z3-src 503.2.1

/*++
Copyright (c) 2015 Microsoft Corporation

Module Name:

    seq_rewriter.cpp

Abstract:

    Basic rewriting rules for sequences constraints.

Authors:

    Nikolaj Bjorner (nbjorner) 2015-12-5
    Murphy Berzish 2017-02-21
    Caleb Stanford 2020-07-07
    Margus Veanes 2021

--*/

#include "util/uint_set.h"
#include "ast/rewriter/seq_rewriter.h"
#include "ast/rewriter/seq_regex_bisim.h"
#include "ast/rewriter/seq_range_collapse.h"
#include "ast/rewriter/seq_regex_witness.h"
#include "ast/arith_decl_plugin.h"
#include "ast/array_decl_plugin.h"
#include "ast/ast_pp.h"
#include "ast/ast_ll_pp.h"
#include "ast/ast_util.h"
#include "ast/well_sorted.h"
#include "ast/rewriter/var_subst.h"
#include "ast/rewriter/expr_safe_replace.h"
#include "params/seq_rewriter_params.hpp"


void seq_rewriter::updt_params(params_ref const & p) {
    seq_rewriter_params sp(p);
    m_coalesce_chars = sp.coalesce_chars();
    m_max_power_expansion = sp.max_power_expansion();
}

void seq_rewriter::get_param_descrs(param_descrs & r) {
    seq_rewriter_params::collect_param_descrs(r);
}

br_status seq_rewriter::mk_bool_app(func_decl* f, unsigned n, expr* const* args, expr_ref& result) {
    switch (f->get_decl_kind()) {
    case OP_AND:
        return mk_bool_app_helper(true, n, args, result);
    case OP_OR:
        return mk_bool_app_helper(false, n, args, result);
    case OP_EQ:
        SASSERT(n == 2);
        // return mk_eq_helper(args[0], args[1], result);
        Z3_fallthrough;
    default:
        return BR_FAILED;
    }
}

br_status seq_rewriter::mk_bool_app_helper(bool is_and, unsigned n, expr* const* args, expr_ref& result) {
    bool found = false;
    expr* arg = nullptr;
    
    for (unsigned i = 0; i < n && !found; ++i) {        
        found = str().is_in_re(args[i]) || (m().is_not(args[i], arg) && str().is_in_re(arg));
    }
    if (!found) return BR_FAILED;
    
    obj_map<expr, expr*> in_re, not_in_re;
    bool found_pair = false;
    
    ptr_buffer<expr> new_args;
    for (unsigned i = 0; i < n; ++i) {
        expr* args_i = args[i];
        expr* x = nullptr, *y = nullptr, *z = nullptr;
        if (str().is_in_re(args_i, x, y) && !str().is_empty(x)) {
            if (in_re.find(x, z)) {				
                in_re[x] = is_and ? re().mk_inter(z, y) : re().mk_union(z, y);
                found_pair = true;
            }
            else {
                in_re.insert(x, y);
                found_pair |= not_in_re.contains(x);
            }
        }
        else if (m().is_not(args_i, arg) && str().is_in_re(arg, x, y) && !str().is_empty(x)) {
            if (not_in_re.find(x, z)) {
                not_in_re[x] = is_and ? re().mk_union(z, y) : re().mk_inter(z, y);
                found_pair = true;
            }
            else {
                not_in_re.insert(x, y);
                found_pair |= in_re.contains(x);
            }
        }
        else 
            new_args.push_back(args_i);
    }
    
    if (!found_pair) {
        return BR_FAILED;
    }
    

    for (auto const & kv : in_re) {
        expr* x = kv.m_key;
        expr* y = kv.m_value;
        expr* z = nullptr;
        if (not_in_re.find(x, z)) {
            expr* z_c = re().mk_complement(z);
            expr* w = is_and ? re().mk_inter(y, z_c) : re().mk_union(y, z_c);
            new_args.push_back(re().mk_in_re(x, w));
        }
        else {
            new_args.push_back(re().mk_in_re(x, y));
        }
    }
    for (auto const& kv : not_in_re) {
        expr* x = kv.m_key;
        expr* y = kv.m_value;
        if (!in_re.contains(x)) {
            new_args.push_back(re().mk_in_re(x, re().mk_complement(y)));
        }
    }
    
    result = is_and ? m().mk_and(new_args) : m().mk_or(new_args);
    return BR_REWRITE_FULL;
}

br_status seq_rewriter::mk_eq_helper(expr* a, expr* b, expr_ref& result) {
    expr* sa = nullptr, *ra = nullptr, *sb = nullptr, *rb = nullptr;
    if (str().is_in_re(b))
        std::swap(a, b);
    if (!str().is_in_re(a, sa, ra))
        return BR_FAILED;
    bool is_not = m().is_not(b, b);
    if (!str().is_in_re(b, sb, rb))
        return BR_FAILED;
    if (sa != sb)
        return BR_FAILED;
    // sa in ra = sb in rb;
    // sa in (ra n rb) u (C(ra) n C(rb))
    if (is_not)
        rb = re().mk_complement(rb);
    auto a_ = re().mk_inter(ra, rb);
    auto b_ = re().mk_complement(ra);
    expr* r = re().mk_union(a_, re().mk_inter(b_, re().mk_complement(rb)));
    result = re().mk_in_re(sa, r);
    return BR_REWRITE_FULL;
}

br_status seq_rewriter::mk_app_core(func_decl * f, unsigned num_args, expr * const * args, expr_ref & result) {
    SASSERT(f->get_family_id() == get_fid());
    br_status st = BR_FAILED;
    switch(f->get_decl_kind()) {
        
    case OP_SEQ_UNIT:
        SASSERT(num_args == 1);
        st = mk_seq_unit(args[0], result);
        break;
    case OP_SEQ_EMPTY:
        return BR_FAILED;
    case OP_RE_PLUS:
        SASSERT(num_args == 1);
        st = mk_re_plus(args[0], result);
        break;
    case OP_RE_STAR:
        SASSERT(num_args == 1);
        st = mk_re_star(args[0], result);
        break;
    case OP_RE_OPTION:
        SASSERT(num_args == 1);
        st = mk_re_opt(args[0], result);
        break;
    case OP_RE_REVERSE:
        SASSERT(num_args == 1);
        st = mk_re_reverse(args[0], result);
        break;
    case OP_RE_DERIVATIVE:
        SASSERT(num_args == 2);
        st = mk_re_derivative(args[0], args[1], result);
        break;
    case OP_RE_CONCAT:
        if (num_args == 1) {
            result = args[0]; 
            st = BR_DONE;
        }
        else {
            SASSERT(num_args == 2);
            st = mk_re_concat(args[0], args[1], result); 
        }
        break;
    case OP_RE_UNION:
        if (num_args == 1) {
            result = args[0]; 
            st = BR_DONE;
        }
        else {
            SASSERT(num_args == 2);
            st = mk_re_union(args[0], args[1], result);
        }
        break;
    case OP_RE_RANGE:
        SASSERT(num_args == 2);
        st = mk_re_range(args[0], args[1], result);
        break;
    case OP_RE_DIFF: 
        if (num_args == 2)
            st = mk_re_diff(args[0], args[1], result);
        else if (num_args == 1) {
            result = args[0];
            st = BR_DONE;
        }          
        break;
    case OP_RE_XOR:
        if (num_args == 2)
            st = mk_re_xor(args[0], args[1], result);
        else if (num_args == 1) {
            result = args[0];
            st = BR_DONE;
        }
        break;
    case OP_RE_INTERSECT:
        if (num_args == 1) {
            result = args[0];
            st = BR_DONE;
        }
        else {
            SASSERT(num_args == 2);
            st = mk_re_inter(args[0], args[1], result);
        }
        break;
    case OP_RE_COMPLEMENT:
        SASSERT(num_args == 1);
        st = mk_re_complement(args[0], result);
        break;
    case OP_RE_LOOP:
        st = mk_re_loop(f, num_args, args, result);
        break;
    case OP_RE_POWER:
        st = mk_re_power(f, args[0], result);
        break;
    case OP_RE_EMPTY_SET:
        return BR_FAILED;    
    case OP_RE_FULL_SEQ_SET:
        return BR_FAILED;    
    case OP_RE_FULL_CHAR_SET:
        return BR_FAILED;    
    case OP_RE_OF_PRED:
        return BR_FAILED;    
    case _OP_SEQ_SKOLEM:
        return BR_FAILED;    
    case OP_SEQ_CONCAT: 
        if (num_args == 1) {
            result = args[0];
            st = BR_DONE;
        }
        else {
            SASSERT(num_args == 2);
            st = mk_seq_concat(args[0], args[1], result);
        }
        break;
    case OP_SEQ_LENGTH:
        SASSERT(num_args == 1);
        st = mk_seq_length(args[0], result);
        break;
    case OP_SEQ_POWER:
        SASSERT(num_args == 2);
        st = mk_seq_power(args[0], args[1], result);
        break;
    case OP_SEQ_EXTRACT:
        SASSERT(num_args == 3);
        st = mk_seq_extract(args[0], args[1], args[2], result);
        break;
    case OP_SEQ_CONTAINS: 
        SASSERT(num_args == 2);
        st = mk_seq_contains(args[0], args[1], result);
        break;
    case OP_SEQ_AT:
        SASSERT(num_args == 2);
        st = mk_seq_at(args[0], args[1], result); 
        break;
    case OP_SEQ_NTH:
        SASSERT(num_args == 2);
        return mk_seq_nth(args[0], args[1], result); 
    case OP_SEQ_NTH_I:
        SASSERT(num_args == 2);
        return mk_seq_nth_i(args[0], args[1], result); 
    case OP_SEQ_PREFIX: 
        SASSERT(num_args == 2);
        st = mk_seq_prefix(args[0], args[1], result);
        break;
    case OP_SEQ_SUFFIX: 
        SASSERT(num_args == 2);
        st = mk_seq_suffix(args[0], args[1], result);
        break;
    case OP_SEQ_INDEX:
        if (num_args == 2) {
            expr_ref arg3(zero(), m());
            result = str().mk_index(args[0], args[1], arg3);
            st = BR_REWRITE1;
        }
        else {
            SASSERT(num_args == 3);
            st = mk_seq_index(args[0], args[1], args[2], result);
        }
        break;
    case OP_SEQ_LAST_INDEX:
        SASSERT(num_args == 2);
        st = mk_seq_last_index(args[0], args[1], result);
        break;
    case OP_SEQ_REPLACE:
        SASSERT(num_args == 3);
        st = mk_seq_replace(args[0], args[1], args[2], result);
        break;
    case OP_SEQ_REPLACE_ALL:
        SASSERT(num_args == 3);
        st = mk_seq_replace_all(args[0], args[1], args[2], result);
        break;
    case OP_SEQ_MAP:
        SASSERT(num_args == 2);
        st = mk_seq_map(args[0], args[1], result);
        break;
    case OP_SEQ_MAPI:
        SASSERT(num_args == 3);
        st = mk_seq_mapi(args[0], args[1], args[2], result);
        break;
    case OP_SEQ_FOLDL:
        SASSERT(num_args == 3);
        st = mk_seq_foldl(args[0], args[1], args[2], result);
        break;
    case OP_SEQ_FOLDLI:
        SASSERT(num_args == 4);
        st = mk_seq_foldli(args[0], args[1], args[2], args[3], result);
        break;
    case OP_SEQ_REPLACE_RE:
        SASSERT(num_args == 3);
        st = mk_seq_replace_re(args[0], args[1], args[2], result);
        break;
    case OP_SEQ_REPLACE_RE_ALL:
        SASSERT(num_args == 3);
        st = mk_seq_replace_re_all(args[0], args[1], args[2], result);
        break;
    case OP_SEQ_TO_RE:
        SASSERT(num_args == 1);
        st = mk_str_to_regexp(args[0], result);
        break;
    case OP_SEQ_IN_RE:
        SASSERT(num_args == 2);
        st = mk_str_in_regexp(args[0], args[1], result);
        break;
    case OP_STRING_LE:
        SASSERT(num_args == 2);
        st = mk_str_le(args[0], args[1], result);
        break;
    case OP_STRING_LT:
        SASSERT(num_args == 2);
        st = mk_str_lt(args[0], args[1], result);
        break;
    case OP_STRING_FROM_CODE:
        SASSERT(num_args == 1);
        st = mk_str_from_code(args[0], result);
        break;
    case OP_STRING_TO_CODE:
        SASSERT(num_args == 1);
        st = mk_str_to_code(args[0], result);
        break;
    case OP_STRING_IS_DIGIT:
        SASSERT(num_args == 1);
        st = mk_str_is_digit(args[0], result);
        break;
    case OP_STRING_CONST:
        st = BR_FAILED;
        if (!m_coalesce_chars) {
            st = mk_str_units(f, result);
        }
        break;
    case OP_STRING_ITOS: 
        SASSERT(num_args == 1);
        st = mk_str_itos(args[0], result);
        break;
    case OP_STRING_STOI: 
        SASSERT(num_args == 1);
        st = mk_str_stoi(args[0], result);
        break;
    case OP_STRING_UBVTOS:
        SASSERT(num_args == 1);
        st = mk_str_ubv2s(args[0], result);
        break;
    case OP_STRING_SBVTOS:
      SASSERT(num_args == 1);
      st = mk_str_sbv2s(args[0], result);
      break;
    case _OP_STRING_CONCAT:
    case _OP_STRING_PREFIX:
    case _OP_STRING_SUFFIX:
    case _OP_STRING_STRCTN:
    case _OP_STRING_LENGTH:
    case _OP_STRING_CHARAT:
    case _OP_STRING_IN_REGEXP: 
    case _OP_STRING_TO_REGEXP: 
    case _OP_STRING_SUBSTR: 
    case _OP_STRING_STRREPL:
    case _OP_STRING_STRIDOF: 
        UNREACHABLE();
    }
    if (st == BR_FAILED) {
        st = lift_ites_throttled(f, num_args, args, result);
    }
    CTRACE(seq_verbose, st != BR_FAILED, tout << expr_ref(m().mk_app(f, num_args, args), m()) << " -> " << result << "\n";);
    SASSERT(st == BR_FAILED || result->get_sort() == f->get_range());
    return st;
}

/*
 * (seq.unit (_ BitVector 8)) ==> String constant
 */
br_status seq_rewriter::mk_seq_unit(expr* e, expr_ref& result) {
    unsigned ch;
    // specifically we want (_ BitVector 8)
    if (m_util.is_const_char(e, ch) && m_coalesce_chars) {
        // convert to string constant
        zstring s(ch);
        TRACE(seq_verbose, tout << "rewrite seq.unit of 8-bit value " << ch << " to string constant \"" << s<< "\"" << std::endl;);
        result = str().mk_string(s);
        return BR_DONE;
    }

    return BR_FAILED;
}

/*
   string + string = string
   (a + b) + c = a + (b + c)
   a + "" = a
   "" + a = a
   string + (string + a) = string + a
*/
expr_ref seq_rewriter::mk_seq_concat(expr* a, expr* b) {
    expr_ref result(m());
    if (BR_FAILED == mk_seq_concat(a, b, result))
        result = str().mk_concat(a, b);
    return result;
}
br_status seq_rewriter::mk_seq_concat(expr* a, expr* b, expr_ref& result) {
    zstring s1, s2;
    expr* c, *d;
    bool isc1 = str().is_string(a, s1) && m_coalesce_chars;
    bool isc2 = str().is_string(b, s2) && m_coalesce_chars;

    // Fast path: neither argument needs restructuring. Falling through to
    // BR_FAILED here (as before) lets the OP_SEQ_CONCAT case in mk_app_core
    // still try lift_ites_throttled on plain, already-flat concatenations.
    if (!(isc1 && isc2) &&
        !str().is_concat(a) &&
        !str().is_empty(a) &&
        !str().is_empty(b) &&
        !(isc1 && str().is_concat(b, c, d) && str().is_string(c, s2)))
        return BR_FAILED;

    if (isc1 && isc2) {
        result = str().mk_string(s1 + s2);
        return BR_DONE;
    }

    // General case: a and/or b need restructuring (one of them is a
    // concatenation, empty, or coalescable with the other). Flatten both
    // into their constituent elements with an explicit worklist instead of
    // recursively decomposing the str.++ tree: a long chain of
    // concatenations (e.g. from repeated quantifier instantiation of
    // `s ++ s`) previously drove mutual recursion between the two
    // mk_seq_concat overloads deep enough to overflow the native stack
    // (#11042). Elements are then coalesced in a single linear pass and the
    // right-associated chain is rebuilt with a plain loop.
    ptr_vector<expr> es, todo;
    todo.push_back(b);
    todo.push_back(a);
    while (!todo.empty()) {
        expr* e = todo.back();
        todo.pop_back();
        expr* e1 = nullptr, *e2 = nullptr;
        if (str().is_concat(e, e1, e2)) {
            todo.push_back(e2);
            todo.push_back(e1);
            continue;
        }
        if (str().is_empty(e))
            continue;
        es.push_back(e);
    }
    if (es.empty()) {
        result = str().mk_empty(a->get_sort());
        return BR_DONE;
    }
    unsigned j = 0;
    for (unsigned i = 0; i < es.size(); ++i) {
        zstring t1, t2;
        if (j > 0 && m_coalesce_chars && str().is_string(es.get(j - 1), t1) && str().is_string(es.get(i), t2))
            es[j - 1] = str().mk_string(t1 + t2);
        else
            es[j++] = es.get(i);
    }
    es.shrink(j);
    expr_ref cur(es.back(), m());
    for (unsigned i = es.size() - 1; i-- > 0; )
        cur = str().mk_concat(es.get(i), cur);
    result = cur;
    return BR_DONE;
}

bool seq_rewriter::mk_seq_reverse(expr* s, expr_ref& result) {
    ptr_vector<expr> elems, todo;
    todo.push_back(s);
    while (!todo.empty()) {
        expr* e = todo.back();
        todo.pop_back();
        if (str().is_concat(e)) {             // str.++ is n-ary; pushing the arguments left
            app* a = to_app(e);               // to right pops them right to left, which is
            for (expr* arg : *a)              // the order the reversed sequence needs
                todo.push_back(arg);
            continue;
        }
        if (str().is_empty(e))
            continue;
        elems.push_back(e);
    }
    zstring zs;
    expr_ref_vector es(m());
    for (expr* e : elems) {
        if (str().is_unit(e))
            es.push_back(e);                  // a one-element sequence is its own reverse
        else if (str().is_string(e, zs))
            es.push_back(str().mk_string(zs.reverse()));
        else
            return false;                     // there is no sequence-level reverse operator,
    }                                         // so a non-concrete element has no reverse here
    result = str().mk_concat(es, s->get_sort());
    return true;
}

br_status seq_rewriter::mk_seq_length(expr* a, expr_ref& result) {
    zstring b;
    rational r;
    m_es.reset();
    str().get_concat(a, m_es);
    unsigned len = 0;
    unsigned j = 0;
    for (expr* e : m_es) {
        auto [bounded, len_e] = min_length(e);
        if (bounded)
            len += len_e;
        else 
            m_es[j++] = e;        
    }
    if (j == 0) {
        result = m_autil.mk_int(len);
        return BR_DONE;
    }
    if (j != m_es.size() || j != 1) {
        expr_ref_vector es(m());        
        for (unsigned i = 0; i < j; ++i) {
            es.push_back(str().mk_length(m_es.get(i)));
        }
        if (len != 0) {
            es.push_back(m_autil.mk_int(len));
        }
        result = m_autil.mk_add(es.size(), es.data());
        return BR_REWRITE2;
    }
    expr* x = nullptr, *y = nullptr, *z = nullptr;
    if (str().is_replace(a, x, y, z) && l_true == eq_length(y, z)) {
        result = str().mk_length(x);
        return BR_REWRITE1;
    }
    if (str().is_map(a, x, y)) {
        result = str().mk_length(y);
        return BR_REWRITE1;
    } 
    if (str().is_mapi(a, x, y, z)) {
        result = str().mk_length(z);
        return BR_REWRITE1;
    } 
    // len(s^n) = n * len(s) for n >= 0
    if (str().is_power(a, x, y) && m_autil.is_numeral(y, r) && r >= 0) {
        result = m_autil.mk_mul(y, str().mk_length(x));
        return BR_REWRITE2;
    }
    // len(extract(x, 0, z)) = min(z, len(x))
    if (str().is_extract(a, x, y, z) && 
        m_autil.is_numeral(y, r) && r.is_zero() &&
        m_autil.is_numeral(z, r) && r >= 0) {
        expr* len_x = str().mk_length(x);
        result = m().mk_ite(m_autil.mk_le(len_x, z), len_x, z);
        return BR_REWRITE_FULL;
    }
    return BR_FAILED;
}

/**
   s^n = "" if n <= 0
   s^n = s ++ ... ++ s (n copies) if n > 0

   Exponents above rewriter.max_power_expansion are left to the theory solver.
*/
br_status seq_rewriter::mk_seq_power(expr* a, expr* b, expr_ref& result) {
    expr* s = nullptr, *k = nullptr;
    rational n, kv;
    if (str().is_empty(a)) {
        result = a;
        return BR_DONE;
    }
    bool is_num = m_autil.is_numeral(b, n);
    if (is_num && !n.is_pos()) {
        result = str().mk_empty(a->get_sort());
        return BR_DONE;
    }
    if (is_num && n.is_one()) {
        result = a;
        return BR_DONE;
    }
    // (s^k)^n = s^(k*n) whenever k or n is a positive numeral
    if (str().is_power(a, s, k) &&
        (is_num || (m_autil.is_numeral(k, kv) && kv.is_pos()))) {
        result = str().mk_power(s, m_autil.mk_mul(k, b));
        return BR_REWRITE_FULL;
    }
    if (!is_num || n > rational(m_max_power_expansion))
        return BR_FAILED;
    unsigned v = n.get_unsigned();
    zstring t;
    if (str().is_string(a, t)) {
        if (!(n * rational(t.length())).is_unsigned())
            return BR_FAILED;
        zstring r = t;
        for (unsigned i = 1; i < v; ++i)
            r += t;
        result = str().mk_string(r);
        return BR_DONE;
    }
    expr_ref_vector es(m());
    for (unsigned i = 0; i < v; ++i)
        es.push_back(a);
    result = str().mk_concat(es, a->get_sort());
    return BR_REWRITE_FULL;
}

/*
*  In general constructs nth(t,0) but if t = substring(s,j,..) then simplifies to nth(s,j)
*  This method assumes that |t| > 0.
*/
expr_ref seq_rewriter::mk_seq_first(expr* t) {
    expr_ref result(m());
    expr* s, * j, * k;
    if (str().is_extract(t, s, j, k))
        result = str().mk_nth_i(s, j);
    else
        result = str().mk_nth_c(t, 0);
    return result;
}

expr_ref seq_rewriter::mk_sub(expr* a, rational const& n) {
    expr* a1, *a2;
    SASSERT(n.is_int());
    rational k;
    if (m_autil.is_sub(a, a1, a2) && m_autil.is_numeral(a2, k))
        return expr_ref(m_autil.mk_sub(a1, m_autil.mk_int(k + n)), m());
    if (m_autil.is_add(a, a1, a2) && m_autil.is_numeral(a2, k))
        return expr_ref(m_autil.mk_add(a1, m_autil.mk_int(k - n)), m());        
    if (m_autil.is_add(a, a1, a2) && m_autil.is_numeral(a1, k))
        return expr_ref(m_autil.mk_add(a2, m_autil.mk_int(k - n)), m());
    return expr_ref(m_autil.mk_sub(a, m_autil.mk_int(n)), m());
}


/*
*  In general constructs substring(t,1,|t|-1) but if t = substring(s,j,k) then simplifies to substring(s,j+1,k-1)
*  This method assumes that |t| > 0.
*/
expr_ref seq_rewriter::mk_seq_rest(expr* t) {
    expr_ref result(m());
    expr* s, * j, * k;
    rational jv;
    if (str().is_extract(t, s, j, k) &&  m_autil.is_numeral(j, jv) && jv >= 0) {
        auto a = m_autil.mk_int(jv + 1);
        result = str().mk_substr(s, a, mk_sub(k, 1));
    }
    else {
        auto a = one();
        result = str().mk_substr(t, a, mk_sub(str().mk_length(t), 1));
    }
    return result;
}

/*
*  In general constructs nth(t,|t|-1) but if t = substring(s,j,|s|-j) j >= 0, then simplifies to nth(s,|s|-1) 
*  This method assumes that |t| > 0.
*/
expr_ref seq_rewriter::mk_seq_last(expr* t) {
    expr_ref result(m());
    expr* s, * j, * k, * s_, * len_s;
    rational jv, i;
    if (str().is_extract(t, s, j, k) &&
        m_autil.is_numeral(j, jv) && jv >= 0 &&
        str().is_len_sub(k, len_s, s_, i) &&
        s == s_ && jv == i) {
        expr_ref lastpos = mk_sub(len_s, 1);
        result = str().mk_nth_i(s, lastpos);
    }
    else {
        expr_ref lhs(str().mk_length(t), m());
        auto rhs = one();
        result = str().mk_nth_i(t, m_autil.mk_sub(lhs, rhs));
    }
    return result;
}

/*
*  In general constructs substring(t,0,|t|-1) 
*  Incorrect comment: "but if t = substring(s,0,k) then simplifies to substring(s,0,k-1). 
*     This method assumes that |t| > 0, thus, if t = substring(s,0,k) then k > 0 so substring(s,0,k-1) is correct."
*  No: if k > |s| then substring(s,0,k) = substring(s,0,k-1)
*/
expr_ref seq_rewriter::mk_seq_butlast(expr* t) {
    auto b = zero();
    auto c = str().mk_length(t);
    auto a = str().mk_substr(t, b, m_autil.mk_sub(c, one()));
    return expr_ref(a, m());
}

/*
    Lift all ite expressions to the top level, safely
    throttled to not blowup the size of the expression.

    Note: this function does not ensure the same BDD form that is
    used in the normal form for derivatives in mk_re_derivative.
*/
br_status seq_rewriter::lift_ites_throttled(func_decl* f, unsigned n, expr* const* args, expr_ref& result) {
    expr* c = nullptr, * t = nullptr, * e = nullptr;
    for (unsigned i = 0; i < n; ++i)
        if (m().is_ite(args[i], c, t, e) &&
            lift_ites_filter(f, args[i]) &&
            (get_depth(t) <= 2 || t->get_ref_count() == 1 ||
                get_depth(e) <= 2 || e->get_ref_count() == 1)) {
            ptr_buffer<expr> new_args;
            for (unsigned j = 0; j < n; ++j) new_args.push_back(args[j]);
            new_args[i] = t;
            expr_ref arg1(m().mk_app(f, new_args), m());
            new_args[i] = e;
            expr_ref arg2(m().mk_app(f, new_args), m());
            result = m().mk_ite(c, arg1, arg2);
            TRACE(seq_verbose, tout << "lifting ite: " << mk_pp(result, m()) << std::endl;);
            return BR_REWRITE2;
        }
    return BR_FAILED;
}

/* returns false iff the ite must not be lifted */
bool seq_rewriter::lift_ites_filter(func_decl* f, expr* ite) {
    // do not lift ites from sequences over regexes
    // for example DO NOT lift to_re(ite(c, s, t)) to ite(c, to_re(s), to_re(t))
    if (u().is_re(f->get_range()) && u().is_seq(ite->get_sort()))
        return false;
    // The following check is intended to avoid lifting cases such as 
    // substring(s,0,ite(c,e1,e2)) ==> ite(c, substring(s,0,e1), substring(s,0,e2))
    // TBD: not sure if this is too restrictive though and may block cases when such lifting is desired
    // if (m_autil.is_int(m().get_sort(ite)) && u().is_seq(f->get_range()))
    //    return false;
    return true;
}


bool seq_rewriter::is_suffix(expr* s, expr* offset, expr* len) {
    expr_ref_vector lens(m());
    rational a, b;
    return 
        get_lengths(len, lens, a) && 
        (a.neg(), m_autil.is_numeral(offset, b) && 
         b.is_pos() && 
         a == b && 
         lens.contains(s));
}

bool seq_rewriter::is_prefix(expr* s, expr* offset, expr* len) {
    expr_ref_vector lens(m());
    rational a, b;
    return 
        get_lengths(len, lens, a) &&
        a < 0 && 
        m_autil.is_numeral(offset, b) && 
        b == 0 && 
        lens.size() == 1 && 
        lens.contains(s);
}

bool seq_rewriter::sign_is_determined(expr* e, sign& s) {
    s = sign_zero;
    if (m_autil.is_add(e)) {
        for (expr* arg : *to_app(e)) {
            sign s1;
            if (!sign_is_determined(arg, s1))
                return false;
            if (s == sign_zero) 
                s = s1;
            else if (s1 == sign_zero)
                continue;
            else if (s1 != s)
                return false;
        }
        return true;
    }
    if (m_autil.is_mul(e)) {
        for (expr* arg : *to_app(e)) {
            sign s1;
            if (!sign_is_determined(arg, s1))
                return false;
            if (s1 == sign_zero) {
                s = sign_zero;
                return true;
            }
            if (s == sign_zero)
                s = s1;
            else if (s != s1)
                s = sign_neg;
            else 
                s = sign_pos;
        }
        return true;
    }
    if (str().is_length(e)) {
        s = sign_pos;
        return true;
    }
    rational r;
    if (m_autil.is_numeral(e, r)) {
        if (r.is_pos())
            s = sign_pos;
        else if (r.is_neg())
            s = sign_neg;
        return true;
    }
    return false;
}

expr_ref seq_rewriter::mk_len(rational const& p, expr_ref_vector const& xs) {
    expr_ref r(m_autil.mk_int(p), m());
    for (expr* e : xs)
        r = m_autil.mk_add(r, str().mk_length(e));
    return r;
}

bool seq_rewriter::extract_pop_suffix(expr_ref_vector const& as, expr* b, expr* c, expr_ref& result) {
    auto len_a1 = min_length(as).second;
    rational pos, len;
    if (!as.empty() && m_autil.is_numeral(b, pos) && 
        m_autil.is_numeral(c, len) && len_a1 >= pos + len && pos >= 0 && len >= 0) {
        unsigned i = 0;
        len_a1 = 0;
        for ( ; i < as.size() && len_a1 < pos + len; ++i) {
            auto len_a2 = min_length(as.get(i)).second;
            len_a1 += len_a2;
        }
        if (i < as.size()) {
            expr* a = str().mk_concat(i, as.data(), as[0]->get_sort());
            result = str().mk_substr(a, b, c);
            return true;
        }
    }
    return false;
}


bool seq_rewriter::extract_push_offset(expr_ref_vector const& as, expr* b, expr* c, expr_ref& result) {
    expr_ref_vector lens(m());
    rational pos1;
    if (get_lengths(b, lens, pos1) && pos1 >= 0) {
        unsigned i = 0;
        for (; i < as.size(); ++i) {
            expr* lhs = as.get(i);
            if (lens.contains(lhs)) {
                lens.erase(lhs);
            }
            else if (str().is_unit(lhs) && pos1.is_pos()) {
                pos1 -= rational(1);
            }
            else {
                break;
            }
        }
        if (i != 0) {
            expr_ref t1(m());
            t1 = str().mk_concat(as.size() - i, as.data() + i, as[0]->get_sort());
            expr_ref t2 = mk_len(pos1, lens);
            result = str().mk_substr(t1, t2, c);
            TRACE(seq, tout << result << "\n";);
            return true;
        }
    }
    return false;
}

bool seq_rewriter::extract_push_length(expr_ref_vector& as, expr* b, expr* c, expr_ref& result) {
    rational pos;
    expr_ref_vector lens(m());
    if (!as.empty() && 
        m_autil.is_numeral(b, pos) && pos.is_zero() && 
        get_lengths(c, lens, pos) && !pos.is_neg()) {
        unsigned i = 0;
        for (; i < as.size(); ++i) {
            expr* lhs = as.get(i);
            if (lens.contains(lhs)) {
                lens.erase(lhs);
            }
            else if (str().is_unit(lhs) && pos.is_pos()) {
                pos -= rational(1);
            }
            else {
                break;
            }
        }
        if (i == as.size()) {
            result = str().mk_concat(as.size(), as.data(), as[0]->get_sort());
            return true;
        }
        else if (i != 0) {
            expr_ref t1(m()), t2(m());
            t1 = str().mk_concat(as.size() - i, as.data() + i, as[0]->get_sort());
            t2 = mk_len(pos, lens);
            result = str().mk_substr(t1, b, t2);
            as[i] = result;
            result = str().mk_concat(i + 1, as.data(), as[0]->get_sort());
            TRACE(seq, tout << result << "\n";);
            return true;
        }
    }
    return false;
}


br_status seq_rewriter::mk_seq_extract(expr* a, expr* b, expr* c, expr_ref& result) {
    zstring s;
    rational pos, len;

    TRACE(seq_verbose, tout << mk_pp(a, m()) << " " << mk_pp(b, m()) << " " << mk_pp(c, m()) << "\n";);
    bool constantBase = str().is_string(a, s);
    bool constantPos = m_autil.is_numeral(b, pos);
    bool constantLen = m_autil.is_numeral(c, len);
    sort* a_sort = a->get_sort();

    sign sg;
    if (sign_is_determined(c, sg) && sg == sign_neg) {
        result = str().mk_empty(a_sort);
        return BR_DONE;
    }
    
    // case 1: pos < 0 or len <= 0
    // rewrite to ""
    if ( (constantPos && pos.is_neg()) || (constantLen && !len.is_pos()) ) {
        result = str().mk_empty(a_sort);
        return BR_DONE;
    }
    // case 1.1: pos >= length(base)
    // rewrite to ""
    if (constantPos && constantBase && pos >= s.length()) {
        result = str().mk_empty(a_sort);
        return BR_DONE;
    }


    rational len_a;
    if (constantPos && max_length(a, len_a) && len_a <= pos) {
        result = str().mk_empty(a_sort);
        return BR_DONE;
    }

    // For integer n >= 0, c*n^k >= n when c >= 1 and k >= 1.
    if (m_autil.is_mul(b)) {
        bool has_base_length = false, all_other_factors_ge_one = true;
        expr* base = nullptr;
        rational factor;
        ptr_buffer<expr> todo;
        obj_hashtable<expr> visited;
        todo.push_back(b);
        while (!todo.empty()) {
            expr* arg = todo.back();
            todo.pop_back();
            if (visited.contains(arg))
                continue;
            visited.insert(arg);
            if (m_autil.is_mul(arg)) {
                for (expr* e : *to_app(arg))
                    todo.push_back(e);
            }
            else if (str().is_length(arg, base) && base == a)
                has_base_length = true;
            else if (!m_autil.is_numeral(arg, factor) || factor < 1) {
                all_other_factors_ge_one = false;
                break;
            }
        }
        if (has_base_length && all_other_factors_ge_one) {
            result = str().mk_empty(a_sort);
            return BR_DONE;
        }
    }
    
    constantPos &= pos.is_unsigned();
    constantLen &= len.is_unsigned();

    if (constantLen && len == 1) {
        result = str().mk_at(a, b);
        return BR_REWRITE1;
    }

    if (constantPos && constantLen && constantBase) {
        unsigned _pos = pos.get_unsigned();
        unsigned _len = len.get_unsigned();
        if (pos + len >= s.length()) 
            result = str().mk_string(s.extract(_pos, s.length()));
        else 
            result = str().mk_string(s.extract(_pos, _len));
        return BR_DONE;
    }


    expr_ref_vector as(m());
    str().get_concat_units(a, as);
    if (as.empty()) {
        result = str().mk_empty(a->get_sort());
        return BR_DONE;
    }

    if (extract_pop_suffix(as, b, c, result))
        return BR_REWRITE1;

    // extract(a + b + c, len(a + b), s) -> extract(c, 0, s)
    // extract(a + b + c, len(a) + len(b), s) -> extract(c, 0, s)
    if (extract_push_offset(as, b, c, result))
        return BR_REWRITE3;

    // extract(a + b + c, 0, len(a) + len(b)) -> c
    if (extract_push_length(as, b, c, result))
        return BR_REWRITE3;
    
    expr* a1 = nullptr, *b1 = nullptr, *c1 = nullptr;
    if (str().is_extract(a, a1, b1, c1) && 
        is_suffix(a1, b1, c1) && is_suffix(a, b, c)) {
        {
            auto _seq939_0 = m_autil.mk_add(b1, b);
            auto _seq939_1 = m_autil.mk_sub(c1, b);
            result = str().mk_substr(a1, _seq939_0, _seq939_1);
        }
        return BR_REWRITE3;
    }
    rational r1, r2;
    if (str().is_extract(a, a1, b1, c1) &&
        m_autil.is_numeral(b1, r1) && r1.is_unsigned() &&
        m_autil.is_numeral(c1, r2) && r2.is_unsigned() &&
        constantPos && constantLen) {
        if (r1 == 0 && r2 >= pos + len) {        
            result = str().mk_substr(a1, b, c);
            return BR_REWRITE1;
        }
        // pos2 <= len1
        // 0 <= pos1
        // extract(extract(x, pos1, len1), pos2, len2)
        // =
        // extract(x, pos1 + pos2, min(len1 - pos2, len2))
        //
        if (r1 >= 0 && pos <= r2) {
            r2 = std::min(r2 - pos, len);
            r1 += pos;
            {
                auto _seq960_0 = m_autil.mk_numeral(r1, true);
                auto _seq960_1 = m_autil.mk_numeral(r2, true);
                result = str().mk_substr(a1, _seq960_0, _seq960_1);
            }
            return BR_REWRITE1;
        }
    }

    if (str().is_extract(a, a1, b1, c1) && 
        is_prefix(a1, b1, c1) && is_prefix(a, b, c)) {
        result = str().mk_substr(a1, b, m_autil.mk_sub(c1, m_autil.mk_sub(str().mk_length(a), c)));
        return BR_REWRITE3;
    }

    if (str().is_extract(a, a1, b1, c1) &&
        is_prefix(a, b, c) && is_suffix(a1, b1, c1)) {
        expr_ref q(m_autil.mk_sub(c, str().mk_length(a)), m());
        expr_ref suffix_length(m_autil.mk_sub(str().mk_length(a1), b1), m());
        result = str().mk_substr(a1, b1, m_autil.mk_add(suffix_length, q));
        return BR_REWRITE3;
    }

    // (extract (extract a p l) 0 (len a)) -> (extract a p l)
    if (str().is_extract(a, a1, b1, c1) && constantPos && pos == 0 && str().is_length(c, b1) && a1 == b1) {
        result = a;
        return BR_DONE;
    }

    // (extract (extract a p l) 0 l) -> (extract a p l)
    if (str().is_extract(a, a1, b1, c1) && constantPos && pos == 0 && c == c1) {
        result = a;
        return BR_DONE;
    }

    // extract(extract(a, 3, 6), 1, len(extract(a, 3, 6)) - 1) -> extract(a, 4, 5)
    if (str().is_extract(a, a1, b1, c1) && is_suffix(a, b, c) && 
        m_autil.is_numeral(c1) && m_autil.is_numeral(b1)) {
        {
            auto _seq993_0 = m_autil.mk_add(b, b1);
            auto _seq993_1 = m_autil.mk_sub(c1, b);
            result = str().mk_substr(a1, _seq993_0, _seq993_1);
        }
        return BR_REWRITE2;
    }
 

    if (!constantPos) 
        return BR_FAILED;

    unsigned offset = 0;
    for (; offset < as.size() && str().is_unit(as.get(offset)) && offset < pos; ++offset) {};
    if (offset == 0 && pos > 0) {
        return BR_FAILED;
    }
    std::function<bool(expr*)> is_unit = [&](expr *e) { return str().is_unit(e); };

    if (pos == 0 && as.forall(is_unit)) {
        result = str().mk_empty(a->get_sort());
        for (unsigned i = 1; i <= as.size(); ++i) {
            auto _seq1011_0 = m_autil.mk_ge(c, m_autil.mk_int(i));
            auto _seq1011_1 = str().mk_concat(i, as.data(), a->get_sort());
            result = m().mk_ite(_seq1011_0, _seq1011_1, result);
        }
        return BR_REWRITE_FULL;
    }
    if (pos == 0 && !constantLen) {
        return BR_FAILED;
    }
    // (extract (++ (unit x) (unit y)) 3 c) = empty
    if (offset == as.size()) {
        result = str().mk_empty(a->get_sort());
        return BR_DONE;
    }
    SASSERT(offset != 0 || pos == 0);
    
    if (constantLen && pos == offset) {
        unsigned _len = len.get_unsigned();
        // (extract (++ (unit a) (unit b) (unit c) x) 1 2) = (++ (unit b) (unit c))
        unsigned i = offset;
        for (; i < as.size() && str().is_unit(as.get(i)) && i - offset < _len; ++i);
        if (i - offset == _len) {
            result = str().mk_concat(_len, as.data() + offset, a->get_sort());
            return BR_DONE;
        }
        if (i == as.size()) {
            result = str().mk_concat(as.size() - offset, as.data() + offset, as[0]->get_sort());
            return BR_DONE;
        }
    }
    if (offset == 0) {
        return BR_FAILED;
    }
    expr_ref position(m());
    position = m_autil.mk_sub(b, m_autil.mk_int(offset));
    result = str().mk_concat(as.size() - offset, as.data() + offset, as[0]->get_sort());
    result = str().mk_substr(result, position, c);
    return BR_REWRITE3;
}

bool seq_rewriter::get_lengths(expr* e, expr_ref_vector& lens, rational& pos) {
    expr* arg = nullptr, *e1 = nullptr, *e2 = nullptr;
    rational pos1;
    if (m_autil.is_add(e)) {
        for (expr* arg1 : *to_app(e)) {
            if (!get_lengths(arg1, lens, pos)) 
                return false;
        }
    }
    else if (str().is_length(e, arg)) {
        lens.push_back(arg);
    }
    else if (m_autil.is_mul(e, e1, e2) && m_autil.is_numeral(e1, pos1) && str().is_length(e2, arg) && 0 <= pos1 && pos1 <= 10) {
        while (pos1 > 0) {
            lens.push_back(arg);
            pos1 -= rational(1);
        }
    }
    else if (m_autil.is_numeral(e, pos1)) {
        pos += pos1;
    }
    else {
        return false;
    }
    return true;
}

bool seq_rewriter::cannot_contain_suffix(expr* a, expr* b) {
    
    if (str().is_unit(a) && str().is_unit(b) && m().are_distinct(a, b)) {
        return true;
    }
    zstring A, B;
    if (str().is_string(a, A) && str().is_string(b, B)) {
        // some prefix of a is a suffix of b
        bool found = false;
        for (unsigned i = 1; !found && i <= A.length(); ++i) {
            found = A.extract(0, i).suffixof(B);
        }
        return !found;
    }

    return false;
}


bool seq_rewriter::cannot_contain_prefix(expr* a, expr* b) {
    
    if (str().is_unit(a) && str().is_unit(b) && m().are_distinct(a, b)) {
        return true;
    }
    zstring A, B;
    if (str().is_string(a, A) && str().is_string(b, B)) {
        // some suffix of a is a prefix of b
        bool found = false;
        for (unsigned i = 0; !found && i < A.length(); ++i) {
            found = A.extract(i, A.length()-i).suffixof(B);
        }
        return !found;
    }

    return false;
}



br_status seq_rewriter::mk_seq_contains(expr* a, expr* b, expr_ref& result) {
    zstring c, d;
    if (str().is_string(a, c) && str().is_string(b, d)) {
        result = m().mk_bool_val(c.contains(d));
        return BR_DONE;
    }
    expr* x = nullptr, *y, *z;
    if (str().is_extract(b, x, y, z) && x == a) {
        result = m().mk_true();
        return BR_DONE;
    }

    // check if subsequence of a is in b.
    expr_ref_vector as(m()), bs(m());
    str().get_concat_units(a, as);
    str().get_concat_units(b, bs);
    
    TRACE(seq, tout << mk_pp(a, m()) << " contains " << mk_pp(b, m()) << "\n";);
   
    if (bs.empty()) {
        result = m().mk_true();
        return BR_DONE;
    }

    if (as.empty()) {
        result = str().mk_is_empty(b);
        return BR_REWRITE2;
    }

    auto [bounded_a, max_a] = max_length(a);
    if (bounded_a && max_a <= 1) {
        result = m().mk_or(str().mk_is_empty(b), m().mk_eq(a, b));
        return BR_REWRITE2;
    }

    for (unsigned i = 0; bs.size() + i <= as.size(); ++i) {
        unsigned j = 0;
        for (; j < bs.size() && as.get(j+i) == bs.get(j); ++j) {};
        if (j == bs.size()) {
            result = m().mk_true();
            return BR_DONE;
        }
    }
    std::function<bool(expr*)> is_value = [&](expr* e) { return m().is_value(e); };
    if (bs.forall(is_value) && as.forall(is_value)) {
        result = m().mk_false();
        return BR_DONE;
    }

    auto [lA, lenA] = min_length(as);
    if (lA) {
        auto lenB = min_length(bs).second;
        if (lenB > lenA) {
            result = m().mk_false();
            return BR_DONE;
        }
    }

    unsigned offs = 0;
    unsigned sz = as.size();
    expr* b0 = bs.get(0);
    expr* bL = bs.get(bs.size()-1);
    for (; offs < as.size() && cannot_contain_prefix(as[offs].get(), b0); ++offs) {}
    for (; sz > offs && cannot_contain_suffix(as.get(sz-1), bL); --sz) {}
    if (offs == sz) {
        result = str().mk_is_empty(b);
        return BR_REWRITE2;
    }
    if (offs > 0 || sz < as.size()) {
        SASSERT(sz > offs);
        result = str().mk_contains(str().mk_concat(sz-offs, as.data()+offs, a->get_sort()), b);
        return BR_REWRITE2;
    }    

    std::function<bool(expr*)> is_unit = [&](expr *e) { return str().is_unit(e); };

    if (bs.forall(is_unit) && as.forall(is_unit)) {
        expr_ref_vector ors(m());
        for (unsigned i = 0; i + bs.size() <= as.size(); ++i) {
            expr_ref_vector ands(m());
            for (unsigned j = 0; j < bs.size(); ++j) {
                ands.push_back(m().mk_eq(as.get(i + j), bs.get(j)));
            }
            ors.push_back(::mk_and(ands));
        }
        result = ::mk_or(ors);
        return BR_REWRITE_FULL;
    }

    if (bs.size() == 1 && bs.forall(is_unit) && as.size() > 1) {
        expr_ref_vector ors(m());        
        for (expr* ai : as) {
            ors.push_back(str().mk_contains(ai, bs.get(0)));
        }
        result = ::mk_or(ors);
        return BR_REWRITE_FULL;
    }
    
    expr_ref ra(a, m());
    if (is_unit(b) && m().is_value(b) && 
        reduce_by_char(ra, b, 4)) {
        result = str().mk_contains(ra, b);
        return BR_REWRITE1;
    }
    return BR_FAILED;
}

bool seq_rewriter::reduce_by_char(expr_ref& r, expr* ch, unsigned depth) {
    expr* x = nullptr, *y = nullptr, *z = nullptr;
    if (str().is_replace(r, x, y, z) && 
        str().is_unit(y) && m().is_value(y) &&
        str().is_unit(z) && m().is_value(z) && 
        ch != y && ch != z) {
        r = x;
        if (depth > 0)
            reduce_by_char(r, ch, depth - 1);
        return true;
    }
    if (depth > 0 && str().is_concat(r)) {
        bool reduced = false;
        expr_ref_vector args(m());
        for (expr* e : *to_app(r)) {
            expr_ref tmp(e, m());
            if (reduce_by_char(tmp, ch, depth - 1))
                reduced = true;
            args.push_back(tmp);
        }
        if (reduced) 
            r = str().mk_concat(args, args.get(0)->get_sort());
        return reduced;
    }
    if (depth > 0 && str().is_extract(r, x, y, z)) {
        expr_ref tmp(x, m());
        if (reduce_by_char(tmp, ch, depth - 1)) {
            r = str().mk_substr(tmp, y, z);
            return true;
        }        
    }        
    return false;
}


/*
 * (str.at s i), constants s/i, i < 0 or i >= |s| ==> (str.at s i) = ""
 */
br_status seq_rewriter::mk_seq_at(expr* a, expr* b, expr_ref& result) {
    zstring c;
    rational r, offset_r, len_r;
    expr* offset, *a1, *len;
    expr_ref_vector lens(m());
    sort* sort_a = a->get_sort();
    if (str().is_extract(a, a1, offset, len) && 
        m_autil.is_numeral(offset, offset_r) && offset_r.is_zero() && 
        m_autil.is_numeral(len, len_r) && m_autil.is_numeral(b, r) && 
        r < len_r) {
        result = str().mk_at(a1, b);
        return BR_REWRITE1;
    }
    if (!get_lengths(b, lens, r)) {
        return BR_FAILED;
    }
    if (lens.empty() && r.is_neg()) {
        result = str().mk_empty(sort_a);
        return BR_DONE;
    } 

    expr* a2 = nullptr, *i2 = nullptr;
    if (lens.empty() && str().is_at(a, a2, i2)) {
        if (r.is_pos()) {
            result = str().mk_empty(sort_a);
        }
        else {
            result = a;
        }
        return BR_DONE;            
    }

    m_lhs.reset();
    str().get_concat_units(a, m_lhs);

    if (m_lhs.empty()) {
        result = str().mk_empty(a->get_sort());
        return BR_DONE;        
    }
    
    unsigned i = 0;
    for (; i < m_lhs.size(); ++i) {
        expr* lhs = m_lhs.get(i);
        if (lens.contains(lhs) && !r.is_neg()) {
            lens.erase(lhs);
        }
        else if (str().is_unit(lhs) && r.is_zero() && lens.empty()) {
            result = lhs;
            return BR_REWRITE1;
        }
        else if (str().is_unit(lhs) && r.is_pos()) {
            r -= rational(1);
        }
        else {
            break;
        }
    }
    if (i == 0) {
        return BR_FAILED;
    }
    if (m_lhs.size() == i) {
        result = str().mk_empty(sort_a);
        return BR_DONE;
    }
    expr_ref pos(m_autil.mk_int(r), m());
    for (expr* rhs : lens) {
        pos = m_autil.mk_add(pos, str().mk_length(rhs));
    }
    result = str().mk_concat(m_lhs.size() - i , m_lhs.data() + i, sort_a);
    result = str().mk_at(result, pos);
    return BR_REWRITE2;   
}

br_status seq_rewriter::mk_seq_nth(expr* a, expr* b, expr_ref& result) {

    rational pos1, pos2;
    expr* s = nullptr, *p = nullptr, *len = nullptr;
    if (str().is_unit(a, s) && m_autil.is_numeral(b, pos1) && pos1.is_zero()) {
        result = s;
        return BR_DONE;
    }
    if (str().is_extract(a, s, p, len) && m_autil.is_numeral(p, pos1) && pos1 > 0) {
        expr_ref_vector lens(m());
        rational pos2;
        /*
         * nth(s[k, |s| - k], b) =
         *   b < 0             -> nth_u(a, b)
         *   b + k < |s|       -> nth(s, b + k)
         *   k >= |s|          -> nth_u(empty, b)
         *   k < |s| <= b + k  -> nth_u(a, b)
         */
        if (get_lengths(len, lens, pos2) && (pos1 == -pos2) && (lens.size() == 1) && (lens.get(0) == s)) {
            expr_ref k(m_autil.mk_int(pos1), m());
            expr_ref case1(str().mk_nth_i(s, m_autil.mk_add(b, k)), m());
            expr_ref case2(str().mk_nth_u(str().mk_empty(s->get_sort()), b), m());
            expr_ref case3(str().mk_nth_u(a, b), m());
            result = case3;
            auto _seq0 = m_autil.mk_add(k, b);
            auto _seq1 = str().mk_length(s);
            result = m().mk_ite(m_autil.mk_lt(_seq0, _seq1), case1, result);
            result = m().mk_ite(m_autil.mk_ge(k, str().mk_length(s)), case2, result);
            result = m().mk_ite(m_autil.mk_lt(b, zero()), case3, result);   
            return BR_REWRITE_FULL;
        }
    }

    auto [bounded_a, len_a] = min_length(a);

    if (bounded_a && m_autil.is_numeral(b, pos1)) {
        if (0 <= pos1 && pos1 < len_a)
            result = str().mk_nth_i(a, b);
        else
            result = str().mk_nth_u(a, b);
        return BR_REWRITE_FULL;
    }    

    expr* la = str().mk_length(a);
    {
        // deterministic evaluation order for guard components
        auto ge0 = m_autil.mk_ge(b, zero());
        auto le_la = m_autil.mk_le(la, b);
        auto not_le = m().mk_not(le_la);
        auto guard = m().mk_and(ge0, not_le);
        auto t1 = str().mk_nth_i(a, b);
        auto e1 = str().mk_nth_u(a, b);
        result = m().mk_ite(guard, t1, e1);
    }
    return BR_REWRITE_FULL;
}


br_status seq_rewriter::mk_seq_nth_i(expr* a, expr* b, expr_ref& result) {
    zstring c;
    rational r;
    if (!m_autil.is_numeral(b, r) || !r.is_unsigned()) {
        return BR_FAILED;
    }
    unsigned offset = r.get_unsigned();

    expr* a2, *i2;
    if (offset == 0 && str().is_at(a, a2, i2) && m_autil.is_numeral(i2, r) && r.is_zero()) {
        result = str().mk_nth_i(a2, i2);
        return BR_REWRITE1;
    }

    expr* f, *s;
    if (str().is_map(a, f, s)) {
        expr* args[2] = { f, str().mk_nth_i(s, b) };
        result = array_util(m()).mk_select(2, args);
        return BR_REWRITE1;
    }

    expr_ref_vector as(m());
    str().get_concat_units(a, as);

    expr* cond = nullptr, *el = nullptr, *th = nullptr;
    unsigned pos = 0;
    for (unsigned i = 0; i < as.size(); ++i) {
        expr* a = as.get(i), *u = nullptr;
        if (str().is_unit(a, u)) {
            if (offset == pos) {
                result = u;
                return BR_DONE;
            }
            ++pos;
            continue;
        }
        else if (m().is_ite(a, cond, th, el)) {
            auto [bounded, len1] = min_length(a);
            if (!bounded)
                break;
            if (pos + len1 <= offset) {
                pos += len1;
                continue;
            }
            expr_ref idx(m());
            idx = m_autil.mk_int(offset - pos);
            th = str().mk_nth_i(th, idx);
            el = str().mk_nth_i(el, idx);
            result = m().mk_ite(cond, th, el);
            return BR_REWRITE2;
        }
        else
            break;
    }
    
    return BR_FAILED;
}

br_status seq_rewriter::mk_seq_last_index(expr* a, expr* b, expr_ref& result) {
    zstring s1, s2;
    bool isc1 = str().is_string(a, s1);
    bool isc2 = str().is_string(b, s2);
    if (isc1 && isc2) {
        int idx = s1.last_indexof(s2);
        result = m_autil.mk_numeral(rational(idx), true);
        return BR_DONE;
    }
    if (a == b) {
        result = m_autil.mk_int(0);
        return BR_DONE;
    }

    if (str().is_empty(b)) {
        result = str().mk_length(a);
        return BR_DONE;
    }

    expr_ref_vector as(m()), bs(m());
    str().get_concat_units(a, as);
    str().get_concat_units(b, bs);

    auto is_suffix = [&](expr_ref_vector const& as, expr_ref_vector const& bs) {
        if (as.size() < bs.size())
            return l_undef;
        for (unsigned j = 0; j < bs.size(); ++j) {
            auto a = as.get(as.size() - j - 1);
            auto b = bs.get(bs.size() - j - 1);
            if (m().are_equal(a, b))
                continue;
            if (m().are_distinct(a, b))
                return l_false;
            return l_undef;
        }
        return l_true;
    };

    switch (compare_lengths(as, bs)) {
    case shorter_c:
        result = minus_one();
        return BR_DONE;
    case same_length_c: {
        expr_ref eq(m().mk_eq(a, b), m());
        auto found = zero();
        auto not_found = minus_one();
        result = m().mk_ite(eq, found, not_found);
        return BR_REWRITE_FULL;
    }
    case longer_c: {
        unsigned i = as.size();
        while (i >= bs.size()) {
            switch (is_suffix(as, bs)) {
            case l_undef:
                return BR_FAILED;
            case l_true: {
                auto _seq0 = str().mk_length(a);
                auto _seq1 = m_autil.mk_int(bs.size() - i);
                result = m_autil.mk_sub(_seq0, _seq1);
                return BR_REWRITE3;
            }
            case l_false:
                as.pop_back();
                --i;
                break;
            }
        }
        break;
    }
    default:
        break;
    }

    return BR_FAILED;
}

/**

  Index of first occurrence of second string in first one starting at 
  the position specified by the third argument.
  (str.indexof s t i), with 0 <= i <= |s| is the position of the first
  occurrence of t in s at or after position i, if any. 
  Otherwise, it is -1. Note that the result is i whenever i is within
  the range [0, |s|] and t is empty.
  (str.indexof String String Int Int)

   indexof(s, b, c) -> -1 if c < 0
   indexof(a, "", 0) -> if a = "" then 0 else -1
   indexof("", b, r) -> if b = "" and r = 0 then 0 else -1
   indexof(a, "", x) -> if 0 <= x <= len(a) then x else - 1
   indexof(unit(x)+a, b, r+1) -> indexof(a, b, r) 
   indexof(unit(x)+a, unit(y)+b, 0) -> indexof(a,unit(y)+b, 0) if x != y
   indexof(substr(x,y,len1), z, len2) -> -1 if len2 > len1
*/
br_status seq_rewriter::mk_seq_index(expr* a, expr* b, expr* c, expr_ref& result) {
    zstring s1, s2;
    rational r;
    bool isc1 = str().is_string(a, s1);
    bool isc2 = str().is_string(b, s2);
    sort* sort_a = a->get_sort();

    if (isc1 && isc2 && m_autil.is_numeral(c, r) && r.is_unsigned()) {
        int idx = s1.indexofu(s2, r.get_unsigned());
        result = m_autil.mk_int(idx);
        return BR_DONE;
    }
    if (m_autil.is_numeral(c, r) && r.is_neg()) {
        result = minus_one();
        return BR_DONE;
    }
    
    if (str().is_empty(b) && m_autil.is_numeral(c, r) && r.is_zero()) {
        result = c;
        return BR_DONE;
    }

    if (str().is_empty(b)) {
        // enforce deterministic evaluation order for bounds checks
        auto a1 = m_autil.mk_le(zero(), c);
        auto b1 = m_autil.mk_le(c, str().mk_length(a));
        auto cond = m().mk_and(a1, b1);
        result = m().mk_ite(cond, c, minus_one());
        return BR_REWRITE2;
    }

    
    if (str().is_empty(a)) {
        expr* emp = str().mk_is_empty(b);
        auto a1 = m().mk_eq(c, zero());
        auto cond = m().mk_and(a1, emp);
        {
            auto then_value = zero();
            auto else_value = minus_one();
            result = m().mk_ite(cond, then_value, else_value);
        }
        return BR_REWRITE2;
    }

    if (a == b) {
        if (m_autil.is_numeral(c, r)) {
            result = r.is_zero() ? zero() : minus_one();            
            return BR_DONE;
        }
        else {
            auto found = zero();
            expr_ref at_zero(m().mk_eq(found, c), m());
            auto not_found = minus_one();
            result = m().mk_ite(at_zero, found, not_found);
            return BR_REWRITE2;
        }
    }
    expr* x = nullptr, *y = nullptr, *len1 = nullptr;
    rational r1, r2;
    if (str().is_extract(a, x, y, len1) && m_autil.is_numeral(len1, r1) && 
        m_autil.is_numeral(c, r2) && r2 > r1) {
        result = minus_one();
        return BR_DONE;
    }

    expr_ref_vector as(m()), bs(m());
    str().get_concat_units(a, as);
    unsigned i = 0;
    if (m_autil.is_numeral(c, r)) {
        i = 0;
        while (r.is_pos() && i < as.size() && str().is_unit(as.get(i))) {
            r -= rational(1);
            ++i;
        }
        if (i > 0) {
            expr_ref a1(m());
            a1 = str().mk_concat(as.size() - i, as.data() + i, sort_a);
            result = str().mk_index(a1, b, m_autil.mk_int(r));
            {
                auto _seq1599_0 = m_autil.mk_ge(result, zero());
                auto _seq1599_1 = m_autil.mk_add(m_autil.mk_int(i), result);
                result = m().mk_ite(_seq1599_0, _seq1599_1, minus_one());
            }
            return BR_REWRITE_FULL;
        }
    }
    bool is_zero = m_autil.is_numeral(c, r) && r.is_zero();
    str().get_concat_units(b, bs);
    i = 0;
    while (is_zero && i < as.size() && 
           0 < bs.size() && 
           str().is_unit(as.get(i)) && 
           str().is_unit(bs.get(0)) &&
           m().are_distinct(as.get(i), bs.get(0))) {
        ++i;
    }
    if (i > 0) {
        result = str().mk_index(
            str().mk_concat(as.size() - i, as.data() + i, sort_a), b, c);
        {
            auto _seq1616_0 = m_autil.mk_ge(result, zero());
            auto _seq1616_1 = m_autil.mk_add(m_autil.mk_int(i), result);
            result = m().mk_ite(_seq1616_0, _seq1616_1, minus_one());
        }
        return BR_REWRITE_FULL;
    }

    switch (compare_lengths(as, bs)) {
    case shorter_c:
        if (is_zero) {
            result = minus_one();
            return BR_DONE;
        }
        break;
    case same_length_c: {
        auto _seqa = m_autil.mk_le(c, minus_one());
        auto _seqb = m().mk_eq(c, zero());
        expr_ref condition(m().mk_eq(a, b), m());
        auto then_value = zero();
        auto else_value = minus_one();
        auto _seqc = m().mk_ite(condition, then_value, else_value);
        {
            auto then_value = minus_one();
            expr_ref else_value(m().mk_ite(_seqb, _seqc, minus_one()), m());
            result = m().mk_ite(_seqa, then_value, else_value);
        }
    }
        return BR_REWRITE_FULL;
    default:
        break;
    }
    if (is_zero && !as.empty() && str().is_unit(as.get(0))) {
        expr_ref a1(str().mk_concat(as.size() - 1, as.data() + 1, as[0]->get_sort()), m());
        expr_ref b1(str().mk_index(a1, b, c), m());
        {
            auto _seq1639_0 = str().mk_prefix(b, a);
            auto _seq1663_0 = m_autil.mk_ge(b1, zero());
            auto _seq1663_1 = m_autil.mk_add(one(), b1);
            auto _seq1639_1 = m().mk_ite(_seq1663_0, _seq1663_1, minus_one());
            result = m().mk_ite(_seq1639_0, zero(), _seq1639_1);
        }
        return BR_REWRITE3;
    }
    expr_ref ra(a, m());
    if (str().is_unit(b) && m().is_value(b) && 
        reduce_by_char(ra, b, 4)) {
        result = str().mk_index(ra, b, c);
        return BR_REWRITE1;
    }

    // Enhancement: walk segments of a, determine which segments must overlap, must not overlap, may overlap.
    return BR_FAILED;
}

seq_rewriter::length_comparison seq_rewriter::compare_lengths(unsigned sza, expr* const* as, unsigned szb, expr* const* bs) {
    unsigned units_a = 0, units_b = 0, k = 0;
    obj_map<expr, unsigned> mults;
    bool b_has_foreign = false;
    for (unsigned i = 0; i < sza; ++i) {
        if (str().is_unit(as[i]))
            units_a++;
        else 
            mults.insert_if_not_there(as[i], 0)++;
    }
    for (unsigned i = 0; i < szb; ++i) {
        if (str().is_unit(bs[i]))
            units_b++;
        else if (mults.find(bs[i], k)) {
            --k;
            if (k == 0) {
                mults.erase(bs[i]);
            }
            else {
                mults.insert(bs[i], k);
            }
        }
        else {
            b_has_foreign = true;
        }
    }
    if (units_a > units_b && !b_has_foreign) {
        return longer_c;
    }
    if (units_a == units_b && !b_has_foreign && mults.empty()) {
        return same_length_c;
    }
    if (units_b > units_a && mults.empty()) {
        return shorter_c;
    }
    return unknown_c;
}


//  (str.replace s t t') is the string obtained by replacing the first occurrence 
//  of t in s, if any, by t'. Note that if t is empty, the result is to prepend
//  t' to s; also, if t does not occur in s then the result is s.

br_status seq_rewriter::mk_seq_replace(expr* a, expr* b, expr* c, expr_ref& result) {
    zstring s1, s2, s3;
    sort* sort_a = a->get_sort();
    if (str().is_string(a, s1) && str().is_string(b, s2) && 
        str().is_string(c, s3)) {
        result = str().mk_string(s1.replace(s2, s3));
        return BR_DONE;
    }
    if (b == c) {
        result = a;
        return BR_DONE;
    }
    if (a == b) {
        result = c;
        return BR_DONE;
    }
    if (str().is_empty(b)) {
        result = str().mk_concat(c, a);
        return BR_REWRITE1;
    }
    if (str().is_empty(a) && str().is_empty(c)) {
        result = a;
        return BR_DONE;
    }

    m_lhs.reset();
    str().get_concat(a, m_lhs);

    // a = "", |b| > 0 -> replace("",b,c) = ""
    if (m_lhs.empty()) {
        str().get_concat(b, m_lhs);
        unsigned len = min_length(m_lhs).second;
        if (len > 0) {
            result = a;
            return BR_DONE;
        }
        return BR_FAILED;
    }

    // a := b + rest
    if (m_lhs.get(0) == b) {
        m_lhs[0] = c;
        result = str().mk_concat(m_lhs.size(), m_lhs.data(), sort_a);
        return BR_REWRITE1;
    }

    // a : a' + rest string, b is string, c is string, a' contains b
    if (str().is_string(b, s2) && str().is_string(c, s3) && 
        str().is_string(m_lhs.get(0), s1) && s1.contains(s2) ) {
        m_lhs[0] = str().mk_string(s1.replace(s2, s3));
        result = str().mk_concat(m_lhs.size(), m_lhs.data(), sort_a);
        return BR_REWRITE1;
    }
    m_lhs.reset();
    m_rhs.reset();
    str().get_concat_units(a, m_lhs);
    str().get_concat_units(b, m_rhs);
    if (m_rhs.empty()) {
        result = str().mk_concat(c, a);
        return BR_REWRITE1;
    }

    // is b a prefix of m_lhs at position i?
    auto compare_at_i = [&](unsigned i) {
        for (unsigned j = 0; j < m_rhs.size() && i + j < m_lhs.size(); ++j) {
            expr* b0 = m_rhs.get(j);
            expr* a0 = m_lhs.get(i + j);
            if (m().are_equal(a0, b0))
                continue;
            if (!str().is_unit(b0) || !str().is_unit(a0)) 
                return l_undef;
            if (m().are_distinct(a0, b0)) 
                return l_false;
            return l_undef;
        }
        return l_true;
    };

    unsigned i = 0;
    for (; i < m_lhs.size(); ++i) {
        lbool cmp = compare_at_i(i);
        if (cmp == l_false && str().is_unit(m_lhs.get(i)))
            continue;
        if (cmp == l_true && m_lhs.size() < i + m_rhs.size()) {
            expr_ref a1(str().mk_concat(i, m_lhs.data(), sort_a), m());
            expr_ref a2(str().mk_concat(m_lhs.size()-i, m_lhs.data()+i, sort_a), m());
            {
                auto _seq1783_0 = m().mk_eq(a2, b);
                auto _seq1783_1 = str().mk_concat(a1, c);
                result = m().mk_ite(_seq1783_0, _seq1783_1, a);
            }
            return BR_REWRITE_FULL;            
        }
        if (cmp == l_true) {
            expr_ref_vector es(m());
            es.append(i, m_lhs.data());
            es.push_back(c);
            es.append(m_lhs.size()-i-m_rhs.size(), m_lhs.data()+i+m_rhs.size());
            result = str().mk_concat(es, sort_a);
            return BR_REWRITE_FULL;        
        }
        break;
    }

    if (i > 0) {
        expr_ref a1(str().mk_concat(i, m_lhs.data(), sort_a), m());
        expr_ref a2(str().mk_concat(m_lhs.size()-i, m_lhs.data()+i, sort_a), m());
        result = str().mk_concat(a1, str().mk_replace(a2, b, c));
        return BR_REWRITE_FULL;        
    }


    return BR_FAILED;
}

br_status seq_rewriter::mk_seq_replace_all(expr* a, expr* b, expr* c, expr_ref& result) {
    if (str().is_empty(b) || b == c) {
        result = a;
        return BR_DONE;
    } 
    if (a == b) {        
        auto _seq1814_0 = str().mk_is_empty(b);
        auto _seq1814_1 = str().mk_empty(a->get_sort());
        result = m().mk_ite(_seq1814_0, _seq1814_1, c);        
        return BR_REWRITE2;
    }
    if (str().is_empty(a) && str().is_empty(c)) {
        result = a;
        return BR_DONE;
    }
    zstring s1, s2;
    expr_ref_vector strs(m());
    if (str().is_string(a, s1) && str().is_string(b, s2)) {
        SASSERT(s2.length() > 0);
        if (s1.length() < s2.length()) {
            result = a;
            return BR_DONE;
        }
        for (unsigned i = 0; i < s1.length(); ++i) {
            if (s1.length() >= s2.length() + i && 
                s2 == s1.extract(i, s2.length())) {
                strs.push_back(c);
                i += s2.length() - 1;
            }
            else 
                strs.push_back(str().mk_unit(str().mk_char(s1, i)));
        }
        result = str().mk_concat(strs, a->get_sort());
        return BR_REWRITE_FULL;
    }
    expr_ref_vector a_vals(m());
    expr_ref_vector b_vals(m());
    if (try_get_unit_values(a, a_vals) && try_get_unit_values(b, b_vals)) {
        replace_all_subvectors(a_vals, b_vals, c, strs);
        result = str().mk_concat(strs, a->get_sort());
        return BR_REWRITE_FULL;
    }

    //TODO: the case when a is a unit or concatenation of units while b is a string 
    //or the other way around -- if that situation is possible at all -- is similar to the above
    return BR_FAILED;
}


/**
 * replace_char("ab", "a", b") = empty
 * replace_char("bc", "a", b") = {"a", "b"}"c"
 * replace_char(R u R', "a", "b") = replace_char(R, "a", "b") u replace_char(R', "a", "b")
 * replace_char(R n R', "a", "b") = replace_char(R, "a", "b") n replace_char(R', "a", "b")
 * replace_char(R*, "a", "b") = replace_char(R, "a", "b")*
 * replace_char(R R', "a", "b") = replace_char(R, "a", "b") replace_char(R', "a", "b")
 */
expr_ref seq_rewriter::re_replace_char(expr *r, unsigned a_ch, unsigned b_ch, expr *a_str, expr *b_str) {
    expr *r1 = nullptr, *r2 = nullptr, *s = nullptr;
    zstring str_val;
    sort *seq_sort = nullptr;

    if (re().is_to_re(r, s) && str().is_string(s, str_val)) {
        seq_sort = s->get_sort();
        expr_ref_vector parts(m());
        for (unsigned i = 0; i < str_val.length(); ++i) {
            if (str_val[i] == a_ch) {
                // replace_all never outputs a_ch, so this position is impossible
                return expr_ref(re().mk_empty(re().mk_re(seq_sort)), m());
            }
            else if (str_val[i] == b_ch) {
                // b in output came from either a or b in x
                auto a_re = re().mk_to_re(a_str);
                auto b_re = re().mk_to_re(b_str);
                parts.push_back(re().mk_union(a_re, b_re));
            }
            else {
                zstring ch(str_val[i]);
                parts.push_back(re().mk_to_re(str().mk_string(ch)));
            }
        }
        if (parts.empty())
            return expr_ref(re().mk_epsilon(seq_sort), m());
        expr_ref result(parts.back(), m());
        for (int i = parts.size() - 1; i-- > 0;)
            result = re().mk_concat(parts.get(i), result);
        return result;
    }

    if (re().is_range(r, r1, r2)) {
        zstring lo_s, hi_s;
        if (str().is_string(r1, lo_s) && str().is_string(r2, hi_s) && lo_s.length() == 1 && hi_s.length() == 1) {
            unsigned lo = lo_s[0], hi = hi_s[0];
            // Build the transformed range:
            // - Remove a_ch from the range (impossible in output)
            // - Replace b_ch with union(a_str, b_str)
            expr_ref_vector parts(m());
            // Characters in [lo, hi] excluding a_ch and b_ch
            if (lo <= hi) {
                // Sub-ranges excluding a_ch and b_ch
                unsigned prev = lo;
                for (unsigned ch = lo; ch <= hi; ++ch) {
                    if (ch == a_ch || ch == b_ch) {
                        if (prev < ch) {
                            zstring prev_z(prev), pred_z(ch - 1);
                            {
                                auto _seq1911_0 = str().mk_string(prev_z);
                                auto _seq1911_1 = str().mk_string(pred_z);
                                parts.push_back(re().mk_range(_seq1911_0, _seq1911_1));
                            }
                        }
                        if (ch == b_ch) {
                            auto _seq1914_0 = re().mk_to_re(a_str);
                            auto _seq1914_1 = re().mk_to_re(b_str);
                            parts.push_back(re().mk_union(_seq1914_0, _seq1914_1));
                        }
                        // a_ch is simply excluded (not added)
                        prev = ch + 1;
                    }
                }
                if (prev <= hi) {
                    zstring prev_z(prev), hi_z(hi);
                    {
                        auto _seq1922_0 = str().mk_string(prev_z);
                        auto _seq1922_1 = str().mk_string(hi_z);
                        parts.push_back(re().mk_range(_seq1922_0, _seq1922_1));
                    }
                }
            }
            if (parts.empty()) {
                sort *re_sort = r->get_sort();
                return expr_ref(re().mk_empty(re_sort), m());
            }
            expr_ref result(parts[0].get(), m());
            for (unsigned i = 1; i < parts.size(); ++i)
                result = re().mk_union(result, parts[i].get());
            return result;
        }
        return expr_ref(r, m());
    }

    if (re().is_union(r, r1, r2)) {
        auto lhs = re_replace_char(r1, a_ch, b_ch, a_str, b_str);
        auto rhs = re_replace_char(r2, a_ch, b_ch, a_str, b_str);
        return expr_ref(re().mk_union(lhs, rhs), m());
    }
    if (re().is_intersection(r, r1, r2)) {
        auto lhs1 = re_replace_char(r1, a_ch, b_ch, a_str, b_str);
        auto rhs1 = re_replace_char(r2, a_ch, b_ch, a_str, b_str);
        return expr_ref(re().mk_inter(lhs1, rhs1), m());
    }
    if (re().is_concat(r, r1, r2)) {
        auto lhs = re_replace_char(r1, a_ch, b_ch, a_str, b_str);
        auto rhs = re_replace_char(r2, a_ch, b_ch, a_str, b_str);
        return expr_ref(re().mk_concat(lhs, rhs), m());
    }
    if (re().is_star(r, r1)) {
        return expr_ref(re().mk_star(re_replace_char(r1, a_ch, b_ch, a_str, b_str)), m());
    }
    if (re().is_plus(r, r1)) {
        return expr_ref(re().mk_plus(re_replace_char(r1, a_ch, b_ch, a_str, b_str)), m());
    }
    if (re().is_opt(r, r1)) {
        return expr_ref(re().mk_opt(re_replace_char(r1, a_ch, b_ch, a_str, b_str)), m());
    }
    unsigned lo, hi;
    if (re().is_loop(r, r1, lo, hi)) {
        return expr_ref(re().mk_loop(re_replace_char(r1, a_ch, b_ch, a_str, b_str), lo, hi), m());
    }
    if (re().is_loop(r, r1, lo)) {
        return expr_ref(re().mk_loop(re_replace_char(r1, a_ch, b_ch, a_str, b_str), lo), m());
    }
    if (re().is_complement(r)) {
        UNREACHABLE();
    }
    // For anything else (full_seq, empty, epsilon, of_pred, etc.), return unchanged
    return expr_ref(r, m());
}

/**
   rewrites for map(f, s):

   map(f, []) = []
   map(f, [x]) = [f(x)]
   map(f, s + t) = map(f, s) + map(f, t)
   len(map(f, s)) = len(s)
   nth_i(map(f,s), i) = f(nth_i(s, i))

 */
br_status seq_rewriter::mk_seq_map(expr* f, expr* seqA, expr_ref& result) {
    if (str().is_empty(seqA)) {
        result = str().mk_empty(str().mk_seq(get_array_range(f->get_sort())));
        return BR_DONE;
    }
    expr* a, *s1, *s2;
    if (str().is_unit(seqA, a)) {
        array_util array(m());
        expr* args[2] = { f, a };
        result = str().mk_unit(array.mk_select(2, args));
        return BR_REWRITE2;
    }
    if (str().is_concat(seqA, s1, s2)) {
        // introduce temporaries to ensure deterministic evaluation order of recursive map calls
        auto m1 = str().mk_map(f, s1);
        auto m2 = str().mk_map(f, s2);
        result = str().mk_concat(m1, m2);
        return BR_REWRITE2;
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_seq_mapi(expr* f, expr* i, expr* seqA, expr_ref& result) {
    if (str().is_empty(seqA)) {
        result = str().mk_empty(str().mk_seq(get_array_range(f->get_sort())));
        return BR_DONE;
    }
    expr* a, *s1, *s2;
    if (str().is_unit(seqA, a)) {
        array_util array(m());
        expr* args[3] = { f, i, a };
        result = str().mk_unit(array.mk_select(3, args));
        return BR_REWRITE2;
    }
    if (str().is_concat(seqA, s1, s2)) {
        expr_ref j(m_autil.mk_add(i, str().mk_length(s1)), m());
        auto left  = str().mk_mapi(f, i, s1);
        auto right = str().mk_mapi(f, j, s2);
        result = str().mk_concat(left, right);
        return BR_REWRITE2;
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_seq_foldl(expr* f, expr* b, expr* seqA, expr_ref& result) {
    if (str().is_empty(seqA)) {
        result = b;
        return BR_DONE;
    }
    expr* a, *s1, *s2;
    if (str().is_unit(seqA, a)) {
        array_util array(m());
        expr* args[3] = { f, b, a };
        result = array.mk_select(3, args);
        return BR_REWRITE1;
    }
    if (str().is_concat(seqA, s1, s2)) {
        result = str().mk_foldl(f, b, s1);
        result = str().mk_foldl(f, result, s2);
        return BR_REWRITE3;
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_seq_foldli(expr* f, expr* i, expr* b, expr* seqA, expr_ref& result) {
    if (str().is_empty(seqA)) {
        result = b;
        return BR_DONE;
    }
    expr* a, *s1, *s2;
    if (str().is_unit(seqA, a)) {
        array_util array(m());
        expr* args[4] = { f, i, b, a };
        result = array.mk_select(4, args);
        return BR_REWRITE1;
    }
    if (str().is_concat(seqA, s1, s2)) {
        expr_ref j(m_autil.mk_add(i, str().mk_length(s1)), m());
        result = str().mk_foldli(f, i, b, s1);
        result = str().mk_foldli(f, j, result, s2);
        return BR_REWRITE3;
    }
    return BR_FAILED;
}

/*
* Returns false if s is not a single unit value or concatenation of unit values.
* Else extracts the units from s into vals and returns true.
*/
bool seq_rewriter::try_get_unit_values(expr* s, expr_ref_vector& vals) {
    expr* h, * t, * v;
    t = s;
    //collect all unit values from s, if not all elements are unit-values then fail
    while (str().is_concat(t, h, t))
        if (str().is_unit(h, v) && m().is_value(v))
            vals.push_back(h);
        else
            return false;
    //add the last element
    if (str().is_unit(t, v) && m().is_value(v))
        vals.push_back(t);
    else
        return false;
    return true;
}

/*
* Replace all subvectors of b in a by c
*/
void seq_rewriter::replace_all_subvectors(expr_ref_vector const& a, expr_ref_vector const& b, expr* c, expr_ref_vector& result) {
    unsigned int i = 0;
    unsigned int k = b.size();
    while (i + k <= a.size()) {
        //if a[i..i+k-1] equals b then replace it by c and inceremnt i by k
        unsigned j = 0;
        while (j < k && b[j] == a[i + j]) 
            ++j;
        if (j < k) //the equality failed
            result.push_back(a[i++]);
        else { //the equality succeeded
            result.push_back(c);
            i += k;
        }
    }
    //add the trailing elements from a 
    while (i < a.size())
        result.push_back(a[i++]);
}

br_status seq_rewriter::mk_seq_replace_re_all(expr* a, expr* b, expr* c, expr_ref& result) {
    return BR_FAILED;
}

br_status seq_rewriter::mk_seq_replace_re(expr* a, expr* b, expr* c, expr_ref& result) {
    return BR_FAILED;
}

br_status seq_rewriter::mk_seq_prefix(expr* a, expr* b, expr_ref& result) {
    TRACE(seq, tout << mk_pp(a, m()) << " " << mk_pp(b, m()) << "\n";);
    zstring s1, s2;
    bool isc1 = str().is_string(a, s1);
    bool isc2 = str().is_string(b, s2);
    sort* sort_a = a->get_sort();
    if (isc1 && isc2) {
        result = m().mk_bool_val(s1.prefixof(s2));
        TRACE(seq, tout << result << "\n";);
        return BR_DONE;
    }
    if (str().is_empty(a)) {
        result = m().mk_true();
        return BR_DONE;
    }
    expr* a1 = str().get_leftmost_concat(a);
    expr* b1 = str().get_leftmost_concat(b);
    isc1 = str().is_string(a1, s1);
    isc2 = str().is_string(b1, s2);
    expr_ref_vector as(m()), bs(m());

    if (a1 != b1 && isc1 && isc2) {
        if (s1.length() <= s2.length()) {
            if (s1.prefixof(s2)) {
                if (a == a1) {
                    result = m().mk_true();
                    TRACE(seq, tout << s1 << " " << s2 << " " << result << "\n";);
                    return BR_DONE;
                }               
                str().get_concat(a, as);
                str().get_concat(b, bs);
                SASSERT(as.size() > 1);
                s2 = s2.extract(s1.length(), s2.length()-s1.length());
                bs[0] = str().mk_string(s2);
                {
                    auto _seq2156_0 = str().mk_concat(as.size() - 1, as.data() + 1, sort_a);
                    auto _seq2156_1 = str().mk_concat(bs.size(), bs.data(), sort_a);
                    result = str().mk_prefix(_seq2156_0, _seq2156_1);
                }
                TRACE(seq, tout << s1 << " " << s2 << " " << result << "\n";);
                return BR_REWRITE_FULL;
            }
            else {
                result = m().mk_false();
                TRACE(seq, tout << s1 << " " << s2 << " " << result << "\n";);
                return BR_DONE;
            }
        }
        else {
            if (s2.prefixof(s1)) {
                if (b == b1) {
                    result = m().mk_false();
                    TRACE(seq, tout << s1 << " " << s2 << " " << result << "\n";);
                    return BR_DONE;
                }
                str().get_concat(a, as);
                str().get_concat(b, bs);
                SASSERT(bs.size() > 1);
                s1 = s1.extract(s2.length(), s1.length() - s2.length());
                as[0] = str().mk_string(s1);
                auto a = str().mk_concat(as.size(), as.data(), sort_a);
                result = str().mk_prefix(a, str().mk_concat(bs.size()-1, bs.data()+1, sort_a));
                TRACE(seq, tout << s1 << " " << s2 << " " << result << "\n";);
                return BR_REWRITE_FULL;                
            }
            else {
                result = m().mk_false();
                TRACE(seq, tout << s1 << " " << s2 << " " << result << "\n";);
                return BR_DONE;
            }
        }        
    }
    str().get_concat_units(a, as);
    str().get_concat_units(b, bs);
    unsigned i = 0;
    expr_ref_vector eqs(m());
    for (; i < as.size() && i < bs.size(); ++i) {
        expr* ai = as.get(i), *bi = bs.get(i);
        if (m().are_equal(ai, bi)) {
            continue;
        }
        if (m().are_distinct(ai, bi)) {
            result = m().mk_false();
            return BR_DONE;
        }
        if (str().is_unit(ai) && str().is_unit(bi)) {
            eqs.push_back(m().mk_eq(ai, bi));
            continue;
        }
        break;
    }
    if (i == as.size()) {
        result = mk_and(eqs);
        TRACE(seq, tout << result << "\n";);
        return BR_REWRITE3;
    }
    SASSERT(i < as.size());
    if (i == bs.size()) {
        for (unsigned j = i; j < as.size(); ++j) {
            eqs.push_back(str().mk_is_empty(as.get(j)));
        }
        result = mk_and(eqs);
        TRACE(seq, tout << result << "\n";);
        return BR_REWRITE3;
    }
    if (i > 0) {
        SASSERT(i < as.size() && i < bs.size());
        a = str().mk_concat(as.size() - i, as.data() + i, sort_a);
        b = str().mk_concat(bs.size() - i, bs.data() + i, sort_a); 
        eqs.push_back(str().mk_prefix(a, b));
        result = mk_and(eqs);
        TRACE(seq, tout << result << "\n";);
        return BR_REWRITE3;
    }

    expr* a2 = nullptr, *a3 = nullptr;    
    if (str().is_replace(a, a1, a2, a3) && a1 == a3 && a2 == b) {
        // TBD: generalize to when a1 is a prefix of a3?
        result = str().mk_prefix(a1, b);
        return BR_DONE;
    }


    auto [bounded_b, len_b] = max_length(b);
    if (bounded_b) {
        auto [bounded_a, len_a] = min_length(a);
        if (len_b <= len_a) {
            result = m().mk_eq(a, b);
            return BR_REWRITE1;
        }
    }

    return BR_FAILED;    
}

br_status seq_rewriter::mk_seq_suffix(expr* a, expr* b, expr_ref& result) {
    if (a == b) {
        result = m().mk_true();
        return BR_DONE;
    }
    sort* sort_a = a->get_sort();
    if (str().is_empty(a)) {
        result = m().mk_true();
        return BR_DONE;
    }
    if (str().is_empty(b)) {
        result = str().mk_is_empty(a);
        return BR_REWRITE3;
    }
    
    expr_ref_vector as(m()), bs(m()), eqs(m());
    str().get_concat_units(a, as);
    str().get_concat_units(b, bs);
    unsigned i = 1, sza = as.size(), szb = bs.size();
    for (; i <= sza && i <= szb; ++i) {
        expr* ai = as.get(sza-i), *bi = bs.get(szb-i);
        if (m().are_equal(ai, bi)) {
            continue;
        }
        if (m().are_distinct(ai, bi)) {
            result = m().mk_false();
            return BR_DONE;
        }
        if (str().is_unit(ai) && str().is_unit(bi)) {
            eqs.push_back(m().mk_eq(ai, bi));
            continue;
        }
        break;
    }
    if (i > sza) {
        result = mk_and(eqs);
        TRACE(seq, tout << result << "\n";);
        return BR_REWRITE3;
    }
    if (i > szb) {
        for (unsigned j = i; j <= sza; ++j) {
            expr* aj = as.get(sza-j);
            eqs.push_back(str().mk_is_empty(aj));
        }
        result = mk_and(eqs);
        TRACE(seq, tout << result << "\n";);
        return BR_REWRITE3;
    }

    if (i > 1) {
        SASSERT(i <= sza && i <= szb);
        a = str().mk_concat(sza - i + 1, as.data(), sort_a);
        b = str().mk_concat(szb - i + 1, bs.data(), sort_a);
        eqs.push_back(str().mk_suffix(a, b));
        result = mk_and(eqs);
        TRACE(seq, tout << result << "\n";);
        return BR_REWRITE3;
    }

    expr* a1 = nullptr, *a2 = nullptr, *a3 = nullptr;    
    if (str().is_replace(a, a1, a2, a3) && a1 == a3 && a2 == b) {
        // TBD: generalize to when a1 is a prefix of a3?
        result = str().mk_suffix(a1, b);
        return BR_DONE;
    }
    auto [bounded_b, len_b] = max_length(b);
    if (bounded_b) {
        auto [bounded_a, len_a] = min_length(a);
        if (len_b <= len_a) {
            result = m().mk_eq(a, b);
            return BR_REWRITE1;
        }
    }

    return BR_FAILED;
}

br_status seq_rewriter::mk_str_units(func_decl* f, expr_ref& result) {
    zstring s;
    VERIFY(str().is_string(f, s));
    expr_ref_vector es(m());
    unsigned sz = s.length();
    for (unsigned j = 0; j < sz; ++j) {
        es.push_back(str().mk_unit(str().mk_char(s, j)));
    }        
    result = str().mk_concat(es, f->get_range());    
    return BR_DONE;
}

br_status seq_rewriter::mk_str_le(expr* a, expr* b, expr_ref& result) {
    result = m().mk_not(str().mk_lex_lt(b, a));
    return BR_REWRITE2;
}

br_status seq_rewriter::mk_str_lt(expr* a, expr* b, expr_ref& result) {
    zstring as, bs;
    if (str().is_empty(b)) {
        result = m().mk_false();
        return BR_DONE;
    }
    if (str().is_empty(a)) {
        result = m().mk_not(m().mk_eq(a, b));
        return BR_REWRITE1;
    }
    if (str().is_string(a, as) && str().is_string(b, bs)) {
        unsigned sz = std::min(as.length(), bs.length());
        for (unsigned i = 0; i < sz; ++i) {
            if (as[i] < bs[i]) {
                result = m().mk_true();
                return BR_DONE;
            }
            if (as[i] > bs[i]) {
                result = m().mk_false();
                return BR_DONE;
            }
        }
        result = m().mk_bool_val(as.length() < bs.length());
        return BR_DONE;
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_str_from_code(expr* a, expr_ref& result) {
    rational r;
    if (m_autil.is_numeral(a, r)) {
        if (r.is_neg() || r > u().max_char()) {
            result = str().mk_string(zstring());
        }
        else {
            unsigned num = r.get_unsigned();
            zstring s(1, &num);
            result = str().mk_string(s);
        }
        return BR_DONE;
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_str_to_code(expr* a, expr_ref& result) {
    zstring s;
    if (str().is_string(a, s)) {
        if (s.length() == 1) 
            result = m_autil.mk_int(s[0]);
        else
            result = minus_one();
        return BR_DONE;
    }    
    return BR_FAILED;
}

br_status seq_rewriter::mk_str_is_digit(expr* a, expr_ref& result) {
    zstring s;
    if (str().is_string(a, s)) {
        if (s.length() == 1 && '0' <= s[0] && s[0] <= '9')
            result = m().mk_true();
        else
            result = m().mk_false();
        return BR_DONE;
    }
    if (str().is_empty(a)) {
        result = m().mk_false();
        return BR_DONE;
    }
    // when a has length > 1 -> false
    // when a is a unit character -> evaluate
   
    return BR_FAILED;
}


br_status seq_rewriter::mk_str_ubv2s(expr* a, expr_ref& result) {
    bv_util bv(m());
    rational val;
    if (bv.is_numeral(a, val)) {
        result = str().mk_string(zstring(val));
        return BR_DONE;
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_str_sbv2s(expr *a, expr_ref &result) {
    bv_util bv(m());
    rational val;
    unsigned bv_size = 0;
    if (bv.is_numeral(a, val, bv_size)) {
        rational r = mod(val, rational::power_of_two(bv_size));
        SASSERT(!r.is_neg());
        if (r >= rational::power_of_two(bv_size - 1)) {
            r -= rational::power_of_two(bv_size);
        }
        result = str().mk_string(zstring(r));
        return BR_DONE;
    }
    
    bv_size = bv.get_bv_size(a);
    {
        auto _seq2449_0 = bv.mk_slt(a, bv.mk_numeral(0, bv_size));
        auto _seq2499_0 = str().mk_string(zstring("-"));
        auto _seq2499_1 = str().mk_ubv2s(bv.mk_bv_neg(a));
        auto _seq2449_1 = str().mk_concat(_seq2499_0, _seq2499_1);
        auto _seq2449_2 = str().mk_ubv2s(a);
        result = m().mk_ite(_seq2449_0, _seq2449_1, _seq2449_2);
    }
    return BR_REWRITE_FULL;
}

br_status seq_rewriter::mk_str_itos(expr* a, expr_ref& result) {
    rational r;
    if (m_autil.is_numeral(a, r)) {
        if (r.is_int() && !r.is_neg()) {
            result = str().mk_string(zstring(r));
        }
        else {
            result = str().mk_string(zstring());
        }
        return BR_DONE;
    }
    // itos(stoi(s)) -> if s = '0' or .... or s = '9' then s else ""
    // when |s| <= 1
    expr* b = nullptr;
    
    if (str().is_stoi(a, b) && max_length(b, r) && r <= 1) {
        expr_ref_vector eqs(m());
        for (unsigned i = 0; i < 10; ++i) {
            zstring s('0' + i);
            eqs.push_back(m().mk_eq(b, str().mk_string(s)));
        }
        result = m().mk_or(eqs);
        result = m().mk_ite(result, b, str().mk_string(zstring()));
        return BR_REWRITE2;
    }
    return BR_FAILED;
}

/**
   \brief rewrite str.to.int according to the rules:
   - if the expression is a string which is a non-empty 
     sequence of digits 0-9 extract the corresponding numeral.
   - if the expression is a string that contains any other character 
     or is empty, produce -1
   - if the expression is int.to.str(x) produce
      ite(x >= 0, x, -1)
     
*/
br_status seq_rewriter::mk_str_stoi(expr* a, expr_ref& result) {
    zstring s;
    if (str().is_string(a, s)) {
        std::string s1 = s.encode();
        if (s1.length() == 0) {
            result = minus_one();
            return BR_DONE;
        } 
        for (unsigned i = 0; i < s1.length(); ++i) {
            if (!('0' <= s1[i] && s1[i] <= '9')) {
                result = minus_one();
                return BR_DONE;
            }
        }
        rational r(s1.c_str());
        result = m_autil.mk_numeral(r, true);
        return BR_DONE;
    }
    expr* b;
    if (str().is_itos(a, b)) {
        auto a = m_autil.mk_ge(b, zero());
        result = m().mk_ite(a, b, minus_one());
        return BR_DONE;
    }
    if (str().is_ubv2s(a, b)) {
        bv_util bv(m());
        result = bv.mk_ubv2int(b);
        return BR_DONE;
    }
    
    expr* c = nullptr, *t = nullptr, *e = nullptr;
    if (m().is_ite(a, c, t, e)) {
        auto a = str().mk_stoi(t);
        result = m().mk_ite(c, a, str().mk_stoi(e));
        return BR_REWRITE_FULL;
    }

    expr* u = nullptr;
    unsigned ch = 0;
    if (str().is_unit(a, u) && m_util.is_const_char(u, ch)) {
        if ('0' <= ch && ch <= '9') {
            result = m_autil.mk_int(ch - '0');
        }
        else {
            result = minus_one();
        }
        return BR_DONE;
    }        

    expr_ref_vector as(m());
    str().get_concat_units(a, as);
    if (as.empty()) {
        result = minus_one();
        return BR_DONE;
    }
    if (str().is_unit(as.back())) {
        // if head = "" then tail else
        // if tail < 0 then tail else 
        // if stoi(head) >= 0 and then stoi(head)*10+tail else -1
        expr_ref tail(str().mk_stoi(as.back()), m());
        expr_ref head(str().mk_concat(as.size() - 1, as.data(), a->get_sort()), m());
        expr_ref stoi_head(str().mk_stoi(head), m());
        {
            auto _seq2559_0 = m_autil.mk_ge(stoi_head, zero());
            auto _seq2559_1 = m_autil.mk_add(m_autil.mk_mul(m_autil.mk_int(10), stoi_head), tail);
            result = m().mk_ite(_seq2559_0, _seq2559_1, minus_one());
        }

        result = m().mk_ite(m_autil.mk_ge(tail, zero()), 
                            result,
                            tail);
        result = m().mk_ite(str().mk_is_empty(head), 
                            tail,
                            result);
        return BR_REWRITE_FULL;
    }
    if (str().is_unit(as.get(0), u) && m_util.is_const_char(u, ch) && '0' == ch) {
        result = str().mk_concat(as.size() - 1, as.data() + 1, as[0]->get_sort());
        {
            auto _seq2573_0 = str().mk_is_empty(result);
            auto _seq2573_1 = str().mk_stoi(result);
            result = m().mk_ite(_seq2573_0, zero(), _seq2573_1);
        }
        return BR_REWRITE_FULL;
    }

    return BR_FAILED;
}

void seq_rewriter::add_next(u_map<expr*>& next, expr_ref_vector& trail, unsigned idx, expr* cond) {
    expr* acc;
    if (!m().is_true(cond) && next.find(idx, acc)) {              
        expr* args[2] = { cond, acc };
        cond = mk_or(m(), 2, args);
    }
    trail.push_back(cond);
    next.insert(idx, cond);   

}


bool seq_rewriter::is_sequence(expr* e, expr_ref_vector& seq) {
    seq.reset();
    zstring s;
    ptr_vector<expr> todo;
    expr *e1, *e2;
    todo.push_back(e);
    while (!todo.empty()) {
        e = todo.back();
        todo.pop_back();
        if (str().is_string(e, s)) {
            for (unsigned i = 0; i < s.length(); ++i) {
                seq.push_back(str().mk_char(s, i));
            }
        }
        else if (str().is_empty(e)) {
            continue;
        }
        else if (str().is_unit(e, e1)) {
            seq.push_back(e1);
        }
        else if (str().is_concat(e, e1, e2)) {
            todo.push_back(e2);
            todo.push_back(e1);
        }
        else {
            return false;
        }
    }
    return true;
}

/*
    s = [head] + tail where head is the first element of s
*/
bool seq_rewriter::get_head_tail(expr* s, expr_ref& head, expr_ref& tail) {
    expr* h = nullptr, *t = nullptr;
    zstring s1;
    if (str().is_unit(s, h)) {
        head = h;
        tail = str().mk_empty(s->get_sort());
        return true;
    }
    if (str().is_string(s, s1) && s1.length() > 0) {
        head = m_util.mk_char(s1[0]);
        tail = str().mk_string(s1.extract(1, s1.length()));
        return true;
    }
    if (str().is_concat(s, h, t) && get_head_tail(h, head, tail)) {
        tail = mk_seq_concat(tail, t);
        return true;
    }
    return false;
}

/*
    s = head + tail where |tail| = 1
*/
bool seq_rewriter::get_head_tail_reversed(expr* s, expr_ref& head, expr_ref& tail) {
    expr* h = nullptr, *t = nullptr;
    zstring s1;
    if (str().is_unit(s, t)) {
        head = str().mk_empty(s->get_sort());
        tail = t;
        return true;
    }
    if (str().is_string(s, s1) && s1.length() > 0) {
        head = str().mk_string(s1.extract(0, s1.length() - 1));
        tail = m_util.mk_char(s1[s1.length() - 1]);
        return true;
    }
    if (str().is_concat(s, h, t) && get_head_tail_reversed(t, head, tail)) {
        head = mk_seq_concat(h, head);
        return true;
    }
    return false;
}

bool seq_rewriter::get_re_head_tail(expr* r, expr_ref& head, expr_ref& tail) {
    expr* r1 = nullptr, *r2 = nullptr;
    if (re().is_concat(r, r1, r2)) {
        head = r1;
        tail = r2;
        return re().min_length(r1) != UINT_MAX && re().max_length(r1) == re().min_length(r1);
    }
    return false;
}

bool seq_rewriter::get_re_head_tail_reversed(expr* r, expr_ref& head, expr_ref& tail) {
    expr* r1 = nullptr, *r2 = nullptr;
    if (re().is_concat(r, r1, r2)) {
        unsigned len = re().min_length(r2);
        if (len != UINT_MAX && re().max_length(r2) == len) {
            if (get_re_head_tail_reversed(r1, head, tail))
                // left associative binding of concat
                tail = mk_re_append(tail, r2);
            else {
                // right associative binding of concat
                head = r1;
                tail = r2;
            }
            return true;
        }
        if (get_re_head_tail_reversed(r2, head, tail)) {
            head = mk_re_append(r1, head);
            return true;
        }
    }
    return false;
}


expr_ref seq_rewriter::re_and(expr* cond, expr* r) {
    expr_ref _cond(cond, m()), _r(r, m());
    if (m().is_true(cond))
        return expr_ref(r, m());    
    expr* re_empty = re().mk_empty(r->get_sort());
    if (m().is_false(cond))
        return expr_ref(re_empty, m());
    return expr_ref(m().mk_ite(cond, r, re_empty), m());
}

expr_ref seq_rewriter::re_predicate(expr* cond, sort* seq_sort) {
    expr_ref re_with_empty(re().mk_to_re(str().mk_empty(seq_sort)), m());
    return re_and(cond, re_with_empty);
}

expr_ref seq_rewriter::is_nullable(expr* r) {
    STRACE(seq_verbose, tout << "is_nullable: "
                               << mk_pp(r, m()) << std::endl;);
    expr_ref result(m_op_cache.find(_OP_RE_IS_NULLABLE, r, nullptr, nullptr), m());
    if (!result) {
        result = m_derive.nullable(r);
        m_op_cache.insert(_OP_RE_IS_NULLABLE, r, nullptr, nullptr, result);        
    }
    STRACE(seq_verbose, tout << "is_nullable result: "
                               << result << std::endl;);
    return result;
}

/*
    Push reverse inwards (whenever possible).
*/
br_status seq_rewriter::mk_re_reverse(expr* r, expr_ref& result) {
    sort* seq_sort = nullptr;
    VERIFY(m_util.is_re(r, seq_sort));
    expr *r1 = nullptr, *r2 = nullptr, *p = nullptr, *s = nullptr;
    expr *s1 = nullptr, *s2 = nullptr;
    zstring zs;
    unsigned lo = 0, hi = 0;
    if (re().is_concat(r, r1, r2)) {
        // deterministic evaluation order for reverse operands
        auto a_rev = mk_reverse(r2);
        auto b_rev = mk_reverse(r1);
        result = mk_re_append(a_rev, b_rev);
        return BR_DONE;
    }
    else if (re().is_star(r, r1)) {
        result = mk_star(mk_reverse(r1));
        return BR_DONE;
    }
    else if (re().is_plus(r, r1)) {
        result = mk_plus(mk_reverse(r1));
        return BR_DONE;
    }
    else if (re().is_union(r, r1, r2)) {
        // ensure deterministic evaluation order of parameters
        auto a = mk_reverse(r1);
        auto b = mk_reverse(r2);
        result = mk_union(a, b);
        return BR_DONE;
    }
    else if (re().is_intersection(r, r1, r2)) {
        auto a = mk_reverse(r1);
        auto b = mk_reverse(r2);
        result = mk_inter(a, b);
        return BR_DONE;
    }
    else if (re().is_diff(r, r1, r2)) {
        auto a = mk_reverse(r1);
        auto b = mk_reverse(r2);
        result = mk_diff(a, b);
        return BR_DONE;
    }
    else if (re().is_xor(r, r1, r2)) {
        auto a = mk_reverse(r1);
        auto b = mk_reverse(r2);
        result = mk_re_xor_simplified(a, b);
        return BR_DONE;
    }
    else if (m().is_ite(r, p, r1, r2)) {
        auto a = mk_reverse(r1);
        auto b = mk_reverse(r2);
        m_br.mk_ite(p, a, b, result);
        return BR_DONE;
    }
    else if (re().is_opt(r, r1)) {
        result = mk_opt(mk_reverse(r1));
        return BR_DONE;
    }
    else if (re().is_complement(r, r1)) {
        result = mk_complement(mk_reverse(r1));
        return BR_DONE;
    }
    else if (re().is_loop(r, r1, lo)) {
        result = mk_loop(mk_reverse(r1), lo);
        return BR_DONE;
    }
    else if (re().is_loop(r, r1, lo, hi)) {
        result = mk_loop(mk_reverse(r1), lo, hi);
        return BR_DONE;
    }
    else if (re().is_reverse(r, r1)) {
        result = r1;
        return BR_DONE;
    }
    else if (re().is_full_seq(r) ||
             re().is_empty(r) ||
             re().is_range(r) ||
             re().is_full_char(r) ||
             re().is_of_pred(r)) {
        result = r;
        return BR_DONE;
    }
    else if (re().is_to_re(r, s) && str().is_string(s, zs)) {
        result = mk_to_re(str().mk_string(zs.reverse()));
        return BR_DONE;
    }
    else if (re().is_to_re(r, s) && str().is_unit(s)) {
        result = r;
        return BR_DONE;
    }
    else if (re().is_to_re(r, s) && str().is_concat(s, s1, s2)) {
        auto a_rev = mk_reverse(mk_to_re(s2));
        auto b_rev = mk_reverse(mk_to_re(s1));
        result = mk_re_append(a_rev, b_rev);
        return BR_DONE;
    }
    else {
        // stuck cases: variable, re().is_derivative, ...
        return BR_FAILED;
    }
}

/*
    Symbolic derivative: seq -> regex -> regex
    seq should be single char

    This is the rewriter entrypoint for computing a derivative.
    Use mk_derivative from seq_decl_plugin instead to create a derivative
    expression without computing it (simplifying).

    This calls mk_derivative, the main logic which builds a derivative
    recursively, but mk_derivative doesn't guarantee full simplification.
    Once the derivative is built, we return BR_REWRITE_FULL so that
    any remaining possible simplification is performed from the bottom up.

    Rewriting also replaces _OP_RE_antimirov_UNION, which is produced
    by is_derivative, with real union.
*/
br_status seq_rewriter::mk_re_derivative(expr* ele, expr* r, expr_ref& result) {
    result = mk_derivative(ele, r);
    // TBD: we may even declare BR_DONE here and potentially miss some simplifications
    // return re().is_derivative(result) ? BR_DONE : BR_REWRITE_FULL;
    return BR_DONE;
}


expr_ref seq_rewriter::mk_derivative(expr* r) {
    auto result = m_derive(seq::derivative_kind::antimirov_t, r);
    TRACE(seq, tout << "Derivative of " << mk_pp(r, m()) << "\nis\n" << result << std::endl;);
    return result;
}

expr_ref seq_rewriter::mk_derivative(expr* ele, expr* r) {
    auto result = m_derive(seq::derivative_kind::antimirov_t, ele, r);
    TRACE(seq,
          tout << "Derivative of " << mk_pp(r, m()) << " w.r.t. " << mk_pp(ele, m()) << "\nis\n" << result << std::endl;);
    return result;
}

expr_ref seq_rewriter::mk_regex_union_normalize(expr* r1, expr* r2) {
    expr_ref _r1(r1, m()), _r2(r2, m());
    expr *a1, *b1, *a2, *b2;
    SASSERT(m_util.is_re(r1));
    SASSERT(m_util.is_re(r2));
    expr_ref result(m());
    std::function<bool(expr*, expr*&, expr*&)> test = [&](expr* t, expr*& a, expr*& b) { return re().is_union(t, a, b); };
    std::function<expr* (expr*, expr*)> compose = [&](expr* r1, expr* r2) { return (is_subset(r1, r2) ? r2 : (is_subset(r2, r1) ? r1 : re().mk_union(r1, r2))); };
    std::function<bool(expr *, expr *)> is_complement = [&](expr *a, expr *b) {
        expr *s;
        if (re().is_complement(a, s) && s == b)
            return true;
        if (re().is_complement(b, s) && s == a)
            return true;
        return false;
    };    
    if (r1 == r2 || re().is_empty(r2) || re().is_full_seq(r1))
        result = r1;
    else if (re().is_empty(r1) || re().is_full_seq(r2))
        result = r2;
    else if (re().is_dot_plus(r1) && re().get_info(r2).min_length > 0)
        result = r1;
    else if (re().is_dot_plus(r2) && re().get_info(r1).min_length > 0)
        result = r2;
    // (R1 \ R2) U (R2 \ R1) = R1 xor R2
    else if (false && re().is_intersection(r1, a1, a2) && re().is_intersection(r2, b1, b2) && 
        is_complement(a1, b2) && is_complement(a2, b1)) {
        result = re().mk_xor(a1, re().mk_complement(a2));
    }   
    else if (false && re().is_intersection(r1, a1, a2) && re().is_intersection(r2, b1, b2) && 
        is_complement(a1, b1) && is_complement(a2, b2)) {
        result = re().mk_xor(a1, re().mk_complement(a2));
    }         
    else {
        // Range ∪ Range: [a,b] ∪ [c,d] = [min(a,c), max(b,d)] when overlapping or adjacent
        unsigned lo1_v = 0, hi1_v = 0, lo2_v = 0, hi2_v = 0;
        if (re().is_range(r1, lo1_v, hi1_v) && re().is_range(r2, lo2_v, hi2_v) &&
            lo2_v <= hi1_v + 1 && lo1_v <= hi2_v + 1) {
            unsigned new_lo = std::min(lo1_v, lo2_v);
            unsigned new_hi = std::max(hi1_v, hi2_v);
            result = re().mk_range(r1->get_sort(), new_lo, new_hi);
        }
        else
            result = merge_regex_sets(r1, r2, re().mk_full_seq(r1->get_sort()), test, compose);
    }

    return result;
}

expr_ref seq_rewriter::mk_regex_inter_normalize(expr* r1, expr* r2) {
    expr_ref _r1(r1, m()), _r2(r2, m());
    SASSERT(m_util.is_re(r1));
    SASSERT(m_util.is_re(r2));
    expr_ref result(m());
    if (re().is_epsilon(r2))
        std::swap(r1, r2);
    std::function<bool(expr*, expr*&, expr*&)> test = [&](expr* t, expr*& a, expr*& b) { return re().is_intersection(t, a, b); };
    std::function<expr* (expr*, expr*)> compose = [&](expr* r1, expr* r2) { return (is_subset(r1, r2) ? r1 : (is_subset(r2, r1) ? r2 : re().mk_inter(r1, r2))); };
    if (r1 == r2 || re().is_empty(r1) || re().is_full_seq(r2))
        result = r1;
    else if (re().is_empty(r2) || re().is_full_seq(r1))
        result = r2;
    else if (re().is_epsilon(r1)) {
        if (re().get_info(r2).nullable == l_true)
            result = r1;
        else if (re().get_info(r2).nullable == l_false)
            result = re().mk_empty(r1->get_sort());
        else
            result = merge_regex_sets(r1, r2, re().mk_empty(r1->get_sort()), test, compose);
    }
    else if (re().is_dot_plus(r1) && re().get_info(r2).min_length > 0)
        result = r2;
    else if (re().is_dot_plus(r2) && re().get_info(r1).min_length > 0)
        result = r1;
    else {
        // Range ∩ Range: [a,b] ∩ [c,d] = [max(a,c), min(b,d)] or empty
        unsigned lo1_v = 0, hi1_v = 0, lo2_v = 0, hi2_v = 0;
        if (re().is_range(r1, lo1_v, hi1_v) && re().is_range(r2, lo2_v, hi2_v)) {
            unsigned new_lo = std::max(lo1_v, lo2_v);
            unsigned new_hi = std::min(hi1_v, hi2_v);
            result = re().mk_range(r1->get_sort(), new_lo, new_hi);
        }
        else
            result = merge_regex_sets(r1, r2, re().mk_empty(r1->get_sort()), test, compose);
    }
    return result;
}

expr_ref seq_rewriter::merge_regex_sets(expr* r1, expr* r2, expr* unit,
    std::function<bool(expr*, expr*&, expr*&)>& test, 
    std::function<expr* (expr*, expr*)>& compose) {
    sort* seq_sort;
    expr_ref result(unit, m());
    expr_ref_vector prefix(m());
    VERIFY(m_util.is_re(r1, seq_sort));
    SASSERT(m_util.is_re(r2));
    SASSERT(r2->get_sort() == r1->get_sort());
    // Ordering of expressions used by merging, 0 means unordered, -1 means e1 < e2, 1 means e2 < e1
    auto compare = [&](expr* x, expr* y) {
        expr* z = nullptr;
        // TODO: consider also the case of A{0,l}++B having the same id as A*++B
        // in which case return 0
        if (x == y)
            return 0;

        unsigned xid = (re().is_complement(x, z) ? z->get_id() : x->get_id());
        unsigned yid = (re().is_complement(y, z) ? z->get_id() : y->get_id());
        SASSERT(xid != yid);
        return (xid < yid ? -1 : 1);
    };
    auto composeresult = [&](expr* suffix) {
        result = suffix;
        while (!prefix.empty()) {
            result = compose(prefix.back(), result);
            prefix.pop_back();
        }
        return result;
    };
    expr* ar = r1;
    expr* br = r2;
    while (true) {        
        if (ar == br)
            return composeresult(ar);

        if (are_complements(ar, br))
            return expr_ref(unit, m());

        expr* a, * ar1, * b, * br1;
        if (test(br, b, br1) && !test(ar, a, ar1))
            std::swap(ar, br);

        // both ar, br are decomposable
        if (test(br, b, br1)) {
            VERIFY(test(ar, a, ar1));
            if (are_complements(a, b))
                return expr_ref(unit, m());
            switch (compare(a, b)) {
            case 0:
                // a == b
                prefix.push_back(a);
                ar = ar1;
                br = br1;
                break;
            case -1:
                // a < b
                prefix.push_back(a);
                ar = ar1;
                break;
            case 1:
                // b < a
                prefix.push_back(b);
                br = br1;
                break;
            default:
                UNREACHABLE();
            }
            continue;
        }

        // ar is decomposable, br is not decomposable
        if (test(ar, a, ar1)) {            
            if (are_complements(a, br))
                return expr_ref(unit, m());
            switch (compare(a, br)) {
            case 0:
                // result = prefix ++ ar
                return composeresult(ar);
            case -1:
                // a < br
                prefix.push_back(a);
                ar = ar1;
                break;
            case 1:
                // br < a, result = prefix ++ (br) ++ ar
                prefix.push_back(br);
                return composeresult(ar);
            default:
                UNREACHABLE();
            }
            continue;
        }

        // neither ar nor br is decomposable
        if (compare(ar, br) == -1)
            std::swap(ar, br);
        // br < ar, result = prefix ++ (br) ++ (ar) 
        prefix.push_back(br);
        return composeresult(ar);
    }
}


/*
    Utility functions to decide char <, ==, !=, and <=.
    Return true if deduced, false if unknown.
*/
bool seq_rewriter::lt_char(expr* ch1, expr* ch2) {
    unsigned u1, u2;
    return u().is_const_char(ch1, u1) &&
           u().is_const_char(ch2, u2) && (u1 < u2);
}
bool seq_rewriter::eq_char(expr* ch1, expr* ch2) {
    return ch1 == ch2;
}
bool seq_rewriter::neq_char(expr* ch1, expr* ch2) {
    unsigned u1, u2;
    return u().is_const_char(ch1, u1) &&
        u().is_const_char(ch2, u2) && (u1 != u2);
}
bool seq_rewriter::le_char(expr* ch1, expr* ch2) {
    return eq_char(ch1, ch2) || lt_char(ch1, ch2);
}

/*
 * pattern match against all ++ "abc" ++ all ++ "def" ++ all regexes.
*/
bool seq_rewriter::is_re_contains_pattern(expr* r, vector<expr_ref_vector>& patterns) {
    expr* r1 = nullptr, *r2 = nullptr, *s = nullptr;
    if (re().is_concat(r, r1, r2) && re().is_full_seq(r1)) {
        r = r2;
        patterns.push_back(expr_ref_vector(m()));
    }
    else {
        return false;
    }
    while (re().is_concat(r, r1, r2)) {
        if (re().is_to_re(r1, s))
            patterns.back().push_back(s);
        else if (re().is_full_seq(r1))
            patterns.push_back(expr_ref_vector(m()));
        else
            return false;
        r = r2;
    }
    return re().is_full_seq(r);
}

/*
 * return true if the sequences p1, p2 cannot overlap in any way.
 * assume |p1| <= |p2|
 *   no suffix of p1 is a prefix of p2
 *   no prefix of p1 is a suffix of p2
 *   p1 is not properly contained in p2
 */
bool seq_rewriter::non_overlap(zstring const& s1, zstring const& s2) const {
    unsigned sz1 = s1.length(), sz2 = s2.length();
    if (sz1 > sz2) 
        return non_overlap(s2, s1);
    auto can_overlap = [&](unsigned start1, unsigned end1, unsigned start2) {
        for (unsigned i = start1; i < end1; ++i) {
            if (s1[i] != s2[start2 + i])
                return false;
        }
        return true;
    };
    for (unsigned i = 1; i < sz1; ++i) 
        if (can_overlap(i, sz1, 0))
            return false;
    for (unsigned j = 0; j + sz1 < sz2; ++j) 
        if (can_overlap(0, sz1, j))
            return false;
    for (unsigned j = sz2 - sz1; j < sz2; ++j) 
        if (can_overlap(0, sz2 - j, j))
            return false;
    return true;
}

bool seq_rewriter::non_overlap(expr_ref_vector const& p1, expr_ref_vector const& p2) const {
    unsigned sz1 = p1.size(), sz2 = p2.size();
    if (sz1 > sz2) 
        return non_overlap(p2, p1);
    if (sz1 == 0 || sz2 == 0)
        return false;
    zstring s1, s2;
    if (sz1 == 1 && sz2 == 1 && str().is_string(p1[0], s1) && str().is_string(p2[0], s2))
        return non_overlap(s1, s2);
    for (expr* e : p1) 
        if (!str().is_unit(e))
            return false;
    for (expr* e : p2) 
        if (!str().is_unit(e))
            return false;
    auto can_overlap = [&](unsigned start1, unsigned end1, unsigned start2) {
        for (unsigned i = start1; i < end1; ++i) {
            if (m().are_distinct(p1[i], p2[start2 + i]))
                return false;
            if (!m().are_equal(p1[i], p2[start2 + i]))
                return true;
        }
        return true;
    };
    for (unsigned i = 1; i < sz1; ++i) 
        if (can_overlap(i, sz1, 0))
            return false;
    for (unsigned j = 0; j + sz1 < sz2; ++j) 
        if (can_overlap(0, sz1, j))
            return false;
    for (unsigned j = sz2 - sz1; j < sz2; ++j) 
        if (can_overlap(0, sz2 - j, j))
            return false;
    return true;
}

/**
  simplify extended contains patterns into simpler membership constraints
       (x ++ "abc" ++ s) in (all ++ "de" ++ all ++ "ee" ++ all ++ "ff" ++ all)
  => 
       ("abc" ++ s) in (all ++ "de" ++ all ++ "ee" ++ all ++ "ff" ++ all)
   or  x in (all ++ "de" ++ all)                & ("abc" ++ s) in (all ++ "ee" ++ all ++ "ff" ++ all)
   or  x in (all ++ "de" ++ all ++ "ee" ++ all) & ("abc" ++ s) in (all ++ "ff" ++ all)
   or  x in (all ++ "de" ++ all ++ "ee" ++ all ++ "ff" ++ all) & .. simplifies to true ..
*/

bool seq_rewriter::rewrite_contains_pattern(expr* a, expr* b, expr_ref& result) {
    vector<expr_ref_vector> patterns;
    expr* x = nullptr, *y = nullptr, *z = nullptr, *u = nullptr;
    if (!str().is_concat(a, x, y))
        return false;
    if (!is_re_contains_pattern(b, patterns)) 
        return false;
    m_lhs.reset();        
    u = y;
    while (str().is_concat(u, z, u) && (str().is_unit(z) || str().is_string(z))) {
        m_lhs.push_back(z);
    }
    for (auto const& p : patterns)
        if (!non_overlap(p, m_lhs))
            return false;

    expr_ref_vector fmls(m());
    sort* rs = b->get_sort();
    expr_ref full(re().mk_full_seq(rs), m()), prefix(m()), suffix(m());
    fmls.push_back(re().mk_in_re(y, b));
    prefix = full;
    for (unsigned i = 0; i < patterns.size(); ++i) {
        for (expr* e : patterns[i])
            prefix = re().mk_concat(prefix, re().mk_to_re(e));
        prefix = re().mk_concat(prefix, full);
        suffix = full;
        for (unsigned j = i + 1; j < patterns.size(); ++j) {
            for (expr* e : patterns[j])
                suffix = re().mk_concat(suffix, re().mk_to_re(e));
            suffix = re().mk_concat(suffix, full);
        }
        {
            auto _seq3217_0 = re().mk_in_re(x, prefix);
            auto _seq3217_1 = re().mk_in_re(y, suffix);
            fmls.push_back(m().mk_and(_seq3217_0, _seq3217_1));
        }
    }
    result = mk_or(fmls);
    return true;    
}

/*
    a in empty -> false
    a in full -> true
    a in (str.to_re a') -> (a == a')
    "" in b -> is_nullable(b)
    (ele + tail) in b -> tail in (derivative e b)
    (head + ele) in b -> head in (right-derivative e b)

Other rewrites:
    s in b1 ++ b2, min_len(b1) = max_len(b2) != UINT_MAX -> 
           (seq.len s) >= min_len(b1) & 
           (seq.extract s 0 min_len(b1)) in b1 & 
           (seq.extract s min_len(b1) (- (seq.len s) min_len(b1))) in b2

    similar for tail of regex

Disabled rewrite:
   s + "ab" + t in all ++ "c" ++ all ++ .... ++ "z" ++ all
   => 
   disjunctions that cover cases where s overlaps given that "ab" does
   not overlap with any of the sequences.
   It is disabled because the solver doesn't handle disjunctions of regexes well.

TBD:
Enable rewrite when R = R1|R2 and derivative cannot make progress: 's in R' ==> 's in R1' | 's in R2'
cannot make progress here means that R1 or R2 starts with an uninterpreted symbol
This will help propagate cases like "abc"X in opt(to_re(X)) to equalities.
*/
br_status seq_rewriter::mk_str_in_regexp(expr* a, expr* b, expr_ref& result) {

    STRACE(seq_verbose, tout << "mk_str_in_regexp: " << mk_pp(a, m())
                               << ", " << mk_pp(b, m()) << std::endl;);

    if (re().is_empty(b)) {
        result = m().mk_false();
        return BR_DONE;
    }
    if (re().is_full_seq(b)) {
        result = m().mk_true();
        return BR_DONE;
    }

    // (str.in_re e (re.range lo hi)) where a bound is not a concrete character.
    // By SMT-LIB semantics (re.range lo hi) is the set of single characters c
    // with lo <= c <= hi when lo and hi are themselves single characters, and
    // the empty language otherwise; so membership is equivalent to lo, hi and
    // e all being single characters with lo <= e <= hi.  The derivative engine
    // only unfolds ranges whose bounds are concrete characters, so without this
    // reduction a range with a symbolic bound is left unsolved (and mk_re_range
    // deliberately keeps such a range symbolic rather than unsoundly collapsing
    // it to re.empty).  Ranges with two concrete single-character bounds keep
    // their existing derivative-based handling.
    {
        expr* rlo = nullptr, *rhi = nullptr;
        if (re().is_range(b, rlo, rhi)) {
            auto concrete_char = [&](expr* e) {
                zstring s;
                expr* ch = nullptr;
                unsigned uc = 0;
                return (str().is_string(e, s) && s.length() == 1) ||
                       (str().is_unit(e, ch) && m_util.is_const_char(ch, uc));
            };
            if (!concrete_char(rlo) || !concrete_char(rhi)) {
                expr_ref_vector conj(m());
                expr_ref lo_length(str().mk_length(rlo), m());
                conj.push_back(m().mk_eq(lo_length, one()));
                {
                    expr_ref lhs(str().mk_length(rhi), m());
                    auto rhs = one();
                    conj.push_back(m().mk_eq(lhs, rhs));
                }
                {
                    expr_ref lhs(str().mk_length(a), m());
                    auto rhs = one();
                    conj.push_back(m().mk_eq(lhs, rhs));
                }
                conj.push_back(str().mk_lex_le(rlo, a));
                conj.push_back(str().mk_lex_le(a, rhi));
                result = m().mk_and(conj);
                return BR_REWRITE_FULL;
            }
        }
    }

    zstring s;
    if (str().is_string(a, s) && re().is_ground(b)) {
        // Just check membership and replace by true/false
        expr_ref r(b, m());
        for (unsigned i = 0; i < s.length(); ++i) {
            if (re().is_empty(r)) {
                result = m().mk_false();
                return BR_DONE;
            }
            unsigned ch = s[i];
            expr_ref new_r = mk_derivative(m_util.mk_char(ch), r);
            r = new_r;
        }

        switch (re().get_info(r).nullable) {
        case l_true:
            result = m().mk_true();
            return BR_DONE;
        case l_false:
            result = m().mk_false();
            return BR_DONE;
        default:
            break;
        }
    }

    
    // replace_all(x, a, b) in R where R is ground, a and b are unit-length strings
    // ==> x in R[b -> {a, b}, a -> empty]
    expr *ra_x = nullptr, *ra_a = nullptr, *ra_b = nullptr;
    zstring sa_val, sb_val;
    if (str().is_replace_all(a, ra_x, ra_a, ra_b) && ra_a == ra_b) {
        result = ra_x;
        return BR_DONE;
    }
    if (str().is_replace_all(a, ra_x, ra_a, ra_b) && str().is_string(ra_a, sa_val) && sa_val.length() == 1 &&
        str().is_string(ra_b, sb_val) && sb_val.length() == 1 && sa_val[0] != sb_val[0] && re().is_ground(b) &&
        re().get_info(b).classical) {
        expr_ref new_re = re_replace_char(b, sa_val[0], sb_val[0], ra_a, ra_b);
        result = re().mk_in_re(ra_x, new_re);
        return BR_REWRITE_FULL;
    }

    expr_ref b_s(m());
    if (lift_str_from_to_re(b, b_s)) {
       result = m_br.mk_eq_rw(a, b_s);
       return BR_REWRITE_FULL;
    }
    expr* c = nullptr, *d = nullptr, *e = nullptr;
    if (re().is_concat(b, c, d) && re().is_to_re(c, e) && re().is_full_seq(d)) {
        result = str().mk_prefix(e, a);
        return BR_REWRITE1;
    }
    if (re().is_concat(b, c, d) && re().is_to_re(d, e) && re().is_full_seq(c)) {
        result = str().mk_suffix(e, a);
        return BR_REWRITE1;
    }
    expr* b1 = nullptr;
    expr* eps = nullptr;
    if (re().is_opt(b, b1) ||
        (re().is_union(b, b1, eps) && re().is_epsilon(eps)) ||
        (re().is_union(b, eps, b1) && re().is_epsilon(eps)))
    {
        // deterministic evaluation order: build sub-expressions first
        auto len_a = str().mk_length(a);
        auto is_empty = m().mk_eq(len_a, zero());
        auto in_b1 = re().mk_in_re(a, b1);
        result = m().mk_ite(is_empty, m().mk_true(), in_b1);
        return BR_REWRITE_FULL;
    }
    if (str().is_empty(a)) {
        result = is_nullable(b);
        if (str().is_in_re(result))
            return BR_DONE;
        else
            return BR_REWRITE_FULL;
    }

    if (!u().can_be_member(a, b)) {
        result = m().mk_false();
        return BR_DONE;
    }

    return BR_FAILED;
}

bool seq_rewriter::has_fixed_length_constraint(expr* a, unsigned& len) {
    unsigned minl = re().min_length(a), maxl = re().max_length(a);
    len = minl;
    return minl == maxl;
}

bool seq_rewriter::lift_str_from_to_re_ite(expr* r, expr_ref& result) {
    expr* cond = nullptr, * then_r = nullptr, * else_r = nullptr;
    expr_ref then_s(m());
    expr_ref else_s(m());
    if (m().is_ite(r, cond, then_r, else_r) &&
        lift_str_from_to_re(then_r, then_s) &&
        lift_str_from_to_re(else_r, else_s)) {
        result = m().mk_ite(cond, then_s, else_s);
        return true;
    }
    return false;
}

bool seq_rewriter::lift_str_from_to_re(expr* r, expr_ref& result)
{
    expr* s = nullptr;
    if (re().is_to_re(r, s)) {
        result = s;
        return true;
    }
    return lift_str_from_to_re_ite(r, result);
}

br_status seq_rewriter::mk_str_to_regexp(expr* a, expr_ref& result) {
    expr* s = nullptr, *i = nullptr;
    if (str().is_at(a, s, i)) {
        expr_ref non_negative(m_autil.mk_ge(i, zero()), m());
        expr_ref below_length(m_autil.mk_lt(i, str().mk_length(s)), m());
        expr_ref valid(m().mk_and(non_negative, below_length), m());
        expr_ref nth(str().mk_unit(str().mk_nth_i(s, i)), m());
        expr_ref non_empty(re().mk_to_re(nth), m());
        expr_ref empty(re().mk_to_re(str().mk_empty(a->get_sort())), m());
        result = m().mk_ite(valid, non_empty, empty);
        return BR_REWRITE_FULL;
    }
    return BR_FAILED;
}

/*
    easy cases:
    .* ++ .* -> .*
    [] ++ r -> []
    r ++ [] -> []
    r ++ "" -> r
    "" ++ r -> r
    . ++ .* -> .+

    to_re and star:
    (str.to_re s1) ++ (str.to_re s2) -> (str.to_re (s1 ++ s2))
    r* ++ r* -> r*
    r* ++ r -> r ++ r*
*/
br_status seq_rewriter::mk_re_concat(expr* a, expr* b, expr_ref& result) {
    auto accepts_empty_word = [&](expr* r) {
        auto info = re().get_info(r);
        return info.interpreted && info.nullable == l_true && info.min_length == 0;
    };
    auto starts_with_full_seq = [&](expr* r) {
        expr* r1 = nullptr, *r2 = nullptr;
        return re().is_full_seq(r) || (re().is_concat(r, r1, r2) && re().is_full_seq(r1));
    };
    auto ends_with_full_seq = [&](expr* r) {
        expr* r1 = nullptr, *r2 = nullptr;
        while (re().is_concat(r, r1, r2))
            r = r2;
        return re().is_full_seq(r);
    };
    auto all_inter_arms_end_with_full_seq = [&](expr* r) {
        ptr_buffer<expr> todo;
        todo.push_back(r);
        while (!todo.empty()) {
            expr* r1 = nullptr, *r2 = nullptr;
            expr* t = todo.back();
            todo.pop_back();
            if (re().is_intersection(t, r1, r2)) {
                todo.push_back(r1);
                todo.push_back(r2);
            }
            else if (!ends_with_full_seq(t)) {
                return false;
            }
        }
        return true;
    };
    if (re().is_full_seq(a) && re().is_full_seq(b)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_seq(a) && accepts_empty_word(b)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_seq(b) && accepts_empty_word(a)) {
        result = b;
        return BR_DONE;
    }
    // Collapse adjacent full_seq factors regardless of concat grouping:
    //   (R ++ Σ*) ++ Σ* → R ++ Σ*    (a ends with Σ*, b is Σ*)
    //   Σ* ++ (Σ* ++ R) → Σ* ++ R    (a is Σ*, b starts with Σ*)
    if (re().is_full_seq(b) && ends_with_full_seq(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_seq(a) && starts_with_full_seq(b)) {
        result = b;
        return BR_DONE;
    }
    expr* u1 = nullptr, *u2 = nullptr;
    if (re().is_full_seq(a) && re().is_union(b, u1, u2) &&
        (starts_with_full_seq(u1) || starts_with_full_seq(u2))) {
        {
            auto _seq3554_0 = mk_regex_concat(a, u1);
            auto _seq3554_1 = mk_regex_concat(a, u2);
            result = mk_regex_union_normalize(_seq3554_0, _seq3554_1);
        }
        return BR_REWRITE2;
    }
    if (re().is_intersection(a, u1, u2) && re().is_full_seq(b) &&
        all_inter_arms_end_with_full_seq(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_empty(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_empty(b)) {
        result = b;
        return BR_DONE;
    }
    if (re().is_epsilon(a)) {
        result = b;
        return BR_DONE;
    }
    if (re().is_epsilon(b)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_char(a) && re().is_full_seq(b)) {
        result = mk_plus(a);
        return BR_DONE;
    }
    if (re().is_full_char(b) && re().is_full_seq(a)) {
        result = mk_plus(b);
        return BR_DONE;
    }
    expr_ref a_str(m());
    expr_ref b_str(m());
    if (lift_str_from_to_re(a, a_str) && lift_str_from_to_re(b, b_str)) {
        result = mk_to_re(mk_seq_concat(a_str, b_str));
        return BR_DONE;
    }
    expr *a1 = nullptr, *a2 = nullptr;
    expr* b1 = nullptr;
    if (re().is_to_re(a, a1) && re().is_to_re(b, b1)) {
        result = mk_to_re(mk_seq_concat(a1, b1));
        return BR_DONE;
    }
    if (re().is_star(a, a1) && re().is_star(b, b1) && a1 == b1) {
        result = a;
        return BR_DONE;
    }
    expr* b2 = nullptr, *b3 = nullptr;
    if (re().is_star(a, a1) && re().is_concat(b, b1, b2) && re().is_star(b1, b3) && a1 == b3) {
        result = b;
        return BR_DONE;
    }
    if (re().is_star(a, a1) && a1 == b) {
        result = mk_re_append(b, a);
        return BR_DONE;
    }
    unsigned lo1, hi1, lo2, hi2;


    if (re().is_loop(a, a1, lo1, hi1) && lo1 <= hi1 &&
        re().is_loop(b, b1, lo2, hi2) && lo2 <= hi2 && a1 == b1 &&
        hi1 <= UINT_MAX - hi2) {
        result = mk_loop(a1, lo1 + lo2, hi1 + hi2);
        return BR_DONE;
    }
    if (re().is_loop(a, a1, lo1) && re().is_loop(b, b1, lo2) && a1 == b1 &&
        lo1 <= UINT_MAX - lo2) {
        result = mk_loop(a1, lo1 + lo2);
        return BR_DONE;
    }
    for (unsigned i = 0; i < 2; ++i) {
        // (loop a lo1) + (loop a lo2 hi2) = (loop a lo1 + lo2) 
        if (re().is_loop(a, a1, lo1) && re().is_loop(b, b1, lo2, hi2) &&
            lo2 <= hi2 && a1 == b1 && lo1 <= UINT_MAX - lo2) {
            result = mk_loop(a1, lo1 + lo2);
            return BR_DONE;
        }
        // (loop a lo1 hi1) + a* = (loop a lo1)
        if (re().is_loop(a, a1, lo1, hi1) && lo1 <= hi1 && re().is_star(b, b1) && a1 == b1) {
            result = mk_loop(a1, lo1);
            return BR_DONE;
        }
        // (loop a lo1) + a* = (loop a lo1)
        if (re().is_loop(a, a1, lo1) && re().is_star(b, b1) && a1 == b1) {
            result = a;
            return BR_DONE;
        }
        // (loop a lo1 hi1) + a = (loop a lo1+1 hi1+1)
        if (re().is_loop(a, a1, lo1, hi1) && lo1 <= hi1 && hi1 < UINT_MAX && a1 == b) {
            result = mk_loop(a1, lo1 + 1, hi1 + 1);
            return BR_DONE;
        }
        std::swap(a, b);
    }
    // Hoist ite out of concat: concat(ite(c, r1, r2), b) → ite(c, concat(r1, b), concat(r2, b))
    expr* c = nullptr;
    if (m().is_ite(a, c, a1, b1)) {
        auto then_r = mk_re_append(a1, b);
        auto else_r = mk_re_append(b1, b);
        m_br.mk_ite(c, then_r, else_r, result);
        return BR_DONE;
    }
    if (m().is_ite(b, c, a1, b1)) {
        auto then_r = mk_re_append(a, a1);
        auto else_r = mk_re_append(a, b1);
        m_br.mk_ite(c, then_r, else_r, result);
        return BR_DONE;
    }
    if (re().is_concat(a, a1, a2)) {
        // Fold a matching loop body before flattening it, and normalize both
        // newly exposed concatenations rather than returning raw children.
        result = mk_re_append(a1, mk_re_append(a2, b));
        return BR_DONE;
    }
    return BR_FAILED;
}

expr_ref seq_rewriter::mk_regex_concat(expr *r, expr *s) {
    return mk_re_append(r, s);
}

bool seq_rewriter::are_complements(expr* r1, expr* r2) const {
    expr* r = nullptr;
    if (re().is_complement(r1, r) && r == r2)
        return true;
    if (re().is_complement(r2, r) && r == r1)
        return true;
    return false;
}

/*
 * basic subset checker.
 */
bool seq_rewriter::is_subset(expr* r1, expr* r2) const {
    return m_subset.is_subset(r1, r2);
}

bool seq_rewriter::try_collapse_re_union(expr* a, expr* b, expr_ref& result) {
    sort* seq_sort = nullptr;
    if (!u().is_re(a->get_sort(), seq_sort))
        return false;
    seq::range_predicate pa(u().max_char()), pb(u().max_char());
    if (!seq::regex_to_range_predicate(u(), a, pa))
        return false;
    if (!seq::regex_to_range_predicate(u(), b, pb))
        return false;
    result = seq::range_predicate_to_regex(u(), pa | pb, seq_sort);
    return true;
}

bool seq_rewriter::try_collapse_re_inter(expr* a, expr* b, expr_ref& result) {
    sort* seq_sort = nullptr;
    if (!u().is_re(a->get_sort(), seq_sort))
        return false;
    seq::range_predicate pa(u().max_char()), pb(u().max_char());
    if (!seq::regex_to_range_predicate(u(), a, pa))
        return false;
    if (!seq::regex_to_range_predicate(u(), b, pb))
        return false;
    result = seq::range_predicate_to_regex(u(), pa & pb, seq_sort);
    return true;
}

br_status seq_rewriter::mk_re_union0(expr* a, expr* b, expr_ref& result) {
    if (a == b) {
        result = a;
        return BR_DONE;
    }
    if (re().is_empty(a)) {
        result = b;
        return BR_DONE;
    }
    if (re().is_empty(b)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_seq(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_seq(b)) {
        result = b;
        return BR_DONE;
    }
    if (re().is_star(a) && re().is_epsilon(b)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_star(b) && re().is_epsilon(a)) {
        result = b;
        return BR_DONE;
    }
    // r ∪ ~r → Σ* (complement absorption)
    if (are_complements(a, b)) {
        result = re().mk_full_seq(a->get_sort());
        return BR_DONE;
    }
    // Hoist ite out of union: union(ite(c, r1, r2), b) → ite(c, union(r1, b), union(r2, b))
    expr *c = nullptr, *r1 = nullptr, *r2 = nullptr;
    if (m().is_ite(a, c, r1, r2)) {
        {
            auto _seq3786_0 = re().mk_union(r1, b);
            auto _seq3786_1 = re().mk_union(r2, b);
            result = m().mk_ite(c, _seq3786_0, _seq3786_1);
        }
        return BR_REWRITE3;
    }
    if (m().is_ite(b, c, r1, r2)) {
        {
            auto _seq3790_0 = re().mk_union(a, r1);
            auto _seq3790_1 = re().mk_union(a, r2);
            result = m().mk_ite(c, _seq3790_0, _seq3790_1);
        }
        return BR_REWRITE3;
    }
    if (try_collapse_re_union(a, b, result))
        return BR_DONE;
    return BR_FAILED;
}

/* Creates a normalized union. */
br_status seq_rewriter::mk_re_union(expr* a, expr* b, expr_ref& result) {
    if (try_collapse_re_union(a, b, result))
        return BR_DONE;
    result = mk_regex_union_normalize(a, b);
    return BR_DONE;
}

/* Creates a normalized complement */
br_status seq_rewriter::mk_re_complement(expr* a, expr_ref& result) {
    expr *e1 = nullptr, *e2 = nullptr;
    if (re().is_intersection(a, e1, e2)) {
        // enforce deterministic evaluation order for nested complement arguments
        auto a1 = mk_complement(e1);
        auto b1 = mk_complement(e2);
        result = mk_union(a1, b1);
        return BR_DONE;
    }
    if (re().is_union(a, e1, e2)) {
        // enforce deterministic evaluation order for nested complement arguments
        auto a1 = mk_complement(e1);
        auto b1 = mk_complement(e2);
        result = mk_inter(a1, b1);
        return BR_DONE;
    }
    if (re().is_empty(a)) {
        result = re().mk_full_seq(a->get_sort());
        return BR_DONE;
    }
    if (re().is_full_seq(a)) {
        result = re().mk_empty(a->get_sort());
        return BR_DONE;
    }
    if (re().is_complement(a, e1)) {
        result = e1;
        return BR_DONE;
    }
    if (re().is_to_re(a, e1) && str().is_empty(e1)) {
        result = mk_plus(re().mk_full_char(a->get_sort()));
        return BR_DONE;
    }
    // Hoist ite out of complement: ~(ite(c, r1, r2)) → ite(c, ~r1, ~r2)
    expr* c = nullptr;
    if (m().is_ite(a, c, e1, e2)) {
        auto then_r = mk_complement(e1);
        auto else_r = mk_complement(e2);
        m_br.mk_ite(c, then_r, else_r, result);
        return BR_DONE;
    }
    return BR_FAILED;
}


br_status seq_rewriter::mk_re_inter0(expr* a, expr* b, expr_ref& result) {
    if (a == b) {
        result = a;
        return BR_DONE;
    }
    if (re().is_empty(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_empty(b)) {
        result = b;
        return BR_DONE;
    }
    if (re().is_full_seq(a)) {
        result = b;
        return BR_DONE;
    }
    if (re().is_full_seq(b)) {
        result = a;
        return BR_DONE;
    }
    // r ∩ ~r → ∅ (complement absorption)
    if (are_complements(a, b)) {
        result = re().mk_empty(a->get_sort());
        return BR_DONE;
    }
    // Hoist ite out of intersection: inter(ite(c, r1, r2), b) → ite(c, inter(r1, b), inter(r2, b))
    expr *c = nullptr, *r1 = nullptr, *r2 = nullptr;
    if (m().is_ite(a, c, r1, r2)) {
        {
            auto _seq3878_0 = re().mk_inter(r1, b);
            auto _seq3878_1 = re().mk_inter(r2, b);
            result = m().mk_ite(c, _seq3878_0, _seq3878_1);
        }
        return BR_REWRITE3;
    }
    if (m().is_ite(b, c, r1, r2)) {
        {
            auto _seq3882_0 = re().mk_inter(a, r1);
            auto _seq3882_1 = re().mk_inter(a, r2);
            result = m().mk_ite(c, _seq3882_0, _seq3882_1);
        }
        return BR_REWRITE3;
    }
    if (try_collapse_re_inter(a, b, result))
        return BR_DONE;
    return BR_FAILED;
}

/* Creates a normalized intersection. */
br_status seq_rewriter::mk_re_inter(expr* a, expr* b, expr_ref& result) {
    if (try_collapse_re_inter(a, b, result))
        return BR_DONE;
    result = mk_regex_inter_normalize(a, b);
    return BR_DONE;
}

br_status seq_rewriter::mk_re_diff(expr* a, expr* b, expr_ref& result) {
    seq::range_predicate pa(u().max_char()), pb(u().max_char());
    sort* seq_sort = nullptr;
    if (u().is_re(a->get_sort(), seq_sort)
        && seq::regex_to_range_predicate(u(), a, pa)
        && seq::regex_to_range_predicate(u(), b, pb)) {
        result = seq::range_predicate_to_regex(u(), pa - pb, seq_sort);
        return BR_DONE;
    }
    result = mk_inter(a, mk_complement(b));
    return BR_DONE;
}

/*
   Symmetric difference / XOR of regexes.
   LANG(a XOR b) = (LANG(a) \ LANG(b)) U (LANG(b) \ LANG(a))

   Equivalence preserving rewrites applied here (paper Section 5):
     r XOR r           = []
     r XOR []          = r
     [] XOR r          = r
     comp(r) XOR comp(s)  = r XOR s
     r XOR comp(s)     = comp(r XOR s)
     comp(r) XOR s     = comp(r XOR s)
     full_seq XOR r    = comp(r)
     r XOR full_seq    = comp(r)
   We also normalize the argument order using expression ids so that the
   structure is canonical for AC.
*/
br_status seq_rewriter::mk_re_xor0(expr* a, expr* b, expr_ref& result) {
    // Reduction-only variant of mk_re_xor for use inside mk_der_op.
    // Avoids any transformation that would create a top-level re.xor
    // node (e.g. AC normalisation or complement absorption), because
    // mk_der_op needs to keep distributing the operation through ITE
    // BDDs. Only structural simplifications that produce a non-XOR
    // result are applied here.
    if (a == b) {
        result = re().mk_empty(a->get_sort());
        return BR_DONE;
    }
    if (re().is_empty(a)) {
        result = b;
        return BR_DONE;
    }
    if (re().is_empty(b)) {
        result = a;
        return BR_DONE;
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_re_xor(expr* a, expr* b, expr_ref& result) {
    if (a == b) {
        result = re().mk_empty(a->get_sort());
        return BR_DONE;
    }
    if (re().is_empty(a)) {
        result = b;
        return BR_DONE;
    }
    if (re().is_empty(b)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_seq(a)) {
        result = mk_complement(b);
        return BR_DONE;
    }
    if (re().is_full_seq(b)) {
        result = mk_complement(a);
        return BR_DONE;
    }
    expr* ra = nullptr, * rb = nullptr;
    bool ca = re().is_complement(a, ra);
    bool cb = re().is_complement(b, rb);
    if (ca && cb) {
        // comp(ra) XOR comp(rb) = ra XOR rb
        result = mk_re_xor_simplified(ra, rb);
        return BR_DONE;
    }
    if (ca) {
        // comp(ra) XOR b = comp(ra XOR b)
        result = mk_complement(mk_re_xor_simplified(ra, b));
        return BR_DONE;
    }
    if (cb) {
        // a XOR comp(rb) = comp(a XOR rb)
        result = mk_complement(mk_re_xor_simplified(a, rb));
        return BR_DONE;
    }
    // Normalize order using expression ids (AC normalization).
    if (a->get_id() > b->get_id()) {
        result = re().mk_xor(b, a);
        return BR_DONE;
    }
    return BR_FAILED;
}


br_status seq_rewriter::mk_re_loop(func_decl* f, unsigned num_args, expr* const* args, expr_ref& result) {
    rational n1, n2;
    unsigned lo, hi, lo2, hi2, np;
    expr* a = nullptr;
    switch (num_args) {
    case 1: 
        np = f->get_num_parameters();
        lo2 = np > 0 ? f->get_parameter(0).get_int() : 0;
        hi2 = np > 1 ? f->get_parameter(1).get_int() : lo2;
        if  (np == 2 && (lo2 > hi2 || hi2 < 0)) {
            result = re().mk_empty(args[0]->get_sort());
            return BR_DONE;
        }
        if (np == 1 && lo2 < 0) {
            result = re().mk_empty(args[0]->get_sort());
            return BR_DONE;
        }
        // (loop a 0 0) = ""
        if (np == 2 && lo2 == 0 && hi2 == 0) {
            result = mk_to_re(str().mk_empty(re().to_seq(args[0]->get_sort())));
            return BR_DONE;
        }
        // Handle zero repetitions before multiplying nested lower bounds:
        // (a{2,})* is not a*, since it cannot accept exactly one a.
        if (np == 1 && lo2 == 0) {
            result = mk_star(args[0]);
            return BR_DONE;
        }
        if (np == 2 && lo2 == 1 && hi2 == 1) {
            result = args[0];
            return BR_DONE;
        }
        // (loop (loop a lo) lo2) = (loop lo*lo2)
        if (re().is_loop(args[0], a, lo) && np == 1 &&
            (lo == 0 || lo2 <= UINT_MAX / lo)) {
            result = mk_loop(a, lo2 * lo);
            return BR_DONE;
        }
        // (loop (loop a l l) h h) = (loop a l*h l*h)
        if (re().is_loop(args[0], a, lo, hi) && np == 2 && lo == hi && lo2 == hi2 &&
            (lo == 0 || lo2 <= UINT_MAX / lo)) {
            result = mk_loop(a, lo2 * lo, hi2 * hi);
            return BR_DONE;
        }
        break;
    case 2:
        if (m_autil.is_numeral(args[1], n1) && n1.is_unsigned()) {
            result = mk_loop(args[0], n1.get_unsigned());
            return BR_DONE;
        }
        if (m_autil.is_numeral(args[1], n1) && n1 < 0) {
            result = re().mk_empty(args[0]->get_sort());
            return BR_DONE;
        }
        break;
    case 3:
        if (m_autil.is_numeral(args[1], n1) && n1.is_unsigned() &&
            m_autil.is_numeral(args[2], n2) && n2.is_unsigned()) {
            result = mk_loop(args[0], n1.get_unsigned(), n2.get_unsigned());
            return BR_DONE;
        }
        break;
    default:
        break;
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_re_power(func_decl* f, expr* a, expr_ref& result) {
    unsigned p = f->get_parameter(0).get_int();
    result = mk_loop(a, p, p);
    return BR_DONE;
}


/*
  a** = a*
  (a* + b)* = (a + b)*
  (a + b*)* = (a + b)*
  (a*b*)*   = (a + b)*
   a+* = a*
   emp* = ""
   all* = all   
   .+* = all
*/
br_status seq_rewriter::mk_re_star(expr* a, expr_ref& result) {
    expr* b, *c, *b1, *c1;
    if (re().is_star(a) || re().is_full_seq(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_char(a)) {
        result = re().mk_full_seq(a->get_sort());
        return BR_DONE;
    }
    if (re().is_empty(a)) {
        sort* seq_sort = nullptr;
        VERIFY(m_util.is_re(a, seq_sort));
        result = mk_to_re(str().mk_empty(seq_sort));
        return BR_DONE;
    }
    if (re().is_to_re(a, b) && str().is_empty(b)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_plus(a, b)) {
        if (re().is_full_char(b))
            result = re().mk_full_seq(a->get_sort());
        else
            result = mk_star(b);
        return BR_DONE;
    }
    if (re().is_union(a, b, c)) {
        if (re().is_star(b, b1)) {
            result = mk_star(mk_union(b1, c));
            return BR_DONE;
        }
        if (re().is_star(c, c1)) {
            result = mk_star(mk_union(b, c1));
            return BR_DONE;
        }
        if (re().is_epsilon(b)) {
            result = mk_star(c);
            return BR_DONE;
        }
        if (re().is_epsilon(c)) {
            result = mk_star(b);
            return BR_DONE;
        }
    }
    if (re().is_concat(a, b, c) &&
        re().is_star(b, b1) && re().is_star(c, c1)) {
        result = mk_star(mk_union(b1, c1));
        return BR_DONE;
    }
    // (Σ*·S)* = () | Σ*·S.
    // Σ*·S is idempotent under concatenation: Σ*·S·Σ*·S = (Σ*·S·Σ*)·S ⊆ Σ*·S,
    // since any prefix is absorbed by Σ*. Hence L·L ⊆ L and L* = () | L.
    // Keeping the flat form avoids a large blowup in the derivative automaton.
    if (re().is_concat(a, b, c) && re().is_full_seq(b)) {
        sort* seq_sort = nullptr;
        VERIFY(m_util.is_re(a, seq_sort));
        result = mk_union(re().mk_epsilon(seq_sort), a);
        return BR_DONE;
    }
    if (m().is_ite(a, c, b1, c1)) {
        if ((re().is_full_char(b1) || re().is_full_seq(b1)) &&
            (re().is_full_char(c1) || re().is_full_seq(c1))) {
            result = re().mk_full_seq(b1->get_sort());
            return BR_REWRITE2;
        }
        auto then_r = mk_star(b1);
        auto else_r = mk_star(c1);
        m_br.mk_ite(c, then_r, else_r, result);
        return BR_DONE;
    }
    return BR_FAILED;
}

/*
 * (re.range c_1 c_n) 
 */
br_status seq_rewriter::mk_re_range(expr* lo, expr* hi, expr_ref& result) {
    zstring slo, shi;
    unsigned clo = 0, chi = 0;
    expr *lo1, *hi1;
    unsigned len = 0;
    bool is_empty = false;
    len = min_length(lo).second;
    if (len > 1)
        is_empty = true;
    len = min_length(hi).second;
    if (len > 1)
        is_empty = true;
    // A bound that is provably of length 0 (e.g. the empty string "") can
    // likewise never be a single character, so the range is empty.  Unlike a
    // symbolic bound, max_length == 0 is a provable emptiness fact, so this is
    // sound (it is never true for a model-dependent bound such as a variable).
    if (max_length(lo) == std::make_pair(true, rational(0)))
        is_empty = true;
    if (max_length(hi) == std::make_pair(true, rational(0)))
        is_empty = true;

    // A provable length constraint (a bound can never be a single character)
    // is the only sound way to conclude emptiness for a possibly-symbolic
    // bound, so decide emptiness here before attempting to read concrete
    // characters.
    if (is_empty) {
        sort* srt = re().mk_re(lo->get_sort());
        result = re().mk_empty(srt);
        return BR_DONE;
    }

    // Try to read concrete single-character bounds.  A bound that is not a
    // syntactic single-character literal is *symbolic* (its value depends on
    // the model), NOT empty: collapsing such a range to re.empty is unsound
    // (e.g. (re.range x x) is {x} whenever x is a single character), so we
    // leave the range unevaluated (BR_FAILED) and let the theory solver
    // reason about it.
    bool has_clo = false, has_chi = false;
    if (str().is_string(lo, slo) && slo.length() == 1) {
        clo = slo[0];
        has_clo = true;
    }
    else if (str().is_unit(lo, lo1) && m_util.is_const_char(lo1, clo))
        has_clo = true;
    if (str().is_string(hi, shi) && shi.length() == 1) {
        chi = shi[0];
        has_chi = true;
    }
    else if (str().is_unit(hi, hi1) && m_util.is_const_char(hi1, chi))
        has_chi = true;

    if (!has_clo || !has_chi)
        return BR_FAILED;

    // Both bounds are concrete characters: an inverted range is empty.
    if (clo > chi) {
        sort* srt = re().mk_re(lo->get_sort());
        result = re().mk_empty(srt);
        return BR_DONE;
    }

    // Singleton: re.range "a" "a" → str.to_re "a"
    if (clo == chi) {
        result = re().mk_to_re(str().mk_string(zstring(clo)));
        return BR_DONE;
    }

    return BR_FAILED;
}

/*
   emp+ = emp
   all+ = all
   a*+ = a*
   a++ = a+
   a+ = aa*
*/
br_status seq_rewriter::mk_re_plus(expr* a, expr_ref& result) {
    // .+ is the canonical nonempty universal language; expanding it would
    // cycle through concat(., .*) and back to .+.
    if (re().is_full_char(a))
        return BR_FAILED;
    if (re().is_empty(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_full_seq(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_epsilon(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_plus(a)) {
        result = a;
        return BR_DONE;
    }
    if (re().is_star(a)) {
        result = a;
        return BR_DONE;
    }

    result = mk_re_append(a, mk_star(a));
    return BR_DONE;
}

br_status seq_rewriter::mk_re_opt(expr* a, expr_ref& result) {
    sort* s = nullptr;
    VERIFY(m_util.is_re(a, s));
    result = mk_union(mk_to_re(str().mk_empty(s)), a);
    return BR_DONE;
}

void seq_rewriter::intersect(unsigned lo, unsigned hi, svector<std::pair<unsigned, unsigned>>& ranges) {
    unsigned j = 0;
    for (unsigned i = 0; i < ranges.size(); ++i) {
        auto [lo1, hi1] = ranges[i];
        if (hi < lo1) 
            break;
        if (hi1 >= lo) 
            ranges[j++] = std::make_pair(std::max(lo1, lo), std::min(hi1, hi));
    }
    ranges.shrink(j);
}

/**
 * Simplify cond using special case rewriting for character equations
 * When elem is uninterpreted compute the simplification of Exists elem . cond
 * if it is possible to solve for elem.
 */
void seq_rewriter::elim_condition(expr* elem, expr_ref& cond) {
    expr_ref_vector conds(m());
    expr_ref_vector conds_range(m());
    flatten_and(cond, conds);
    expr* lhs = nullptr, *rhs = nullptr, *e1 = nullptr; 
    bool all_ranges = false;
    if (u().is_char(elem)) {
        all_ranges = true;
        unsigned ch = 0, ch2 = 0;
        svector<std::pair<unsigned, unsigned>> ranges, ranges1;
        ranges.push_back(std::make_pair(0, u().max_char()));
        auto exclude_range = [&](unsigned lower, unsigned upper) {
            SASSERT(lower <= upper);
            if (lower == 0) {
                if (upper == u().max_char())
                    ranges.reset();
                else
                    intersect(upper + 1, u().max_char(), ranges);
            }
            else if (upper == u().max_char())
                intersect(0, lower - 1, ranges);
            else {
                // not(lower <= e <= upper) iff ((0 <= e <= lower-1) or (upper+1 <= e <= max))
                // Note that this transformation is correct only when lower <= upper
                ranges1.reset();
                ranges1.append(ranges);
                intersect(0, lower - 1, ranges);
                intersect(upper + 1, u().max_char(), ranges1);
                ranges.append(ranges1);
            }
        };
        bool negated = false;
        for (expr* e : conds) {
            if (u().is_char_const_range(elem, e, ch, ch2, negated)) {
                if (ch > ch2) {
                    if (negated)
                        // !(ch <= elem <= ch2) is trivially true
                        continue;
                    else
                        // (ch <= elem <= ch2) is trivially false
                        ranges.reset();
                }
                else if (negated)
                    exclude_range(ch, ch2);
                else
                    intersect(ch, ch2, ranges);
                conds_range.push_back(e);
            }
            // trivially true conditions
            else if (m().is_true(e) || (m().is_eq(e, lhs, rhs) && lhs == rhs))
                continue;
            else if (m().is_not(e, e1) && m().is_eq(e1, lhs, rhs) && u().is_const_char(lhs, ch) && u().is_const_char(rhs, ch2) && ch != ch2)
                continue;
            else if (u().is_char_le(e, lhs, rhs) && u().is_const_char(lhs, ch) && u().is_const_char(rhs, ch2) && ch <= ch2)
                continue;
            else if (m().is_not(e, e1) && u().is_char_le(e1, lhs, rhs) && u().is_const_char(lhs, ch) && u().is_const_char(rhs, ch2) && ch > ch2)
                continue;
            // trivially false conditions
            else if (m().is_false(e) || (m().is_not(e, e1) && m().is_eq(e1, lhs, rhs) && lhs == rhs))
                ranges.reset();
            else if (u().is_char_le(e, lhs, rhs) && u().is_const_char(lhs, ch) && u().is_const_char(rhs, ch2) && ch > ch2)
                ranges.reset();
            else if (m().is_not(e, e1) && u().is_char_le(e1, lhs, rhs) && u().is_const_char(lhs, ch) && u().is_const_char(rhs, ch2) && ch <= ch2)
                ranges.reset();
            else {
                all_ranges = false;
                break;
            }
            if (ranges.empty())
                break;
        }
        if (all_ranges) {
            if (ranges.empty()) {
                cond = m().mk_false();
                return;
            }
            if (is_uninterp_const(elem)) {
                cond = m().mk_true();
                return;
            }
            conds.set(conds_range);
        }
    }
            
    expr* solution = nullptr;
    for (expr* e : conds) {
        if (!m().is_eq(e, lhs, rhs)) 
            continue;
        if (rhs == elem)
            std::swap(lhs, rhs);
        if (lhs != elem)
            continue;
        solution = rhs;
        break;        
    }
    if (solution) {
        expr_safe_replace rep(m());
        rep.insert(elem, solution);
        rep(cond);
        if (!is_uninterp_const(elem)) { 
            // ensure deterministic evaluation order when augmenting condition
            auto eq_sol = m().mk_eq(elem, solution);
            cond = m().mk_and(eq_sol, cond);
        }
    }    
    else if (all_ranges) {
        if (conds.empty())
            // all ranges were removed as trivially true
            cond = m().mk_true();
        else if (conds.size() == 1)
            cond = conds.get(0);
        else
            cond = m().mk_and(conds);
    }
}


br_status seq_rewriter::reduce_re_is_empty(expr* r, expr_ref& result) {
    expr* r1, *r2, *r3, *r4;
    zstring s1, s2;
    unsigned lo, hi;
    auto eq_empty = [&](expr* r) { return m().mk_eq(r, re().mk_empty(r->get_sort())); };
    if (re().is_union(r, r1, r2)) {
        expr_ref lhs(eq_empty(r1), m());
        expr_ref rhs(eq_empty(r2), m());
        result = m().mk_and(lhs, rhs);
        return BR_REWRITE2;
    }
    if (re().is_star(r) ||
        re().is_to_re(r) ||
        re().is_full_char(r) ||
        re().is_full_seq(r)) {
        result = m().mk_false();
        return BR_DONE;
    }
    if (re().is_concat(r, r1, r2)) {
        expr_ref lhs(eq_empty(r1), m());
        expr_ref rhs(eq_empty(r2), m());
        result = m().mk_or(lhs, rhs);
        return BR_REWRITE2;
    }
    else if (re().is_range(r, r1, r2) && 
             str().is_string(r1, s1) && str().is_string(r2, s2) && 
             s1.length() == 1 && s2.length() == 1) {
        result = m().mk_bool_val(s1[0] > s2[0]);
        return BR_DONE;
    }
    else if (re().is_range(r, r1, r2) && 
             str().is_string(r1, s1) && s1.length() != 1) {
        result = m().mk_true();
        return BR_DONE;
    }
    else if (re().is_range(r, r1, r2) && 
             str().is_string(r2, s2) && s2.length() != 1) {
        result = m().mk_true();
        return BR_DONE;
    }
    else if ((re().is_loop(r, r1, lo) ||
              re().is_loop(r, r1, lo, hi)) && lo == 0) {
        result = m().mk_false();
        return BR_DONE;
    }
    else if (re().is_loop(r, r1, lo) ||
             (re().is_loop(r, r1, lo, hi) && lo <= hi)) {
        result = eq_empty(r1);
        return BR_REWRITE1;
    }
    // Partial DNF expansion:
    else if (re().is_intersection(r, r1, r2) && re().is_union(r1, r3, r4)) {
        // enforce deterministic order for nested intersections inside union
        auto a1 = re().mk_inter(r3, r2);
        auto b1 = re().mk_inter(r4, r2);
        result = eq_empty(re().mk_union(a1, b1));
        return BR_REWRITE3;
    }
    else if (re().is_intersection(r, r1, r2) && re().is_union(r2, r3, r4)) {
        auto a1 = re().mk_inter(r3, r1);
        auto b1 = re().mk_inter(r4, r1);
        result = eq_empty(re().mk_union(a1, b1));
        return BR_REWRITE3;
    }
    return BR_FAILED;
}

br_status seq_rewriter::reduce_re_eq(expr* l, expr* r, expr_ref& result) {
    if (re().is_empty(l)) {
        std::swap(l, r);
    }
    if (re().is_empty(r)) {
        return reduce_re_is_empty(l, result);
    }
    if (l == r) {
        result = m().mk_true();
        return BR_DONE;
    }
    /*
     * Try the union-find bisimulation procedure for ground regex equality.
     * Guarded against re-entry because the bisim may construct equalities
     * indirectly.  On l_undef the rewriter falls through to the existing
     * axiomatisation path.
     */
    if (!m_in_bisim && re().is_ground(l) && re().is_ground(r)) {
        flet<bool> _block(m_in_bisim, true);
        seq::regex_bisim bisim(*this);
        switch (bisim.are_equivalent(l, r)) {
        case l_true:
            result = m().mk_true();
            return BR_DONE;
        case l_false:
            result = m().mk_false();
            return BR_DONE;
        case l_undef:
            break;
        }
    }
    return BR_FAILED;
}

br_status seq_rewriter::mk_le_core(expr * l, expr * r, expr_ref & result) {

    return BR_FAILED;

    // k <= len(x) -> true  if k <= 0
    rational n;
    if (str().is_length(r) && m_autil.is_numeral(l, n) && n <= 0) {
        result = m().mk_true();
        return BR_DONE;
    } 
    // len(x) <= 0 -> x = ""
    // len(x) <= k -> false if k < 0
    expr* e = nullptr;
    if (str().is_length(l, e) && m_autil.is_numeral(r, n)) {
        if (n == 0)
            result = str().mk_is_empty(e);
        else if (n < 0)
            result = m().mk_false();
        else
            return BR_FAILED;
        return BR_REWRITE1;
    } 
    return BR_FAILED;
}

br_status seq_rewriter::mk_eq_core(expr * l, expr * r, expr_ref & result) {
    TRACE(seq, tout << mk_pp(l, m()) << " == " << mk_pp(r, m()) << "\n");
    expr_ref_vector res(m());
    expr_ref_pair_vector new_eqs(m());
    if (m_util.is_re(l)) {
        return reduce_re_eq(l, r, result);
    }
    bool changed = false;
    if (reduce_eq_empty(l, r, result)) 
        return BR_REWRITE_FULL;
    // s^n = s^k <=> s = "" or (n <= 0 and k <= 0) or (0 < n = k)
    expr* s1 = nullptr, *n1 = nullptr, *s2 = nullptr, *n2 = nullptr;
    if (str().is_power(l, s1, n1) && str().is_power(r, s2, n2) && s1 == s2) {
        expr_ref emp(str().mk_empty(s1->get_sort()), m());
        expr_ref_vector fmls(m());
        fmls.push_back(m().mk_eq(s1, emp));
        expr_ref n1_nonpos(m_autil.mk_le(n1, zero()), m());
        expr_ref n2_nonpos(m_autil.mk_le(n2, zero()), m());
        fmls.push_back(m().mk_and(n1_nonpos, n2_nonpos));
        expr_ref n1_pos(m_autil.mk_lt(zero(), n1), m());
        expr_ref n2_pos(m_autil.mk_lt(zero(), n2), m());
        expr_ref eq(m().mk_eq(n1, n2), m());
        fmls.push_back(m().mk_and(n1_pos, n2_pos, eq));
        result = m().mk_or(fmls);
        return BR_REWRITE_FULL;
    }

    // a, b are unit-length ground strings => replace_all(x, a, b) in re.to_re(s)
    {
        expr *ra_x = nullptr, *ra_a = nullptr, *ra_b = nullptr;
        zstring sa_val, sb_val, s_val;
        expr *str_side = nullptr, *ra_side = nullptr;
        if (str().is_replace_all(l))
            ra_side = l, str_side = r;
        else if (str().is_replace_all(r))
            ra_side = r, str_side = l;
        if (ra_side && str_side && 
            str().is_replace_all(ra_side, ra_x, ra_a, ra_b) && str().is_string(ra_a, sa_val) &&
            sa_val.length() == 1 && 
            str().is_string(ra_b, sb_val) && sb_val.length() == 1 &&
            str().is_string(str_side, s_val)) {
            result = re().mk_in_re(ra_side, re().mk_to_re(str_side));
            return BR_REWRITE_FULL;
        }
    }

#if 0
    if (reduce_arith_eq(l, r, res) || reduce_arith_eq(r, l, res)) {
        result = mk_and(res);
        TRACE(seq_verbose, tout << result << "\n";);
        return BR_REWRITE3;        
    }

    if (reduce_extract(l, r, res)) {
        result = mk_and(res);
        TRACE(seq_verbose, tout << result << "\n";);
        return BR_REWRITE3;        
    }
#endif

    if (!reduce_eq(l, r, new_eqs, changed)) {
        result = m().mk_false();
        TRACE(seq_verbose, tout << result << "\n";);
        return BR_DONE;
    }
    if (!changed) {
        return BR_FAILED;
    }
    for (auto const& [lhs, rhs] : new_eqs) {
        res.push_back(m().mk_eq(lhs, rhs));
    }
    result = mk_and(res);
    TRACE(seq_verbose, tout << result << "\n";);
    return BR_REWRITE3;
}

void seq_rewriter::remove_empty_and_concats(expr_ref_vector& es) {
    unsigned j = 0;
    bool has_concat = false;
    for (expr* e : es) {
        has_concat |= str().is_concat(e);
        if (!str().is_empty(e))
            es[j++] = e;
    }
    es.shrink(j);
    if (has_concat) {
        expr_ref_vector fs(m());
        for (expr* e : es) 
            str().get_concat(e, fs);
        es.swap(fs);
    }
}

void seq_rewriter::remove_leading(unsigned n, expr_ref_vector& es) {
    SASSERT(n <= es.size());
    if (n == 0)
        return;
    for (unsigned i = n; i < es.size(); ++i) {
        es[i-n] = es.get(i);
    }
    es.shrink(es.size() - n);
}

bool seq_rewriter::reduce_back(expr_ref_vector& ls, expr_ref_vector& rs, expr_ref_pair_vector& new_eqs) {    
    expr* a, *b;    
    zstring s, s1, s2;
    while (true) {
        if (ls.empty() || rs.empty()) {
            break;
        }
        expr* l = ls.back();
        expr* r = rs.back();            
        if (str().is_unit(r) && str().is_string(l)) {
            std::swap(l, r);
            ls.swap(rs);
        }
        if (l == r) {
            ls.pop_back();
            rs.pop_back();
        }
        else if(str().is_unit(l, a) &&
                str().is_unit(r, b)) {
            if (m().are_distinct(a, b)) {
                return false;
            }
            new_eqs.push_back(a, b);
            ls.pop_back();
            rs.pop_back();
        }
        else if (str().is_unit(l, a) && str().is_string(r, s)) {
            SASSERT(s.length() > 0);
            
            app_ref ch(str().mk_char(s, s.length()-1), m());
            SASSERT(ch->get_sort() == a->get_sort());
            new_eqs.push_back(ch, a);
            ls.pop_back();
            if (s.length() == 1) {
                rs.pop_back();
            }
            else {
                expr_ref s2(str().mk_string(s.extract(0, s.length()-1)), m());
                rs[rs.size()-1] = s2;
            }
        }
        else if (str().is_string(l, s1) && str().is_string(r, s2)) {
            unsigned min_l = std::min(s1.length(), s2.length());
            for (unsigned i = 0; i < min_l; ++i) {
                if (s1[s1.length()-i-1] != s2[s2.length()-i-1]) {
                    return false;
                }
            }
            ls.pop_back();          
            rs.pop_back();
            if (min_l < s1.length()) {
                ls.push_back(str().mk_string(s1.extract(0, s1.length()-min_l)));
            }
            if (min_l < s2.length()) {
                rs.push_back(str().mk_string(s2.extract(0, s2.length()-min_l)));
            }        
        }
        else {
            break;
        }
    }
    return true;
}

bool seq_rewriter::reduce_front(expr_ref_vector& ls, expr_ref_vector& rs, expr_ref_pair_vector& new_eqs) {    
    expr* a, *b;    
    zstring s, s1, s2;
    unsigned head1 = 0, head2 = 0;
    while (true) {
        if (head1 == ls.size() || head2 == rs.size()) {
            break;
        }
        SASSERT(head1 < ls.size() && head2 < rs.size());

        expr* l = ls.get(head1);
        expr* r = rs.get(head2);
        if (str().is_unit(r) && str().is_string(l)) {
            std::swap(l, r);
            ls.swap(rs);
            std::swap(head1, head2);
        }
        if (l == r) {
            ++head1;
            ++head2;
        }
        else if(str().is_unit(l, a) &&
                str().is_unit(r, b)) {
            if (m().are_distinct(a, b)) {
                return false;
            }
            new_eqs.push_back(a, b);
            ++head1;
            ++head2;
        }
        else if (str().is_unit(l, a) && str().is_string(r, s)) {
            SASSERT(s.length() > 0);
            app* ch = str().mk_char(s, 0);
            SASSERT(ch->get_sort() == a->get_sort());
            new_eqs.push_back(ch, a);
            ++head1;
            if (s.length() == 1) {
                ++head2;
            }
            else {
                expr_ref s2(str().mk_string(s.extract(1, s.length()-1)), m());
                rs[head2] = s2;
            }            
        }
        else if (str().is_string(l, s1) &&
                 str().is_string(r, s2)) {
            TRACE(seq, tout << s1 << " - " << s2 << " " << s1.length() << " " << s2.length() << "\n";);
            unsigned min_l = std::min(s1.length(), s2.length());
            for (unsigned i = 0; i < min_l; ++i) {
                if (s1[i] != s2[i]) {
                    TRACE(seq, tout << "different at position " << i << " " << s1[i] << " " << s2[i] << "\n";);
                    return false;
                }
            }
            if (min_l == s1.length()) {
                ++head1;            
            }
            else {
                ls[head1] = str().mk_string(s1.extract(min_l, s1.length()-min_l));
            }
            if (min_l == s2.length()) {
                ++head2;            
            }
            else {
                rs[head2] = str().mk_string(s2.extract(min_l, s2.length()-min_l));
            }
        }
        else {
            break;
        }
    }
    remove_leading(head1, ls);
    remove_leading(head2, rs);
    return true;
}

bool seq_rewriter::split_bag(expr_ref_vector &ls, expr_ref_vector &rs, expr_ref_pair_vector &new_eqs) {
    auto eq = [](auto &a, auto &b) {
        if (a.size() != b.size())
            return false;
        for (auto [k, v] : a)
            if (!b.contains(k) || b[k] != v)
                return false;
        return true;
    };
    obj_map<expr, unsigned> ls_bag, rs_bag;
    for (unsigned i = ls.size(), j = rs.size(); i-- > 0 && j-- > 0 && (i > 0 || j > 0);) {
        ls_bag.insert_if_not_there(ls.get(i), 0)++;
        rs_bag.insert_if_not_there(rs.get(j), 0)++;
        if (eq(ls_bag, rs_bag)) {
            auto l = str().mk_concat(ls.size() - i, ls.data() + i, ls.get(i)->get_sort());
            auto r = str().mk_concat(rs.size() - j, rs.data() + j, rs.get(j)->get_sort());
            new_eqs.push_back(l, r);
            ls_bag.reset();
            rs_bag.reset();
            ls.shrink(i);
            rs.shrink(j);
        }
    }
    ls_bag.reset();
    rs_bag.reset();
    unsigned start = 0;
    for (unsigned i = 0; i + 1 < ls.size() && i + 1 < rs.size(); ++i) {
        ls_bag.insert_if_not_there(ls.get(i), 0)++;
        rs_bag.insert_if_not_there(rs.get(i), 0)++;
        if (eq(ls_bag, rs_bag)) {
            auto l = str().mk_concat(i + 1 - start, ls.data() + start, ls.get(i)->get_sort());
            auto r = str().mk_concat(i + 1 - start, rs.data() + start, rs.get(i)->get_sort());
            new_eqs.push_back(l, r);
            ls_bag.reset();
            rs_bag.reset();
            start = i + 1;
        }
    }
    if (start > 0) {
        remove_leading(start, ls);
        remove_leading(start, rs);
    }
    return true;
}

/**
   \brief simplify equality ls = rs
   - New equalities are inserted into eqs.
   - Last remaining equalities that cannot be simplified further are kept in ls, rs
   - returns false if equality is unsatisfiable   
   - sets change to true if some simplification occurred
*/
bool seq_rewriter::reduce_eq(expr_ref_vector& ls, expr_ref_vector& rs, expr_ref_pair_vector& eqs, bool& change) {
    TRACE(seq_verbose, tout << ls << "\n"; tout << rs << "\n";);
    unsigned hash_l = ls.hash();
    unsigned hash_r = rs.hash();
    unsigned sz_eqs = eqs.size();
    remove_empty_and_concats(ls);
    remove_empty_and_concats(rs);
    return 
        reduce_back(ls, rs, eqs) && 
        reduce_front(ls, rs, eqs) &&
        reduce_itos(ls, rs, eqs) &&
        reduce_itos(rs, ls, eqs) &&
        reduce_value_clash(ls, rs, eqs) && 
        reduce_by_length(ls, rs, eqs) &&
        reduce_subsequence(ls, rs, eqs) &&
        reduce_non_overlap(ls, rs, eqs) && 
        reduce_non_overlap(rs, ls, eqs) && 
        split_bag(ls, rs, eqs) &&
        (change = (hash_l != ls.hash() || hash_r != rs.hash() || eqs.size() != sz_eqs), 
         true);
}

bool seq_rewriter::reduce_eq(expr* l, expr* r, expr_ref_pair_vector& new_eqs, bool& changed) {
    m_lhs.reset();
    m_rhs.reset();
    str().get_concat(l, m_lhs);
    str().get_concat(r, m_rhs);
    bool change = false;
    if (reduce_eq(m_lhs, m_rhs, new_eqs, change)) {
        if (!change) {
            new_eqs.push_back(l, r);
        }
        else {
            add_seqs(m_lhs, m_rhs, new_eqs);
        }
        changed |= change;
        return true;
    }
    else {
        TRACE(seq, tout << mk_bounded_pp(l, m()) << " != " << mk_bounded_pp(r, m()) << "\n";);
        return false;
    }
}

bool seq_rewriter::reduce_arith_eq(expr* l, expr* r, expr_ref_vector& res) {
    expr* s = nullptr, *sub = nullptr, *idx = nullptr;
    rational i, n;
    if (str().is_index(l, s, sub, idx) && m_autil.is_numeral(idx, i) && m_autil.is_numeral(r, n)) {
        if (n == 0 && i == 0) {
            res.push_back(str().mk_prefix(sub, s));
            return true;
        }
    }
    return false;
}

void seq_rewriter::add_seqs(expr_ref_vector const& ls, expr_ref_vector const& rs, expr_ref_pair_vector& eqs) {
    if (!ls.empty() || !rs.empty()) {
        sort * s = (ls.empty() ? rs[0] : ls[0])->get_sort();
        expr_ref lhs(str().mk_concat(ls, s), m());
        expr_ref rhs(str().mk_concat(rs, s), m());
        eqs.push_back(lhs, rhs);
    }
}


bool seq_rewriter::reduce_contains(expr* a, expr* b, expr_ref_vector& disj) {
    m_lhs.reset();
    str().get_concat(a, m_lhs);
    TRACE(seq, tout << expr_ref(a, m()) << " " << expr_ref(b, m()) << "\n";);
    sort* sort_a = a->get_sort();
    zstring s;
    for (unsigned i = 0; i < m_lhs.size(); ++i) {
        expr* e = m_lhs.get(i);
        if (str().is_empty(e)) {
            continue;
        }

        if (str().is_string(e, s)) {
            unsigned sz = s.length();
            expr_ref_vector es(m());
            for (unsigned j = 0; j < sz; ++j) {
                es.push_back(str().mk_unit(str().mk_char(s, j)));
            }
            es.append(m_lhs.size() - i, m_lhs.data() + i);
            for (unsigned j = 0; j < sz; ++j) {
                disj.push_back(str().mk_prefix(b, str().mk_concat(es.size() - j, es.data() + j, sort_a)));
            }
            continue;
        }
        if (str().is_unit(e)) {
            disj.push_back(str().mk_prefix(b, str().mk_concat(m_lhs.size() - i, m_lhs.data() + i, sort_a)));
            continue;
        }

        if (str().is_string(b, s)) {
            expr* all = re().mk_full_seq(re().mk_re(b->get_sort()));
            {
                auto _seq4840_0 = str().mk_concat(m_lhs.size() - i, m_lhs.data() + i, sort_a);
                auto _seq4840_1 = re().mk_concat(all, re().mk_concat(re().mk_to_re(b), all));
                disj.push_back(re().mk_in_re(_seq4840_0, _seq4840_1));
            }
            return true;
        }

        if (i == 0) {
            return false;
        }
        disj.push_back(str().mk_contains(str().mk_concat(m_lhs.size() - i, m_lhs.data() + i, sort_a), b));
        return true;
    }
    disj.push_back(str().mk_is_empty(b));
    return true;
}


expr* seq_rewriter::concat_non_empty(expr_ref_vector& es) {
    sort* s = es[0]->get_sort();
    unsigned j = 0;
    for (expr* e : es) {
        if (str().is_unit(e) || str().is_string(e) || m().is_ite(e))
            es[j++] = e;
    }
    es.shrink(j);
    return str().mk_concat(es, s);
}

/**
  \brief assign the non-unit and non-string elements to the empty sequence.
  If all is true, then return false if there is a unit or non-empty substring.
*/

bool seq_rewriter::set_empty(unsigned sz, expr* const* es, bool all, expr_ref_pair_vector& eqs) {
    zstring s;
    expr* emp = nullptr;
    for (unsigned i = 0; i < sz; ++i) {
        auto [bounded, len] = min_length(es[i]);
        if (len > 0) {
            if (all)
                return false;
            continue;
        }
        if (bounded && len == 0)
            continue;
        emp = emp?emp:str().mk_empty(es[i]->get_sort());
        eqs.push_back(emp, es[i]);        
    }
    return true;
}

lbool seq_rewriter::eq_length(expr* x, expr* y) {
    auto [bounded_x, xl] = min_length(x);
    if (!bounded_x)
        return l_undef;
    auto [bounded_y, yl] = min_length(y);
    if (!bounded_y)
        return l_undef;
    return xl == yl ? l_true : l_false;
}

/***
    \brief extract the minimal length of the sequence.
    Return true if the minimal length is equal to the 
    maximal length (the sequence is bounded).
*/

std::pair<bool, unsigned> seq_rewriter::min_length(unsigned sz, expr* const* ss) {
    ptr_buffer<expr> es, sub;
    for (unsigned i = 0; i < sz; ++i)
        es.push_back(ss[i]);
    obj_map<expr, std::pair<bool, unsigned>> cache;
    
    zstring s;
    unsigned len = 0;
    bool bounded = true;

    if (sz == 0)
        return { bounded, len };
    auto visit = [&](expr* e) {
        expr* c, *th, *el;
        if (cache.contains(e))
            return true;
        if (str().is_unit(e)) {
            cache.insert(e, { true, 1u });
            return true;
        }
        else if (str().is_empty(e)) {
            cache.insert(e, { true, 0u });
            return true;
        }
        else if (str().is_string(e, s)) {
            cache.insert(e, { true, s.length() });
            return true;
        }
        else if (str().is_concat(e)) {
            bool visited = true;
            bool is_valid = true;
            unsigned count = 0u;
            for (expr* arg : *to_app(e)) {
                std::pair<bool, unsigned> r;
                if (cache.find(arg, r)) {
                    auto [r_valid, r_count] = r;
                    is_valid &= r_valid;
                    count += r_count;
                }
                else {
                    sub.push_back(arg);
                    visited = false;
                }
            }
            if (visited)
                cache.insert(e, { is_valid, count });
            return visited;
        }
        else if (m().is_ite(e, c, th, el)) {
            unsigned subsz = sub.size();
            std::pair<bool, unsigned> r1, r2;
            if (!cache.find(th, r1))
                sub.push_back(th);
            if (!cache.find(el, r2))
                sub.push_back(el);
            if (subsz != sub.size())
                return false;
            auto [r1_valid, r1_count] = r1;
            auto [r2_valid, r2_count] = r2;
            cache.insert(e, { r1_valid && r2_valid && r1_count == r2_count, std::min(r1_count, r2_count)});
            return true;
        }
        else {
            cache.insert(e, { false, 0u });
            return true;
        }
    };
    while (!es.empty()) {
        expr* c, *th, *el;
        expr* e = es.back();
        es.pop_back();
        if (str().is_unit(e)) 
            len += 1;
        else if (str().is_empty(e)) 
            continue;
        else if (str().is_string(e, s)) 
            len += s.length();
        else if (str().is_concat(e)) 
            for (expr* arg : *to_app(e))
                es.push_back(arg);
        else if (m().is_ite(e, c, th, el)) {
            sub.push_back(th);
            sub.push_back(el);
            while (!sub.empty()) {
                e = sub.back();
                if (visit(e)) 
                    sub.pop_back();
            }
            auto [bounded1, len1] = cache[th];
            auto [bounded2, len2] = cache[el];
            if (!bounded1 || !bounded2 || len1 != len2)
                bounded = false;
            len += std::min(len1, len2);
        }
        else
            bounded = false;
    }
    return { bounded, len };    
}

std::pair<bool, unsigned> seq_rewriter::min_length(expr* e) {
    return min_length(1, &e);
}

std::pair<bool, unsigned> seq_rewriter::min_length(expr_ref_vector const& es) {
    return min_length(es.size(), es.data());
}

std::pair<bool, rational> seq_rewriter::max_length(expr* e) {
    ptr_buffer<expr> es;
    es.push_back(e);
    rational len(0);
    zstring s;
    expr* s1 = nullptr, *i = nullptr, *l = nullptr;
    rational n;

    while (!es.empty()) {
        e = es.back();
        es.pop_back();
        if (str().is_unit(e)) 
            len += 1;
        else if (str().is_at(e))
            len += 1;
        else if (str().is_string(e, s))
            len += rational(s.length());
        else if (str().is_extract(e, s1, i, l) && m_autil.is_numeral(l, n) && !n.is_neg())
            len += n;
        else if (str().is_empty(e))
            continue;
        else if (str().is_concat(e)) {
            for (expr* arg : *to_app(e))
                es.push_back(arg);
        }
        else
            return std::make_pair(false, len);
    }
    return std::make_pair(true, len);
}


bool seq_rewriter::is_string(unsigned n, expr* const* es, zstring& s) const {
    zstring s1;
    expr* e;
    unsigned ch;
    for (unsigned i = 0; i < n; ++i) {
        if (str().is_string(es[i], s1)) {
            s = s + s1;
        }
        else if (str().is_unit(es[i], e) && m_util.is_const_char(e, ch)) {
            s = s + zstring(ch);
        }
        else {
            return false;
        }
    }
    return true;
}

expr_ref seq_rewriter::mk_length(expr* s) {
    expr_ref result(m());
    if (BR_FAILED == mk_seq_length(s, result))
        result = str().mk_length(s);
    return result;
}


/**
 * itos(n) = <numeric string> -> n = numeric
 */

bool seq_rewriter::reduce_itos(expr_ref_vector& ls, expr_ref_vector& rs,
                              expr_ref_pair_vector& eqs) {
    expr* n = nullptr;
    zstring s;
    if (ls.size() == 1 && 
        str().is_itos(ls.get(0), n) &&
        is_string(rs.size(), rs.data(), s)) {
        std::string s1 = s.encode();
        for (auto c : s1) {
            if (!('0' <= c && c <= '9'))
                return false;
        }
        if (s1.size() > 1 && s1[0] == '0')
            return false;
        try {
            rational r(s1.c_str());
            if (s1 == r.to_string()) {
                eqs.push_back(n, m_autil.mk_numeral(r, true));
                ls.reset();
                rs.reset();
                return true;
            }
        }
        catch (...)
        { }
    }
    return true;
}

bool seq_rewriter::reduce_eq_empty(expr* l, expr* r, expr_ref& result) {
    if (str().is_empty(r))
        std::swap(l, r);
    if (str().is_length(r))
        std::swap(l, r);
#if 0
    rational n;
    if (str().is_length(l) && m_autil.is_numeral(r, n) && n.is_zero()) {
        VERIFY(str().is_length(l, l));
        result = m().mk_eq(l, str().mk_empty(l->get_sort()));
        return true;
    }
#endif
    if (!str().is_empty(l))
        return false;
    expr* s = nullptr, *offset = nullptr, *len = nullptr;
    if (str().is_extract(r, s, offset, len)) {
        expr_ref len_s(str().mk_length(s), m()); 
        expr_ref_vector fmls(m());
        fmls.push_back(m_autil.mk_lt(offset, zero()));
        fmls.push_back(m().mk_eq(s, l));
        fmls.push_back(m_autil.mk_le(len, zero()));
        fmls.push_back(m_autil.mk_le(len_s, offset));
        result = m().mk_or(fmls);
        return true;
    }
    if (str().is_itos(r, s)) {
        result = m_autil.mk_lt(s, zero());
        return true;
    }
    // s^n = "" <=> n <= 0 or s = ""
    expr* base = nullptr, *n = nullptr;
    if (str().is_power(r, base, n)) {
        expr_ref_vector fmls(m());
        fmls.push_back(m_autil.mk_le(n, zero()));
        fmls.push_back(m().mk_eq(base, l));
        result = m().mk_or(fmls);
        return true;
    }
    // at(s, offset) = "" <=> len(s) <= offset or offset < 0
    if (str().is_at(r, s, offset)) {
        expr_ref len_s(str().mk_length(s), m());
        {
            auto _seq5138_0 = m_autil.mk_le(len_s, offset);
            auto _seq5138_1 = m_autil.mk_lt(offset, zero());
            result = m().mk_or(_seq5138_0, _seq5138_1);
        }
        return true;
    }
    return false;
}

bool seq_rewriter::has_var(expr_ref_vector const& es) {
    for (expr* e : es) {
        auto [bounded, len] = min_length(e);
        if (len == 0)
            return true;
    }
    return false;
}


bool seq_rewriter::reduce_by_length(expr_ref_vector& ls, expr_ref_vector& rs,
                                    expr_ref_pair_vector& eqs) {

    if (ls.empty() && rs.empty())
        return true;

    auto [bounded1, len1] = min_length(ls);
    auto [bounded2, len2] = min_length(rs);

    if (bounded1 && len1 < len2) 
        return false;
    if (bounded2 && len2 < len1) 
        return false;
    if (bounded1 && len1 == len2 && len1 > 0 && has_var(rs)) {
        if (!set_empty(rs.size(), rs.data(), false, eqs))
            return false;
        expr_ref lhs(concat_non_empty(ls), m());
        expr_ref rhs(concat_non_empty(rs), m());
        eqs.push_back(lhs, rhs);
        ls.reset(); 
        rs.reset();
    }
    else if (bounded2 && len1 == len2 && len1 > 0 && has_var(ls))  {
        if (!set_empty(ls.size(), ls.data(), false, eqs))
            return false;
        expr_ref lhs(concat_non_empty(ls), m());
        expr_ref rhs(concat_non_empty(rs), m());
        eqs.push_back(lhs, rhs);
        ls.reset(); 
        rs.reset();
    }
    return true;
}

/**
   reduce for the case where rs = a constant string, 
   ls contains a substring that matches no substring of rs.
 */
bool seq_rewriter::reduce_non_overlap(expr_ref_vector& ls, expr_ref_vector& rs, expr_ref_pair_vector& eqs) {
    for (expr* u : rs)
        if (!str().is_unit(u))
            return true;
    expr_ref_vector pattern(m());
    for (expr* x : ls) {
        if (str().is_unit(x))
            pattern.push_back(x);
        else if (!pattern.empty()) {
            if (non_overlap(pattern, rs))
                return false;
            pattern.reset();
        }
    }
    if (!pattern.empty() && non_overlap(pattern, rs))
        return false;
    return true;
}


/**
 * partial check for value clash.
 * checks that elements that do not occur in 
 * other sequence are non-values.
 * The check could be extended to take non-value 
 * characters (units) into account.
 */
bool seq_rewriter::reduce_value_clash(expr_ref_vector& ls, expr_ref_vector& rs, expr_ref_pair_vector& eqs) {
    ptr_buffer<expr> es;

    if (ls.empty() || rs.empty())
        return true;
    es.append(ls.size(), ls.data());
    auto remove = [&](expr* r) {
        for (unsigned i = 0; i < es.size(); ++i) {
            if (r == es[i]) {
                es[i] = es.back();
                es.pop_back();
                return true;
            }
        }
        return false;                
    };
    auto is_unit_value = [&](expr* r) {
        return m().is_value(r) && str().is_unit(r);
    };
    for (expr* r : rs) {
        if (remove(r))
            continue;
        if (!is_unit_value(r))
            return true;
    }
    if (es.empty())
        return true;
    for (expr* e : es)
        if (!is_unit_value(e))
            return true;
    return false;
}

bool seq_rewriter::reduce_extract(expr* l, expr* r, expr_ref_vector& res) {
    expr* sub = nullptr, *p = nullptr, *ln = nullptr;
    m_es.reset();
    str().get_concat(r, m_es);
    rational pos, len;
    if (str().is_extract(l, sub, p, ln) &&
        m_autil.is_numeral(p, pos) && m_autil.is_numeral(ln, len) &&
        0 <= pos &&
        0 <= len && 
        all_of(m_es, [&](expr* e) { return str().is_unit(e); })) {

        if (len == m_es.size()) {
            expr_ref_vector result(m());
            for (unsigned i = 0; i < pos.get_unsigned(); ++i) 
                result.push_back(str().mk_unit(str().mk_nth_i(sub, m_autil.mk_int(i))));
            result.append(m_es);
            res.push_back(str().mk_prefix(str().mk_concat(result, sub->get_sort()), sub));
            return true;
        }
        
    }
    return false;
}


bool seq_rewriter::reduce_subsequence(expr_ref_vector& ls, expr_ref_vector& rs, expr_ref_pair_vector& eqs) {

    if (ls.size() > rs.size()) 
        ls.swap(rs);

    if (ls.size() == rs.size())
        return true;

    if (ls.empty() && rs.size() == 1)
        return true;
    
    uint_set rpos;
    for (expr* x : ls) {
        unsigned j = 0;
        bool is_unit = str().is_unit(x);
        for (expr* y : rs) {
            if (!rpos.contains(j) && (x == y || (is_unit && str().is_unit(y)))) {
                rpos.insert(j);
                break;
            }
            ++j;
        }
        if (j == rs.size())
            return true;
    }
    // if we reach here, then every element of l is contained in r in some position.
    // or each non-unit in l is matched by a non-unit in r, and otherwise, the non-units match up.
    unsigned i = 0, j = 0;
    for (expr* y : rs) {
        if (rpos.contains(i)) {
            rs[j++] = y;
        }
        else if (!set_empty(1, &y, true, eqs)) {
            return false;
        }
        ++i;
    }
    if (j == rs.size()) {
        return true;
    }
    rs.shrink(j);
    SASSERT(ls.size() == rs.size());
    if (!ls.empty()) {
        sort* srt = ls[0]->get_sort();
        expr_ref lhs(str().mk_concat(ls, srt), m());
        expr_ref rhs(str().mk_concat(rs, srt), m());
        eqs.push_back(lhs, rhs);
        ls.reset();
        rs.reset();
        TRACE(seq, tout << "subsequence " << eqs << "\n";);
    }
    return true;
} 

seq_rewriter::op_cache::op_cache(ast_manager& m):
    m_trail(m)
{}

expr* seq_rewriter::op_cache::find(decl_kind op, expr* a, expr* b, expr* c) {
    op_entry e(op, a, b, c, nullptr);
    m_table.find(e, e);

    return e.r;
}

void seq_rewriter::op_cache::insert(decl_kind op, expr* a, expr* b, expr* c, expr* r) {
    cleanup();
    if (a) m_trail.push_back(a);
    if (b) m_trail.push_back(b);
    if (c) m_trail.push_back(c);
    if (r) m_trail.push_back(r);
    m_table.insert(op_entry(op, a, b, c, r));
}

void seq_rewriter::op_cache::cleanup() {
    if (m_table.size() >= m_max_cache_size) {
        m_trail.reset();
        m_table.reset();
        STRACE(seq_regex, tout << "Op cache reset!" << std::endl;);
        STRACE(seq_regex_brief, tout << "(OP CACHE RESET) ";);
        STRACE(seq_verbose, tout << "Derivative op cache reset" << std::endl;);
    }
}

lbool seq_rewriter::some_string_in_re(expr* r, zstring& s) {
    seq::regex_witness rw(*this);
    return rw.get_witness(r, s);
}