
                new_args.push_back(m_util.mk_numeral(a, true));
            continue;
        }
        expr * pp = get_power_product(arg, a);
        a /= g;
        SASSERT(a.is_int());
        if (!a.is_zero()) {
            if (a.is_one())
                new_args.push_back(pp);
            else
                new_args.push_back(m_util.mk_mul(m_util.mk_numeral(a, true), pp));
        }
    }
    switch (new_args.size()) {
    case 0: result = m_util.mk_numeral(numeral(0), true); return true;
    case 1: result = new_args[0]; return true;
    default: result = m_util.mk_add(new_args.size(), new_args.data()); return true;
    }
}

bool arith_rewriter::is_bound(expr * arg1, expr * arg2, op_kind kind, expr_ref & result) {
    numeral b, c;
    if (!is_add(arg1) && !m_util.is_mod(arg1) && is_numeral(arg2, c)) {
        numeral a;
        bool r = false;
        expr * pp = get_power_product(arg1, a);
        if (a.is_neg()) {
            a.neg();
            c.neg();
            kind = inv(kind);
            r = true;
        }
        if (a.is_zero())
            return false;
        if (!a.is_one())
            r = true;
        if (!r)
            return false;
        c /= a;
        bool is_int = m_util.is_int(arg1);
        if (is_int && !c.is_int()) {
            switch (kind) {
            case LE: c = floor(c); break;
            case GE: c = ceil(c); break;
            case EQ: result = m.mk_false(); return true;
            }
        }
        expr_ref k(m_util.mk_numeral(c, is_int), m);
        switch (kind) {
        case LE: result = m_util.mk_le(pp, k); return true;
        case GE: result = m_util.mk_ge(pp, k); return true;
        case EQ: result = m_util.mk_eq(pp, k); return true;
        }
    }
    expr* t1, *t2;
    bool is_int = false;
    if (m_util.is_mod(arg2)) {
        std::swap(arg1, arg2);
        switch (kind) {
        case LE: kind = GE; break;
        case GE: kind = LE; break;
        case EQ: break;
        }
    }

    if (m_util.is_numeral(arg2, c, is_int) && is_int && 
        m_util.is_mod(arg1, t1, t2) && m_util.is_numeral(t2, b, is_int) && !b.is_zero()) {
        //  mod x b <= c = false if c < 0, b != 0, true if c >= b, b != 0
        if (c.is_neg()) {
            switch (kind) {
            case EQ:
            case LE: result = m.mk_false(); return true;
            case GE: result = m.mk_true(); return true;
            }
        }                    
        if (c.is_zero() && kind == GE) {
            result = m.mk_true(); 
            return true;
        }
        if (c.is_pos() && c >= abs(b)) {
            switch (kind) {
            case LE: result = m.mk_true(); return true;
            case EQ:
            case GE: result = m.mk_false(); return true;
            }
        }
        // mod x b <= b - 1
        if (c + rational::one() == abs(b) && kind == LE) {
            result = m.mk_true();
            return true;
        }
    }

    return false;
}


bool arith_rewriter::is_non_negative(expr* e) {
    rational r; 
    auto is_even_power = [&](expr* e) {
        expr* n = nullptr, *p = nullptr;
        unsigned pu;
        return m_util.is_power(e, n, p) && m_util.is_unsigned(p, pu) && (pu % 2 == 0);
    };
    auto is_power_of_positive = [&](expr* e) {
        expr* n = nullptr, * p = nullptr;
        return m_util.is_power(e, n, p) && m_util.is_numeral(n, r) && r.is_pos(); 
    };
    if (is_even_power(e)) 
        return true;
    if (is_power_of_positive(e))
        return true;
    if (seq().str.is_length(e))
        return true;
    if (!m_util.is_mul(e)) 
        return false;
    expr_mark mark;
    ptr_buffer<expr> args;
    flat_mul(e, args);
    bool sign = false;
    for (expr* arg : args) {
        if (is_even_power(arg))
            continue;
        if (is_power_of_positive(arg))
            continue;
        if (seq().str.is_length(e))
            continue;
        if (m_util.is_numeral(arg, r)) {
            if (r.is_neg()) 
                sign = !sign;
            continue;
        }
        mark.mark(arg, !mark.is_marked(arg));
    }
    if (sign)
        return false;
    for (expr* arg : args) 
        if (mark.is_marked(arg)) 
            return false;
    return true;
}

/**
 * perform static analysis on arg1 to determine a non-negative lower bound.
 * a + b + r1 <= r2 -> false if r1 > r2 and a >= 0, b >= 0
 * a + b + r1 >= r2 -> false if r1 < r2 and a <= 0, b <= 0
 * a + b + r1 >= r1 -> a = 0 and b = 0 if a >= 0, b >= 0 where a, b are multipliers
*/
br_status arith_rewriter::is_separated(expr* arg1, expr* arg2, op_kind kind, expr_ref& result) {
    if (kind != LE && kind != GE)
        return BR_FAILED;
    rational bound(0), r1, r2;
    expr_ref narg(m);
    bool has_bound = true;
    if (!m_util.is_numeral(arg2, r2))
        return BR_FAILED;
    auto update_bound = [&](expr* arg) {
        if (m_util.is_numeral(arg, r1)) {
            bound += r1;
            return;
        }
        if (kind == LE && is_non_negative(arg))
            return;
        if (kind == GE && is_neg_poly(arg, narg) && is_non_negative(narg))
            return;
        has_bound = false;
    };
    if (m_util.is_add(arg1)) {
        for (expr* arg : *to_app(arg1)) {
            update_bound(arg);
        }
    }
    else {
        update_bound(arg1);
    }
    if (!has_bound)
        return BR_FAILED;

    if (kind == LE && r1 < r2)
        return BR_FAILED;
    if (kind == GE && r1 > r2)
        return BR_FAILED;
    if (kind == LE && r1 > r2) { 
        result = m.mk_false();
        return BR_DONE;
    }
    if (kind == GE && r1 < r2) { 
        result = m.mk_false();
        return BR_DONE;
    }

    SASSERT(r1 == r2);
    expr_ref zero(m_util.mk_numeral(rational(0), arg1->get_sort()), m);

    if (r1.is_zero() && m_util.is_mul(arg1)) {
        expr_ref_buffer eqs(m);
        ptr_buffer<expr> args;
        flat_mul(arg1, args);
        for (expr* arg : args) {
            if (m_util.is_numeral(arg))
                continue;
            eqs.push_back(m.mk_eq(arg, zero));
        }
        result = m.mk_or(eqs);
        return BR_REWRITE2;
    }

    if (kind == LE && m_util.is_add(arg1)) {
        expr_ref_buffer leqs(m);
        for (expr* arg : *to_app(arg1)) {
            if (!m_util.is_numeral(arg))
                leqs.push_back(m_util.mk_le(arg, zero));
        }
        result = m.mk_and(leqs);
        return BR_REWRITE2;
    } 

    if (kind == GE && m_util.is_add(arg1)) {
        expr_ref_buffer geqs(m);
        for (expr* arg : *to_app(arg1)) {
            if (!m_util.is_numeral(arg))
                geqs.push_back(m_util.mk_ge(arg, zero));
        }
        result = m.mk_and(geqs);
        return BR_REWRITE2;
    }
        
    return BR_FAILED;
}

bool arith_rewriter::elim_to_real_var(expr * var, expr_ref & new_var) {
    numeral val;
    if (m_util.is_numeral(var, val)) {
        if (!val.is_int())
            return false;
        new_var = m_util.mk_numeral(val, true);
        return true;
    }
    else if (m_util.is_to_real(var)) {
        new_var = to_app(var)->get_arg(0);
        return true;
    }
    return false;
}

bool arith_rewriter::elim_to_real_mon(expr * monomial, expr_ref & new_monomial) {
    if (m_util.is_mul(monomial)) {
        expr_ref_buffer new_vars(m);
        expr_ref new_var(m);
        unsigned num = to_app(monomial)->get_num_args();
        for (unsigned i = 0; i < num; i++) {
            if (!elim_to_real_var(to_app(monomial)->get_arg(i), new_var))
                return false;
            new_vars.push_back(new_var);
        }
        new_monomial = m_util.mk_mul(new_vars.size(), new_vars.data());
        return true;
    }
    else {
        return elim_to_real_var(monomial, new_monomial);
    }
}

bool arith_rewriter::elim_to_real_pol(expr * p, expr_ref & new_p) {
    if (m_util.is_add(p)) {
        expr_ref_buffer new_monomials(m);
        expr_ref new_monomial(m);
        for (expr* arg : *to_app(p)) {
            if (!elim_to_real_mon(arg, new_monomial))
                return false;
            new_monomials.push_back(new_monomial);
        }
        new_p = m_util.mk_add(new_monomials.size(), new_monomials.data());
        return true;
    }
    else {
        return elim_to_real_mon(p, new_p);
    }
}

bool arith_rewriter::elim_to_real(expr * arg1, expr * arg2, expr_ref & new_arg1, expr_ref & new_arg2) {
    if (!m_util.is_real(arg1))
        return false;
    return elim_to_real_pol(arg1, new_arg1) && elim_to_real_pol(arg2, new_arg2);
}

bool arith_rewriter::is_reduce_power_target(expr * arg, bool is_eq) {
    unsigned sz;
    expr * const * args;
    if (m_util.is_mul(arg)) {
        sz = to_app(arg)->get_num_args();
        args = to_app(arg)->get_args();
    }
    else {
        sz = 1;
        args = &arg;
    }
    for (unsigned i = 0; i < sz; i++) {
        expr * arg = args[i];
        expr* arg0, *arg1;
        if (m_util.is_power(arg, arg0, arg1)) {
            rational k;
            if (m_util.is_numeral(arg1, k) && k.is_int() && ((is_eq && k > rational(1)) || (!is_eq && k > rational(2))))
                return true;
        }
    }
    return false;
}

expr * arith_rewriter::reduce_power(expr * arg, bool is_eq) {
    if (is_zero(arg))
        return arg;
    unsigned sz;
    expr * const * args;
    if (m_util.is_mul(arg)) {
        sz = to_app(arg)->get_num_args();
        args = to_app(arg)->get_args();
    }
    else {
        sz = 1;
        args = &arg;
    }
    ptr_buffer<expr> new_args;
    rational k;
    for (unsigned i = 0; i < sz; i++) {
        expr * arg = args[i];
        expr * arg0, *arg1;
        if (m_util.is_power(arg, arg0, arg1) && m_util.is_numeral(arg1, k) && k.is_int() &&
            ((is_eq && k > rational(1)) || (!is_eq && k > rational(2)))) {
            if (is_eq || !k.is_even()) {
                if (m_util.is_int(arg0))
                    arg0 = m_util.mk_to_real(arg0);
                new_args.push_back(arg0);
            }
            else {
                new_args.push_back(m_util.mk_power(arg0, m_util.mk_real(2)));
            }
        }
        else {
            new_args.push_back(arg);
        }
    }
    SASSERT(new_args.size() >= 1);
    if (new_args.size() == 1)
        return new_args[0];
    else
        return m_util.mk_mul(new_args.size(), new_args.data());
}

br_status arith_rewriter::reduce_power(expr * arg1, expr * arg2, op_kind kind, expr_ref & result) {
    expr * new_arg1 = reduce_power(arg1, kind == EQ);
    expr * new_arg2 = reduce_power(arg2, kind == EQ);
    switch (kind) {
    case LE: result = m_util.mk_le(new_arg1, new_arg2); return BR_REWRITE1;
    case GE: result = m_util.mk_ge(new_arg1, new_arg2); return BR_REWRITE1;
    default: result = m.mk_eq(new_arg1, new_arg2); return BR_REWRITE1;
    }
}

bool arith_rewriter::is_mul_factor(expr* s, expr* t) {
    if (m_util.is_mul(t))
        return any_of(*to_app(t), [&](expr* m) { return is_mul_factor(s, m); });
    return s == t;
}

bool arith_rewriter::is_add_factor(expr* s, expr* t) {
    if (m_util.is_add(t))
        return all_of(*to_app(t), [&](expr* f) { return is_add_factor(s, f); });
    return is_mul_factor(s, t);
}

expr_ref arith_rewriter::remove_factor(expr* s, expr* t) {

    if (m_util.is_mul(t)) {
        ptr_buffer<expr> r;
        r.push_back(t);
        for (unsigned i = 0; i < r.size(); ++i) {
            expr* arg = r[i];
            if (m_util.is_mul(arg)) {
                r.append(to_app(arg)->get_num_args(), to_app(arg)->get_args());
                r[i] = r.back();
                r.pop_back();
                --i;
                continue;
            }
            if (s == arg) {
                r[i] = r.back();
                r.pop_back();
                break;
            }
        }
        switch (r.size()) {
        case 0:
            return expr_ref(m_util.mk_numeral(rational(1), m_util.is_int(t)), m);
        case 1:
            return expr_ref(r[0], m);
        default:
            return expr_ref(m_util.mk_mul(r.size(), r.data()), m);
        }
    }
    if (m_util.is_add(t)) {
        expr_ref_vector sum(m);
        sum.push_back(t);
        for (unsigned i = 0; i < sum.size(); ++i) {
            expr* arg = sum.get(i);
            if (m_util.is_add(arg)) {
                sum.append(to_app(arg)->get_num_args(), to_app(arg)->get_args());
                sum[i] = sum.back();
                sum.pop_back();
                --i;
                continue;
            }
            sum[i] = remove_factor(s, arg);
        }
        if (sum.size() == 1)
            return expr_ref(sum.get(0), m);
        else
            return expr_ref(m_util.mk_add(sum.size(), sum.data()), m);
    }
    else {
        SASSERT(s == t);
        return expr_ref(m_util.mk_numeral(rational(1), m_util.is_int(t)), m);
    }
}


void arith_rewriter::get_nl_muls(expr* t, ptr_buffer<expr>& muls) {
    if (m_util.is_mul(t)) {
        for (expr* arg : *to_app(t))
            get_nl_muls(arg, muls);
    }
    else if (!m_util.is_numeral(t))
        muls.push_back(t);    
}

expr* arith_rewriter::find_nl_factor(expr* t) {
    ptr_buffer<expr> sum, muls;
    sum.push_back(t);

    for (unsigned i = 0; i < sum.size(); ++i) {
        expr* arg = sum[i];
        if (m_util.is_add(arg)) 
            sum.append(to_app(arg)->get_num_args(), to_app(arg)->get_args());        
        else if (m_util.is_mul(arg)) {
            muls.reset();
            get_nl_muls(arg, muls);
            if (muls.size() <= 1)
                continue;
            for (auto m : muls) {
                if (is_add_factor(m, t))
                    return m;
            }
            return nullptr;
        }
    }
    return nullptr;
}

br_status arith_rewriter::factor_le_ge_eq(expr * arg1, expr * arg2, op_kind kind, expr_ref & result) {
    
    if (is_zero(arg2)) {
        expr* f = find_nl_factor(arg1);
        if (!f)
            return BR_FAILED;
        expr_ref f2 = remove_factor(f, arg1);
        expr* z = m_util.mk_numeral(rational(0), m_util.is_int(arg1));
        result = m.mk_or(m_util.mk_eq(f, z), m_util.mk_eq(f2, z));
        switch (kind) {
        case EQ: 
            break;
        case GE:
            result = m.mk_or(m.mk_iff(m_util.mk_ge(f, z), m_util.mk_ge(f2, z)), result);
            break;
        case LE:
            result = m.mk_or(m.mk_not(m.mk_iff(m_util.mk_ge(f, z), m_util.mk_ge(f2, z))), result);
            break;            
        }
        return BR_REWRITE3;                    
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_le_ge_eq_core(expr * arg1, expr * arg2, op_kind kind, expr_ref & result) {
    expr *orig_arg1 = arg1, *orig_arg2 = arg2;
    expr_ref new_arg1(m);
    expr_ref new_arg2(m);
    if ((is_zero(arg1) && is_reduce_power_target(arg2, kind == EQ)) ||
        (is_zero(arg2) && is_reduce_power_target(arg1, kind == EQ)))
        return reduce_power(arg1, arg2, kind, result);
    br_status st = cancel_monomials(arg1, arg2, m_arith_ineq_lhs || m_arith_lhs, new_arg1, new_arg2);
    TRACE(mk_le_bug, tout << "st: " << st << " " << new_arg1 << " " << new_arg2 << "\n";);
    if (st != BR_FAILED) {
        arg1 = new_arg1;
        arg2 = new_arg2;
    }
    expr_ref new_new_arg1(m);
    expr_ref new_new_arg2(m);
    if (m_elim_to_real && elim_to_real(arg1, arg2, new_new_arg1, new_new_arg2)) {
        arg1 = new_new_arg1;
        arg2 = new_new_arg2;
        CTRACE(elim_to_real, m_elim_to_real, tout << "after_elim_to_real\n" << mk_ismt2_pp(arg1, m) << "\n" << mk_ismt2_pp(arg2, m) << "\n";);
        if (st == BR_FAILED)
            st = BR_DONE;
    }
    numeral a1, a2;
    if (is_numeral(arg1, a1) && is_numeral(arg2, a2)) {
        switch (kind) {
        case LE: result = a1 <= a2 ? m.mk_true() : m.mk_false(); return BR_DONE;
        case GE: result = a1 >= a2 ? m.mk_true() : m.mk_false(); return BR_DONE;
        default: result = a1 == a2 ? m.mk_true() : m.mk_false(); return BR_DONE;
        }
    }

#define ANUM_LE_GE_EQ() {                                                               \
    switch (kind) {                                                                     \
    case LE: result = am.le(v1, v2) ? m.mk_true() : m.mk_false(); return BR_DONE; \
    case GE: result = am.ge(v1, v2) ? m.mk_true() : m.mk_false(); return BR_DONE; \
    default: result = am.eq(v1, v2) ? m.mk_true() : m.mk_false(); return BR_DONE; \
    }                                                                                   \
}

    if (m_anum_simp) {
        auto& am = m_util.am();
        scoped_anum v1(am), v2(am);
        if (is_algebraic_numeral(arg1, v1) && is_algebraic_numeral(arg2, v2)) {
            ANUM_LE_GE_EQ();            
        }
    }
    br_status st1 = is_separated(arg1, arg2, kind, result);
    if (st1 != BR_FAILED)
        return st1;
    if (is_bound(arg1, arg2, kind, result))
        return BR_DONE;
    if (is_bound(arg2, arg1, inv(kind), result))
        return BR_DONE;
    bool is_int = m_util.is_int(arg1);
    if (is_int && m_gcd_rounding) {
        bool first = true;
        numeral g;
        unsigned num_consts = 0;
        get_coeffs_gcd(arg1, g, first, num_consts);
        TRACE(arith_rewriter_gcd, tout << "[step1] g: " << g << ", num_consts: " << num_consts << "\n";);
        if ((first || !g.is_one()) && num_consts <= 1)
            get_coeffs_gcd(arg2, g, first, num_consts);
        TRACE(arith_rewriter_gcd, tout << "[step2] g: " << g << ", num_consts: " << num_consts << "\n";);
        g = abs(g);
        if (!first && !g.is_one() && num_consts <= 1) {
            bool is_sat = div_polynomial(arg1, g, (kind == LE ? CT_CEIL : (kind == GE ? CT_FLOOR : CT_FALSE)), new_arg1);
            if (!is_sat) {
                result = m.mk_false();
                return BR_DONE;
            }
            is_sat = div_polynomial(arg2, g, (kind == LE ? CT_FLOOR : (kind == GE ? CT_CEIL : CT_FALSE)), new_arg2);
            if (!is_sat) {
                result = m.mk_false();
                return BR_DONE;
            }
            arg1 = new_arg1.get();
            arg2 = new_arg2.get();
            st = BR_DONE;
        }
    }
    expr* c = nullptr, *t = nullptr, *e = nullptr;
    if (m.is_ite(arg1, c, t, e) && is_numeral(t, a1) && is_numeral(arg2, a2)) {
        expr_ref a(m.mk_not(c), m);
        switch (kind) {
        case LE: result = a1 <= a2 ? m.mk_or(c, m_util.mk_le(e, arg2)) : m.mk_and(a, m_util.mk_le(e, arg2)); return BR_REWRITE2;
        case GE: result = a1 >= a2 ? m.mk_or(c, m_util.mk_ge(e, arg2)) : m.mk_and(a, m_util.mk_ge(e, arg2)); return BR_REWRITE2;
        case EQ: result = a1 == a2 ? m.mk_or(c, m.mk_eq(e, arg2))    : m.mk_and(a, m_util.mk_eq(e, arg2)); return BR_REWRITE2;
        }
    }
    if (m.is_ite(arg1, c, t, e) && is_numeral(e, a1) && is_numeral(arg2, a2)) {
        expr_ref a(m.mk_not(c), m);
        switch (kind) {
        case LE: result = a1 <= a2 ? m.mk_or(a, m_util.mk_le(t, arg2)) : m.mk_and(c, m_util.mk_le(t, arg2)); return BR_REWRITE2;
        case GE: result = a1 >= a2 ? m.mk_or(a, m_util.mk_ge(t, arg2)) : m.mk_and(c, m_util.mk_ge(t, arg2)); return BR_REWRITE2;
        case EQ: result = a1 == a2 ? m.mk_or(a, m.mk_eq(t, arg2))    : m.mk_and(c, m_util.mk_eq(t, arg2)); return BR_REWRITE2;
        }
    }
    if (m.is_ite(arg1, c, t, e) && arg1->get_ref_count() == 1) {
        switch (kind) {
        case LE:
        {
            auto a = m_util.mk_le(t, arg2);
            result = m.mk_ite(c, a, m_util.mk_le(e, arg2)); return BR_REWRITE2;
        }
        case GE: {
            auto a = m_util.mk_ge(t, arg2);
            result = m.mk_ite(c, a, m_util.mk_ge(e, arg2)); return BR_REWRITE2;
        }
        case EQ:{
            auto a = m.mk_eq(t, arg2);
            result = m.mk_ite(c, a, m.mk_eq(e, arg2)); return BR_REWRITE2;
        }
        }
    }
    if (m_util.is_to_int(arg2) && is_numeral(arg1)) {        
        kind = inv(kind);
        std::swap(arg1, arg2);
    } 
    if (m_util.is_to_int(arg1, t) && is_numeral(arg2, a2)) {        
        switch (kind) {
        case LE: 
            result = m_util.mk_lt(t, m_util.mk_numeral(a2+1, false)); 
            return BR_REWRITE1;
        case GE: 
            result = m_util.mk_ge(t, m_util.mk_numeral(a2, false)); 
            return BR_REWRITE1;
        case EQ: 
            result = m_util.mk_ge(t, m_util.mk_numeral(a2, false));
            result = m.mk_and(m_util.mk_lt(t, m_util.mk_numeral(a2+1, false)), result);
            return BR_REWRITE3;
        }        
    }
    if ((m_arith_lhs || m_arith_ineq_lhs) && is_numeral(arg2, a2) && is_neg_poly(arg1, new_arg1)) {
        a2.neg();
        new_arg2 = m_util.mk_numeral(a2, m_util.is_int(new_arg1));
        switch (kind) {
        case LE: result = m_util.mk_ge(new_arg1, new_arg2); return BR_DONE;
        case GE: result = m_util.mk_le(new_arg1, new_arg2); return BR_DONE;
        case EQ: result = m_util.mk_eq(new_arg1, new_arg2); return BR_DONE;
        }
    }
    else if (st == BR_DONE && arg1 == orig_arg1 && arg2 == orig_arg2) {
        // Nothing new; return BR_FAILED to avoid rewriting loops.
        return BR_FAILED;
    }
    else if (st != BR_FAILED) {
        switch (kind) {
        case LE: result = m_util.mk_le(arg1, arg2); return BR_DONE;
        case GE: result = m_util.mk_ge(arg1, arg2); return BR_DONE;
        default: result = m.mk_eq(arg1, arg2); return BR_DONE;
        }
    }
    return factor_le_ge_eq(arg1, arg2, kind, result);
}


br_status arith_rewriter::mk_le_core(expr * arg1, expr * arg2, expr_ref & result) {
    return mk_le_ge_eq_core(arg1, arg2, LE, result);
}

br_status arith_rewriter::mk_lt_core(expr * arg1, expr * arg2, expr_ref & result) {
    result = m.mk_not(m_util.mk_le(arg2, arg1));
    return BR_REWRITE2;
}

br_status arith_rewriter::mk_ge_core(expr * arg1, expr * arg2, expr_ref & result) {
    return mk_le_ge_eq_core(arg1, arg2, GE, result);
}

br_status arith_rewriter::mk_gt_core(expr * arg1, expr * arg2, expr_ref & result) {
    result = m.mk_not(m_util.mk_le(arg1, arg2));
    return BR_REWRITE2;
}

bool arith_rewriter::is_arith_term(expr * n) const {
    return n->get_kind() == AST_APP && to_app(n)->get_family_id() == get_fid();
}

br_status arith_rewriter::mk_eq_core(expr * arg1, expr * arg2, expr_ref & result) {
    br_status st = BR_FAILED;
    if (m_eq2ineq) {
        result = m.mk_and(m_util.mk_le(arg1, arg2), m_util.mk_ge(arg1, arg2));
        st = BR_REWRITE2;
    }
    else if (m_arith_lhs || is_arith_term(arg1) || is_arith_term(arg2)) {
        st = mk_le_ge_eq_core(arg1, arg2, EQ, result);
    }

    if (st == BR_FAILED && mk_eq_mod(arg1, arg2, result)) 
        st = BR_REWRITE2;
    return st;
}

br_status arith_rewriter::mk_and_core(unsigned n, expr* const* args, expr_ref& result) {
    if (n <= 1)
        return BR_FAILED;
    expr* x, * y, * z, * u;
    rational a, b;
    if (m_util.is_le(args[0], x, y) && m_util.is_numeral(x, a)) {
        expr* arg0 = args[0];
        ptr_buffer<expr> rest;
        for (unsigned i = 1; i < n; ++i) {
            if (m_util.is_le(args[i], z, u) && u == y && m_util.is_numeral(z, b)) {
                if (b > a)
                    arg0 = args[i];
            }
            else
                rest.push_back(args[i]);
        }
        if (rest.size() < n - 1) {
            rest.push_back(arg0);
            result = m.mk_and(rest);
            return BR_REWRITE1;
        }
    }
    return BR_FAILED;
}

bool arith_rewriter::mk_eq_mod(expr* arg1, expr* arg2, expr_ref& result) {
    expr* x = nullptr, *y = nullptr, *z = nullptr, *u = nullptr;
    rational p, k, l;
    // match k*u mod p = l, where k, p, l are integers
    if (m_util.is_mod(arg1, x, y) && m_util.is_numeral(y, p) &&
        m_util.is_mul(x, z, u) && m_util.is_numeral(z, k) &&
        m_util.is_numeral(arg2, l) && 0 <= l && l < p) {
        // a*p + k*b = g
        rational a, b;
        rational g = gcd(p, k, a, b);
        if (g == 1) {
            expr_ref nb(m_util.mk_numeral(b, true), m);
            result = m.mk_eq(m_util.mk_mod(u, y),
                             m_util.mk_mod(m_util.mk_mul(nb, arg2), y));
            return true;            
        }
    }
    return false;
}

expr_ref arith_rewriter::neg_monomial(expr* e) {
    expr_ref_vector args(m);
    rational a1;
    if (m_util.is_numeral(e, a1)) 
        args.push_back(m_util.mk_numeral(-a1, e->get_sort()));
    else if (m_util.is_irrational_algebraic_numeral(e)) {
        auto& n = m_util.to_irrational_algebraic_numeral(e);
        auto& am = m_util.am();
        scoped_anum new_n(am);
        am.set(new_n, n);
        am.neg(new_n);
        args.push_back(m_util.mk_numeral(am, new_n, m_util.is_int(e)));
    }
    else if (is_app(e) && m_util.is_mul(e)) {
        if (is_numeral(to_app(e)->get_arg(0), a1)) {
            if (!a1.is_minus_one()) {
                args.push_back(m_util.mk_numeral(-a1, m_util.is_int(e)));
            }
            args.append(to_app(e)->get_num_args() - 1, to_app(e)->get_args() + 1);
        }
        else {
            args.push_back(m_util.mk_numeral(rational(-1), m_util.is_int(e)));
            args.push_back(e);
        }
    }
    else {
        args.push_back(m_util.mk_numeral(rational(-1), m_util.is_int(e)));
        args.push_back(e);
    }
    if (args.size() == 1) {
        return expr_ref(args.back(), m);
    }
    else {
        return expr_ref(m_util.mk_mul(args.size(), args.data()), m);
    }
}

bool arith_rewriter::is_neg_poly(expr* t, expr_ref& neg) {
    rational r;
    if (m_util.is_mul(t) && is_numeral(to_app(t)->get_arg(0), r) && r.is_neg()) {
        neg = neg_monomial(t);
        return true;
    }

    if (!m_util.is_add(t)) {
        return false;
    }
    expr * t2 = to_app(t)->get_arg(0);

    if (m_util.is_mul(t2) && is_numeral(to_app(t2)->get_arg(0), r) && r.is_neg()) {
        expr_ref_vector args1(m);
        for (expr* e1 : *to_app(t)) {
            args1.push_back(neg_monomial(e1));
        }       
        neg = m_util.mk_add(args1.size(), args1.data());      
        return true;
    }    
    return false;
}

bool arith_rewriter::is_anum_simp_target(unsigned num_args, expr * const * args) {
    if (!m_anum_simp)
        return false;
    unsigned num_irrat = 0;
    unsigned num_rat   = 0;
    for (unsigned i = 0; i < num_args; i++) {
        if (m_util.is_numeral(args[i])) {
            num_rat++;
            if (num_irrat > 0)
                return true;
        }
        if (m_util.is_irrational_algebraic_numeral(args[i]) &&
            m_util.am().degree(m_util.to_irrational_algebraic_numeral(args[i])) <= m_max_degree) {
            num_irrat++;
            if (num_irrat > 1 || num_rat > 0)
                return true;
        }
    }
    return false;
}

bool arith_rewriter::is_algebraic_numeral(expr* n, scoped_anum& a) {
    auto& am = m_util.am();
    expr* x, *y;
    rational r;
    if (m_util.is_mul(n, x, y)) {
        scoped_anum ax(am), ay(am);
        if (is_algebraic_numeral(x, ax) && is_algebraic_numeral(y, ay)) {
            am.mul(ax, ay, a);
            return true;
        }
    }
    else if (m_util.is_add(n, x, y)) {
        scoped_anum ax(am), ay(am);
        if (is_algebraic_numeral(x, ax) && is_algebraic_numeral(y, ay)) {
            am.add(ax, ay, a);
            return true;
        }
    }
    else if (m_util.is_numeral(n, r)) {
        am.set(a, r.to_mpq());
        return true;
    }    
    else if (m_util.is_irrational_algebraic_numeral(n)) {
        am.set(a, m_util.to_irrational_algebraic_numeral(n));
        return true;
    }
    return false;
}


br_status arith_rewriter::mk_add_core(unsigned num_args, expr * const * args, expr_ref & result) {
    if (is_anum_simp_target(num_args, args)) {
        expr_ref_buffer new_args(m);
        anum_manager & am = m_util.am();
        scoped_anum r(am);
        scoped_anum arg(am);
        rational rarg;
        am.set(r, 0);
        for (unsigned i = 0; i < num_args; i ++) {
            unsigned d = am.degree(r);
            if (d > 1 && d > m_max_degree) {
                new_args.push_back(m_util.mk_numeral(am, r, false));
                am.set(r, 0);
            }

            if (m_util.is_numeral(args[i], rarg)) {
                am.set(arg, rarg.to_mpq());
                am.add(r, arg, r);
                continue;
            }

            if (m_util.is_irrational_algebraic_numeral(args[i])) {
                anum const & irarg = m_util.to_irrational_algebraic_numeral(args[i]);
                if (am.degree(irarg) <= m_max_degree) {
                    am.add(r, irarg, r);
                    continue;
                }
            }

            new_args.push_back(args[i]);
        }

        if (new_args.empty()) {
            result = m_util.mk_numeral(am, r, false);
            return BR_DONE;
        }

        new_args.push_back(m_util.mk_numeral(am, r, false));
        br_status st = poly_rewriter<arith_rewriter_core>::mk_add_core(new_args.size(), new_args.data(), result);
        if (st == BR_FAILED) {
            result = m.mk_app(get_fid(), OP_ADD, new_args.size(), new_args.data());
            return BR_DONE;
        }
        return st;
    }
    else {
        return poly_rewriter<arith_rewriter_core>::mk_add_core(num_args, args, result);
    }
}

br_status arith_rewriter::mk_mul_core(unsigned num_args, expr * const * args, expr_ref & result) {
    if (is_anum_simp_target(num_args, args)) {
        expr_ref_buffer new_args(m);
        anum_manager & am = m_util.am();
        scoped_anum r(am);
        scoped_anum arg(am);
        rational rarg;
        am.set(r, 1);
        for (unsigned i = 0; i < num_args; i ++) {
            unsigned d = am.degree(r);
            if (d > 1 && d > m_max_degree) {
                new_args.push_back(m_util.mk_numeral(am, r, false));
                am.set(r, 1);
            }

            if (m_util.is_numeral(args[i], rarg)) {
                am.set(arg, rarg.to_mpq());
                am.mul(r, arg, r);
                continue;
            }
            if (m_util.is_irrational_algebraic_numeral(args[i])) {
                anum const & irarg = m_util.to_irrational_algebraic_numeral(args[i]);
                if (am.degree(irarg) <= m_max_degree) {
                    am.mul(r, irarg, r);
                    continue;
                }
            }

            new_args.push_back(args[i]);
        }

        if (new_args.empty()) {
            result = m_util.mk_numeral(am, r, false);
            return BR_DONE;
        }
        new_args.push_back(m_util.mk_numeral(am, r, false));

        br_status st = poly_rewriter<arith_rewriter_core>::mk_mul_core(new_args.size(), new_args.data(), result);
        if (st == BR_FAILED) {
            result = m.mk_app(get_fid(), OP_MUL, new_args.size(), new_args.data());
            return BR_DONE;
        }
        return st;
    }
    else {
        return poly_rewriter<arith_rewriter_core>::mk_mul_core(num_args, args, result);
    }
}

br_status arith_rewriter::mk_div_irrat_rat(expr * arg1, expr * arg2, expr_ref & result) {
    SASSERT(m_util.is_real(arg1));
    SASSERT(m_util.is_irrational_algebraic_numeral(arg1));
    SASSERT(m_util.is_numeral(arg2));
    anum_manager & am = m_util.am();
    anum const & val1  = m_util.to_irrational_algebraic_numeral(arg1);
    rational rval2;
    VERIFY(m_util.is_numeral(arg2, rval2));
    if (rval2.is_zero())
        return BR_FAILED;
    scoped_anum val2(am);
    am.set(val2, rval2.to_mpq());
    scoped_anum r(am);
    am.div(val1, val2, r);
    result = m_util.mk_numeral(am, r, false);
    return BR_DONE;
}

br_status arith_rewriter::mk_div_rat_irrat(expr * arg1, expr * arg2, expr_ref & result) {
    SASSERT(m_util.is_real(arg1));
    SASSERT(m_util.is_numeral(arg1));
    SASSERT(m_util.is_irrational_algebraic_numeral(arg2));
    anum_manager & am = m_util.am();
    rational rval1;
    VERIFY(m_util.is_numeral(arg1, rval1));
    scoped_anum val1(am);
    am.set(val1, rval1.to_mpq());
    anum const & val2  = m_util.to_irrational_algebraic_numeral(arg2);
    scoped_anum r(am);
    am.div(val1, val2, r);
    result = m_util.mk_numeral(am, r, false);
    return BR_DONE;
}

br_status arith_rewriter::mk_div_irrat_irrat(expr * arg1, expr * arg2, expr_ref & result) {
    SASSERT(m_util.is_real(arg1));
    SASSERT(m_util.is_irrational_algebraic_numeral(arg1));
    SASSERT(m_util.is_irrational_algebraic_numeral(arg2));
    anum_manager & am = m_util.am();
    anum const & val1  = m_util.to_irrational_algebraic_numeral(arg1);
    if (am.degree(val1) > m_max_degree)
        return BR_FAILED;
    anum const & val2  = m_util.to_irrational_algebraic_numeral(arg2);
    if (am.degree(val2) > m_max_degree)
        return BR_FAILED;
    scoped_anum r(am);
    am.div(val1, val2, r);
    result = m_util.mk_numeral(am, r.get(), false);
    return BR_DONE;
}

br_status arith_rewriter::mk_div_core(expr * arg1, expr * arg2, expr_ref & result) {
    if (m_anum_simp) {
        if (m_util.is_irrational_algebraic_numeral(arg1) && m_util.is_numeral(arg2))
            return mk_div_irrat_rat(arg1, arg2, result);
        if (m_util.is_irrational_algebraic_numeral(arg1) && m_util.is_irrational_algebraic_numeral(arg2))
            return mk_div_irrat_irrat(arg1, arg2, result);
        if (m_util.is_irrational_algebraic_numeral(arg2) && m_util.is_numeral(arg1))
            return mk_div_rat_irrat(arg1, arg2, result);
    }
    set_curr_sort(arg1->get_sort());
    numeral v1, v2;
    bool is_int;
    if (m_util.is_numeral(arg2, v2, is_int)) {
        SASSERT(!is_int);
        if (v2.is_zero()) {
            return BR_FAILED;
        }
        else if (m_util.is_numeral(arg1, v1, is_int)) {
            result = m_util.mk_numeral(v1/v2, false);
            return BR_DONE;
        }
        else {
            numeral k(1);
            k /= v2;
            result = m.mk_app(get_fid(), OP_MUL,
                                m_util.mk_numeral(k, false),
                                arg1);
            return BR_REWRITE1;
        }
    }

#if 0
    if (!m_util.is_int(arg1)) {
        // (/ (* v1 b) (* v2 d)) --> (* v1/v2 (/ b d))
        expr * a, * b, * c, * d;
        if (m_util.is_mul(arg1, a, b) && m_util.is_numeral(a, v1)) {
            // do nothing arg1 is of the form v1 * b
        }
        else {
            v1 = rational(1);
            b  = arg1;
        }
        if (m_util.is_mul(arg2, c, d) && m_util.is_numeral(c, v2)) {
            // do nothing arg2 is of the form v2 * d
        }
        else {
            v2 = rational(1);
            d  = arg2;
        }
        TRACE(div_bug, tout << "v1: " << v1 << ", v2: " << v2 << "\n";);
        if (!v1.is_one() || !v2.is_one()) {
            v1 /= v2;
            result = m_util.mk_mul(m_util.mk_numeral(v1, false),
                                   m_util.mk_div(b, d));
            expr_ref z(m_util.mk_real(0), m);
            result = m.mk_ite(m.mk_eq(d, z), m_util.mk_div(arg1, z), result);
            return BR_REWRITE2;
        }
    }
#endif

    return BR_FAILED;
}

br_status arith_rewriter::mk_idivides(unsigned k, expr * arg, expr_ref & result) {
    result = m.mk_eq(m_util.mk_mod(arg, m_util.mk_int(k)), m_util.mk_int(0));
    return BR_REWRITE2;
}

br_status arith_rewriter::mk_idiv_core(expr * arg1, expr * arg2, expr_ref & result) {
    set_curr_sort(arg1->get_sort());
    numeral v1, v2;
    bool is_int;
    bool is_num1 = m_util.is_numeral(arg1, v1, is_int);
    bool is_num2 = m_util.is_numeral(arg2, v2, is_int);
    if (is_num1 && is_num2 && !v2.is_zero()) {
        result = m_util.mk_numeral(div(v1, v2), is_int);
        return BR_DONE;
    }
    if (is_num2 && v2.is_one()) {
        result = arg1; 
        return BR_DONE; 
    } 
    if (is_num2 && v2.is_minus_one()) {
        result = m_util.mk_mul(m_util.mk_int(-1), arg1); 
        return BR_REWRITE1; 
    }
    if (is_num2 && v2.is_zero()) { 
        return BR_FAILED; 
    } 
    if (arg1 == arg2) { 
        expr_ref zero(m_util.mk_int(0), m); 
        result = m.mk_ite(m.mk_eq(arg1, zero), m_util.mk_idiv(zero, zero), m_util.mk_int(1)); 
        return BR_REWRITE3; 
    } 
    if (is_num2 && v2.is_pos() && m_util.is_add(arg1)) { 
        expr_ref_buffer args(m);
        bool change = false;
        rational add(0);
        for (expr* arg : *to_app(arg1)) {
            rational arg_v;
            if (m_util.is_numeral(arg, arg_v) && arg_v.is_pos() && mod(arg_v, v2) != arg_v) {
                change = true;
                args.push_back(m_util.mk_numeral(mod(arg_v, v2), true));
                add += div(arg_v, v2);
            }
            else {
                args.push_back(arg);
            }
        }
        if (change) {
            result = m_util.mk_idiv(m.mk_app(to_app(arg1)->get_decl(), args.size(), args.data()), arg2);
            result = m_util.mk_add(m_util.mk_numeral(add, true), result);
            TRACE(div_bug, tout << "mk_div result: " << result << "\n";);
            return BR_REWRITE3;
        }
    } 
    if (get_divides(arg1, arg2, result)) { 
        expr_ref zero(m_util.mk_int(0), m); 
        result = m.mk_ite(m.mk_eq(zero, arg2), m_util.mk_idiv(arg1, zero), result);
        return BR_REWRITE_FULL; 
    }
#if 0
    expr* x = nullptr, *y = nullptr, *z = nullptr;
    if (is_num2 && m_util.is_idiv(arg1, x, y) && m_util.is_numeral(y, v1) && v1 > 0 && v2 > 0) {
        result = m_util.mk_idiv(x, m_util.mk_numeral(v1*v2, is_int));
        return BR_DONE;
    }
#endif
    return BR_FAILED;
}

//  
// implement div ab ac = floor( ab / ac) = floor (b / c) = div b c 
//
bool arith_rewriter::get_divides(expr* num, expr* den, expr_ref& result) { 
    expr_fast_mark1 mark; 
    rational num_r(1), den_r(1); 
    expr* num_e = nullptr, *den_e = nullptr; 
    ptr_buffer<expr> args1, args2; 
    flat_mul(num, args1); 
    flat_mul(den, args2); 
    for (expr * arg : args1) { 
        mark.mark(arg); 
        if (m_util.is_numeral(arg, num_r)) num_e = arg; 
    } 
    for (expr* arg : args2) { 
        // don't remove divisor on (div (* -1 x) (* -1 y)) because rewriting would diverge. 
        if (mark.is_marked(arg) && (!m_util.is_numeral(arg, num_r) || !num_r.is_minus_one())) { 
            result = remove_divisor(arg, num, den); 
            return true; 
        } 
        if (m_util.is_numeral(arg, den_r)) den_e = arg; 
    } 
    rational g = gcd(num_r, den_r); 
    if (!g.is_one()) { 
        SASSERT(g.is_pos()); 
        // replace num_e, den_e by their gcd reduction. 
        for (unsigned i = 0; i < args1.size(); ++i) { 
            if (args1[i] == num_e) { 
                args1[i] = m_util.mk_numeral(num_r / g, true); 
                break; 
            } 
        } 
        for (unsigned i = 0; i < args2.size(); ++i) { 
            if (args2[i] == den_e) { 
                args2[i] = m_util.mk_numeral(den_r / g, true); 
                break; 
            } 
        } 
        num = m_util.mk_mul(args1.size(), args1.data()); 
        den = m_util.mk_mul(args2.size(), args2.data()); 
        result = m_util.mk_idiv(num, den); 
        return true; 
    } 
    return false; 
} 


expr_ref arith_rewriter::remove_divisor(expr* arg, expr* num, expr* den) { 
    ptr_buffer<expr> args1, args2; 
    flat_mul(num, args1); 
    flat_mul(den, args2); 
    remove_divisor(arg, args1); 
    remove_divisor(arg, args2); 
    expr_ref zero(m_util.mk_int(0), m); 
    num = args1.empty() ? m_util.mk_int(1) : m_util.mk_mul(args1.size(), args1.data()); 
    den = args2.empty() ? m_util.mk_int(1) : m_util.mk_mul(args2.size(), args2.data()); 
    expr_ref d(m_util.mk_idiv(num, den), m);
    expr_ref nd(m_util.mk_idiv(m_util.mk_uminus(num), m_util.mk_uminus(den)), m);
    return expr_ref(m.mk_ite(m.mk_eq(zero, arg), 
                               m_util.mk_idiv(zero, zero), 
                               m.mk_ite(m_util.mk_ge(arg, zero), 
                                          d,
                                          nd)),
                    m);
} 
 
void arith_rewriter::flat_mul(expr* e, ptr_buffer<expr>& args) { 
    args.push_back(e); 
    for (unsigned i = 0; i < args.size(); ++i) { 
        e = args[i]; 
        if (m_util.is_mul(e)) { 
            args.append(to_app(e)->get_num_args(), to_app(e)->get_args()); 
            args[i] = args.back(); 
            args.shrink(args.size()-1); 
            --i; 
        }                 
    } 
} 
 
void arith_rewriter::remove_divisor(expr* d, ptr_buffer<expr>& args) { 
    for (unsigned i = 0; i < args.size(); ++i) { 
        if (args[i] == d) { 
            args[i] = args.back(); 
            args.shrink(args.size()-1); 
            return; 
        } 
    } 
    UNREACHABLE(); 
} 

static rational symmod(rational const& a, rational const& b) {
    rational r = mod(a, b);
    if (2*r > b) r -= b;
    return r;
}

br_status arith_rewriter::mk_mod_core(expr * arg1, expr * arg2, expr_ref & result) {
    set_curr_sort(arg1->get_sort());
    numeral v1, v2;
    bool is_int;
    bool is_num1 = m_util.is_numeral(arg1, v1, is_int);
    bool is_num2 = m_util.is_numeral(arg2, v2, is_int);

    if (is_num1 && is_num2 && !v2.is_zero()) {
        result = m_util.mk_numeral(mod(v1, v2), is_int);
        return BR_DONE;
    }

    if (is_num2 && is_int && (v2.is_one() || v2.is_minus_one())) {
        result = m_util.mk_numeral(numeral(0), true);
        return BR_DONE;
    }

    if (arg1 == arg2 && !is_num2) {
        expr_ref zero(m_util.mk_int(0), m);
        result = m.mk_ite(m.mk_eq(arg2, zero), m_util.mk_mod(zero, zero), zero);
        return BR_DONE;
    }

    // mod is idempotent on non-zero modulus.
    expr* t1, *t2;
    if (m_util.is_mod(arg1, t1, t2) && t2 == arg2 && is_num2 && is_int && !v2.is_zero()) {
        result = arg1;
        return BR_DONE;
    }

    // propagate mod inside only if there is something to reduce.
    if (is_num2 && is_int && v2.is_pos() && (is_add(arg1) || is_mul(arg1))) {
        TRACE(mod_bug, tout << "mk_mod:\n" << mk_ismt2_pp(arg1, m) << "\n" << mk_ismt2_pp(arg2, m) << "\n";);
        expr_ref_buffer args(m);
        bool change = false;
        for (expr* arg : *to_app(arg1)) {
            rational arg_v;
            if (m_util.is_numeral(arg, arg_v) && mod(arg_v, v2) != arg_v) {
                change = true;
                args.push_back(m_util.mk_numeral(mod(arg_v, v2), true));
            }
            else if (m_util.is_mod(arg, t1, t2) && t2 == arg2) {
                change = true;
                args.push_back(t1);
            }
            else if (m_util.is_mul(arg, t1, t2) && m_util.is_numeral(t1, arg_v) && symmod(arg_v, v2) != arg_v) {
                change = true;
                args.push_back(m_util.mk_mul(m_util.mk_numeral(symmod(arg_v, v2), true), t2));
            }
            else {
                args.push_back(arg);
            }
        }
        if (change) {
            result = m_util.mk_mod(m.mk_app(to_app(arg1)->get_decl(), args.size(), args.data()), arg2);
            TRACE(mod_bug, tout << "mk_mod result: " << mk_ismt2_pp(result, m) << "\n";);
            return BR_REWRITE3;
        }
    }

    expr* x = nullptr, * y = nullptr, * z = nullptr;
    if (is_num2 && v2.is_pos() && m_util.is_mul(arg1, x, y) && m_util.is_numeral(x, v1, is_int) && v1 > 0 && divides(v1, v2)) {
        result = m_util.mk_mul(m_util.mk_int(v1), m_util.mk_mod(y, m_util.mk_int(v2/v1)));        
        return BR_REWRITE1;
    }

    // mod x -y = mod x y
    if (m_util.is_mul(arg2, t1, t2) && m_util.is_numeral(t1, v1) && v1 == -1) {
        result = m_util.mk_mod(arg1, t2);
        return BR_REWRITE1;
    }

    if (m.is_ite(arg2, x, y, z)) {
        expr_ref mod1(m_util.mk_mod(arg1, y), m);
        expr_ref mod2(m_util.mk_mod(arg1, z), m);
        result = m.mk_ite(x, mod1, mod2);
        return BR_REWRITE3;
    }

    return BR_FAILED;
}

bool arith_rewriter::get_range(expr* e, rational& lo, rational& hi) {
    expr* x, *y;
    rational r;
    if (m_util.is_idiv(e, x, y) && m_util.is_numeral(y, r) && get_range(x, lo, hi) && 0 <= lo && r > 0) {
        lo = div(lo, r);
        hi = div(hi, r);
        return true;
    }
    if (m_util.is_mod(e, x, y) && m_util.is_numeral(y, r) && r > 0) {
        lo = 0;
        hi = r - 1;
        return true;
    }
    if (m_util.is_numeral(e, r)) {
        lo = hi = r;
        return true;
    }
    return false;
}


br_status arith_rewriter::mk_rem_core(expr * arg1, expr * arg2, expr_ref & result) {
    set_curr_sort(arg1->get_sort());
    numeral v1, v2;
    bool is_int;
    if (m_util.is_numeral(arg1, v1, is_int) && m_util.is_numeral(arg2, v2, is_int) && !v2.is_zero()) {
        numeral m = mod(v1, v2);
        //
        // rem(v1,v2) = if v2 >= 0 then mod(v1,v2) else -mod(v1,v2)
        //
        if (v2.is_neg()) {
            m.neg();
        }
        result = m_util.mk_numeral(m, is_int);
        return BR_DONE;
    }
    else if (m_util.is_numeral(arg2, v2, is_int) && is_int && v2.is_one()) {
        result = m_util.mk_numeral(numeral(0), true);
        return BR_DONE;
    }
    else if (m_util.is_numeral(arg2, v2, is_int) && is_int && !v2.is_zero()) {
        if (is_add(arg1) || is_mul(arg1)) {
            return BR_FAILED;
        }
        else {
            if (v2.is_neg()) {
                result = m_util.mk_uminus(m_util.mk_mod(arg1, arg2));
                return BR_REWRITE2;
            }
            else {
                result = m_util.mk_mod(arg1, arg2);
                return BR_REWRITE1;
            }
        }
    }
    else if (m_elim_rem) {
        expr * mod = m_util.mk_mod(arg1, arg2);
        result = m.mk_ite(m_util.mk_ge(arg2, m_util.mk_numeral(rational(0), true)),
                            mod,
                            m_util.mk_uminus(mod));
        TRACE(elim_rem, tout << "result: " << mk_ismt2_pp(result, m) << "\n";);
        return BR_REWRITE3;
    }
    return BR_FAILED;
}

expr* arith_rewriter_core::coerce(expr* x, sort* s) {
    if (m_util.is_int(x) && m_util.is_real(s))
        return m_util.mk_to_real(x);
    if (m_util.is_real(x) && m_util.is_int(s))
        return m_util.mk_to_int(x);
    return x;
}

app* arith_rewriter_core::mk_power(expr* x, rational const& r, sort* s) { 
    SASSERT(r.is_unsigned() && r.is_pos());
    bool is_int = m_util.is_int(x);
    app* y = m_util.mk_power(x, m_util.mk_numeral(r, is_int));
    if (m_util.is_int(s))
        y = m_util.mk_to_int(y);
    return y;
}

br_status arith_rewriter::mk_shl_core(unsigned sz, expr* arg1, expr* arg2, expr_ref& result) {
    numeral x, y, N;
    bool is_num_x = m_util.is_numeral(arg1, x);
    bool is_num_y = m_util.is_numeral(arg2, y);
    N = rational::power_of_two(sz);
    if (is_num_x) 
        x = mod(x, N);
    if (is_num_y)
        y = mod(y, N);
    if (is_num_x && is_num_y) {
        if (y >= sz) 
            result = m_util.mk_int(0);
        else 
            result = m_util.mk_int(mod(x * rational::power_of_two(y.get_unsigned()), N));
        return BR_DONE;
    }
    if (is_num_y) {
        if (y >= sz) 
            result = m_util.mk_int(0);
        else 
            result = m_util.mk_mod(m_util.mk_mul(arg1, m_util.mk_int(rational::power_of_two(y.get_unsigned()))), m_util.mk_int(N));
        return BR_REWRITE1;
    }
    if (is_num_x && x == 0) {
        result = m_util.mk_int(0);
        return BR_DONE;
    }        
    return BR_FAILED;
}
br_status arith_rewriter::mk_ashr_core(unsigned sz, expr* arg1, expr* arg2, expr_ref& result) {
    numeral x, y, N;
    bool is_num_x = m_util.is_numeral(arg1, x);
    bool is_num_y = m_util.is_numeral(arg2, y);
    N = rational::power_of_two(sz);
    if (is_num_x) 
        x = mod(x, N);
    if (is_num_y)
        y = mod(y, N);
    if (is_num_x && x == 0) {
        result = m_util.mk_int(0);
        return BR_DONE;
    }
    if (is_num_x && is_num_y) {
        bool signx = x >= N/2;
        rational d = div(x, rational::power_of_two(y.get_unsigned()));
        SASSERT(y >= 0);
        if (signx) {
            if (y >= sz)
                result = m_util.mk_int(N-1);
            else
                result = m_util.mk_int(d);
        }
        else {
            if (y >= sz) 
                result = m_util.mk_int(0);
            else 
                result = m_util.mk_int(mod(d - rational::power_of_two(sz - y.get_unsigned()), N));
        }
        return BR_DONE;
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_lshr_core(unsigned sz, expr* arg1, expr* arg2, expr_ref& result) {
    numeral x, y, N;
    bool is_num_x = m_util.is_numeral(arg1, x);
    bool is_num_y = m_util.is_numeral(arg2, y);
    N = rational::power_of_two(sz);
    if (is_num_x) 
        x = mod(x, N);
    if (is_num_y)
        y = mod(y, N);
    if (is_num_x && x == 0) {
        result = m_util.mk_int(0);
        return BR_DONE;
    }
    if (is_num_y && y == 0) {
        result = arg1;
        return BR_DONE;
    }
    if (is_num_x && is_num_y) {
        if (y >= sz)
            result = m_util.mk_int(0);
        else {
            rational d = div(x, rational::power_of_two(y.get_unsigned()));
            result = m_util.mk_int(d);
        }
        return BR_DONE;
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_band_core(unsigned sz, expr* arg1, expr* arg2, expr_ref& result) {
    numeral x, y, N;
    bool is_num_x = m_util.is_numeral(arg1, x);
    bool is_num_y = m_util.is_numeral(arg2, y);
    N = rational::power_of_two(sz);
    if (is_num_x) 
        x = mod(x, N);      
    if (is_num_y)
        y = mod(y, N);
    if (is_num_x && x.is_zero()) {
        result = m_util.mk_int(0);
        return BR_DONE;
    }
    if (is_num_y && y.is_zero()) {
        result = m_util.mk_int(0);
        return BR_DONE;
    }
    if (is_num_x && is_num_y) {
        rational r(0);
        for (unsigned i = 0; i < sz; ++i)
            if (x.get_bit(i) && y.get_bit(i))
                r += rational::power_of_two(i);
        result = m_util.mk_int(r);
        return BR_DONE;
    }
    if (is_num_x && (x + 1).is_power_of_two()) {
        result = m_util.mk_mod(arg2, m_util.mk_int(x + 1));
        return BR_REWRITE1;
    }
    if (is_num_y && (y + 1).is_power_of_two()) {
        result = m_util.mk_mod(arg1, m_util.mk_int(y + 1));
        return BR_REWRITE1;
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_power_core(expr * arg1, expr * arg2, expr_ref & result) {
    numeral x, y;
    bool is_num_x    = m_util.is_numeral(arg1, x);
    bool is_num_y    = m_util.is_numeral(arg2, y);
    auto ensure_real = [&](expr* e) { return m_util.is_int(e) ? m_util.mk_to_real(e) : e;  };

    TRACE(arith, tout << mk_bounded_pp(arg1, m) << " " << mk_bounded_pp(arg2, m) << "\n";);
    if (is_num_x && x.is_one()) {
        result = m_util.mk_numeral(x, false);
        return BR_DONE;
    }

    if (is_num_y && y.is_one()) {
        result = ensure_real(arg1);
        return BR_REWRITE1;
    }

    if (is_num_x && is_num_y) {
        if (x.is_zero() && y.is_zero())
            return BR_FAILED;

        if (y.is_zero()) {
            result = m_util.mk_numeral(rational(1), false);
            return BR_DONE;
        }

        if (x.is_zero()) {
            result = m_util.mk_numeral(x, false);
            return BR_DONE;
        }

        if (y.is_unsigned() && y.get_unsigned() <= m_max_degree) {
            x = power(x, y.get_unsigned());
            result = m_util.mk_numeral(x, false);
            return BR_DONE;
        }

        if ((-y).is_unsigned() && (-y).get_unsigned() <= m_max_degree) {
            x = power(rational(1)/x, (-y).get_unsigned());
            result = m_util.mk_numeral(x, false);
            return BR_DONE;
        }

        if (y.is_minus_one()) {
            result = m_util.mk_numeral(rational(1) / x, false);
            return BR_DONE;
        }
    }

    expr* arg10, *arg11;
    if (m_util.is_power(arg1, arg10, arg11) && is_num_y && y.is_int() && !y.is_zero()) {
        // (^ (^ t y2) y) --> (^ t (* y2 y))  If y2 > 0 && y != 0 && y and y2 are integers
        rational y2;
        if (m_util.is_numeral(arg11, y2) && y2.is_int() && y2.is_pos()) {
            result = m_util.mk_power(ensure_real(arg10), m_util.mk_numeral(y*y2, false));
            return BR_REWRITE2;
        }
    }

    if (is_num_y && y.is_minus_one()) {        
        result = m_util.mk_div(m_util.mk_real(1), ensure_real(arg1));
        result = m.mk_ite(m.mk_eq(arg1, m_util.mk_numeral(rational(0), m_util.is_int(arg1))),
                            m_util.mk_real(0),
                            result);        
        return BR_REWRITE2;
    }

    if (is_num_y && y.is_neg()) {
        // (^ t -k) --> (^ (/ 1 t) k)
        result = m_util.mk_power(m_util.mk_div(m_util.mk_numeral(rational(1), false), arg1),
                                 m_util.mk_numeral(-y, false));
        result = m.mk_ite(m.mk_eq(arg1, m_util.mk_numeral(rational(0), m_util.is_int(arg1))),
                            m_util.mk_real(0),
                            result);
        return BR_REWRITE3;
    }

    if (is_num_y && !y.is_int() && !numerator(y).is_one()) {
        // (^ t (/ p q)) --> (^ (^ t (/ 1 q)) p)
        result = m_util.mk_power(m_util.mk_power(ensure_real(arg1), m_util.mk_numeral(rational(1)/denominator(y), false)),
                                 m_util.mk_numeral(numerator(y), false));
        return BR_REWRITE3;
    }

    if ((m_expand_power || (m_som && is_app(arg1) && to_app(arg1)->get_family_id() == get_fid())) &&
        is_num_y && y.is_unsigned() && 1 < y.get_unsigned() && y.get_unsigned() <= m_max_degree) {
        ptr_buffer<expr> args;
        unsigned k = y.get_unsigned();
        for (unsigned i = 0; i < k; i++) {
            args.push_back(arg1);
        }
        result = ensure_real(m_util.mk_mul(args.size(), args.data()));
        return BR_REWRITE2;
    }

    if (!is_num_y)
        return BR_FAILED;

    bool is_irrat_x = m_util.is_irrational_algebraic_numeral(arg1);

    if (!is_num_x && !is_irrat_x)
        return BR_FAILED;

    if (y.is_zero()) {
        return BR_FAILED;
    }

    rational num_y = numerator(y);
    rational den_y = denominator(y);
    bool is_neg_y  = false;
    if (num_y.is_neg()) {
        num_y.neg();
        is_neg_y = true;
    }
    SASSERT(num_y.is_pos());
    SASSERT(den_y.is_pos());

    if (!num_y.is_unsigned() || !den_y.is_unsigned())
        return BR_FAILED;

    unsigned u_num_y = num_y.get_unsigned();
    unsigned u_den_y = den_y.get_unsigned();

    if (u_num_y > m_max_degree || u_den_y > m_max_degree)
        return BR_FAILED;

    if (is_num_x) {
        rational xk, r;
        xk = power(x, u_num_y);
        if (xk.is_neg() && u_den_y % 2 == 0) {
            return BR_FAILED;
        }
        if (xk.root(u_den_y, r)) {
            if (is_neg_y)
                r = rational(1)/r;
            result = m_util.mk_numeral(r, false);
            return BR_DONE;
        }
        if (m_anum_simp) {
            anum_manager & am = m_util.am();
            scoped_anum r(am);
            am.set(r, xk.to_mpq());
            am.root(r, u_den_y, r);
            if (is_neg_y)
                am.inv(r);
            result = m_util.mk_numeral(am, r, false);
            return BR_DONE;
        }
        return BR_FAILED;
    }

    SASSERT(is_irrat_x);
    if (!m_anum_simp)
        return BR_FAILED;

    anum const & val  = m_util.to_irrational_algebraic_numeral(arg1);
    anum_manager & am = m_util.am();
    if (am.degree(val) > m_max_degree)
        return BR_FAILED;
    scoped_anum r(am);
    am.power(val, u_num_y, r);
    am.root(r, u_den_y, r);
    if (is_neg_y)
        am.inv(r);
    result = m_util.mk_numeral(am, r, false);
    return BR_DONE;
}

br_status arith_rewriter::mk_to_int_core(expr * arg, expr_ref & result) {
    numeral a;
    expr* x = nullptr;
    if (m_util.convert_int_numerals_to_real())
        return BR_FAILED;

    if (m_util.is_numeral(arg, a)) {
        result = m_util.mk_numeral(floor(a), true);
        return BR_DONE;
    }

    if (m_util.is_to_real(arg, x)) {
        result = x;
        return BR_DONE;
    }

    if (m_util.is_add(arg) || m_util.is_mul(arg) || m_util.is_power(arg)) {
        // Try to apply simplifications such as:
        //    (to_int (+ 1.0 (to_real x)) y) --> (+ 1 x (to_int y))
        
        expr_ref_buffer int_args(m), real_args(m);
        for (expr* c : *to_app(arg)) {
            if (m_util.is_numeral(c, a) && a.is_int()) {
                int_args.push_back(m_util.mk_numeral(a, true));
            }
            else if (m_util.is_to_real(c, x)) {
                int_args.push_back(x);
            }
            else {
                real_args.push_back(c);
            }
        }
        if (real_args.empty() && m_util.is_power(arg))
            return BR_FAILED;
        
        if (real_args.empty()) {
            result = m.mk_app(get_fid(), to_app(arg)->get_decl()->get_decl_kind(), int_args.size(), int_args.data());
            return BR_REWRITE1;
        }
        if (!int_args.empty() && m_util.is_add(arg)) {
            decl_kind k = to_app(arg)->get_decl()->get_decl_kind();
            expr_ref t1(m.mk_app(get_fid(), k, int_args.size(), int_args.data()), m);
            expr_ref t2(m.mk_app(get_fid(), k, real_args.size(), real_args.data()), m);
            int_args.reset();
            int_args.push_back(t1);
            int_args.push_back(m_util.mk_to_int(t2));
            result = m.mk_app(get_fid(), k, int_args.size(), int_args.data());
            return BR_REWRITE3;
        }
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_to_real_core(expr * arg, expr_ref & result) {
    numeral a;
    if (m_util.is_numeral(arg, a)) {
        result = m_util.mk_numeral(a, false);
        return BR_DONE;
    }
    // push to_real over OP_ADD, OP_MUL
    if (m_push_to_real) {
        if (m_util.is_add(arg) || m_util.is_mul(arg)) {
            ptr_buffer<expr> new_args;
            for (expr* e : *to_app(arg))
                new_args.push_back(m_util.mk_to_real(e));            
            if (m_util.is_add(arg))
                result = m.mk_app(get_fid(), OP_ADD, new_args.size(), new_args.data());
            else
                result = m.mk_app(get_fid(), OP_MUL, new_args.size(), new_args.data());
            return BR_REWRITE2;
        }
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_is_int(expr * arg, expr_ref & result) {
    numeral n;
    
    if (m_util.is_numeral(arg, n)) {
        result = n.is_int() ? m.mk_true() : m.mk_false();
        return BR_DONE;
    }
    
    if (m_util.is_to_real(arg)) {
        result = m.mk_true();
        return BR_DONE;
    }

    ptr_buffer<expr> todo;
    todo.push_back(arg);
    expr_fast_mark1 mark; 
    for (unsigned i = 0; i < todo.size(); ++i) {
        expr* e = todo[i];
        if (mark.is_marked(e))
            continue;
        mark.mark(e, true);
        if (m_util.is_to_real(e))
            continue;
        if (m_util.is_numeral(e, n)) {
            if (n.is_int())
                continue;
            goto bail;
        }
        if (m_util.is_mul(e) || m_util.is_add(e) || m_util.is_sub(e) || m_util.is_uminus(e)) {
            for (expr* a : *to_app(e))
                todo.push_back(a);
            continue;
        }
        goto bail;
    }
    result = m.mk_true();
    return BR_DONE;

 bail:
    result = m.mk_eq(m.mk_app(get_fid(), OP_TO_REAL,
                              m.mk_app(get_fid(), OP_TO_INT, arg)),
                     arg);
    return BR_REWRITE3;
}

br_status arith_rewriter::mk_abs_core(expr * arg, expr_ref & result) {
    result = m.mk_ite(m_util.mk_ge(arg, m_util.mk_numeral(rational(0), m_util.is_int(arg))), arg, m_util.mk_uminus(arg));
    return BR_REWRITE2;
}


// Return true if t is of the form c*Pi where c is a numeral.
// Store c into k
bool arith_rewriter::is_pi_multiple(expr * t, rational & k) {
    if (m_util.is_pi(t)) {
        k = rational(1);
        return true;
    }
    expr * a, * b;
    return m_util.is_mul(t, a, b) && m_util.is_pi(b) && m_util.is_numeral(a, k);
}

// Return true if t is of the form (+ s c*Pi) where c is a numeral.
// Store c into k, and c*Pi into m.
bool arith_rewriter::is_pi_offset(expr * t, rational & k, expr * & m) {
    if (m_util.is_add(t)) {
        for (expr* arg : *to_app(t)) 
            if (is_pi_multiple(arg, k)) {
                m = arg;
                return true;
            }        
    }
    return false;
}

// Return true if t is of the form 2*pi*to_real(s).
bool arith_rewriter::is_2_pi_integer(expr * t) {
    expr * a, * m, * b, * c;
    rational k;
    return
        m_util.is_mul(t, a, m) &&
        m_util.is_numeral(a, k) &&
        k.is_int() &&
        mod(k, rational(2)).is_zero() &&
        m_util.is_mul(m, b, c) &&
        ((m_util.is_pi(b) && m_util.is_to_real(c)) || (m_util.is_to_real(b) && m_util.is_pi(c)));
}

// Return true if t is of the form s + 2*pi*to_real(s).
// Store 2*pi*to_real(s) into m.
bool arith_rewriter::is_2_pi_integer_offset(expr * t, expr * & m) {
    if (m_util.is_add(t)) {
        for (expr* arg : *to_app(t))
            if (is_2_pi_integer(arg)) {
                m = arg;
                return true;
            }        
    }
    return false;
}

// Return true if t is of the form pi*to_real(s).
bool arith_rewriter::is_pi_integer(expr * t) {
    expr * a, * b;
    if (m_util.is_mul(t, a, b)) {
        rational k;
        if (m_util.is_numeral(a, k)) {
            if (!k.is_int())
                return false;
            expr * c, * d;
            if (!m_util.is_mul(b, c, d))
                return false;
            a = c;
            b = d;
        }
        TRACE(tan, tout << "is_pi_integer " << mk_ismt2_pp(t, m) << "\n";
              tout << "a: " << mk_ismt2_pp(a, m) << "\n";
              tout << "b: " << mk_ismt2_pp(b, m) << "\n";);
        return
            (m_util.is_pi(a) && m_util.is_to_real(b)) ||
            (m_util.is_to_real(a) && m_util.is_pi(b));
    }
    return false;
}

// Return true if t is of the form s + pi*to_real(s).
// Store 2*pi*to_real(s) into m.
bool arith_rewriter::is_pi_integer_offset(expr * t, expr * & m) {
    if (m_util.is_add(t)) {
        for (expr* arg : *to_app(t))
            if (is_pi_integer(arg)) {
                m = arg;
                return true;
            }        
    }
    return false;
}

app * arith_rewriter::mk_sqrt(rational const & k) {
    return m_util.mk_power(m_util.mk_numeral(k, false), m_util.mk_numeral(rational(1, 2), false));
}

// Return a constant representing sin(k * pi).
// Return 0 if failed.
expr * arith_rewriter::mk_sin_value(rational const & k) {
    rational k_prime = mod(floor(k), rational(2)) + k - floor(k);
    TRACE(sine, tout << "k: " << k << ", k_prime: " << k_prime << "\n";);
    SASSERT(k_prime >= rational(0) && k_prime < rational(2));
    bool     neg = false;
    if (k_prime >= rational(1)) {
        neg     = true;
        k_prime = k_prime - rational(1);
    }
    SASSERT(k_prime >= rational(0) && k_prime < rational(1));
    if (k_prime.is_zero() || k_prime.is_one()) {
        // sin(0) == sin(pi) == 0
        return m_util.mk_numeral(rational(0), false);
    }
    if (k_prime == rational(1, 2)) {
        // sin(pi/2) == 1,  sin(3/2 pi) == -1
        return m_util.mk_numeral(rational(neg ? -1 : 1), false);
    }
    if (k_prime == rational(1, 6) || k_prime == rational(5, 6)) {
        // sin(pi/6)   == sin(5/6 pi)  == 1/2
        // sin(7 pi/6) == sin(11/6 pi) == -1/2
        return m_util.mk_numeral(rational(neg ? -1 : 1, 2), false);
    }
    if (k_prime == rational(1, 4) || k_prime == rational(3, 4)) {
        // sin(pi/4)   == sin(3/4 pi) ==   Sqrt(1/2)
        // sin(5/4 pi) == sin(7/4 pi) == - Sqrt(1/2)
        expr * result = mk_sqrt(rational(1, 2));
        return neg ? m_util.mk_uminus(result) : result;
    }
    if (k_prime == rational(1, 3) || k_prime == rational(2, 3)) {
        // sin(pi/3)   == sin(2/3 pi) ==   Sqrt(3)/2
        // sin(4/3 pi) == sin(5/3 pi) == - Sqrt(3)/2
        expr * result = m_util.mk_div(mk_sqrt(rational(3)), m_util.mk_numeral(rational(2), false));
        return neg ? m_util.mk_uminus(result) : result;
    }
    if (k_prime == rational(1, 12) || k_prime == rational(11, 12)) {
        // sin(1/12 pi)  == sin(11/12 pi)  ==  [sqrt(6) - sqrt(2)]/4
        // sin(13/12 pi) == sin(23/12 pi)  == -[sqrt(6) - sqrt(2)]/4
        expr * result = m_util.mk_div(m_util.mk_sub(mk_sqrt(rational(6)), mk_sqrt(rational(2))), m_util.mk_numeral(rational(4), false));
        return neg ? m_util.mk_uminus(result) : result;
    }
    if (k_prime == rational(5, 12) || k_prime == rational(7, 12)) {
        // sin(5/12 pi)  == sin(7/12 pi)   == [sqrt(6) + sqrt(2)]/4
        // sin(17/12 pi) == sin(19/12 pi)  == -[sqrt(6) + sqrt(2)]/4
        expr * result = m_util.mk_div(m_util.mk_add(mk_sqrt(rational(6)), mk_sqrt(rational(2))), m_util.mk_numeral(rational(4), false));
        return neg ? m_util.mk_uminus(result) : result;
    }
    return nullptr;
}

br_status arith_rewriter::mk_sin_core(expr * arg, expr_ref & result) {
    expr * m, *x;
    if (m_util.is_asin(arg, x)) {
        // sin(asin(x)) == x
        result = x;
        return BR_DONE;
    }
    if (m_util.is_acos(arg, x)) {
        // sin(acos(x)) == sqrt(1 - x^2)
        result = m_util.mk_power(m_util.mk_sub(m_util.mk_real(1), m_util.mk_mul(x,x)), m_util.mk_numeral(rational(1,2), false));
        return BR_REWRITE_FULL;
    }
    rational k;
    if (is_numeral(arg, k) && k.is_zero()) {
        // sin(0) == 0
        result = arg;
        return BR_DONE;
    }

    if (is_pi_multiple(arg, k)) {
        result = mk_sin_value(k);
        if (result.get() != nullptr)
            return BR_REWRITE_FULL;
    }

    if (is_pi_offset(arg, k, m)) {
        rational k_prime = mod(floor(k), rational(2)) + k - floor(k);
        SASSERT(k_prime >= rational(0) && k_prime < rational(2));
        if (k_prime.is_zero()) {
            // sin(x + 2*n*pi) == sin(x)
            result = m_util.mk_sin(m_util.mk_sub(arg, m));
            return BR_REWRITE2;
        }
        if (k_prime == rational(1, 2)) {
            // sin(x + pi/2 + 2*n*pi) == cos(x)
            result = m_util.mk_cos(m_util.mk_sub(arg, m));
            return BR_REWRITE2;
        }
        if (k_prime.is_one()) {
            // sin(x + pi + 2*n*pi) == -sin(x)
            result = m_util.mk_uminus(m_util.mk_sin(m_util.mk_sub(arg, m)));
            return BR_REWRITE3;
        }
        if (k_prime == rational(3, 2)) {
            // sin(x + 3/2*pi + 2*n*pi) == -cos(x)
            result = m_util.mk_uminus(m_util.mk_cos(m_util.mk_sub(arg, m)));
            return BR_REWRITE3;
        }
    }

    if (is_2_pi_integer_offset(arg, m)) {
        // sin(x + 2*pi*to_real(a)) == sin(x)
        result = m_util.mk_sin(m_util.mk_sub(arg, m));
        return BR_REWRITE2;
    }

    return BR_FAILED;
}

br_status arith_rewriter::mk_cos_core(expr * arg, expr_ref & result) {
    expr* x;
    if (m_util.is_acos(arg, x)) {
        // cos(acos(x)) == x
        result = x;
        return BR_DONE;
    }
    if (m_util.is_asin(arg, x)) {
        // cos(asin(x)) == ...
    }

    rational k;
    if (is_numeral(arg, k) && k.is_zero()) {
        // cos(0) == 1
        result = m_util.mk_numeral(rational(1), false);
        return BR_DONE;
    }

    if (is_pi_multiple(arg, k)) {
        k = k + rational(1, 2);
        result = mk_sin_value(k);
        if (result.get() != nullptr)
            return BR_REWRITE_FULL;
    }

    expr * m;
    if (is_pi_offset(arg, k, m)) {
        rational k_prime = mod(floor(k), rational(2)) + k - floor(k);
        SASSERT(k_prime >= rational(0) && k_prime < rational(2));
        if (k_prime.is_zero()) {
            // cos(x + 2*n*pi) == cos(x)
            result = m_util.mk_cos(m_util.mk_sub(arg, m));
            return BR_REWRITE2;
        }
        if (k_prime == rational(1, 2)) {
            // cos(x + pi/2 + 2*n*pi) == -sin(x)
            result = m_util.mk_uminus(m_util.mk_sin(m_util.mk_sub(arg, m)));
            return BR_REWRITE3;
        }
        if (k_prime.is_one()) {
            // cos(x + pi + 2*n*pi) == -cos(x)
            result = m_util.mk_uminus(m_util.mk_cos(m_util.mk_sub(arg, m)));
            return BR_REWRITE3;
        }
        if (k_prime == rational(3, 2)) {
            // cos(x + 3/2*pi + 2*n*pi) == sin(x)
            result = m_util.mk_sin(m_util.mk_sub(arg, m));
            return BR_REWRITE2;
        }
    }

    if (is_2_pi_integer_offset(arg, m)) {
        // cos(x + 2*pi*to_real(a)) == cos(x)
        result = m_util.mk_cos(m_util.mk_sub(arg, m));
        return BR_REWRITE2;
    }

    return BR_FAILED;
}

br_status arith_rewriter::mk_tan_core(expr * arg, expr_ref & result) {
    expr* x;
    if (m_util.is_atan(arg, x)) {
        // tan(atan(x)) == x
        result = x;
        return BR_DONE;
    }

    rational k;
    if (is_numeral(arg, k) && k.is_zero()) {
        // tan(0) == 0
        result = arg;
        return BR_DONE;
    }

    if (is_pi_multiple(arg, k)) {
        expr_ref n(m), d(m);
        n = mk_sin_value(k);
        if (n.get() == nullptr)
            goto end;
        if (is_zero(n)) {
            result = n;
            return BR_DONE;
        }
        k = k + rational(1, 2);
        d = mk_sin_value(k);
        SASSERT(d.get() != 0);
        if (is_zero(d)) {
            goto end;
        }
        result = m_util.mk_div(n, d);
        return BR_REWRITE_FULL;
    }

    expr * m;
    if (is_pi_offset(arg, k, m)) {
        rational k_prime = k - floor(k);
        SASSERT(k_prime >= rational(0) && k_prime < rational(1));
        if (k_prime.is_zero()) {
            // tan(x + n*pi) == tan(x)
            result = m_util.mk_tan(m_util.mk_sub(arg, m));
            return BR_REWRITE2;
        }
    }

    if (is_pi_integer_offset(arg, m)) {
        // tan(x + pi*to_real(a)) == tan(x)
        result = m_util.mk_tan(m_util.mk_sub(arg, m));
        return BR_REWRITE2;
    }

 end:
    if (m_expand_tan) {
        result = m_util.mk_div(m_util.mk_sin(arg), m_util.mk_cos(arg));
        return BR_REWRITE2;
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_asin_core(expr * arg, expr_ref & result) {
    // Remark: we assume that ForAll x : asin(-x) == asin(x).
    // Mathematica uses this as an axiom. Although asin is an underspecified function for x < -1 or x > 1.
    // Actually, in Mathematica, asin(x) is a total function that returns a complex number for x < -1 or x > 1.
    rational k;
    if (is_numeral(arg, k)) {
        if (k.is_zero()) {
            result = arg;
            return BR_DONE;
        }
        if (k < rational(-1)) {
            // asin(-2) == -asin(2)
            // asin(-3) == -asin(3)
            k.neg();
            result = m_util.mk_uminus(m_util.mk_asin(m_util.mk_numeral(k, false)));
            return BR_REWRITE2;
        }

        if (k > rational(1))
            return BR_FAILED;

        bool neg = false;
        if (k.is_neg()) {
            neg = true;
            k.neg();
        }

        if (k.is_one()) {
            // asin(1)  == pi/2
            // asin(-1) == -pi/2
            result = m_util.mk_mul(m_util.mk_numeral(rational(neg ? -1 : 1, 2), false), m_util.mk_pi());
            return BR_REWRITE2;
        }

        if (k == rational(1, 2)) {
            // asin(1/2)  == pi/6
            // asin(-1/2) == -pi/6
            result = m_util.mk_mul(m_util.mk_numeral(rational(neg ? -1 : 1, 6), false), m_util.mk_pi());
            return BR_REWRITE2;
        }
    }

    expr * t;
    if (m_util.is_times_minus_one(arg, t)) {
        // See comment above
        // asin(-x) ==> -asin(x)
        result = m_util.mk_uminus(m_util.mk_asin(t));
        return BR_REWRITE2;
    }

    return BR_FAILED;
}

br_status arith_rewriter::mk_acos_core(expr * arg, expr_ref & result) {
    rational k;
    if (is_numeral(arg, k)) {
        if (k.is_zero()) {
            // acos(0) = pi/2
            result = m_util.mk_mul(m_util.mk_numeral(rational(1, 2), false), m_util.mk_pi());
            return BR_REWRITE2;
        }
        if (k.is_one()) {
            // acos(1) = 0
            result = m_util.mk_numeral(rational(0), false);
            return BR_DONE;
        }
        if (k.is_minus_one()) {
            // acos(-1) = pi
            result = m_util.mk_pi();
            return BR_DONE;
        }
        if (k == rational(1, 2)) {
            // acos(1/2) = pi/3
            result = m_util.mk_mul(m_util.mk_numeral(rational(1, 3), false), m_util.mk_pi());
            return BR_REWRITE2;
        }
        if (k == rational(-1, 2)) {
            // acos(-1/2) = 2/3 pi
            result = m_util.mk_mul(m_util.mk_numeral(rational(2, 3), false), m_util.mk_pi());
            return BR_REWRITE2;
        }
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_atan_core(expr * arg, expr_ref & result) {
    rational k;
    if (is_numeral(arg, k)) {
        if (k.is_zero()) {
            result = arg;
            return BR_DONE;
        }

        if (k.is_one()) {
            // atan(1)  == pi/4
            result = m_util.mk_mul(m_util.mk_numeral(rational(1, 4), false), m_util.mk_pi());
            return BR_REWRITE2;
        }

        if (k.is_minus_one()) {
            // atan(-1) == -pi/4
            result = m_util.mk_mul(m_util.mk_numeral(rational(-1, 4), false), m_util.mk_pi());
            return BR_REWRITE2;
        }

        if (k < rational(-1)) {
            // atan(-2) == -tan(2)
            // atan(-3) == -tan(3)
            k.neg();
            result = m_util.mk_uminus(m_util.mk_atan(m_util.mk_numeral(k, false)));
            return BR_REWRITE2;
        }
        return BR_FAILED;
    }

    expr * t;
    if (m_util.is_times_minus_one(arg, t)) {
        // atan(-x) ==> -atan(x)
        result = m_util.mk_uminus(m_util.mk_atan(t));
        return BR_REWRITE2;
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_sinh_core(expr * arg, expr_ref & result) {
    expr* x;
    if (m_util.is_asinh(arg, x)) {
        // sinh(asinh(x)) == x
        result = x;
        return BR_DONE;
    }
    expr * t;
    if (m_util.is_times_minus_one(arg, t)) {
        // sinh(-t) == -sinh(t)
        result = m_util.mk_uminus(m_util.mk_sinh(t));
        return BR_REWRITE2;
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_cosh_core(expr * arg, expr_ref & result) {
    expr* t;
    if (m_util.is_acosh(arg, t)) {
        // cosh(acosh(t)) == t
        result = t;
        return BR_DONE;
    }
    if (m_util.is_times_minus_one(arg, t)) {
        // cosh(-t) == cosh
        result = m_util.mk_cosh(t);
        return BR_DONE;
    }
    return BR_FAILED;
}

br_status arith_rewriter::mk_tanh_core(expr * arg, expr_ref & result) {
    expr * t;
    if (m_util.is_atanh(arg, t)) {
        // tanh(atanh(t)) == t
        result = t;
        return BR_DONE;
    }
    if (m_util.is_times_minus_one(arg, t)) {
        // tanh(-t) == -tanh(t)
        result = m_util.mk_uminus(m_util.mk_tanh(t));
        return BR_REWRITE2;
    }
    return BR_FAILED;
}

template class poly_rewriter<arith_rewriter_core>;