z3/src/smt/seq/seq_nielsen.cpp

/*++
Copyright (c) 2026 Microsoft Corporation

Module Name:

    seq_nielsen.cpp

Abstract:

    Nielsen graph implementation for string constraint solving.

    Ports the constraint types and Nielsen graph structures from
    ZIPT (https://github.com/CEisenhofer/ZIPT/tree/parikh/ZIPT/Constraints)

Author:

    Clemens Eisenhofer 2026-03-02
    Nikolaj Bjorner (nbjorner) 2026-03-02

NSB review:

   ostrich\substring2b.smt2, ostrich\substring.smt2
   - We are missing rewrites for unit(x) = unit('a') that would eliminate x by a.


--*/

#include "smt/seq/seq_nielsen.h"
#include "smt/seq/seq_parikh.h"
#include "smt/seq/seq_regex.h"
#include "ast/arith_decl_plugin.h"
#include "ast/ast_pp.h"
#include "ast/ast_util.h"
#include "ast/rewriter/seq_rewriter.h"
#include "ast/rewriter/th_rewriter.h"
#include "ast/rewriter/seq_skolem.h"
#include "ast/rewriter/var_subst.h"
#include "util/statistics.h"
#include <algorithm>
#include <complex>
#include <cstdlib>
#include <set>
#include <stack>
#include <unordered_map>
#include <vector>

namespace seq {

    void deps_to_lits(dep_manager &dep_mgr, const dep_tracker deps,
        svector<enode_pair> &eqs, svector<sat::literal> &lits) {
        
        vector<dep_source, false> vs;
        dep_mgr.linearize(deps, vs);
        for (dep_source const &d : vs) {
            if (std::holds_alternative<enode_pair>(d)) {
                eqs.push_back(std::get<enode_pair>(d));
            }
            else if (std::holds_alternative<sat::literal>(d))
                lits.push_back(std::get<sat::literal>(d));
            else
                UNREACHABLE();
        }
    }

    // Normalize an arithmetic expression using th_rewriter.
    // Simplifies e.g. (n - 1 + 1) to n, preventing unbounded growth
    // of power exponents during unwind/merge cycles.
    static expr_ref normalize_arith(ast_manager &m, expr *e) {
        expr_ref result(e, m);
        th_rewriter rw(m);
        rw(result);
        return result;
    }

    std::pair<euf::snode const*, euf::snode const*> split_membership(euf::snode const *str, euf::snode const *regex, euf::sgraph& sg, unsigned threshold, split_set& result) {
        seq_util& seq = sg.get_seq_util();
        ast_manager& m = sg.get_manager();
        euf::snode const* first = str->first();
        SASSERT(first);
        SASSERT(!first->is_char());     // constants are consumed earlier

        // Choose the factorization boundary so the tail starts with the
        // LONGEST run of concrete characters c — this gives the split-engine
        // lookahead oracle the most pruning information.  head = u' (tokens
        // before the run), tail = c · u''' (tokens from the run onward).
        euf::snode_vector toks;
        str->collect_tokens(toks);
        const unsigned total = toks.size();
        unsigned run_start = 0, run_len = 0;
        for (unsigned i = 0; i < total; ) {
            if (!toks[i]->is_char()) { ++i; continue; }
            unsigned j = i;
            while (j < total && toks[j]->is_char()) ++j;
            if (j - i > run_len) { run_len = j - i; run_start = i; }
            i = j;
        }
        // No constant run → fall back to splitting off the first token.
        const unsigned p = run_len == 0 ? 1 : run_start;
        SASSERT(p >= 1);
        euf::snode const* head = p == 1 ? first : sg.drop_right(str, total - p);
        euf::snode const* tail = sg.drop_left(str, p);
        SASSERT(head && tail);

        // Build the constant lookahead c and (if non-empty) an oracle that
        // prunes splits whose ∇ cannot match c.
        zstring c;
        for (unsigned i = 0; i < run_len; ++i) {
            expr* ch; unsigned cv;
            VERIFY(seq.str.is_unit(toks[run_start + i]->get_expr(), ch));
            VERIFY(seq.is_const_char(ch, cv));
            c = c + zstring(cv);
        }
        split_oracle oracle;
        if (!c.empty())
            oracle = [&sg, c](expr*, expr* n) { return split_lookahead_viable(n, sg, c); };

        // Decompose the regex into a split-set via the shared seq_split engine
        // (sigma from the paper): head ∈ Δ ∧ tail ∈ ∇ for each ⟨Δ,∇⟩, with the
        // lookahead oracle pruning non-viable ∇ during generation.
        seq_rewriter rw(m);
        if (!rw.split(regex->get_expr(), result, threshold, split_mode::strong, oracle)) {
            result.clear();
            return { nullptr, nullptr };
        }

        rw.simplify_split(result);

        // Eagerly consume the constant run c from the tail by taking the c-derivative
        // of each ∇:  tail = c·u''' ∈ ∇  ⟺  u''' ∈ δ_c(∇)  (Brzozowski).
        // This makes progress — the returned tail becomes u''' (c consumed) — and
        // drops any split whose ∇ cannot start with c, since there δ_c(∇) = ∅
        // (e.g. the star rule's ⟨ε,ε⟩: δ_c(ε) = ∅ for non-empty c).  This is sound
        // because ∇ is now a complete top-level component (no factor appended).
        if (!c.empty()) {
            unsigned w = 0;
            for (unsigned i = 0; i < result.size(); ++i) {
                euf::snode const* d = sg.mk(result[i].m_n);
                for (unsigned k = 0; d && !d->is_fail() && k < c.length(); ++k) {
                    d = sg.brzozowski_deriv(d, sg.mk_char(c[k]));
                }
                SASSERT(d);
                if (d->is_fail())
                    continue;   // ∇ can't start with c → infeasible split, drop
                result[w++] = split_pair(result[i].m_d, d->get_expr(), m);
            }
            result.shrink(w);
            tail = sg.drop_left(str, run_start + run_len);   // u''' (c consumed)
        }

        return { head, tail };
    }

    bool split_lookahead_viable(expr* n_regex, euf::sgraph& sg, zstring const& c) {
        euf::snode const* cur = sg.mk(n_regex);
        SASSERT(cur);
        for (unsigned i = 0; i < c.length(); i++) {
            if (sg.re_nullable(cur) == l_true)
                return true;            // N accepts the prefix c[0..i) → a suffix completes it
            cur = sg.brzozowski_deriv(cur, sg.mk_char(c[i]));
            SASSERT(cur);
            if (cur->is_fail())
                return false;           // N went (syntactically) dead before reaching c
        }
        return !cur->is_fail();
    }

    // Directional helpers mirroring ZIPT's fwd flag:
    // fwd=true  -> left-to-right (prefix/head)
    // fwd=false -> right-to-left (suffix/tail)
    static euf::snode const* dir_token(euf::snode const* s, const bool fwd) {
        if (!s)
            return nullptr;
        return fwd ? s->first() : s->last();
    }

    static euf::snode const* dir_drop(euf::sgraph &sg, euf::snode const* s, const unsigned count, const bool fwd) {
        if (!s || count == 0)
            return s;
        return fwd ? sg.drop_left(s, count) : sg.drop_right(s, count);
    }

    static euf::snode const* dir_concat(euf::sgraph &sg, euf::snode const* a, euf::snode const* b, const bool fwd) {
        if (!a)
            return b;
        if (!b)
            return a;
        return fwd ? sg.mk_concat(a, b) : sg.mk_concat(b, a);
    }

    static void collect_tokens_dir(euf::snode const* s, const bool fwd, euf::snode_vector &toks) {
        toks.reset();
        if (!s)
            return;
        s->collect_tokens(toks);
        if (!fwd)
            toks.reverse();
    }

    // Right-derivative helper used by backward str_mem simplification:
    // dR(re, c) = reverse( derivative(c, reverse(re)) ).
    static euf::snode const* reverse_brzozowski_deriv(euf::sgraph &sg, euf::snode const* re, euf::snode const* elem) {
        if (!re || !elem || !re->get_expr() || !elem->get_expr())
            return nullptr;
        ast_manager &m = sg.get_manager();
        seq_util &seq = sg.get_seq_util();
        seq_rewriter rw(m);

        expr *elem_expr = elem->get_expr();
        expr *ch = nullptr;
        if (seq.str.is_unit(elem_expr, ch))
            elem_expr = ch;

        const expr_ref re_rev(seq.re.mk_reverse(re->get_expr()), m);
        const expr_ref d = rw.mk_derivative(elem_expr, re_rev);
        if (!d.get())
            return nullptr;
        expr_ref result(seq.re.mk_reverse(d), m);
        th_rewriter tr(m);
        tr(result);
        return sg.mk(result);
    }

    // -----------------------------------------------
    // str_eq
    // -----------------------------------------------

    void str_eq::sort() {
        if (m_lhs && m_rhs && m_lhs->id() > m_rhs->id()) {
            std::swap(m_lhs, m_rhs);
        }
        SASSERT(!m_lhs || !m_rhs || m_lhs->id() <= m_rhs->id());
    }

    bool str_eq::is_trivial() const {
        return m_lhs == m_rhs || (m_lhs && m_rhs && m_lhs->is_empty() && m_rhs->is_empty());
    }

    bool str_eq::contains_var(euf::snode const* var) const {
        if (!var)
            return false;
        // check if var appears in the token list of lhs or rhs
        if (m_lhs) {
            euf::snode_vector tokens;
            m_lhs->collect_tokens(tokens);
            for (euf::snode const* t : tokens) {
                if (t == var)
                    return true;
            }
        }
        if (m_rhs) {
            euf::snode_vector tokens;
            m_rhs->collect_tokens(tokens);
            for (euf::snode const* t : tokens) {
                if (t == var)
                    return true;
            }
        }
        return false;
    }

    // -----------------------------------------------
    // str_mem
    // -----------------------------------------------

    bool str_mem::is_primitive() const {
        // A still-unresolved symbolic-derivative residual (ite) is not a settled
        // primitive — apply_regex_if_split must resolve it first.
        return m_str && m_str->length() == 1 && m_str->is_var() && m_regex->is_ground()
            && m_regex->kind() != euf::snode_kind::s_ite;
    }

    bool str_mem::is_trivial(nielsen_node const* n) const {
        if (!(m_str && m_regex))
            return false;
        if (m_kind == mem_kind::no_loop)
            // guard: discharged ⇒ Σ* (accepts all); ε has no non-empty lap-prefix.
            return m_discharged || m_str->is_empty();
        if (!m_str->is_empty())
            return false;
        if (m_kind == mem_kind::stab_view)
            // ε ∈ stab(root,Q) iff current state ≡ root (i.e. root ∈ F={root}).
            return m_regex == m_root;
        return n->graph().sg().re_nullable(m_regex) == l_true;
    }

    bool str_mem::is_contradiction(nielsen_node const* n) const {
        if (!(m_str && m_regex && m_str->is_empty()))
            return false;
        if (m_kind == mem_kind::no_loop)
            return false; // guard acceptance is always true on the empty word
        if (m_kind == mem_kind::stab_view)
            return m_regex != m_root; // ε ∉ stab(root,Q) when state ≢ root
        return n->graph().sg().re_nullable(m_regex) == l_false;
    }

    bool str_mem::contains_var(euf::snode const* var) const {
        SASSERT(var);
        if (m_str) {
            euf::snode_vector tokens;
            m_str->collect_tokens(tokens);
            return any_of(tokens, [var](auto t) { return t == var; });
        }
        return false;
    }

    // -----------------------------------------------
    // nielsen_subst
    // -----------------------------------------------

    bool nielsen_subst::is_eliminating() const {
        SASSERT(m_var && m_replacement);
        // check if var appears in replacement
        euf::snode_vector tokens;
        m_replacement->collect_tokens(tokens);
        return all_of(tokens, [this](auto t) { return t != m_var; });
    }

    bool nielsen_subst::is_char_subst() const {
        SASSERT(m_var && m_replacement);
        SASSERT(!m_var->is_unit() || m_replacement->is_char_or_unit());
        return m_var->is_unit();
    }

    // -----------------------------------------------
    // nielsen_edge
    // -----------------------------------------------

    nielsen_edge::nielsen_edge(nielsen_node* src, nielsen_node* tgt, const char* rule, const bool is_progress):
        m_src(src), m_tgt(tgt), m_rule_name(rule),
        m_is_progress(is_progress) { }

    void nielsen_edge::add_subst(nielsen_subst const& s) {
        m_subst.push_back(s);
    }

    // -----------------------------------------------
    // nielsen_node
    // -----------------------------------------------

    nielsen_node::nielsen_node(nielsen_graph& graph, const unsigned id):
        m_id(id), m_graph(graph), m_is_progress(true) { }

    void nielsen_node::clone_from(nielsen_node const& parent) {
        m_str_eq.reset();
        m_str_deq.reset();
        m_str_mem.reset();
        m_constraints.reset();
        m_char_ranges.reset();
        m_str_eq.append(parent.m_str_eq);
        m_str_deq.append(parent.m_str_deq);
        m_str_mem.append(parent.m_str_mem);
        m_constraints.append(parent.m_constraints);

        // clone character ranges
        for (auto const &[k, v] : parent.m_char_ranges)
            m_char_ranges.insert(k, std::make_pair(v.first.clone(), v.second));

        // clone regex occurrence tracking
        m_regex_occurrence = parent.m_regex_occurrence;

        SASSERT(m_str_eq.size() == parent.m_str_eq.size());
        SASSERT(m_str_deq.size() == parent.m_str_deq.size());
        SASSERT(m_str_mem.size() == parent.m_str_mem.size());
        SASSERT(m_constraints.size() == parent.m_constraints.size());
    }

    bool nielsen_node::track_regex_occurrence(unsigned regex_id, unsigned mem_id) {
        const auto key = std::make_pair(regex_id, mem_id);
        const auto it = m_regex_occurrence.find(key);
        if (it != m_regex_occurrence.end()) {
            // Already seen — cycle detected
            return true;
        }
        m_regex_occurrence[key] = m_id;
        return false;
    }

    void nielsen_node::add_str_eq(str_eq const& eq) {
        SASSERT(eq.m_lhs != nullptr);
        SASSERT(eq.m_rhs != nullptr);
        if (eq.is_trivial())
            return;
        m_str_eq.push_back(eq);
    }

    void nielsen_node::add_str_deq(str_deq const& deq) {
        SASSERT(deq.m_lhs != nullptr);
        SASSERT(deq.m_rhs != nullptr);
        m_str_deq.push_back(deq);
    }

    void nielsen_node::add_str_mem(str_mem const& mem) {
        SASSERT(mem.m_str != nullptr);
        SASSERT(mem.m_regex != nullptr);
        if (mem.is_trivial(this))
            return;
        m_str_mem.push_back(mem);
    }

    bool nielsen_node::lower_bound(expr *e, rational &lo, dep_tracker &dep) {
        literal_vector lits;
        enode_pair_vector eqs;
        if (m_graph.a.is_numeral(e, lo))
            return true;
        if (!m_graph.m_context_solver.lower_bound(e, lo, lits, eqs))
            return false;
        for (auto lit : lits)
            dep = m_graph.dep_mgr().mk_join(dep, m_graph.dep_mgr().mk_leaf(lit));
        for (auto eq : eqs)
            dep = m_graph.dep_mgr().mk_join(dep, m_graph.dep_mgr().mk_leaf(eq));

        const expr_ref lo_expr(m_graph.a.mk_int(lo), m_graph.m);
        m_graph.add_le_dependency(dep, this, lo_expr, e);
        return true;
    }

    bool nielsen_node::upper_bound(expr *e, rational &up, dep_tracker &dep) {
        literal_vector lits;
        enode_pair_vector eqs;
        if (m_graph.a.is_numeral(e, up))
            return true;
        if (!m_graph.m_context_solver.upper_bound(e, up, lits, eqs))
            return false;
        for (auto lit : lits)
            dep = m_graph.dep_mgr().mk_join(dep, m_graph.dep_mgr().mk_leaf(lit));
        for (auto eq : eqs)
            dep = m_graph.dep_mgr().mk_join(dep, m_graph.dep_mgr().mk_leaf(eq));
        const expr_ref up_expr(m_graph.a.mk_int(up), m_graph.m);
        m_graph.add_le_dependency(dep, this, e, up_expr);
        return true;
    }

    void nielsen_node::add_constraint(constraint const &c) {
        auto& m = graph().get_manager();
        if (m.is_true(c.fml))
            return;
        // TODO: Is it possible that we miss a conflict if we decompose?
        if (m.is_and(c.fml)) {
            // We have to add all - even if some of it conflict
            // [otw. we would leave the node partially initialized]
            for (const auto f : *to_app(c.fml)) {
                add_constraint(constraint(f, c.dep, m));
            }
            return;
        }
        m_constraints.push_back(c);
    }

    void nielsen_node::apply_subst(euf::sgraph& sg, nielsen_subst const& s) {
        SASSERT(!s.m_var->is_char_or_unit() || s.m_replacement->is_char_or_unit());
        SASSERT(s.m_var);
        SASSERT(s.m_replacement != nullptr);
        for (auto &eq : m_str_eq) {
            const auto new_lhs = sg.subst(eq.m_lhs, s.m_var, s.m_replacement);
            const auto new_rhs = sg.subst(eq.m_rhs, s.m_var, s.m_replacement);
            if (new_lhs != eq.m_lhs || new_rhs != eq.m_rhs) {
                 eq.m_lhs = new_lhs;
                 eq.m_rhs = new_rhs;
                 eq.m_dep = m_graph.dep_mgr().mk_join(eq.m_dep, s.m_dep);
                 eq.sort();
            }
        }

        for (auto &deq : m_str_deq) {
            const auto new_lhs = sg.subst(deq.m_lhs, s.m_var, s.m_replacement);
            const auto new_rhs = sg.subst(deq.m_rhs, s.m_var, s.m_replacement);
            if (new_lhs != deq.m_lhs || new_rhs != deq.m_rhs) {
                 deq.m_lhs = new_lhs;
                 deq.m_rhs = new_rhs;
                 deq.m_dep = m_graph.dep_mgr().mk_join(deq.m_dep, s.m_dep);
                 deq.sort();
            }
        }

        for (auto &mem : m_str_mem) {
            const auto new_str = sg.subst(mem.m_str, s.m_var, s.m_replacement);
            const auto new_regex = sg.subst(mem.m_regex, s.m_var, s.m_replacement);
            if (new_str != mem.m_str || new_regex != mem.m_regex) {
                mem.m_str = new_str;
                mem.m_regex = new_regex;
                mem.m_dep = m_graph.dep_mgr().mk_join(mem.m_dep, s.m_dep);
            }
        }

        const unsigned var_id = s.m_var->id();

        ast_manager& m = graph().get_manager();

        if (s.is_char_subst()) {
            expr* var_c_expr = s.m_var->arg0()->get_expr();
            expr* repl_c_expr = s.m_replacement->arg0()->get_expr();
            add_constraint(
                constraint(m.mk_eq(var_c_expr, repl_c_expr), s.m_dep, m));

            if (m_char_ranges.contains(var_id)) {
                const auto range = m_char_ranges.find(var_id); // copy exactly
                m_char_ranges.remove(var_id);
                add_char_range(s.m_replacement, range.first, m_graph.dep_mgr().mk_join(range.second, s.m_dep));
            }
        }
    }

    void nielsen_node::add_char_range(euf::snode const* sym_char, char_set const& range, dep_tracker dep) {
        if (sym_char->is_char()) {
            // for a concrete character just check if it matches
            const expr * val = sym_char->get_expr();
            unsigned ch;
            expr* ch_expr;
            VERIFY(graph().seq().str.is_unit(val, ch_expr));
            VERIFY(graph().seq().is_const_char(ch_expr, ch));
            for (unsigned i = 0; i < range.ranges().size(); i++) {
                if (range.ranges()[i].contains(ch))
                    return; // matches, no conflict
            }
            set_conflict(backtrack_reason::character_range, dep);
            set_general_conflict();
            return;
        }
        SASSERT(sym_char && sym_char->is_unit());
        const unsigned id = sym_char->id();
        if (m_char_ranges.contains(id)) {
            auto& existing = m_char_ranges.find(id);
            char_set inter = existing.first.intersect_with(range);
            existing = std::make_pair(inter, graph().dep_mgr().mk_join(existing.second, dep));
            if (inter.is_empty()) {
                set_conflict(backtrack_reason::character_range, existing.second);
                set_general_conflict();
            }
        }
        else
            m_char_ranges.insert(id, std::make_pair(range.clone(), dep));

        auto& ranges = range.ranges();
        auto& m = graph().get_manager();
        const auto & seq = graph().seq();
        expr* var = sym_char->get_expr();
        SASSERT(seq.str.is_unit(var));
        var = to_app(var)->get_arg(0);
        ptr_vector<expr> cases;
        cases.reserve(ranges.size());

        for (unsigned i = 0; i < ranges.size(); ++i) {
            cases[i] = m.mk_and(
                seq.mk_le(seq.mk_char(ranges[i].m_lo), var),
                seq.mk_le(var, seq.mk_char(ranges[i].m_hi - 1)));
        }
        add_constraint(constraint(m.mk_or(cases), dep, m));
    }
    // -----------------------------------------------
    // nielsen_graph
    // -----------------------------------------------

    nielsen_graph::nielsen_graph(euf::sgraph &sg, sub_solver_i &solver, context_solver_i &ctx_solver)
        : m(sg.get_manager()), a(sg.get_manager()), m_seq(sg.get_seq_util()), m_sg(sg), m_rw(m), m_sk(m, m_rw),
          m_length_solver(solver), m_context_solver(ctx_solver), m_partial_dfa_pin(sg.get_manager()),
          m_parikh(alloc(seq_parikh, sg)), m_seq_regex(alloc(seq::seq_regex, sg)) {
    }

    nielsen_graph::~nielsen_graph() {
        dealloc(m_parikh);
        dealloc(m_seq_regex);
        reset();
    }

    bool nielsen_graph::projection_state_in_Q(expr *state, unsigned nu) {
        if (!state || nu == 0)
            return false;
        const unsigned sid = state->get_id();
        // r ∈ Q_nu iff r is incident to a partial-DFA edge whose extraction
        // index lies in [1, nu] (the "edges marked ≤ ν" subautomaton of the
        // implementation-aspects section of the paper).
        auto incident = [&](std::unordered_map<unsigned, unsigned_vector> const &adj) {
            auto it = adj.find(sid);
            if (it == adj.end())
                return false;
            for (const unsigned edge_idx : it->second) {
                if (edge_idx >= m_partial_dfa_edges.size())
                    continue;
                const unsigned pidx = m_partial_dfa_edges[edge_idx].m_projection_idx;
                if (pidx != 0 && pidx <= nu)
                    return true;
            }
            return false;
        };
        return incident(m_partial_dfa_out) || incident(m_partial_dfa_in);
    }

    nielsen_node *nielsen_graph::mk_node() {
        const unsigned id = m_nodes.size();
        nielsen_node *n = alloc(nielsen_node, *this, id);
        m_nodes.push_back(n);
        SASSERT(n->id() == m_nodes.size() - 1);
        return n;
    }

    nielsen_node *nielsen_graph::mk_child(nielsen_node *parent) {
        nielsen_node *child = mk_node();
        child->clone_from(*parent);
        child->m_parent_ic_count = parent->constraints().size();
        return child;
    }

    nielsen_edge *nielsen_graph::mk_edge(nielsen_node *src, nielsen_node *tgt, const char *rule,
                                         const bool is_progress) {
        SASSERT(src != nullptr);
        SASSERT(tgt != nullptr);
        nielsen_edge *e = alloc(nielsen_edge, src, tgt, rule, is_progress);
        m_edges.push_back(e);
        src->add_outgoing(e);
        return e;
    }

    void nielsen_graph::add_str_eq(euf::snode const* lhs, euf::snode const* rhs, smt::enode *l, smt::enode *r) const {
        const dep_tracker dep = m_dep_mgr.mk_leaf(enode_pair(l, r));
        str_eq eq(lhs, rhs, dep);
        eq.sort();
        // check if root node contains this equation already
        if (std::ranges::any_of(m_root->str_eqs(),
                                [&](const str_eq &e) { return e.m_lhs == eq.m_lhs && e.m_rhs == eq.m_rhs; }))
            // already present, no need to add again
            return;
        m_root->add_str_eq(eq);
    }

    void nielsen_graph::add_str_deq(euf::snode const* lhs, euf::snode const* rhs, sat::literal l) const {
        const dep_tracker dep = m_dep_mgr.mk_leaf(l);
        str_deq deq(lhs, rhs, dep);
        // check if root node contains this equation already
        if (std::ranges::any_of(m_root->str_deqs(),
                                [&](const str_deq &e) { return e.m_lhs == deq.m_lhs && e.m_rhs == deq.m_rhs; }))
            // already present, no need to add again
            return;
        m_root->add_str_deq(deq);
    }

    void nielsen_graph::add_str_mem(euf::snode const* str, euf::snode const* regex, sat::literal l) const {
        // check if root node contains this membership constraint already
        if (std::ranges::any_of(m_root->str_mems(),
                                [&](const str_mem &e) { return e.m_regex == regex && e.m_str == str; }))
            // already present, no need to add again
            return;
        const dep_tracker dep = m_dep_mgr.mk_leaf(l);
        m_root->add_str_mem(str_mem(str, regex, dep));
    }

    // test-friendly overloads (no external dependency tracking)
    void nielsen_graph::add_str_eq(euf::snode const* lhs, euf::snode const* rhs) {
        const dep_tracker dep = m_dep_mgr.mk_leaf(enode_pair(nullptr, nullptr));
        str_eq eq(lhs, rhs, dep);
        eq.sort();
        m_root->add_str_eq(eq);
    }

    void nielsen_graph::add_str_deq(euf::snode const* lhs, euf::snode const* rhs) {
        const dep_tracker dep = m_dep_mgr.mk_leaf(enode_pair(nullptr, nullptr));
        str_deq deq(lhs, rhs, dep);
        m_root->add_str_deq(deq);
    }

    void nielsen_graph::add_str_mem(euf::snode const* str, euf::snode const* regex) {
        const dep_tracker dep = nullptr;
        m_root->add_str_mem(str_mem(str, regex, dep));
    }

    void nielsen_graph::reset() {
        for (nielsen_node *n : m_nodes) {
            dealloc(n);
        }
        for (nielsen_edge *e : m_edges) {
            dealloc(e);
        }
        m_nodes.reset();
        m_edges.reset();
        m_root = nullptr;
        m_sat_node = nullptr;
        m_sat_path.reset();
        m_depth_bound = 0;
        m_fresh_cnt = 0;
        m_root_constraints_asserted = false;
        // m_mod_cnt.reset();
        m_partial_dfa_edges.reset();
        m_partial_dfa_out.clear();
        m_partial_dfa_in.clear();
        m_partial_dfa_edge_index.clear();
        m_partial_dfa_pin.reset();
        m_projection_extract_idx = 0;
        m_explored_automaton.reset();
        m_unsat_node_cache.clear();
        m_num_cache_hits = 0;
        // m_length_trail.reset();
        // m_length_info.reset();
        m_dep_mgr.reset();
        m_length_solver.reset();
        SASSERT(m_nodes.empty());
        SASSERT(m_edges.empty());
        SASSERT(m_root == nullptr);
        SASSERT(m_sat_node == nullptr);
    }

    void nielsen_graph::add_le_dependency(const dep_tracker dep, nielsen_node *n, expr *lhs, expr *rhs) const {
        SASSERT(lhs);
        SASSERT(rhs);
        const expr_ref le(a.mk_le(lhs, rhs), m);
        // just assume it to be correct
        // Just add the constraint - we do not have to recompute it
        // [also it is on the set of side-conditions if we assert a satisfied node]
        n->add_constraint(constraint(le, dep, m));
    }

    // -----------------------------------------------------------------------
    // nielsen_node: simplify_and_init
    // -----------------------------------------------------------------------

    bool nielsen_node::check_empty_side_conflict(euf::sgraph& sg, euf::snode const* non_empty_side,
                                                 dep_tracker const& dep) {
        euf::snode_vector tokens;
        non_empty_side->collect_tokens(tokens);
        const bool has_char = std::ranges::any_of(tokens, [](euf::snode const* t){ return t->is_char(); });
        const bool all_eliminable = std::ranges::all_of(tokens, [](euf::snode const* t){
            return t->is_var() || t->is_power();
        });
        if (has_char || !all_eliminable) {
            set_general_conflict();
            set_conflict(backtrack_reason::symbol_clash, dep);
            return true;
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Power simplification helpers (mirrors ZIPT's MergePowers,
    // SimplifyPowerElim/CommPower, SimplifyPowerSingle)
    // -----------------------------------------------------------------------

    // Check if exponent b equals exponent a + diff for some rational constant diff.
    // Uses syntactic matching on Z3 expression structure: pointer equality
    // detects shared sub-expressions created during ConstNumUnwinding.
    //
    static bool get_const_power_diff(expr* b, expr* a, arith_util& arith, rational& diff) {
        if (a == b) { diff = rational(0); return true; }
        expr* x = nullptr, *y = nullptr;
        rational val;
        // b = (+ a k) ?
        if (arith.is_add(b, x, y)) {
            if (x == a && arith.is_numeral(y, val)) { diff = val; return true; }
            if (y == a && arith.is_numeral(x, val)) { diff = val; return true; }
        }
        // a = (+ b k) → diff = -k
        if (arith.is_add(a, x, y)) {
            if (x == b && arith.is_numeral(y, val)) { diff = -val; return true; }
            if (y == b && arith.is_numeral(x, val)) { diff = -val; return true; }
        }
        // b = (- a k) → diff = -k
        if (arith.is_sub(b, x, y) && x == a && arith.is_numeral(y, val)) { diff = -val; return true; }
        // a = (- b k) → diff = k
        if (arith.is_sub(a, x, y) && x == b && arith.is_numeral(y, val)) { diff = val; return true; }
        return false;
    }

    // Get the base expression of a power snode.
    static expr* get_power_base_expr(euf::snode const* power, seq_util& seq) {
        if (!power || !power->is_power()) return nullptr;
        const expr * e = power->get_expr();
        expr* base = nullptr, *exp = nullptr;
        return (e && seq.str.is_power(e, base, exp)) ? base : nullptr;
    }

    // Get the exponent expression of a power snode.
    static expr* get_power_exp_expr(euf::snode const* power, seq_util& seq) {
        if (!power->is_power()) return nullptr;
        const expr * e = power->get_expr();
        expr* base = nullptr, *exp = nullptr;
        return (e && seq.str.is_power(e, base, exp)) ? exp : nullptr;
    }

    // Merge adjacent tokens with the same power base on one side of an equation.
    // Handles: char(c) · power(c^e) → power(c^(e+1)),
    //          power(c^e) · char(c) → power(c^(e+1)),
    //          power(c^e1) · power(c^e2) → power(c^(e1+e2)).
    // Returns new snode if merging happened, nullptr otherwise.
    static euf::snode const* merge_adjacent_powers(euf::sgraph& sg, euf::snode const* side) {
        if (!side || side->is_empty() || side->is_token())
            return nullptr;

        euf::snode_vector tokens;
        side->collect_tokens(tokens);
        if (tokens.size() < 2)
            return nullptr;

        ast_manager& m = sg.get_manager();
        arith_util arith(m);
        seq_util& seq = sg.get_seq_util();

        bool merged = false;
        euf::snode_vector result;

        unsigned i = 0;
        while (i < tokens.size()) {
            euf::snode const* tok = tokens[i];

            // Case 1: current is a power token — absorb following same-base tokens.
            // Skip at leading position (i == 0) to keep exponents small: CommPower
            // cross-side cancellation works better with unmerged leading powers
            // (e.g., w^k trivially ≤ 1+k, but w^(2k) vs 1+k requires k ≥ 1).
            if (tok->is_power() && i > 0) {
                expr* base_e = get_power_base_expr(tok, seq);
                expr* exp_acc = get_power_exp_expr(tok, seq);
                if (!base_e || !exp_acc) { result.push_back(tok); ++i; continue; }

                bool local_merged = false;
                unsigned j = i + 1;
                while (j < tokens.size()) {
                    euf::snode const* next = tokens[j];
                    if (next->is_power()) {
                        const expr * nb = get_power_base_expr(next, seq);
                        if (nb == base_e) {
                            exp_acc = arith.mk_add(exp_acc, get_power_exp_expr(next, seq));
                            local_merged = true; ++j; continue;
                        }
                    }
                    if (next->is_char() && next->get_expr() == base_e) {
                        exp_acc = arith.mk_add(exp_acc, arith.mk_int(1));
                        local_merged = true; ++j; continue;
                    }
                    break;
                }
                if (local_merged) {
                    merged = true;
                    expr_ref norm_exp = normalize_arith(m, exp_acc);
                    expr_ref new_pow(seq.str.mk_power(base_e, norm_exp), m);
                    result.push_back(sg.mk(new_pow));
                }
                else
                    result.push_back(tok);
                i = j;
                continue;
            }

            // Case 2: current is a char — check if next is a same-base power.
            // Skip at leading position (i == 0) to avoid undoing power unwinding:
            // unwind produces u · u^(n-1); merging it back to u^n creates an infinite cycle.
            if (i > 0 && tok->is_char() && tok->get_expr() && i + 1 < tokens.size()) {
                euf::snode const* next = tokens[i + 1];
                if (next->is_power() && get_power_base_expr(next, seq) == tok->get_expr()) {
                    expr* base_e = tok->get_expr();
                    // Use same arg order as Case 1: add(exp, 1), not add(1, exp),
                    // so that merging "c · c^e" and "c^e · c" both produce add(e, 1)
                    // and the resulting power expression is hash-consed identically.
                    expr* exp_acc = arith.mk_add(get_power_exp_expr(next, seq), arith.mk_int(1));
                    unsigned j = i + 2;
                    while (j < tokens.size()) {
                        euf::snode const* further = tokens[j];
                        if (further->is_power() && get_power_base_expr(further, seq) == base_e) {
                            exp_acc = arith.mk_add(exp_acc, get_power_exp_expr(further, seq));
                            ++j; continue;
                        }
                        if (further->is_char() && further->get_expr() == base_e) {
                            exp_acc = arith.mk_add(exp_acc, arith.mk_int(1));
                            ++j; continue;
                        }
                        break;
                    }
                    merged = true;
                    expr_ref norm_exp = normalize_arith(m, exp_acc);
                    expr_ref new_pow(seq.str.mk_power(base_e, norm_exp), m);
                    result.push_back(sg.mk(new_pow));
                    i = j;
                    continue;
                }
            }

            result.push_back(tok);
            ++i;
        }

        if (!merged)
            return nullptr;

        euf::snode const* rebuilt = nullptr;
        for (const auto tok : result)
            rebuilt = rebuilt ? sg.mk_concat(rebuilt, tok) : tok;
        if (!rebuilt)
            rebuilt = sg.mk_empty_seq(side->get_sort());
        return rebuilt;
    }

    // Simplify constant-exponent powers: base^0 → ε, base^1 → base.
    // Returns new snode if any simplification happened, nullptr otherwise.
    static euf::snode const* simplify_const_powers(nielsen_node* node, euf::sgraph& sg, euf::snode const* side, dep_tracker& dep) {
        dep = nullptr;
        SASSERT(side);
        if (side->is_empty())
            return nullptr;

        euf::snode_vector tokens;
        side->collect_tokens(tokens);

        seq_util& seq = sg.get_seq_util();

        bool simplified = false;
        euf::snode_vector result;

        for (euf::snode const* tok : tokens) {
            if (tok->is_power()) {
                expr* exp_e = get_power_exp_expr(tok, seq);
                rational ub;
                dep_tracker ub_dep = nullptr;
                if (exp_e && node->upper_bound(exp_e, ub, ub_dep)) {
                    if (ub.is_zero()) {
                        // base^0 → ε (skip this token entirely)
                        dep = node->graph().dep_mgr().mk_join(dep, ub_dep);
                        simplified = true;
                        continue;
                    }
                    if (ub.is_one()) {
                        // base^1 → base
                        euf::snode const* base_sn = tok->arg0();
                        if (base_sn) {
                            dep = node->graph().dep_mgr().mk_join(dep, ub_dep);
                            result.push_back(base_sn);
                            simplified = true;
                            continue;
                        }
                    }
                }
            }
            result.push_back(tok);
        }

        if (!simplified)
            return nullptr;

        euf::snode const* rebuilt = nullptr;
        for (euf::snode const* tok : result) {
            rebuilt = rebuilt ? sg.mk_concat(rebuilt, tok) : tok;
        }
        if (!rebuilt)
            rebuilt = sg.mk_empty_seq(side->get_sort());
        return rebuilt;
    }

    // CommPower: count how many times a power's base pattern appears in
    // the directional prefix of the other side (fwd=true: left prefix,
    // fwd=false: right suffix). Mirrors ZIPT's CommPower helper.
    // Returns (count_expr, num_tokens_consumed).  count_expr is nullptr
    // when no complete base-pattern match is found.
    static std::pair<expr_ref, unsigned> comm_power(
            euf::snode const* base_sn, euf::snode const* side, ast_manager& m, seq_util& seq, const bool fwd) {
        arith_util arith(m);
        euf::snode_vector base_tokens, side_tokens;
        collect_tokens_dir(base_sn, fwd, base_tokens);
        collect_tokens_dir(side, fwd, side_tokens);
        if (base_tokens.empty() || side_tokens.empty())
            return {expr_ref(nullptr, m), 0};

        expr* sum = nullptr;
        unsigned pos = 0;
        expr* last_stable_sum = nullptr;
        unsigned last_stable_idx = 0;

        unsigned i = 0;
        for (; i < side_tokens.size(); i++) {
            euf::snode const* t = side_tokens[i];
            if (pos == 0) {
                last_stable_idx = i;
                last_stable_sum = sum;
            }
            // Case 1: direct token match with base pattern
            if (pos < base_tokens.size() && t == base_tokens[pos]) {
                pos++;
                if (pos >= base_tokens.size()) {
                    pos = 0;
                    sum = sum ? arith.mk_add(sum, arith.mk_int(1))
                              : arith.mk_int(1);
                }
                continue;
            }
            // Case 2: power token whose base matches our base pattern (at pos == 0)
            // Skip at leading position (i == 0) to avoid undoing power unwinding:
            // unwind produces u · u^(n-1); merging it back to u^n creates an infinite cycle.
            if (i > 0 && t->is_power()) {
                euf::snode const* pow_base = t->arg0();
                if (pow_base) {
                    euf::snode_vector pb_tokens;
                    collect_tokens_dir(pow_base, fwd, pb_tokens);
                    if (pb_tokens.size() == base_tokens.size()) {
                        bool match = true;
                        for (unsigned j = 0; j < pb_tokens.size() && match; j++)
                            match = (pb_tokens[j] == base_tokens[j]);
                        if (match) {
                            expr* pow_exp = get_power_exp_expr(t, seq);
                            if (pow_exp) {
                                sum = sum ? arith.mk_add(sum, pow_exp) : pow_exp;
                                continue;
                            }
                        }
                    }
                }
            }
            break;
        }
        // After loop: i = break index or side_tokens.size()
        if (pos == 0) {
            last_stable_idx = i;
            last_stable_sum = sum;
        }
        return {expr_ref(last_stable_sum, m), last_stable_idx};
    }

    euf::snode const* nielsen_graph::to_partial_label_regex(euf::snode const* label) const {
        SASSERT(label && label->get_expr());

        expr* e = label->get_expr();
        if (m_seq.is_re(e) || m_seq.re.is_full_char(e))
            return label;
        if (m_seq.is_char(e)) {
            const expr_ref u(m_seq.str.mk_unit(e), m);
            return m_sg.mk(m_seq.re.mk_to_re(u));
        }
        if (m_seq.str.is_string(e) || m_seq.str.is_unit(e))
            return m_sg.mk(m_seq.re.mk_to_re(e));

        std::cout << "Unexpected: " << mk_pp(label->get_expr(), m) << std::endl;
        UNREACHABLE();
        return nullptr;
    }

    void nielsen_graph::record_partial_derivative_edge(euf::snode const* src_re, euf::snode const* label, euf::snode const* dst_re) {
        SASSERT(src_re && dst_re);
        if (!src_re->is_ground() || !dst_re->is_ground())
            return;
        if (src_re->is_fail() || dst_re->is_fail())
            return;

        //euf::snode const* label_re = to_partial_label_regex(label);
        //SASSERT(label_re);
        //if (!m_seq.is_re(label_re->get_expr()) || !label_re->is_ground())
        //    return;

        expr* src_e   = src_re->get_expr();
        //expr* label_e = label_re->get_expr();
        expr* dst_e   = dst_re->get_expr();

        // Deduplicate transitions by (src, dst) only — NOT by label.  The
        // projection operator is parameterized by the explored state set Q
        // (the Brzozowski automaton is deterministic, so the only a-transition
        // out of a state is to δ_a(state)); edge labels are never consulted by
        // projection_state_in_Q / collect_scc_for_projection / the projection
        // derivative.  Keying by label would record the SAME transition twice
        // when it is discovered once as a concrete char (to_re "a") and once as
        // a minterm range ([a-a]), spuriously inflating the SCC edge count and
        // re-triggering cycle decomposition as the derivation walks the cycle.
        partial_dfa_edge_key key{ src_e->get_id(), 0, dst_e->get_id() };
        if (m_partial_dfa_edge_index.contains(key))
            return;

        // Pin each expression so the egraph cannot release it on pop while
        // we still reference it from the cache.  Bumping the ref count of an
        // already-pinned expression is harmless; the wasted slot is bounded
        // by the unique-edge count.
        m_partial_dfa_pin.push_back(src_e);
        //m_partial_dfa_pin.push_back(label_e);
        m_partial_dfa_pin.push_back(dst_e);

        unsigned edge_idx = m_partial_dfa_edges.size();
        m_partial_dfa_edge_index.emplace(key, edge_idx);

        partial_dfa_edge e;
        e.m_src   = src_e;
        //e.m_label = label_e;
        e.m_dst   = dst_e;
        m_partial_dfa_edges.push_back(e);

        m_partial_dfa_out[src_e->get_id()].push_back(edge_idx);
        m_partial_dfa_in[dst_e->get_id()].push_back(edge_idx);
    }

    bool nielsen_graph::collect_scc_for_projection(euf::snode const* root_re, uint_set& scc) const {
        scc.reset();
        if (!root_re || !root_re->get_expr())
            return false;

        // scc, fwd, bwd contain expression ids (matching the keys in
        // m_partial_dfa_out / m_partial_dfa_in).  Expression ids are stable
        // across sgraph pops because we pin them in m_partial_dfa_pin.
        const unsigned root_id = root_re->get_expr()->get_id();
        uint_set fwd, bwd;
        unsigned_vector stack;

        stack.push_back(root_id);
        while (!stack.empty()) {
            unsigned s = stack.back();
            stack.pop_back();
            if (fwd.contains(s))
                continue;
            fwd.insert(s);
            auto it = m_partial_dfa_out.find(s);
            if (it == m_partial_dfa_out.end())
                continue;
            for (const unsigned edge_idx : it->second) {
                if (edge_idx >= m_partial_dfa_edges.size())
                    continue;
                partial_dfa_edge const& e = m_partial_dfa_edges[edge_idx];
                if (e.m_dst)
                    stack.push_back(e.m_dst->get_id());
            }
        }

        stack.push_back(root_id);
        while (!stack.empty()) {
            unsigned s = stack.back();
            stack.pop_back();
            if (bwd.contains(s))
                continue;
            bwd.insert(s);
            auto it = m_partial_dfa_in.find(s);
            if (it == m_partial_dfa_in.end())
                continue;
            for (const unsigned edge_idx : it->second) {
                if (edge_idx >= m_partial_dfa_edges.size())
                    continue;
                partial_dfa_edge const& e = m_partial_dfa_edges[edge_idx];
                if (e.m_src)
                    stack.push_back(e.m_src->get_id());
            }
        }

        for (const unsigned s : fwd) {
            if (bwd.contains(s))
                scc.insert(s);
        }

        if (!scc.contains(root_id))
            return false;

        if (scc.num_elems() > 1)
            return true;

        const auto it = m_partial_dfa_out.find(root_id);
        if (it == m_partial_dfa_out.end())
            return false;
        for (const unsigned edge_idx : it->second) {
            if (edge_idx >= m_partial_dfa_edges.size())
                continue;
            partial_dfa_edge const& e = m_partial_dfa_edges[edge_idx];
            if (e.m_dst && e.m_dst->get_id() == root_id)
                return true;
        }
        return false;
    }

    unsigned nielsen_graph::mark_scc_projection_edges(uint_set const& scc) {
        // scc contains expr ids (see collect_scc_for_projection).
        //
        // Returns the number of edges *newly* added to the projection coverage
        // by this call (edges that were previously unmarked).  Crucially, the
        // monotone extraction index is bumped ONLY when there is something new
        // to mark.  This makes "the explored SCC has grown" a *global* property
        // of the SCC's edge set rather than a per-entry-state one: re-visiting
        // an already-fully-marked SCC from a different state (e.g. a derivative
        // state br of r, which shares the SCC {r, br}) marks nothing new and is
        // therefore not treated as a fresh cycle to decompose.  Without this,
        // each state of the cycle would trigger its own redundant decomposition
        // as the derivation walks around the SCC.
        unsigned newly_marked = 0;
        for (unsigned src_id : scc) {
            auto it = m_partial_dfa_out.find(src_id);
            if (it == m_partial_dfa_out.end())
                continue;
            for (const unsigned edge_idx : it->second) {
                if (edge_idx >= m_partial_dfa_edges.size())
                    continue;
                partial_dfa_edge const& e = m_partial_dfa_edges[edge_idx];
                if (!e.m_dst || !scc.contains(e.m_dst->get_id()))
                    continue;
                if (e.m_projection_idx == 0)
                    ++newly_marked;
            }
        }
        if (newly_marked == 0)
            return 0;

        ++m_projection_extract_idx;
        const unsigned extract_idx = m_projection_extract_idx;
        for (unsigned src_id : scc) {
            auto it = m_partial_dfa_out.find(src_id);
            if (it == m_partial_dfa_out.end())
                continue;
            for (const unsigned edge_idx : it->second) {
                if (edge_idx >= m_partial_dfa_edges.size())
                    continue;
                partial_dfa_edge& e = m_partial_dfa_edges[edge_idx];
                if (!e.m_dst || !scc.contains(e.m_dst->get_id()))
                    continue;
                if (e.m_projection_idx == 0)
                    e.m_projection_idx = extract_idx;
            }
        }
        return newly_marked;
    }

    euf::snode const* nielsen_graph::get_slice(euf::snode const* v, expr* left, expr* right) {
        SASSERT(v && v->get_expr() && left && right);
        SASSERT(v->is_var());

        expr_ref new_arg(v->get_expr(), m);
        expr_ref new_l(left, m), new_r(right, m);
        expr* arg, *l, *r;

        if (m_sk.is_slice(new_arg, arg, l, r)) {
            new_l = a.mk_add(left, l);
            m_rw(new_l);
            new_r = a.mk_add(right, r);
            m_rw(new_r);
            new_arg = arg;
        }
        expr_ref slice = m_sk.mk_slice(new_arg, new_l, new_r);
        return m_sg.mk(slice);
    }


    euf::snode const* nielsen_graph::get_tail(euf::snode const* v, expr* cnt, const bool fwd) {
        if (fwd)
            return get_slice(v, cnt, a.mk_int(0));
        return get_slice(v, a.mk_int(0), cnt);
    }

    euf::snode const* nielsen_graph::get_tail(euf::snode const* v, const unsigned cnt, const bool fwd) {
        return get_tail(v, a.mk_int(cnt), fwd);
    }

    simplify_result nielsen_node::simplify_and_init(ptr_vector<nielsen_edge> const& cur_path) {
        if (m_is_extended)
            return simplify_result::proceed;

        euf::sgraph& sg = m_graph.sg();
        ast_manager& m = sg.get_manager();
        seq_util& seq = this->graph().seq();
        bool changed = true;

        //if (id() == 9) {
        //    std::string dot = graph().to_dot();
        //    std::cout << dot << std::endl;
        //}

        // DON'T add rules here that add new constraints or apply substitutions
        // add them to apply_det_modifier instead

        while (changed) {
            changed = false;

            // pass 1: remove trivially satisfied equalities and memberships
            unsigned wi = 0;
            for (unsigned i = 0; i < m_str_eq.size(); ++i) {
                str_eq& eq = m_str_eq[i];
                if (eq.is_trivial())
                    continue;
                m_str_eq[wi++] = eq;
            }
            if (wi < m_str_eq.size()) {
                m_str_eq.shrink(wi);
                changed = true;
            }

            unsigned wj = 0;
            for (unsigned j = 0; j < m_str_mem.size(); ++j) {
                str_mem& mem = m_str_mem[j];
                if (mem.is_trivial(this))
                    continue;
                m_str_mem[wj++] = mem;
            }
            if (wj < m_str_mem.size()) {
                m_str_mem.shrink(wj);
                changed = true;
            }

            auto cannot_be_empty = [&](euf::snode const* side) {
                euf::snode_vector tokens;
                side->collect_tokens(tokens);
                const bool has_char = std::ranges::any_of(tokens, [](euf::snode const* t){ return t->is_char(); });
                const bool all_eliminable = std::ranges::all_of(tokens, [](euf::snode const* t){
                    return t->is_var() || t->is_power();
                });
                return has_char || !all_eliminable;
            };

            unsigned wk = 0;
            for (unsigned k = 0; k < m_str_deq.size(); ++k) {
                str_deq& deq = m_str_deq[k];

                if (deq.m_lhs == deq.m_rhs || (deq.m_lhs && deq.m_rhs && deq.m_lhs->is_empty() && deq.m_rhs->is_empty())) {
                    set_general_conflict();
                    set_conflict(backtrack_reason::symbol_clash, deq.m_dep);
                    return simplify_result::conflict;
                }
                if (deq.m_lhs->length() == 1 && deq.m_rhs->length() == 1) {
                    expr* l, *r;
                    if (graph().seq().str.is_unit(deq.m_lhs->get_expr(), l) &&
                        graph().seq().str.is_unit(deq.m_rhs->get_expr(), r)) {

                        add_constraint(constraint(m.mk_not(m.mk_eq(l, r)), deq.m_dep, m));
                        continue;
                    }
                }

                if (deq.m_lhs->is_empty() && !deq.m_rhs->is_empty()) {
                    if (cannot_be_empty(deq.m_rhs)) continue;
                }
                else if (deq.m_rhs->is_empty() && !deq.m_lhs->is_empty()) {
                    if (cannot_be_empty(deq.m_lhs)) continue;
                }

                // simplify constant-exponent powers
                dep_tracker lhs_pow_dep = nullptr;
                if (euf::snode const* s = simplify_const_powers(this, sg, deq.m_lhs, lhs_pow_dep)) {
                    deq.m_lhs = s;
                    deq.m_dep = m_graph.dep_mgr().mk_join(deq.m_dep, lhs_pow_dep);
                    changed = true;
                }
                dep_tracker rhs_pow_dep = nullptr;
                if (euf::snode const* s = simplify_const_powers(this, sg, deq.m_rhs, rhs_pow_dep)) {
                    deq.m_rhs = s;
                    deq.m_dep = m_graph.dep_mgr().mk_join(deq.m_dep, rhs_pow_dep);
                    changed = true;
                }

                // prefix/suffix cancellation
                {
                    euf::snode_vector lhs_toks, rhs_toks;
                    deq.m_lhs->collect_tokens(lhs_toks);
                    deq.m_rhs->collect_tokens(rhs_toks);

                    unsigned prefix = 0;
                    while (prefix < lhs_toks.size() && prefix < rhs_toks.size()) {
                        euf::snode const* lt = lhs_toks[prefix];
                        euf::snode const* rt = rhs_toks[prefix];
                        if (m.are_equal(lt->get_expr(), rt->get_expr())) {
                            ++prefix;
                        }
                        else if (sg.are_unit_distinct(lt, rt)) {
                            prefix = static_cast<unsigned>(-1);
                            break;
                        }
                        else
                            break;
                    }
                    if (prefix == static_cast<unsigned>(-1)) {
                        continue;
                    }

                    unsigned lsz = lhs_toks.size(), rsz = rhs_toks.size();
                    unsigned suffix = 0;
                    while (suffix < lsz - prefix && suffix < rsz - prefix) {
                        euf::snode const* lt = lhs_toks[lsz - 1 - suffix];
                        euf::snode const* rt = rhs_toks[rsz - 1 - suffix];
                        if (m.are_equal(lt->get_expr(), rt->get_expr())) {
                            ++suffix;
                        } else if (sg.are_unit_distinct(lt, rt)) {
                            suffix = static_cast<unsigned>(-1);
                            break;
                        }
                        else
                            break;
                    }
                    if (suffix == static_cast<unsigned>(-1)) {
                        continue;
                    }

                    if (prefix > 0 || suffix > 0) {
                        deq.m_lhs = sg.drop_left(sg.drop_right(deq.m_lhs, suffix), prefix);
                        deq.m_rhs = sg.drop_left(sg.drop_right(deq.m_rhs, suffix), prefix);
                        changed = true;
                    }
                }

                if (deq.m_lhs == deq.m_rhs || (deq.m_lhs->is_empty() && deq.m_rhs->is_empty())) {
                    set_general_conflict();
                    set_conflict(backtrack_reason::symbol_clash, deq.m_dep);
                    return simplify_result::conflict;
                }

                m_str_deq[wk++] = deq;
            }
            if (wk < m_str_deq.size()) {
                m_str_deq.shrink(wk);
                changed = true;
            }

            // pass 2: detect symbol clashes, empty-propagation, and prefix cancellation
            for (str_eq& eq : m_str_eq) {
                SASSERT(eq.well_formed());

                // one side empty, the other not empty => conflict check
                // (the actual substitution is done in apply_det_modifier)
                if (eq.m_lhs->is_empty() && !eq.m_rhs->is_empty()) {
                    if (check_empty_side_conflict(sg, eq.m_rhs, eq.m_dep))
                        return simplify_result::conflict;
                    continue;
                }
                if (eq.m_rhs->is_empty() && !eq.m_lhs->is_empty()) {
                    if (check_empty_side_conflict(sg, eq.m_lhs, eq.m_dep))
                        return simplify_result::conflict;
                    continue;
                }

                // prefix/suffix cancellation: strip common leading and trailing tokens.
                // Same char or same variable on both sides can always be cancelled.
                // Two different concrete characters is a symbol clash.
                {
                    euf::snode_vector lhs_toks, rhs_toks;
                    eq.m_lhs->collect_tokens(lhs_toks);
                    eq.m_rhs->collect_tokens(rhs_toks);

                    // --- prefix ---
                    unsigned prefix = 0;
                    while (prefix < lhs_toks.size() && prefix < rhs_toks.size()) {
                        euf::snode const* lt = lhs_toks[prefix];
                        euf::snode const* rt = rhs_toks[prefix];
                        if (m.are_equal(lt->get_expr(), rt->get_expr())) {
                            ++prefix;
                        }
                        else if (sg.are_unit_distinct(lt, rt)) {
                            set_general_conflict();
                            set_conflict(backtrack_reason::symbol_clash, eq.m_dep);
                            return simplify_result::conflict;
                        }
                        else
                            break;
                    }

                    // --- suffix (only among the tokens not already consumed by prefix) ---
                    unsigned lsz = lhs_toks.size(), rsz = rhs_toks.size();
                    unsigned suffix = 0;
                    while (suffix < lsz - prefix && suffix < rsz - prefix) {
                        euf::snode const* lt = lhs_toks[lsz - 1 - suffix];
                        euf::snode const* rt = rhs_toks[rsz - 1 - suffix];
                        if (m.are_equal(lt->get_expr(), rt->get_expr())) {
                            ++suffix;
                        } else if (sg.are_unit_distinct(lt, rt)) {
                            set_general_conflict();
                            set_conflict(backtrack_reason::symbol_clash, eq.m_dep);
                            return simplify_result::conflict;
                        }
                        else
                            break;
                    }

                    if (prefix > 0 || suffix > 0) {
                        eq.m_lhs = sg.drop_left(sg.drop_right(eq.m_lhs, suffix), prefix);
                        eq.m_rhs = sg.drop_left(sg.drop_right(eq.m_rhs, suffix), prefix);
                        changed = true;
                    }
                }
            }

            // pass 3: power simplification (mirrors ZIPT's LcpCompression +
            // SimplifyPowerElim + SimplifyPowerSingle)
            for (str_eq& eq : m_str_eq) {
                SASSERT(eq.well_formed());
                if (eq.is_trivial())
                    continue;

                // 3a: simplify constant-exponent powers (base^0 → ε, base^1 → base)
                dep_tracker lhs_pow_dep = nullptr;
                if (euf::snode const* s = simplify_const_powers(this, sg, eq.m_lhs, lhs_pow_dep)) {
                    eq.m_lhs = s;
                    eq.m_dep = m_graph.dep_mgr().mk_join(eq.m_dep, lhs_pow_dep);
                    changed = true;
                }
                dep_tracker rhs_pow_dep = nullptr;
                if (euf::snode const* s = simplify_const_powers(this, sg, eq.m_rhs, rhs_pow_dep)) {
                    eq.m_rhs = s;
                    eq.m_dep = m_graph.dep_mgr().mk_join(eq.m_dep, rhs_pow_dep);
                    changed = true;
                }

                // 3b: merge adjacent same-base tokens into combined powers
                if (euf::snode const* s = merge_adjacent_powers(sg, eq.m_lhs))
                    { eq.m_lhs = s; changed = true; }
                if (euf::snode const* s = merge_adjacent_powers(sg, eq.m_rhs))
                    { eq.m_rhs = s; changed = true; }

                // 3c: CommPower-based power elimination — when one side starts
                // with a power w^p, count base-pattern occurrences c on the
                // other side's prefix.  If we can determine the ordering between
                // p and c, cancel the matched portion.  Mirrors ZIPT's
                // SimplifyPowerElim calling CommPower.
                // Spec: CommPower cancellation.
                //   Given: pow_side = w^p · rest_pow  and  other_side = w^c · rest_other
                //   where c is the number of times the base pattern w occurs in the
                //   directional prefix of other_side.
                //   - If p ≤ c: pow_side := rest_pow,            other_side := w^(c-p) · rest_other
                //   - If c ≤ p: pow_side := w^(p-c) · rest_pow,  other_side := rest_other
                //   - If p = c: both reduce completely (handled by both conditions above).
                SASSERT(eq.well_formed());
                bool comm_changed = false;
                for (int side = 0; side < 2 && !comm_changed; ++side) {
                    euf::snode const*& pow_side = side == 0 ? eq.m_lhs : eq.m_rhs;
                    euf::snode const*& other_side = side == 0 ? eq.m_rhs : eq.m_lhs;
                    if (!pow_side || !other_side)
                        continue;
                    for (unsigned od = 0; od < 2 && !comm_changed; ++od) {
                        bool fwd = od == 0;
                        euf::snode const* end_tok = dir_token(pow_side, fwd);
                        if (!end_tok || !end_tok->is_power())
                            continue;
                        euf::snode const* base_sn = end_tok->arg0();
                        expr* pow_exp = get_power_exp_expr(end_tok, seq);
                        if (!base_sn || !pow_exp)
                            continue;

                        auto [count, consumed] =
                            comm_power(base_sn, other_side, m, seq, fwd);
                        if (!count.get() || consumed == 0)
                            continue;

                        expr_ref norm_count = normalize_arith(m, count);
                        bool pow_le_count = false, count_le_pow = false;
                        dep_tracker pow_le_dep = nullptr, count_le_dep = nullptr;
                        rational diff;
                        if (get_const_power_diff(norm_count, pow_exp, m_graph.a, diff)) {
                            count_le_pow = diff.is_nonpos();
                            pow_le_count = diff.is_nonneg();
                        }
                        else if (!cur_path.empty()) {
                            pow_le_count = m_graph.check_lp_le(pow_exp, norm_count, this, pow_le_dep);
                            count_le_pow = m_graph.check_lp_le(norm_count, pow_exp, this, count_le_dep);
                        }
                        if (!pow_le_count && !count_le_pow)
                            continue;

                        eq.m_dep = m_graph.dep_mgr().mk_join(eq.m_dep, pow_le_dep);
                        eq.m_dep = m_graph.dep_mgr().mk_join(eq.m_dep, count_le_dep);

                        pow_side = dir_drop(sg, pow_side, 1, fwd);
                        other_side = dir_drop(sg, other_side, consumed, fwd);
                        expr* base_e = get_power_base_expr(end_tok, seq);
                        if (pow_le_count && count_le_pow) {
                            // equal: both cancel completely
                        }
                        else if (pow_le_count) {
                            // pow <= count: remainder goes to other_side
                            expr_ref rem = normalize_arith(m, m_graph.a.mk_sub(norm_count, pow_exp));
                            expr_ref pw(seq.str.mk_power(base_e, rem), m);
                            other_side = dir_concat(sg, sg.mk(pw), other_side, fwd);
                        }
                        else {
                            // count <= pow: remainder goes to pow_side
                            expr_ref rem = normalize_arith(m, m_graph.a.mk_sub(pow_exp, norm_count));
                            expr_ref pw(seq.str.mk_power(base_e, rem), m);
                            pow_side = dir_concat(sg, sg.mk(pw), pow_side, fwd);
                        }
                        comm_changed = true;
                    }
                }
                if (comm_changed)
                    changed = true;

                // After any change in passes 3a–3c, restart the while loop
                // before running 3d/3e.  This lets 3a simplify new constant-
                // exponent powers (e.g. base^1 → base created by 3c) before
                // 3e attempts LP-based elimination which would introduce a
                // needless fresh variable.
                if (changed)
                    continue;

                // 3d: power prefix elimination — when both sides start with a
                // power of the same base, cancel the common power prefix.
                // (Subsumed by 3c for many cases, but handles same-base-power
                // pairs that CommPower may miss when both leading tokens are powers.)
                SASSERT(eq.well_formed());
                for (unsigned od = 0; od < 2 && !changed; ++od) {
                    bool fwd = (od == 0);
                    euf::snode const* lh = dir_token(eq.m_lhs, fwd);
                    euf::snode const* rh = dir_token(eq.m_rhs, fwd);
                    if (!(lh && rh && lh->is_power() && rh->is_power()))
                        continue;
                    expr* lb = get_power_base_expr(lh, seq);
                    expr* rb = get_power_base_expr(rh, seq);
                    if (!(lb && rb && lb == rb))
                        continue;
                    expr* lp = get_power_exp_expr(lh, seq);
                    expr* rp = get_power_exp_expr(rh, seq);
                    rational diff;
                    if (lp && rp && get_const_power_diff(rp, lp, m_graph.a, diff)) {
                        // rp = lp + diff (constant difference)
                        eq.m_lhs = dir_drop(sg, eq.m_lhs, 1, fwd);
                        eq.m_rhs = dir_drop(sg, eq.m_rhs, 1, fwd);
                        if (diff.is_pos()) {
                            // rp > lp: put base^diff on RHS (direction-aware prepend/append)
                            expr_ref de(m_graph.a.mk_int(diff), m);
                            expr_ref pw(seq.str.mk_power(lb, de), m);
                            eq.m_rhs = dir_concat(sg, sg.mk(pw), eq.m_rhs, fwd);
                        }
                        else if (diff.is_neg()) {
                            // lp > rp: put base^(-diff) on LHS
                            expr_ref de(m_graph.a.mk_int(-diff), m);
                            expr_ref pw(seq.str.mk_power(lb, de), m);
                            eq.m_lhs = dir_concat(sg, sg.mk(pw), eq.m_lhs, fwd);
                        }
                        // diff == 0: both powers cancel completely
                        changed = true;
                    }
                    // 3e: LP-aware power directional elimination
                    else if (lp && rp && !cur_path.empty()) {
                        dep_tracker lp_le_dep = nullptr, rp_le_dep = nullptr;
                        bool lp_le_rp = m_graph.check_lp_le(lp, rp, this, lp_le_dep);
                        bool rp_le_lp = m_graph.check_lp_le(rp, lp, this, rp_le_dep);
                        if (lp_le_rp || rp_le_lp) {
                            if (lp_le_rp)
                                eq.m_dep = m_graph.dep_mgr().mk_join(eq.m_dep, lp_le_dep);
                            if (rp_le_lp)
                                eq.m_dep = m_graph.dep_mgr().mk_join(eq.m_dep, rp_le_dep);
                            expr* smaller_exp = lp_le_rp ? lp : rp;
                            expr* larger_exp  = lp_le_rp ? rp : lp;
                            eq.m_lhs = dir_drop(sg, eq.m_lhs, 1, fwd);
                            eq.m_rhs = dir_drop(sg, eq.m_rhs, 1, fwd);
                            if (lp_le_rp && rp_le_lp) {
                                // both ≤ -> equal -> both cancel completely
                                add_constraint(m_graph.mk_constraint(m.mk_eq(lp, rp), eq.m_dep));
                            }
                            else {
                                // we only know for sure that one is smaller than the other
                                expr_ref d(m_graph.a.mk_sub(larger_exp, smaller_exp), m);
                                expr_ref zero(m_graph.a.mk_int(0), m);
                                add_constraint(m_graph.mk_constraint(m_graph.a.mk_ge(d, zero), eq.m_dep));
                                expr_ref pw(seq.str.mk_power(lb, d), m);
                                euf::snode const*& larger_side = lp_le_rp ? eq.m_rhs : eq.m_lhs;
                                larger_side = dir_concat(sg, sg.mk(pw), larger_side, fwd);
                            }
                            changed = true;
                        }
                    }
                }
            }
        }

        // consume concrete characters from str_mem via Brzozowski derivatives
        // in both directions (left-to-right, then right-to-left), mirroring ZIPT.
        for (str_mem& mem : m_str_mem) {
            SASSERT(mem.well_formed());
            if (mem.is_primitive() || !mem.is_plain())
                continue;
            for (unsigned od = 0; od < 2; ++od) {
                bool fwd = od == 0;
                while (!mem.m_str->is_empty()) {
                    euf::snode const* tok = dir_token(mem.m_str, fwd);
                    if (!tok || !tok->is_char_or_unit())
                        break;
                    euf::snode const* src_re = mem.m_regex;
                    euf::snode const* deriv = fwd
                        ? sg.brzozowski_deriv(mem.m_regex, tok)
                        : reverse_brzozowski_deriv(sg, mem.m_regex, tok);
                    TRACE(seq, tout << mem_pp(mem, m) << " d: " << spp(deriv, m) << "\n");
                    if (!deriv)
                        break;
                    if (deriv->is_fail()) {
                        set_general_conflict();
                        set_conflict(backtrack_reason::regex, mem.m_dep);
                        return simplify_result::conflict;
                    }
                    if (fwd) {
                        if (tok->is_char()) {
                            // concrete char: record single edge directly
                            m_graph.record_partial_derivative_edge(src_re, tok, deriv);
                        } else if (src_re->is_ground()) {
                            // symbolic unit: record all concrete minterm edges for src_re
                            // so cycle_decomp can detect SCCs lazily (mirrors old concrete
                            // per-minterm var_split behaviour).
                            euf::snode_vector mts;
                            sg.compute_minterms(src_re, mts);
                            for (euf::snode const* mt : mts) {
                                euf::snode const* mt_deriv = sg.brzozowski_deriv(src_re, mt);
                                if (mt_deriv && !mt_deriv->is_fail() && mt_deriv->is_ground())
                                    m_graph.record_partial_derivative_edge(src_re, mt, mt_deriv);
                            }
                        }
                    }
                    mem.m_str = dir_drop(sg, mem.m_str, 1, fwd);
                    mem.m_regex = deriv;
                }
            }
        }

        // consume symbolic characters via uniform derivatives
        for (str_mem& mem : m_str_mem) {
            SASSERT(mem.well_formed());
            if (mem.is_primitive() || !mem.is_plain())
                continue;
            while (mem.m_str && !mem.m_str->is_empty()) {

                // TODO: generalize this to work for reverse derivative as well.
                euf::snode const* tok = mem.m_str->first();
                if (!tok || !tok->is_char_or_unit())
                    break;

                euf::snode const* src_re = mem.m_regex;

                euf::snode const* next = nullptr;
                {
                    seq_rewriter rw(m);
                    expr_ref d(rw.mk_derivative(mem.m_regex->get_expr()), m);

                    // Extract the inner char expression from seq.unit(?inner)
                    expr *inner_char = tok->arg0()->get_expr();

                    // substitute the inner char for the derivative variable in d
                    var_subst vs(m);
                    d = vs(d, inner_char);

                    th_rewriter thrw(m);
                    thrw(d);

                    // Record concrete minterm edges for src_re so cycle_decomp can
                    // detect SCCs lazily (mirrors the incremental DFA building from
                    // concrete char consumption in the loop above).
                    if (src_re->is_ground()) {
                        euf::snode_vector mts;
                        sg.compute_minterms(src_re, mts);
                        for (euf::snode const* mt : mts) {
                            euf::snode const* mt_deriv = sg.brzozowski_deriv(src_re, mt);
                            if (mt_deriv && !mt_deriv->is_fail())
                                m_graph.record_partial_derivative_edge(src_re, mt, mt_deriv);
                        }
                    }

                    next = sg.mk(d);
                }
                if (next->is_fail()) {
                    TRACE(seq, tout << "empty regex" << spp(mem.m_regex, m) << "\n");
                    set_general_conflict();
                    set_conflict(backtrack_reason::regex, mem.m_dep);
                    return simplify_result::conflict;
                }

                mem.m_str = sg.drop_left(mem.m_str, 1);
                mem.m_regex = next;
            }
        }

        // consume leading characters of view / guard memberships (Section 3.3).
        // m_regex is the current (plain) derivative state; we gate on whether it
        // lies in Q_ν (projection_state_in_Q) and step with the ordinary
        // derivative, keeping the view/guard annotation.
        for (str_mem& mem : m_str_mem) {
            SASSERT(mem.well_formed());
            if (mem.is_plain())
                continue;
            if (consume_view_guard(mem))
                return simplify_result::conflict;
        }

        // check for regex memberships that are immediately infeasible
        for (str_mem& mem : m_str_mem) {
            if (mem.is_contradiction(this)) {
                TRACE(seq, tout << "contradiction " << mem_pp(mem, m) << "\n");
                set_general_conflict();
                set_conflict(backtrack_reason::regex, mem.m_dep);
                return simplify_result::conflict;
            }
        }

        // remove trivial membership constraints once again
        unsigned wj = 0;
        for (unsigned j = 0; j < m_str_mem.size(); ++j) {
            str_mem& mem = m_str_mem[j];
            if (mem.is_trivial(this))
                continue;
            m_str_mem[wj++] = mem;
        }
        if (wj < m_str_mem.size())
            m_str_mem.shrink(wj);

        // Regex widening: for each remaining str_mem, overapproximate
        // the string by replacing variables with their regex intersection
        // and check if the result intersected with the target regex is empty.
        // Detects infeasible constraints that would otherwise require
        // expensive exploration. Mirrors ZIPT NielsenNode.CheckRegexWidening.
        SASSERT(m_graph.m_seq_regex);
        for (str_mem const& mem : m_str_mem) {
            SASSERT(mem.well_formed());
            if (mem.is_primitive())
                continue;
            dep_tracker dep = mem.m_dep;
            if (m_graph.check_regex_widening(*this, mem, dep)) {
                set_general_conflict();
                set_conflict(backtrack_reason::regex_widening, dep);
                return simplify_result::conflict;
            }
        }

        if (is_satisfied()) {
            // pass 1 removed all trivial str_eq entries; is_satisfied() requires
            // the remainder to be trivial, so the vector must be empty here.
            SASSERT(m_str_eq.empty());
            return simplify_result::satisfied;
        }
        return simplify_result::proceed;
    }

    bool nielsen_node::consume_view_guard(str_mem& mem) {
        SASSERT(!mem.is_plain());
        euf::sgraph& sg = m_graph.sg();
        ast_manager& m = sg.get_manager();
        seq_util& seq = m_graph.seq();

        auto set_regex_conflict = [&]() {
            set_general_conflict();
            set_conflict(backtrack_reason::regex, mem.m_dep);
        };

        while (mem.m_str && !mem.m_str->is_empty()) {
            euf::snode const* tok = mem.m_str->first();
            if (!tok || !tok->is_char_or_unit())
                break; // leading token is a variable/power — nothing to consume yet
            euf::snode const* c = mem.m_regex;
            // The gate tests the CURRENT (plain) state c against Q_ν.  An ite
            // state means a previous symbolic step has not been resolved yet;
            // leave it for apply_regex_if_split.
            if (!c->is_ground() || c->kind() == euf::snode_kind::s_ite)
                break;
            const bool in_Q = m_graph.projection_state_in_Q(c->get_expr(), mem.m_nu);
            if (!in_Q) {
                if (mem.is_guard()) {
                    // The run left Q: no lap from the start can complete within Q
                    // anymore, so the guard is discharged (accepts every suffix).
                    mem.m_discharged = true;
                    return false;
                }
                // view: a^{-1} L_{Q,F}(c) = ∅ when c ∉ Q.
                set_regex_conflict();
                return true;
            }
            // Step with brzozowski_deriv for BOTH concrete and symbolic tokens.
            // This is essential: the partial-DFA states (and m_root) are produced
            // by brzozowski_deriv, so its canonicalization must be used here too —
            // otherwise the guard's resolved state never equals m_root by snode
            // identity and laps never close.  For a symbolic unit it yields a
            // canonical ite residual that apply_regex_if_split later resolves.
            euf::snode const* next = sg.brzozowski_deriv(c, tok);
            if (!next)
                break;
            mem.m_str = sg.drop_left(mem.m_str, 1);
            mem.m_regex = next;
            if (next->is_fail()) {
                // view: derivative collapsed to ∅ — unsatisfiable.
                // guard: the lap can never close through ∅; treat as discharged.
                if (mem.is_guard()) { mem.m_discharged = true; return false; }
                set_regex_conflict();
                return true;
            }
            if (next->is_ground() && next->kind() != euf::snode_kind::s_ite) {
                // concrete next state resolved immediately
                if (mem.is_guard() && next == mem.m_root) {
                    // a non-empty prefix completed a lap r→…→r within Q.
                    set_regex_conflict();
                    return true;
                }
            }
            else {
                // symbolic ite residual: defer to apply_regex_if_split, which
                // resolves the character and (for guards) detects a lap landing
                // back on the root.
                break;
            }
        }
        return false;
    }

    bool nielsen_node::is_satisfied() const {
        if (!m_str_deq.empty() || !m_str_eq.empty())
            return false;
        if (any_of(m_str_mem, [](auto const &m) { return !m.is_primitive();}))
            return false;
        return true;
    }

    euf::snode const* nielsen_graph::mk_rewrite(expr* e) const {
        expr_ref er(e, m);
        th_rewriter rw(m);
        rw(er);
        return m_sg.mk(er);
    }

    // -----------------------------------------------------------------------
    // nielsen_graph: search
    // -----------------------------------------------------------------------

    void nielsen_graph::apply_parikh_to_node(nielsen_node& node) const {
        if (!m_parikh_enabled || node.m_parikh_applied)
            return;
        node.m_parikh_applied = true;

        // Generate modular length constraints (len(str) = min_len + stride·k, etc.)
        // and append them to the node's integer constraint list.
        m_parikh->apply_to_node(node);

        // Lightweight feasibility pre-check: does the Parikh modular constraint
        // contradict the variable's current integer bounds?  If so, mark this
        // node as a Parikh-image conflict immediately (avoids a solver call).
        dep_tracker parikh_dep = nullptr;
        if (!node.is_currently_conflict() && m_parikh->check_parikh_conflict(node, parikh_dep) != nullptr) {
            node.set_general_conflict();
            node.set_conflict(backtrack_reason::parikh_image, parikh_dep);
        }
    }

    void nielsen_graph::assert_root_constraints_to_solver() {
        if (m_root_constraints_asserted)
            return;
        m_root_constraints_asserted = true;
        // Constraint.Shared: assert all root-level length/Parikh constraints
        // to m_solver at the base level (no push/pop). These include:
        //   - len(lhs) = len(rhs) for each non-trivial string equality
        //   - len(str) >= min_len and len(str) <= max_len for each regex membership
        //   - len(x) >= 0 for each variable appearing in the root constraints
        // Making these visible to the solver before the DFS allows arithmetic
        // pruning at every node, not just the root.
        vector<length_constraint> constraints;
        generate_length_constraints(constraints);
        for (auto const& lc : constraints) {
            m_root->add_constraint(lc.to_constraint());
        }
    }

    template<typename T> class scoped_push {
        T &t;
    public:
        scoped_push(T& t) : t(t) {
            t.push_scope();
        }
        ~scoped_push() {
            t.pop_scope(1);
        }
    };

    nielsen_graph::search_result nielsen_graph::solve() {
        SASSERT(m_root);

        try {
            ++m_stats.m_num_solve_calls;
            clear_sat_node();

            TRACE(seq, tout << "Solve call " << m_stats.m_num_solve_calls << "\n");

            // Constraint.Shared: assert root-level length/Parikh constraints to the
            // solver at the base level, so they are visible during all feasibility checks.
            assert_root_constraints_to_solver();

            // Iterative deepening: increment by 1 on each failure.
            // m_max_search_depth == 0 means unlimited; otherwise stop when bound exceeds it.
            m_depth_bound = 3;
            while (true) {
                if (!m.inc()) {
#ifdef Z3DEBUG
                    // Examining the Nielsen graph is probably the best way of debugging
                    const std::string dot = to_dot();
                    IF_VERBOSE(1, verbose_stream() << dot << "\n";);
                    IF_VERBOSE(1, display(verbose_stream()));
#endif
                    break;
                }
                if (m_max_search_depth > 0 && m_depth_bound > m_max_search_depth)
                    break;
                ptr_vector<nielsen_edge> cur_path;
                // scoped_push _scoped_push(m_dep_mgr); // gc dependencies after search
                SASSERT(!m_root->is_currently_conflict());
                const search_result r = search_dfs(m_root, cur_path); // the main search loop
                IF_VERBOSE(1, verbose_stream()
                                  << " depth_bound=" << m_depth_bound << " dfs_nodes=" << m_stats.m_num_dfs_nodes
                                  << " max_depth=" << m_stats.m_max_depth << " extensions=" << m_stats.m_num_extensions
                                  << " arith_prune=" << m_stats.m_num_arith_infeasible << " result="
                                  << (r == search_result::sat     ? "SAT"
                                      : r == search_result::unsat ? "UNSAT"
                                                                  : "UNKNOWN")
                                  << "\n";);
                if (r == search_result::sat) {
                    IF_VERBOSE(1,
                        verbose_stream() << "side constraints: \n";
                        for (auto const &c : cur_path.back()->side_constraints()) {
                            verbose_stream() << "  side constraint: " << c.fml << "\n";
                        });
                    ++m_stats.m_num_sat;
                    SASSERT(m_sat_node != nullptr);
                    return r;
                }
                if (r == search_result::unsat) {
                    ++m_stats.m_num_unsat;
                    const auto deps = collect_conflict_deps();
                    m_dep_mgr.linearize(deps, m_conflict_sources);
                    TRACE(seq, display(tout, m_root));
                    return r;
                }
                // depth limit hit – double the bound and retry
                m_depth_bound *= 2;
                SASSERT(m_depth_bound < INT_MAX);
            }
            ++m_stats.m_num_unknown;
            return search_result::unknown;
        }
        catch(const std::exception&) {
#ifdef Z3DEBUG
            std::string dot = to_dot();
#endif
            throw;
        }
    }

    // ---- Transposition-table helpers (node memoization) ----------------------

    static bool reason_is_string_only(backtrack_reason r) {
        switch (r) {
        case backtrack_reason::regex:
        case backtrack_reason::regex_widening:
        case backtrack_reason::symbol_clash:
        case backtrack_reason::character_range:
            return true;
        default:
            // arithmetic, parikh_image, external, children_failed, unevaluated
            return false;
        }
    }

    bool nielsen_graph::node_unsat_string_only(nielsen_node const* n) const {
        if (n->m_reason == backtrack_reason::children_failed)
            return n->m_unsat_cacheable; // set when the children_failed result was formed
        return reason_is_string_only(n->m_reason);
    }

    std::vector<unsigned> nielsen_graph::compute_node_signature(nielsen_node const* n) const {
        std::vector<unsigned> sig;
        // string equalities (order-independent)
        {
            std::vector<std::pair<unsigned,unsigned>> v;
            for (auto const& e : n->str_eqs())
                v.emplace_back(e.m_lhs->id(), e.m_rhs->id());
            std::sort(v.begin(), v.end());
            sig.push_back(static_cast<unsigned>(v.size()));
            for (auto const& [a,b] : v) { sig.push_back(a); sig.push_back(b); }
        }
        sig.push_back(UINT_MAX); // section separator
        // string disequalities
        {
            std::vector<std::pair<unsigned,unsigned>> v;
            for (auto const& d : n->str_deqs())
                v.emplace_back(d.m_lhs->id(), d.m_rhs->id());
            std::sort(v.begin(), v.end());
            sig.push_back(static_cast<unsigned>(v.size()));
            for (auto const& [a,b] : v) { sig.push_back(a); sig.push_back(b); }
        }
        sig.push_back(UINT_MAX);
        // regex memberships incl. view/guard metadata
        {
            std::vector<std::vector<unsigned>> v;
            for (auto const& mm : n->str_mems())
                v.push_back({ mm.m_str->id(), mm.m_regex->id(),
                              static_cast<unsigned>(mm.m_kind),
                              mm.m_root ? mm.m_root->id() : UINT_MAX,
                              mm.m_nu, mm.m_discharged ? 1u : 0u });
            std::sort(v.begin(), v.end());
            sig.push_back(static_cast<unsigned>(v.size()));
            for (auto const& a : v) for (unsigned x : a) sig.push_back(x);
        }
        sig.push_back(UINT_MAX);
        // character-range constraints (per symbolic unit)
        {
            std::vector<std::vector<unsigned>> v;
            for (auto const& [uid, cr] : n->char_ranges()) {
                std::vector<unsigned> entry;
                entry.push_back(uid);
                for (auto const& rg : cr.first.ranges()) { entry.push_back(rg.m_lo); entry.push_back(rg.m_hi); }
                v.push_back(std::move(entry));
            }
            std::sort(v.begin(), v.end());
            sig.push_back(static_cast<unsigned>(v.size()));
            for (auto const& e : v) { sig.push_back(static_cast<unsigned>(e.size())); for (unsigned x : e) sig.push_back(x); }
        }
        return sig;
    }

    dep_tracker nielsen_graph::node_all_deps(nielsen_node const* n) {
        dep_tracker d = nullptr;
        for (auto const& e : n->str_eqs())  d = m_dep_mgr.mk_join(d, e.m_dep);
        for (auto const& q : n->str_deqs()) d = m_dep_mgr.mk_join(d, q.m_dep);
        for (auto const& mm : n->str_mems()) d = m_dep_mgr.mk_join(d, mm.m_dep);
        for (auto const& [uid, cr] : n->char_ranges()) d = m_dep_mgr.mk_join(d, cr.second);
        return d;
    }

    nielsen_graph::search_result nielsen_graph::search_dfs(nielsen_node* node,
        ptr_vector<nielsen_edge>& cur_path, const unsigned depth) {

        ++m_stats.m_num_dfs_nodes;
        // std::cout << m_stats.m_num_dfs_nodes << std::endl;
        // depth is NOT necessarily the length of the path
        // Reason: Progress nodes are not counted towards the depth limit
        // Otw. problems with a lot of variables would barely terminate
        SASSERT(depth <= cur_path.size());
        m_stats.m_max_depth = std::max(m_stats.m_max_depth, depth);

        if (node->is_general_conflict()) {
            ++m_stats.m_num_simplify_conflict;
            return search_result::unsat;
        }

        // check for external cancellation (timeout, user interrupt)
        if (!m.inc())
            return search_result::unknown;

#ifdef Z3DEBUG
        if (m_stats.m_num_dfs_nodes % 50 == 0) {
            std::string dot = to_dot();
            std::cout << "";
        }
#endif

        // check DFS node budget (0 = unlimited)
        if (m_max_nodes > 0 && m_stats.m_num_dfs_nodes > m_max_nodes)
            return search_result::unknown;

        // we might need to tell the SAT solver about the new integer inequalities
        // that might have been added by an extension step
        assert_node_side_constraints(node);

        if (node->is_currently_conflict()) {
            ++m_stats.m_num_simplify_conflict;
            return search_result::unsat;
        }

        // simplify constraints (idempotent after first call)
        const simplify_result sr = node->simplify_and_init(cur_path);

        if (sr == simplify_result::conflict || node->is_general_conflict()) {
            SASSERT(node->is_general_conflict());
            ++m_stats.m_num_simplify_conflict;
            node->set_general_conflict();
            return search_result::unsat;
        }

        // Transposition-table lookup.  The node is now canonical (post-simplify).
        // If an equivalent node was already proven UNSAT for string/regex-only
        // reasons, this node is unsat too — independently of its (integer) side
        // constraints — so we prune without re-exploring its subtree.  We derive
        // the conflict from this node's own constraint deps (a sound over-approx).
        {
            const std::vector<unsigned> sig = compute_node_signature(node);
            if (m_unsat_node_cache.contains(sig)) {
                node->set_conflict(backtrack_reason::regex, node_all_deps(node));
                node->set_general_conflict();
                node->m_unsat_cacheable = true;
                ++m_stats.m_num_simplify_conflict;
                ++m_num_cache_hits;
                return search_result::unsat;
            }
        }

        // Apply Parikh image filter: generate modular length constraints and
        // perform a lightweight feasibility pre-check.  The filter is guarded
        // internally (m_parikh_applied) so it only runs once per node.
        // Note: Parikh filtering is skipped for satisfied nodes (returned above);
        // a fully satisfied node has no remaining memberships to filter.
        apply_parikh_to_node(*node);

        if (node->is_general_conflict()) {
            ++m_stats.m_num_simplify_conflict;
            return search_result::unsat;
        }

        // Assert any new int_constraints added during simplify_and_init for this
        // node into the current solver scope. Constraints inherited from the parent
        // (indices 0..m_parent_ic_count-1) are already present at the enclosing
        // scope level; only the newly-added tail needs to be asserted here.
        // Also generate per-node |LHS| = |RHS| length constraints for descendant
        // equations (root constraints are already at the base level).
        generate_node_length_constraints(node);
        assert_node_side_constraints(node);

        if (node->is_currently_conflict()) {
            ++m_stats.m_num_simplify_conflict;
            return search_result::unsat;
        }

        // integer feasibility check: the solver now holds all path constraints
        // incrementally; just query the solver directly
        if (!cur_path.empty() && !check_int_feasibility()) {
            const dep_tracker deps = get_subsolver_dependency(node);
            node->set_conflict(backtrack_reason::arithmetic, deps);
            node->set_general_conflict();

            ++m_stats.m_num_arith_infeasible;
            return search_result::unsat;
        }

        SASSERT(sr != simplify_result::satisfied || node->is_satisfied());
        SASSERT(!node->is_currently_conflict());

        if (node->is_satisfied()) {
            // Before declaring SAT, check leaf-node regex feasibility:
            // for each variable with multiple regex constraints, verify
            // that the intersection of all its regexes is non-empty.
            // Mirrors ZIPT NielsenNode.CheckRegex.
            dep_tracker dep = nullptr;
            if (!check_leaf_regex(*node, dep)) {
                node->set_general_conflict();
                node->set_conflict(backtrack_reason::regex, dep);
                // string-only conflict (empty intersection) → memoize.
                node->m_unsat_cacheable = true;
                m_unsat_node_cache.insert(compute_node_signature(node));
                return search_result::unsat;
            }
            assert_node_side_constraints(node);
            // We need to have everything asserted before reporting SAT
            // (otw. the outer solver might assume false-assigned literals to be true)
            if (node->is_currently_conflict()) {
                ++m_stats.m_num_simplify_conflict;
                return search_result::unsat;
            }
            m_sat_node = node;
            m_sat_path = cur_path;
            return search_result::sat;
        }

        if (node->is_currently_conflict())
            return search_result::unsat;

        // depth bound check
        if (depth >= m_depth_bound)
            return search_result::unknown;

        SASSERT(!node->is_currently_conflict());

        // generate extensions only once per node; children persist across runs
        if (!node->is_extended()) {
            const bool ext = generate_extensions(node);
            IF_VERBOSE(1, display(verbose_stream(), node));
            CTRACE(seq, !ext, display(tout, node) << to_dot() << "\n");
            if (!ext) {
                std::cout << "No extensions generated for node " << node->id() << ", but not satisfied or conflict?!"
                          << std::endl;
                node->to_html(std::cout, m);
                std::cout << std::endl;
                display(std::cout, node);
            }
            VERIFY(ext);

            if (node->is_currently_conflict())
                // in rare cases, trying to extend can make a complicated conflict visible
                return search_result::unsat;

            node->set_extended(true);
            ++m_stats.m_num_extensions;
        }

        // explore children
        bool any_unknown = false;
        bool all_general_conflict = true;
        for (nielsen_edge *e : node->outgoing()) {
            cur_path.push_back(e);
            // Push a solver scope for this edge and assert its side integer
            // constraints.  The child's own new constraints will be asserted
            // inside the recursive call (above).  On return, pop the scope so
            // that backtracking removes those assertions.
            m_length_solver.push();

            // Lazily compute substitution length constraints (|x| = |u|) on first
            // traversal. This must happen before asserting side_constraints
            if (!e->len_constraints_computed()) {
                add_subst_length_constraints(e);
                e->set_len_constraints_computed(true);

                for (const auto& sc : e->side_constraints()) {
                    e->tgt()->add_constraint(sc);
                }
            }

            const auto new_depth = depth + (e->is_progress() ? 0 : 1);
            const search_result r = search_dfs(e->tgt(), cur_path, new_depth);

            m_length_solver.pop(1);
            if (r == search_result::sat)
                return search_result::sat;
            cur_path.pop_back();
            if (r == search_result::unknown)
                any_unknown = true;
            if (!e->tgt()->is_general_conflict())
                all_general_conflict = false;
        }

        if (all_general_conflict) {
            SASSERT(!any_unknown);
            // mark it such that we do not have to reconsider it even after a hot-restart
            node->set_general_conflict();
        }
        if (!any_unknown) {
            node->set_child_conflict();
            // Memoize this internal UNSAT iff its whole subtree closed for
            // string/regex-only reasons (no child relied on arithmetic / Parikh /
            // external context).  Then a same-signature node found via any other
            // path is pruned by the lookup above.
            bool all_string_only = true;
            for (nielsen_edge* e : node->outgoing()) {
                if (!node_unsat_string_only(e->tgt())) {
                    all_string_only = false;
                    break;
                }
            }
            node->m_unsat_cacheable = all_string_only;
            if (all_string_only)
                m_unsat_node_cache.insert(compute_node_signature(node));
            return search_result::unsat;
        }
        return search_result::unknown;
    }

    // Returns true if variable snode `var` appears anywhere in the token list of `n`.
    static bool snode_contains_var(euf::snode const* n, euf::snode const* var) {
        SASSERT(n && var);
        euf::snode_vector tokens;
        n->collect_tokens(tokens);
        return any_of(tokens, [var](auto const &t) { return t == var; });
    }

    bool nielsen_graph::apply_det_modifier(nielsen_node* node) {
        // resist the temptation to add rules that "simplify" primitive membership constraints!
        // pretty much all of them could cause divergence!
        // e.g., x \in aa* => don't apply substitution x / ax even though it looks "safe" to do
        // there might be another constraint x \in a* and they would just push the "a" back and forth!

        for (unsigned eq_idx = 0; eq_idx < node->str_eqs().size(); ++eq_idx) {
            str_eq const& eq = node->str_eqs()[eq_idx];
            if (eq.is_trivial())
                continue; // We should have simplified it away before
            auto l = eq.m_lhs, r = eq.m_rhs;
            if (!l || !r)
                continue;

            // 0. empty side propagation
            if (l->is_empty() || r->is_empty()) {
                euf::snode const* non_empty_side = l->is_empty() ? r : l;
                nielsen_node* child = mk_child(node);
                nielsen_edge* e = mk_edge(node, child, "det", true);
                euf::snode_vector tokens;
                non_empty_side->collect_tokens(tokens);

                auto& eqs = child->str_eqs();
                eqs[eq_idx] = eqs.back();
                eqs.pop_back();

                for (euf::snode const* t : tokens) {
                    if (t->is_var()) {
                        nielsen_subst s(t, m_sg.mk_empty_seq(t->get_sort()), eq.m_dep);
                        e->add_subst(s);
                        child->apply_subst(m_sg, s);
                    } else if (t->is_power()) {
                        expr* expr_node = t->get_expr();
                        expr* pow_base = nullptr, *pow_exp = nullptr;
                        if (seq().str.is_power(expr_node, pow_base, pow_exp) && pow_exp)
                            e->add_side_constraint(mk_constraint(a.mk_eq(pow_exp, a.mk_int(0)), eq.m_dep));
                        nielsen_subst s(t, m_sg.mk_empty_seq(t->get_sort()), eq.m_dep);
                        e->add_subst(s);
                        child->apply_subst(m_sg, s);
                    }
                }
                return true;
            }

            // 1. unit equalities produced by unit-unit prefix/suffix splits
            {
                euf::snode_vector lhs_toks, rhs_toks;
                l->collect_tokens(lhs_toks);
                r->collect_tokens(rhs_toks);

                // --- prefix ---
                unsigned prefix = 0;
                while (prefix < lhs_toks.size() && prefix < rhs_toks.size()) {
                    euf::snode const* lt = lhs_toks[prefix];
                    euf::snode const* rt = rhs_toks[prefix];
                    if (m.are_equal(lt->get_expr(), rt->get_expr()))
                        ++prefix;
                    else if (m_sg.are_unit_distinct(lt, rt))
                        break;
                    else if (lt->is_char_or_unit() && rt->is_char_or_unit()) {
                        nielsen_node* child = mk_child(node);
                        nielsen_edge* e = mk_edge(node, child, "det", true);

                        if (lt->is_char()) {
                            std::swap(lt, rt);
                            std::swap(l, r);
                        }
                        SASSERT(lt->is_unit());

                        euf::snode const* lhs_rest = m_sg.drop_left(l, prefix + 1);
                        euf::snode const* rhs_rest = m_sg.drop_left(r, prefix + 1);

                        auto& eqs = child->str_eqs();
                        eqs[eq_idx] = eqs.back();
                        eqs.pop_back();

                        if (lt->is_char())
                            std::swap(lt, rt);
                        nielsen_subst subst(lt, rt, eq.m_dep);
                        e->add_subst(subst);
                        child->apply_subst(m_sg, subst);

                        if (!lhs_rest->is_empty() || !rhs_rest->is_empty())
                            eqs.push_back(str_eq(lhs_rest, rhs_rest, eq.m_dep));
                        return true;
                    }
                    else
                        break;
                }

                // --- suffix ---
                unsigned lsz = lhs_toks.size(), rsz = rhs_toks.size();
                unsigned suffix = 0;
                while (suffix < lsz - prefix && suffix < rsz - prefix) {
                    euf::snode const* lt = lhs_toks[lsz - 1 - suffix];
                    euf::snode const* rt = rhs_toks[rsz - 1 - suffix];
                    if (m.are_equal(lt->get_expr(), rt->get_expr()))
                        ++suffix;
                    else if (m_sg.are_unit_distinct(lt, rt))
                        break;
                    else if (lt->is_char_or_unit() && rt->is_char_or_unit()) {
                        nielsen_node* child = mk_child(node);
                        nielsen_edge* e = mk_edge(node, child, "det", true);

                        euf::snode const* lhs_rest = m_sg.drop_right(l, suffix + 1);
                        euf::snode const* rhs_rest = m_sg.drop_right(r, suffix + 1);

                        auto& eqs = child->str_eqs();
                        eqs[eq_idx] = eqs.back();
                        eqs.pop_back();

                        if (lt->is_char())
                            std::swap(lt, rt);
                        nielsen_subst subst(lt, rt, eq.m_dep);
                        e->add_subst(subst);
                        child->apply_subst(m_sg, subst);

                        if (!lhs_rest->is_empty() || !rhs_rest->is_empty())
                            eqs.push_back(str_eq(lhs_rest, rhs_rest, eq.m_dep));
                        return true;
                    }
                    else
                        break;
                }
            }

            // 2. power-character directional inconsistency
            for (unsigned od = 0; od < 2; ++od) {
                bool fwd = (od == 0);
                euf::snode const* lh = dir_token(l, fwd);
                euf::snode const* rh = dir_token(r, fwd);
                for (int side = 0; side < 2; ++side) {
                    euf::snode const* pow_head = (side == 0) ? lh : rh;
                    euf::snode const* other_head = (side == 0) ? rh : lh;
                    if (!pow_head || !pow_head->is_power() || !other_head || !other_head->is_char())
                        continue;
                    euf::snode const* base_sn = pow_head->arg0();
                    if (!base_sn) continue;
                    euf::snode const* base_head = dir_token(base_sn, fwd);
                    if (!base_head || !base_head->is_char()) continue;
                    if (m.are_equal(base_head->get_expr(), other_head->get_expr())) continue;
                    // Directional base/head mismatch -> force exponent 0 and power -> ε.
                    nielsen_node* child = mk_child(node);
                    nielsen_edge* e = mk_edge(node, child, "det", true);
                    nielsen_subst s(pow_head, m_sg.mk_empty_seq(pow_head->get_sort()), eq.m_dep);
                    e->add_subst(s);
                    child->apply_subst(m_sg, s);
                    expr* pow_exp = get_power_exp_expr(pow_head, m_seq);
                    if (pow_exp) {
                        expr* zero = a.mk_int(0);
                        e->add_side_constraint(mk_constraint(a.mk_eq(pow_exp, zero), eq.m_dep));
                    }
                    return true;
                }
            }

            // 3. variable-character look-ahead substitution
            for (unsigned od = 0; od < 2; ++od) {
                bool fwd = od == 0;

                euf::snode const* var_side = nullptr;
                euf::snode const* char_side = nullptr;
                euf::snode const* lhead = dir_token(l, fwd);
                euf::snode const* rhead = dir_token(r, fwd);
                if (!lhead || !rhead) continue;

                if (lhead->is_var() && rhead->is_char()) {
                    var_side = l;
                    char_side = r;
                }
                else if (rhead->is_var() && lhead->is_char()) {
                    var_side = r;
                    char_side = l;
                }
                else
                    continue;

                euf::snode_vector var_toks, char_toks;
                collect_tokens_dir(var_side, fwd, var_toks);
                collect_tokens_dir(char_side, fwd, char_toks);
                if (var_toks.size() <= 1 || char_toks.empty())
                    continue;

                euf::snode const* var_node = var_toks[0];
                SASSERT(var_node->is_var());

                unsigned i = 0;
                for (; i < char_toks.size() && char_toks[i]->is_char(); ++i) {
                    unsigned j1 = 1;
                    unsigned j2 = i;
                    bool failed = false;

                    while (j1 < var_toks.size() && j2 < char_toks.size()) {
                        euf::snode const* st1 = var_toks[j1];
                        euf::snode const* st2 = char_toks[j2];

                        if (!st2->is_char()) break;
                        if (st1->is_char()) {
                            if (st1->id() == st2->id()) {
                                j1++; j2++;
                                continue;
                            }
                            failed = true; break;
                        }
                        if (st1->id() != var_node->id()) break;

                        bool inner_indet = false;
                        for (unsigned l_idx = 0; j2 < char_toks.size() && l_idx < i; ++l_idx) {
                            st2 = char_toks[j2];
                            if (!st2->is_char()) {
                                inner_indet = true; break;
                            }
                            if (st2->id() == char_toks[l_idx]->id()) {
                                j2++; continue;
                            }
                            failed = true; break;
                        }
                        if (inner_indet || failed) break;
                        j1++;
                    }

                    if (failed) continue;
                    break;
                }

                if (i == 0) continue;

                bool skip_dir = false;
                euf::snode const* next_var = nullptr;
                for (unsigned k = i; k < char_toks.size(); ++k) {
                    euf::snode const* t = char_toks[k];
                    if (t->is_power()) {
                        skip_dir = true;
                        break;
                    }
                    if (t->is_var()) {
                        next_var = t;
                        break;
                    }
                }
                if (skip_dir) continue;

                if (next_var) {
                    u_map<unsigned> dep_edges;
                    for (str_eq const& other_eq : node->str_eqs()) {
                        if (other_eq.is_trivial() || !other_eq.m_lhs || !other_eq.m_rhs)
                            continue;
                        euf::snode const* lh2 = dir_token(other_eq.m_lhs, fwd);
                        euf::snode const* rh2 = dir_token(other_eq.m_rhs, fwd);
                        if (!lh2 || !rh2) continue;

                        auto record_dep = [&](euf::snode const* head_var, euf::snode const* other_side) {
                            euf::snode_vector other_toks;
                            collect_tokens_dir(other_side, fwd, other_toks);
                            for (unsigned idx = 0; idx < other_toks.size(); ++idx) {
                                if (other_toks[idx]->is_var()) {
                                    if (!dep_edges.contains(head_var->id()))
                                        dep_edges.insert(head_var->id(), other_toks[idx]->id());
                                    return;
                                }
                            }
                        };

                        if (lh2->is_var() && rh2->is_var()) {
                            if (!dep_edges.contains(lh2->id()))
                                dep_edges.insert(lh2->id(), rh2->id());
                            if (!dep_edges.contains(rh2->id()))
                                dep_edges.insert(rh2->id(), lh2->id());
                        }
                        else if (lh2->is_var() && !rh2->is_var()) record_dep(lh2, other_eq.m_rhs);
                        else if (rh2->is_var() && !lh2->is_var()) record_dep(rh2, other_eq.m_lhs);
                    }

                    uint_set visited;
                    svector<unsigned> worklist;
                    worklist.push_back(next_var->id());
                    bool cycle_found = false;
                    while (!worklist.empty() && !cycle_found) {
                        unsigned cur = worklist.back();
                        worklist.pop_back();
                        if (cur == var_node->id()) {
                            cycle_found = true; break;
                        }
                        if (visited.contains(cur)) continue;
                        visited.insert(cur);
                        unsigned dep_id;
                        if (dep_edges.find(cur, dep_id))
                            worklist.push_back(dep_id);
                    }
                    if (cycle_found) continue;
                }

                euf::snode const* prefix_sn = char_toks[0];
                for (unsigned j = 1; j < i; ++j) {
                    prefix_sn = dir_concat(m_sg, prefix_sn, char_toks[j], fwd);
                }
                euf::snode const* tail = get_tail(var_node, compute_length_expr(prefix_sn).get(), fwd);
                euf::snode const* replacement = dir_concat(m_sg, prefix_sn, tail, fwd);
                nielsen_subst s(var_node, replacement, eq.m_dep);
                nielsen_node* child = mk_child(node);
                nielsen_edge* e = mk_edge(node, child, "det", true);
                e->add_side_constraint(mk_constraint(a.mk_ge(compute_length_expr(tail), a.mk_int(0)), eq.m_dep));
                e->add_subst(s);
                child->apply_subst(m_sg, s);
                return true;
            }

            // variable definition: x = t where x is a single var and x ∉ vars(t)
            // → deterministically substitute x → t throughout the node
            euf::snode const* var = nullptr;
            euf::snode const* def;

            if (l->is_var() && !snode_contains_var(r, l)) {
                var = l;
                def = r;
            }
            else if (r->is_var() && !snode_contains_var(l, r)) {
                var = r;
                def = l;
            }
            else if (l->is_unit() && r->is_char_or_unit()) {
                var = l;
                def = r;
            }
            else if (r->is_unit() && l->is_char_or_unit()) {
                var = r;
                def = l;
            }

            if (var) {
                nielsen_node* child = mk_child(node);
                nielsen_edge* e = mk_edge(node, child, "det", true);
                nielsen_subst s(var, def, eq.m_dep);
                e->add_subst(s);
                child->apply_subst(m_sg, s);
                return true;
            }
        }
        return false;
    }

    bool nielsen_graph::apply_const_nielsen(nielsen_node* node) {
        for (str_eq const& eq : node->str_eqs()) {
            if (eq.is_trivial())
                continue;
            SASSERT(eq.well_formed());
            for (unsigned od = 0; od < 2; ++od) {
                const bool fwd = (od == 0);
                euf::snode const* lhead = dir_token(eq.m_lhs, fwd);
                euf::snode const* rhead = dir_token(eq.m_rhs, fwd);
                if (!lhead || !rhead)
                    continue;

                // char vs var: branch 1: var -> ε, branch 2: var -> char·var   (depending on direction)
                euf::snode const* char_head = lhead->is_char_or_unit() ? lhead : (rhead->is_char_or_unit() ? rhead : nullptr);
                euf::snode const* var_head = lhead->is_var() ? lhead : (rhead->is_var() ? rhead : nullptr);
                if (char_head && var_head) {
                    nielsen_node* child = mk_child(node);
                    nielsen_edge* e = mk_edge(node, child, "nielsen const 0", true);
                    const nielsen_subst s1(var_head, m_sg.mk_empty_seq(var_head->get_sort()), eq.m_dep);
                    e->add_subst(s1);
                    child->apply_subst(m_sg, s1);


                    euf::snode const* tail = get_tail(var_head, a.mk_int(1), fwd);
                    euf::snode const* replacement = dir_concat(m_sg, char_head, tail, fwd);
                    child = mk_child(node);
                    e = mk_edge(node, child, "nielsen const &gt;", false);
                    e->add_side_constraint(mk_constraint(a.mk_ge(compute_length_expr(tail), a.mk_int(0)), eq.m_dep));
                    const nielsen_subst s2(var_head, replacement, eq.m_dep);
                    e->add_subst(s2);
                    child->apply_subst(m_sg, s2);
                    return true;
                }
            }
        }
        return false;
    }

    bool nielsen_graph::apply_var_nielsen(nielsen_node* node) {
        for (str_eq const& eq : node->str_eqs()) {
            if (eq.is_trivial())
                continue;
            SASSERT(eq.well_formed());
            for (unsigned od = 0; od < 2; ++od) {
                const bool fwd = od == 0;
                euf::snode const* lhead = dir_token(eq.m_lhs, fwd);
                euf::snode const* rhead = dir_token(eq.m_rhs, fwd);
                SASSERT(lhead && rhead);
                SASSERT(lhead->id() != rhead->id());
                if (!lhead->is_var() || !rhead->is_var())
                    continue;

                // x·A = y·B where x,y are distinct variables (classic Nielsen)
                // child 1: x → ε (progress)
                {
                    nielsen_node* child = mk_child(node);
                    nielsen_edge* e = mk_edge(node, child, "nielsen var =l", true);
                    const nielsen_subst s(lhead, m_sg.mk_empty_seq(lhead->get_sort()), eq.m_dep);
                    e->add_subst(s);
                    child->apply_subst(m_sg, s);
                }
                // child 2: y → ε (progress)
                {
                    nielsen_node* child = mk_child(node);
                    nielsen_edge* e = mk_edge(node, child, "nielsen var =r", true);
                    const nielsen_subst s(rhead, m_sg.mk_empty_seq(rhead->get_sort()), eq.m_dep);
                    e->add_subst(s);
                    child->apply_subst(m_sg, s);
                }
                // child 3: x → y·x (no progress)
                {
                    euf::snode const* replacement = dir_concat(m_sg, rhead, get_tail(lhead, compute_length_expr(rhead).get(), fwd), fwd);
                    nielsen_node* child = mk_child(node);
                    nielsen_edge* e = mk_edge(node, child, "nielsen var &gt;", false);
                    const nielsen_subst s(lhead, replacement, eq.m_dep);
                    e->add_subst(s);
                    child->apply_subst(m_sg, s);
                }
                // child 4: y → x·y (no progress)
                {
                    euf::snode const* replacement = dir_concat(m_sg, lhead, get_tail(rhead, compute_length_expr(lhead).get(), fwd), fwd);
                    nielsen_node* child = mk_child(node);
                    nielsen_edge* e = mk_edge(node, child, "nielsen var &lt;", false);
                    const nielsen_subst s(rhead, replacement, eq.m_dep);
                    e->add_subst(s);
                    child->apply_subst(m_sg, s);
                }
                return true;
            }
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // EqSplit helpers: token length classification
    // -----------------------------------------------------------------------

    bool nielsen_graph::token_has_variable_length(euf::snode const* tok) const {
        // Chars and units have known constant length 1.
        if (tok->is_char_or_unit())
            return false;
        // Variables and powers have symbolic/unknown length.
        if (tok->is_var() || tok->is_power())
            return true;
        // For s_var string literals: check if it's a string literal (known constant length).
        if (tok->get_expr()) {
            zstring s;
            if (m_seq.str.is_string(tok->get_expr(), s))
                return false;
        }
        // Everything else is treated as variable length.
        return true;
    }

    unsigned nielsen_graph::token_const_length(euf::snode const* tok) const {
        if (tok->is_char_or_unit())
            return 1;
        if (tok->get_expr()) {
            zstring s;
            if (m_seq.str.is_string(tok->get_expr(), s))
                return s.length();
        }
        return 0;
    }

    // -----------------------------------------------------------------------
    // EqSplit: find_eq_split_point
    // Port of ZIPT's StrEq.SplitEq algorithm.
    //
    // Walks tokens from each side tracking two accumulators:
    //   - lhs_has_symbolic / rhs_has_symbolic : whether a variable-length token
    //     has been consumed on that side since the last reset
    //   - const_diff : net constant-length difference (LHS constants − RHS constants)
    //
    // A potential split point arises when both accumulators are "zero"
    // (no outstanding symbolic contributions on either side), meaning
    // the prefix lengths are determined up to a constant offset (const_diff).
    //
    // Among all such split points, we pick the one minimising |const_diff|
    // (the padding amount). We also require having seen at least one variable-
    // length token before accepting a split, so that the split is non-trivial.
    // -----------------------------------------------------------------------

    bool nielsen_graph::find_eq_split_point(
            euf::snode_vector const& lhs_toks,
            euf::snode_vector const& rhs_toks,
            unsigned& out_lhs_idx,
            unsigned& out_rhs_idx,
            int& out_padding) const {
        const unsigned lhs_len = lhs_toks.size();
        const unsigned rhs_len = rhs_toks.size();
        if (lhs_len <= 1 || rhs_len <= 1)
            return false;

        // Initialize accumulators with the first token on each side (index 0).
        bool lhs_has_symbolic = token_has_variable_length(lhs_toks[0]);
        bool rhs_has_symbolic = token_has_variable_length(rhs_toks[0]);
        int const_diff = 0;
        if (!lhs_has_symbolic)
            const_diff += (int)token_const_length(lhs_toks[0]);
        if (!rhs_has_symbolic)
            const_diff -= (int)token_const_length(rhs_toks[0]);

        bool seen_variable = lhs_has_symbolic || rhs_has_symbolic;

        // Best confirmed split point.
        bool has_best = false;
        unsigned best_lhs = 0, best_rhs = 0;
        int best_padding = 0;

        // Pending split (needs confirmation by a subsequent variable token).
        bool has_pending = false;
        unsigned pending_lhs = 0, pending_rhs = 0;
        int pending_padding = 0;

        unsigned li = 1, ri = 1;

        while (li < lhs_len || ri < rhs_len) {
            const bool lhs_zero = !lhs_has_symbolic;
            const bool rhs_zero = !rhs_has_symbolic;

            // Potential split point: both symbolic accumulators are zero.
            if (seen_variable && lhs_zero && rhs_zero) {
                if (!has_pending || std::abs(const_diff) < std::abs(pending_padding)) {
                    has_pending = true;
                    pending_padding = const_diff;
                    pending_lhs = li;
                    pending_rhs = ri;
                }
            }

            // Decide which side to consume from.
            // Prefer consuming from the side whose symbolic accumulator is zero
            // (i.e., that side has no outstanding variable contributions).
            bool consume_lhs;
            if (lhs_zero && !rhs_zero)
                consume_lhs = true;
            else if (!lhs_zero && rhs_zero)
                consume_lhs = false;
            else if (lhs_zero && rhs_zero)
                consume_lhs = (const_diff <= 0);  // consume from side with less accumulated constant
            else
                consume_lhs = (li <= ri);  // both non-zero: round-robin

            if (consume_lhs) {
                if (li >= lhs_len) break;
                euf::snode const* tok = lhs_toks[li++];
                const bool is_var = token_has_variable_length(tok);
                if (is_var) {
                    // Confirm any pending split point.
                    if (has_pending && (!has_best || std::abs(pending_padding) < std::abs(best_padding))) {
                        has_best = true;
                        best_padding = pending_padding;
                        best_lhs = pending_lhs;
                        best_rhs = pending_rhs;
                        has_pending = false;
                    }
                    seen_variable = true;
                    lhs_has_symbolic = true;
                }
                else
                    const_diff += (int)token_const_length(tok);
            }
            else {
                if (ri >= rhs_len)
                    break;
                euf::snode const* tok = rhs_toks[ri++];
                const bool is_var = token_has_variable_length(tok);
                if (is_var) {
                    if (has_pending && (!has_best || std::abs(pending_padding) < std::abs(best_padding))) {
                        has_best = true;
                        best_padding = pending_padding;
                        best_lhs = pending_lhs;
                        best_rhs = pending_rhs;
                        has_pending = false;
                    }
                    seen_variable = true;
                    rhs_has_symbolic = true;
                }
                else
                    const_diff -= (int)token_const_length(tok);
            }
        }

        if (!has_best)
            return false;

        // A split at the very start or very end is degenerate — skip.
        if ((best_lhs == 0 && best_rhs == 0) ||
            (best_lhs == lhs_len && best_rhs == rhs_len))
            return false;

        out_lhs_idx = best_lhs;
        out_rhs_idx = best_rhs;
        out_padding = best_padding;
        return true;
    }

    // -----------------------------------------------------------------------
    // apply_eq_split — Port of ZIPT's EqSplitModifier.Apply
    //
    // For a regex-free equation LHS = RHS, finds a split point and decomposes
    // into two shorter equations with optional padding variable:
    //
    //   eq1: LHS[0..lhsIdx] · [pad if padding<0] = [pad if padding>0] · RHS[0..rhsIdx]
    //   eq2: LHS[lhsIdx..] · [pad if padding>0] = [pad if padding<0] · RHS[rhsIdx..]
    //
    // Creates a single progress child.
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_eq_split(nielsen_node* node) {

        for (unsigned eq_idx = 0; eq_idx < node->str_eqs().size(); ++eq_idx) {
            str_eq const& eq = node->str_eqs()[eq_idx];
            SASSERT(eq.well_formed());
            if (eq.is_trivial())
                continue;
            // EqSplit only applies to regex-free equations.
            if (!eq.m_lhs->is_regex_free() || !eq.m_rhs->is_regex_free())
                continue;

            euf::snode_vector lhs_toks, rhs_toks;
            eq.m_lhs->collect_tokens(lhs_toks);
            eq.m_rhs->collect_tokens(rhs_toks);
            if (lhs_toks.empty() || rhs_toks.empty())
                continue;

            unsigned split_lhs = 0, split_rhs = 0;
            int padding = 0;
            if (!find_eq_split_point(lhs_toks, rhs_toks, split_lhs, split_rhs, padding))
                continue;

            // Split the equation sides into prefix / suffix.
            euf::snode const* lhs_prefix = m_sg.drop_right(eq.m_lhs, lhs_toks.size() - split_lhs);
            euf::snode const* lhs_suffix = m_sg.drop_left(eq.m_lhs, split_lhs);
            euf::snode const* rhs_prefix = m_sg.drop_right(eq.m_rhs, rhs_toks.size() - split_rhs);
            euf::snode const* rhs_suffix = m_sg.drop_left(eq.m_rhs, split_rhs);

            // Build the two new equations, incorporating a padding variable if needed.
            euf::snode const* eq1_lhs = lhs_prefix;
            euf::snode const* eq1_rhs = rhs_prefix;
            euf::snode const* eq2_lhs = lhs_suffix;
            euf::snode const* eq2_rhs = rhs_suffix;
            th_rewriter rw(m);


            euf::snode const* pad = nullptr;
            if (padding != 0) {
                // NSB review: can we represent pad_var using a string function?
                expr *pad_var = m_sk.mk("eq-split", a.mk_int(padding), eq.m_lhs->get_expr(),
                                                 eq.m_rhs->get_expr(), eq.m_lhs->get_sort());
                pad = m_sg.mk(pad_var);
                if (padding > 0) {
                    // LHS prefix is longer by |padding| constants.
                    // Prepend pad to RHS prefix, append pad to LHS suffix.
                    eq1_rhs = m_sg.mk_concat(pad, rhs_prefix);
                    eq2_lhs = m_sg.mk_concat(lhs_suffix, pad);
                }
                else {
                    // RHS prefix is longer by |padding| constants.
                    // Append pad to LHS prefix, prepend pad to RHS suffix.
                    eq1_lhs = m_sg.mk_concat(lhs_prefix, pad);
                    eq2_rhs = m_sg.mk_concat(pad, rhs_suffix);
                }
            }

            // Create single progress child.
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "eq split", true);

            // Remove the original equation and add the two new ones.
            auto& eqs = child->str_eqs();
            eqs[eq_idx] = eqs.back();
            eqs.pop_back();
            eqs.push_back(str_eq(eq1_lhs, eq1_rhs, eq.m_dep));
            eqs.push_back(str_eq(eq2_lhs, eq2_rhs, eq.m_dep));

            // Int constraints on the edge.
            // 1) len(pad) = |padding|  (if padding variable was created)
            if (pad && pad->get_expr()) {
                const expr_ref len_pad(m_seq.str.mk_length(pad->get_expr()), m);
                const expr_ref abs_pad(a.mk_int(std::abs(padding)), m);
                e->add_side_constraint(mk_constraint(m.mk_eq(len_pad, abs_pad), eq.m_dep));
            }
            // 2) len(eq1_lhs) = len(eq1_rhs)
            const expr_ref l1 = compute_length_expr(eq1_lhs);
            const expr_ref r1 = compute_length_expr(eq1_rhs);
            e->add_side_constraint(mk_constraint(m.mk_eq(l1, r1), eq.m_dep));

            // 3) len(eq2_lhs) = len(eq2_rhs)
            const expr_ref l2 = compute_length_expr(eq2_lhs);
            const expr_ref r2 = compute_length_expr(eq2_rhs);
            e->add_side_constraint(mk_constraint(m.mk_eq(l2, r2), eq.m_dep));

            return true;
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // nielsen_graph: mk_fresh_var
    // -----------------------------------------------------------------------

    euf::snode const* nielsen_graph::mk_fresh_var(sort* s) {
        ++m_stats.m_num_fresh_vars;
        const std::string name = "v!" + std::to_string(m_fresh_cnt++);
        return m_sg.mk_var(symbol(name.c_str()), s);
    }

    bool nielsen_graph::generate_extensions(nielsen_node *node) {
        SASSERT(node != nullptr);
        SASSERT(!node->is_currently_conflict());
        // The first modifier that produces edges is used and returned immediately.

        // Priority 1: deterministic modifiers (single child, always progress)
        if (apply_det_modifier(node))
            return ++m_stats.m_mod_det, true;

        // Priority 2: PowerEpsilon - power → ε via base=ε or n=0
        if (apply_power_epsilon(node))
            return ++m_stats.m_mod_power_epsilon, true;

        // Priority 3: NumCmp - length comparison branching for power tokens
        if (apply_num_cmp(node))
            return ++m_stats.m_mod_num_cmp, true;

        // Priority 3b: SplitPowerElim - CommPower-based branching when
        // one side has a power and the other side has same-base occurrences
        // but ordering is unknown.
        if (apply_split_power_elim(node))
            return ++m_stats.m_mod_split_power_elim, true;

        // Priority 4: ConstNumUnwinding - power vs constant: n=0 or peel
        if (apply_const_num_unwinding(node))
            return ++m_stats.m_mod_const_num_unwinding, true;

        // Priority 5: EqSplit - split equations into two (single progress child)
        if (apply_eq_split(node))
            return ++m_stats.m_mod_eq_split, true;

        // Priority 5b: CycleSubsumption - eliminate leading variable subsumed by stabilizer
        if (apply_cycle_subsumption(node))
            return ++m_stats.m_mod_cycle_subsumption, true;

        // Priority 6: CycleDecomp - stabilizer introduction for regex cycles using partial DFA projection
        if (apply_cycle_decomposition(node))
            return ++m_stats.m_mod_star_intr, true;

        // Priority 7: GPowerIntr - ground power introduction
        if (apply_gpower_intr(node))
            return ++m_stats.m_mod_gpower_intr, true;

        // Priority 8: Regex Factorization
        if (apply_regex_factorization(node))
            return ++m_stats.m_mod_regex_factorization, true;

        // Priority 8b: ConstNielsen - char vs var (2 children)
        if (apply_const_nielsen(node))
            return ++m_stats.m_mod_const_nielsen, true;

        // Priority 9: RegexIfSplit - split str_mem s ∈ ite(c,th,el) by branching on c
        if (apply_regex_if_split(node))
            return ++m_stats.m_mod_regex_if_split, true;

        // Priority 9b: SignatureSplit - heuristic string equation splitting
        if (m_signature_split && apply_signature_split(node))
            return ++m_stats.m_mod_signature_split, true;

        // Priority 10: RegexVarSplit - split str_mem by minterms
        if (apply_regex_var_split(node))
            return ++m_stats.m_mod_regex_var_split, true;

        // Priority 11: PowerSplit - power unwinding with bounded prefix
        if (apply_power_split(node))
            return ++m_stats.m_mod_power_split, true;

        // Priority 12: VarNielsen - var vs var, all progress (classic Nielsen)
        if (apply_var_nielsen(node))
            return ++m_stats.m_mod_var_nielsen, true;

        // Priority 13: VarNumUnwinding - variable power unwinding for equality constraints
        if (apply_var_num_unwinding_eq(node))
            return ++m_stats.m_mod_var_num_unwinding_eq, true;

        // Priority 14: variable power unwinding for membership constraints
        if (apply_var_num_unwinding_mem(node))
            return ++m_stats.m_mod_var_num_unwinding_mem, true;

        // let's unwindind a disequality
        if (axiomatize_diseq(node))
            return ++m_stats.m_ax_diseq, true;

        return false;
    }

    // -----------------------------------------------------------------------
    // Helper: find a power token in any str_eq
    // -----------------------------------------------------------------------

    euf::snode const* nielsen_graph::find_power_token(nielsen_node* node) {
        for (str_eq const& eq : node->str_eqs()) {
            if (eq.is_trivial())
                continue;
            SASSERT(eq.well_formed());
            euf::snode_vector toks;
            eq.m_lhs->collect_tokens(toks);
            for (euf::snode const* t : toks) {
                if (t->is_power())
                    return t;
            }
            toks.reset();
            eq.m_rhs->collect_tokens(toks);
            for (euf::snode const* t : toks) {
                if (t->is_power())
                    return t;
            }
        }
        return nullptr;
    }

    // -----------------------------------------------------------------------
    // Helper: find a power token facing a constant (char) head
    // Returns true if found, sets power, other_head, eq_out.
    // -----------------------------------------------------------------------

    bool nielsen_graph::find_power_vs_non_var(nielsen_node* node,
                                            euf::snode const*& power,
                                            euf::snode const*& other_head,
                                            str_eq const*& eq_out,
                                            bool& fwd) {
        for (str_eq const& eq : node->str_eqs()) {
            if (eq.is_trivial())
                continue;
            for (unsigned od = 0; od < 2; ++od) {
                const bool local_fwd = (od == 0);
                euf::snode const* lhead = dir_token(eq.m_lhs, local_fwd);
                euf::snode const* rhead = dir_token(eq.m_rhs, local_fwd);
                // Match power vs any non-variable, non-empty token (char, unit,
                // power with different base, etc.).
                // Same-base power vs power is handled by NumCmp (priority 3).
                // Power vs variable is handled by PowerSplit (priority 11).
                // Power vs empty is handled by PowerEpsilon (priority 2).
                if (lhead && lhead->is_power() && rhead && !rhead->is_var() && !rhead->is_empty()) {
                    power = lhead;
                    other_head = rhead;
                    eq_out = &eq;
                    fwd = local_fwd;
                    return true;
                }
                if (rhead && rhead->is_power() && lhead && !lhead->is_var() && !lhead->is_empty()) {
                    power = rhead;
                    other_head = lhead;
                    eq_out = &eq;
                    fwd = local_fwd;
                    return true;
                }
            }
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Helper: find a power token facing a variable head
    // -----------------------------------------------------------------------

    bool nielsen_graph::find_power_vs_var(nielsen_node* node,
                                          euf::snode const*& power,
                                          euf::snode const*& var_head,
                                          str_eq const*& eq_out,
                                          bool& fwd) {
        for (str_eq const& eq : node->str_eqs()) {
            SASSERT(eq.well_formed() && !eq.is_trivial());

            for (unsigned od = 0; od < 2; ++od) {
                const bool local_fwd = (od == 0);
                euf::snode const* lhead = dir_token(eq.m_lhs, local_fwd);
                euf::snode const* rhead = dir_token(eq.m_rhs, local_fwd);
                if (lhead && lhead->is_power() && rhead && rhead->is_var()) {
                    power = lhead;
                    var_head = rhead;
                    eq_out = &eq;
                    fwd = local_fwd;
                    return true;
                }
                if (rhead && rhead->is_power() && lhead && lhead->is_var()) {
                    power = rhead;
                    var_head = lhead;
                    eq_out = &eq;
                    fwd = local_fwd;
                    return true;
                }
            }
        }
        return false;
    }

    bool nielsen_graph::find_power_vs_var(nielsen_node* node,
                                          euf::snode const*& power,
                                          str_mem const*& mem_out,
                                          bool& fwd) {
        for (str_mem const& mem : node->str_mems()) {
            if (mem.is_trivial(node)) {
                std::cout << "Trivial mem: " << mem_pp(mem, node->graph().get_manager()) << std::endl;
            }
            SASSERT(mem.well_formed() && !mem.is_trivial(node));

            for (unsigned od = 0; od < 2; ++od) {
                const bool local_fwd = (od == 0);
                euf::snode const* lhead = dir_token(mem.m_str, local_fwd);
                if (lhead && lhead->is_power()) {
                    power = lhead;
                    mem_out = &mem;
                    fwd = local_fwd;
                    return true;
                }
            }
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_power_epsilon
    // Fires only when one side of an equation is empty and the other side
    // starts with a power token u^n.  In that case, branch:
    //   (1) base u → ε (base is empty, so u^n = ε)
    //   (2) u^n → ε (the power is zero, replace power with empty)
    // mirrors ZIPT's PowerEpsilonModifier (which requires LHS.IsEmpty())
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_power_epsilon(nielsen_node* node) {
        // Match only when one equation side is empty and the other starts
        // with a power.  This mirrors ZIPT where PowerEpsilonModifier is
        // constructed only inside the "if (LHS.IsEmpty())" branch of
        // ExtendDir.  When both sides are non-empty and a power faces a
        // constant, ConstNumUnwinding (priority 4) handles it with both
        // n=0 and n≥1 branches.
        euf::snode const* power = nullptr;
        dep_tracker dep = m_dep_mgr.mk_empty();
        for (str_eq const& eq : node->str_eqs()) {
            if (eq.is_trivial())
                continue;
            SASSERT(eq.well_formed());
            if (eq.m_lhs->is_empty() && eq.m_rhs->first() && eq.m_rhs->first()->is_power()) {
                power = eq.m_rhs->first();
                dep = eq.m_dep;
                break;
            }
            if (eq.m_rhs->is_empty() && eq.m_lhs->first() && eq.m_lhs->first()->is_power()) {
                power = eq.m_lhs->first();
                dep = eq.m_dep;
                break;
            }
        }
        if (!power)
            return false;

        SASSERT(power->is_power() && power->num_args() >= 1);
        euf::snode const* base = power->arg0();

        nielsen_node* child;
        nielsen_edge* e;

        // Branch 1: base → ε (if base is a variable, substitute it to empty)
        // This makes u^n = ε^n = ε for any n.
        if (base->is_var()) {
            child = mk_child(node);
            e = mk_edge(node, child, "power power 0", true);
            const nielsen_subst s1(base, m_sg.mk_empty_seq(base->get_sort()), dep);
            e->add_subst(s1);
            child->apply_subst(m_sg, s1);
        }

        // Branch 2: replace the power token itself with ε (n = 0 semantics)
        child = mk_child(node);
        e = mk_edge(node, child, "power base 0", true);
        const nielsen_subst s2(power, m_sg.mk_empty_seq(power->get_sort()), dep);
        e->add_subst(s2);
        child->apply_subst(m_sg, s2);

        return true;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_num_cmp
    // For equations involving two power tokens u^m and u^n with the same base,
    // branch on the numeric relationship: m <= n vs n < m.
    // Generates proper integer side constraints for each branch.
    // mirrors ZIPT's NumCmpModifier
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_num_cmp(nielsen_node* node) {

        // Look for two directional endpoint power tokens with the same base.
        for (str_eq const& eq : node->str_eqs()) {
            SASSERT(eq.well_formed());
            if (eq.is_trivial())
                continue;
            for (unsigned od = 0; od < 2; ++od) {
                const bool fwd = (od == 0);
                euf::snode const* lhead = dir_token(eq.m_lhs, fwd);
                euf::snode const* rhead = dir_token(eq.m_rhs, fwd);
                if (!lhead || !rhead)
                    continue;
                if (!lhead->is_power() || !rhead->is_power())
                    continue;
                if (lhead->num_args() < 1 || rhead->num_args() < 1)
                    continue;
                // same base: compare the two powers
                if (lhead->arg0() != rhead->arg0())
                    continue;

                // Skip if the exponents differ by a constant — simplify_and_init's
                // directional power elimination already handles that case.
                expr* exp_m = get_power_exponent(lhead);
                expr* exp_n = get_power_exponent(rhead);
                if (!exp_m || !exp_n)
                    continue;
                rational diff;
                SASSERT(!get_const_power_diff(exp_n, exp_m, a, diff)); // handled by simplification

                // Branch 1 (explored first): n < m  (add constraint c ≥ p + 1)
                {
                    nielsen_edge *e = mk_edge(node, mk_child(node), "power cmp <", true);
                    const expr_ref n_plus_1(a.mk_add(exp_n, a.mk_int(1)), m);
                    e->add_side_constraint(mk_constraint(a.mk_ge(exp_m, n_plus_1), eq.m_dep));
                }
                // Branch 2 (explored second): m <= n  (add constraint p ≥ c)
                {
                    nielsen_edge *e = mk_edge(node, mk_child(node), "power cmp >=", true);
                    e->add_side_constraint(mk_constraint(a.mk_ge(exp_n, exp_m), eq.m_dep));
                }
                return true;
            }
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_split_power_elim
    // When one side starts with a power w^p, call CommPower on the other
    // side to count base-pattern occurrences c. If c > 0 and the ordering
    // between p and c cannot be determined, create two branches:
    //   Branch 1: p < c   (add constraint c ≥ p + 1)
    //   Branch 2: c ≤ p   (add constraint p ≥ c)
    // After branching, simplify_and_init's CommPower pass (3c) can resolve
    // the ordering deterministically and cancel the matched portion.
    // mirrors ZIPT's SplitPowerElim + NumCmpModifier
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_split_power_elim(nielsen_node* node) {

        for (str_eq const& eq : node->str_eqs()) {
            SASSERT(eq.well_formed());
            if (eq.is_trivial())
                continue;

            for (int side = 0; side < 2; ++side) {
                euf::snode const* pow_side   = (side == 0) ? eq.m_lhs : eq.m_rhs;
                euf::snode const* other_side = (side == 0) ? eq.m_rhs : eq.m_lhs;
                // NB: Shuvendu - this test is always false
                if (!pow_side || !other_side)
                    continue;

                for (unsigned od = 0; od < 2; ++od) {
                    const bool fwd = od == 0;
                    euf::snode const* end_tok = dir_token(pow_side, fwd);
                    if (!end_tok || !end_tok->is_power())
                        continue;
                    euf::snode const* base_sn = end_tok->arg0();
                    expr* pow_exp = get_power_exp_expr(end_tok, m_seq);
                    // NB: Shuvendu - this test is also redundant
                    if (!base_sn || !pow_exp)
                        continue;

                    auto [count, consumed] = comm_power(base_sn, other_side, m, m_seq, fwd);
                    if (!count.get() || consumed == 0)
                        continue;

                    expr_ref norm_count = normalize_arith(m, count);

                    // Skip if ordering is already deterministic — simplify_and_init
                    // pass 3c should have handled it.
                    rational diff;
                    if (get_const_power_diff(norm_count, pow_exp, a, diff))
                        continue;

                    // Branch 1: pow_exp < count (i.e., count >= pow_exp + 1)
                    {
                        nielsen_edge *e = mk_edge(node, mk_child(node), "power elim &gt;", true);
                        const expr_ref pow_plus1(a.mk_add(pow_exp, a.mk_int(1)), m);
                        e->add_side_constraint(mk_constraint(a.mk_ge(norm_count, pow_plus1), eq.m_dep));
                    }
                    // Branch 2: count <= pow_exp (i.e., pow_exp >= count)
                    {
                        nielsen_edge *e = mk_edge(node, mk_child(node), "power elim &le;", true);
                        e->add_side_constraint(mk_constraint(a.mk_ge(pow_exp, norm_count), eq.m_dep));
                    }
                    return true;
                }
            }
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_const_num_unwinding
    // For a power token u^n facing a constant (char) head,
    // branch: (1) n = 0 → u^n = ε, (2) n >= 1 → peel one u from power.
    // Generates integer side constraints for each branch.
    // mirrors ZIPT's ConstNumUnwindingModifier
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_const_num_unwinding(nielsen_node* node) {

        euf::snode const* power = nullptr;
        euf::snode const* other_head = nullptr;
        str_eq const *eq = nullptr;
        bool fwd = true;
        if (!find_power_vs_non_var(node, power, other_head, eq, fwd))
            return false;

        SASSERT(power->is_power() && power->num_args() >= 1);
        euf::snode const* base = power->arg0();
        expr *exp_n = get_power_exponent(power);
        expr *zero = a.mk_int(0);
        expr *one = a.mk_int(1);

        // Branch 1 (explored first): n = 0 → replace power with ε (progress)
        // Side constraint: n = 0
        nielsen_node *child = mk_child(node);
        nielsen_edge *e = mk_edge(node, child, "unwinding 0", true);
        const nielsen_subst s1(power, m_sg.mk_empty_seq(power->get_sort()), eq->m_dep);
        e->add_subst(s1);
        child->apply_subst(m_sg, s1);
        if (exp_n)
            e->add_side_constraint(mk_constraint(a.mk_eq(exp_n, zero), eq->m_dep));

        // Branch 2 (explored second): n >= 1 → peel one u: replace u^n with u · u^(n-1)
        // Side constraint: n >= 1
        // Create a proper nested power base^(n-1) instead of a fresh string variable.
        // This preserves power structure so that simplify_and_init can merge and
        // cancel adjacent same-base powers (mirroring ZIPT's SimplifyPowerUnwind).
        // Explored first because the n≥1 branch is typically more productive
        // for SAT instances (preserves power structure).
        const seq_util &seq = m_sg.get_seq_util();
        expr *power_e = power->get_expr();
        SASSERT(power_e);
        expr *base_expr = to_app(power_e)->get_arg(0);
        const expr_ref n_minus_1 = normalize_arith(m, a.mk_sub(exp_n, one));
        const expr_ref nested_pow(seq.str.mk_power(base_expr, n_minus_1), m);
        euf::snode const* nested_power_snode = m_sg.mk(nested_pow);

        euf::snode const* replacement = dir_concat(m_sg, base, nested_power_snode, fwd);
        child = mk_child(node);
        e = mk_edge(node, child, "unwinding &gt;", true);
        const nielsen_subst s2(power, replacement, eq->m_dep);
        e->add_subst(s2);
        child->apply_subst(m_sg, s2);
        if (exp_n)
            e->add_side_constraint(mk_constraint(a.mk_ge(exp_n, one), eq->m_dep));

        return true;
    }

    // -----------------------------------------------------------------------
    // Helper: ensure_automaton_explored  (lazy, on-demand, cached)
    // Records the *complete* reachable Brzozowski automaton of root_re into the
    // partial DFA — but only once per regex component.  This is the lazy
    // Q-completion: instead of re-running a fixed-depth BFS on every cycle
    // decomposition (the old precompute_partial_dfa(R, depth), which re-explored
    // R's automaton on each of the ~hundreds of decompositions and, with a
    // shallow depth, left Q incomplete so guards discharged prematurely and laps
    // never closed), we explore each component once to saturation and remember
    // every state we visited.  A later decomposition whose head was reached by an
    // earlier exploration finds it already covered and does no work.
    //
    // Completeness matters: the cycle guard / stabilizer view gate on whether the
    // current state lies in Q (projection_state_in_Q), and that is only sound as
    // a *bound* when Q contains the whole SCC of the head.  The BFS is bounded by
    // the finite reachable automaton (Brzozowski states modulo ACI), so it
    // terminates; m.inc() keeps it responsive to the resource limit.
    // -----------------------------------------------------------------------

    void nielsen_graph::ensure_automaton_explored(euf::snode const* root_re) {
        SASSERT(root_re);
        if (!root_re->is_ground())
            return;
        if (m_explored_automaton.contains(root_re->get_expr()->get_id()))
            return;

        svector<euf::snode const*> queue;
        queue.push_back(root_re);

        while (!queue.empty()) {
            if (!m.inc())
                return; // resource limit: leave Q partial (sound under-approx)
            euf::snode const* re = queue.back();
            queue.pop_back();
            const unsigned re_eid = re->get_expr()->get_id();
            if (m_explored_automaton.contains(re_eid))
                continue; // already explored (here or in a previous component)
            m_explored_automaton.insert(re_eid);

            euf::snode_vector mts;
            m_sg.compute_minterms(re, mts);
            for (euf::snode const* mt : mts) {
                euf::snode const* deriv = m_sg.brzozowski_deriv(re, mt);
                if (!deriv || deriv->is_fail())
                    continue;
                record_partial_derivative_edge(re, mt, deriv);
                if (deriv->is_ground() && !m_explored_automaton.contains(deriv->get_expr()->get_id()))
                    queue.push_back(deriv);
            }
        }
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_cycle_subsumption  (paper Section "Cycle Subsumption")
    // For a membership x·u ∈ R (u≠ε) with L(⊓Reg_x) ⊆ stab(R,Q_ν), drop the
    // leading x: replace x·u ∈ R by u ∈ R.  The inclusion is decided as the
    // product-emptiness test  L(⊓Reg_x) ∩ ~stab(R,Q_ν) = ∅  (Section 3.3),
    // adding one co-view component for ~stab.
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_cycle_subsumption(nielsen_node* node) {
        for (unsigned mi = 0; mi < node->str_mems().size(); ++mi) {
            str_mem const& mem = node->str_mems()[mi];
            SASSERT(mem.well_formed());
            if (!mem.is_plain() || mem.is_primitive())
                continue;
            euf::snode const* first = mem.m_str->first();
            SASSERT(first);
            if (!first->is_var())
                continue;
            euf::snode const* R = mem.m_regex;

            // R must lie on a detected cycle with a marked SCC snapshot.
            uint_set scc;
            if (!collect_scc_for_projection(R, scc))
                continue;
            const unsigned nu = m_projection_extract_idx;
            if (nu == 0)
                continue;

            // Decide  L(⊓Reg_x) ⊆ stab(R,Q_ν)  as  ⊓Reg_x ∩ ~stab = ∅.
            vector<prod_comp> comps;
            dep_tracker x_dep = nullptr;
            collect_var_components(first, *node, comps, x_dep);
            comps.push_back(prod_comp::mk_view(R, R, nu, /*complemented*/ true));
            if (check_product_emptiness(comps, 5000) != l_true)
                continue;

            // Subsume: replace x·u ∈ R with u ∈ R.
            euf::snode const* tail = m_sg.drop_first(mem.m_str);
            SASSERT(tail);

            nielsen_node* child = mk_child(node);
            mk_edge(node, child, "cycle subs", true);

            for (auto& cm : child->str_mems()) {
                if (cm == mem) {
                    cm.m_str = tail;
                    cm.m_dep = m_dep_mgr.mk_join(cm.m_dep, x_dep);
                    break;
                }
            }

            TRACE(seq, tout << "cycle_subsumption: dropped x=" << mk_pp(first->get_expr(), m)
                            << " from " << mk_pp(mem.m_str->get_expr(), m)
                            << " ∈ " << mk_pp(R->get_expr(), m) << " nu=" << nu << "\n");
            return true;
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_cycle_decomposition  (paper Section "Cycle Decomposition")
    //
    // For a membership x·u ∈ R whose leading variable x sits on a detected cycle
    // (R lies in an SCC of the partial DFA) and that does not already carry a
    // matching cycle guard for the current SCC snapshot, split
    //   x → x'·x''
    // and attach the two *view*/*guard* primitive constraints (Section 3.3):
    //   x' ∈ stab(R, Q_ν)            -- stabilizer view  (F = {R})
    //   noloop(x'', R, Q_ν)          -- cycle guard      (two-mode monitor)
    // The leading x' is immediately subsumed (its only constraint is the view,
    // and stab ⊆ stab trivially), so it is dropped from the primary constraint.
    //
    // Unlike the old projection-operator encoding, the view and guard are kept
    // as *constraint metadata* over the plain state R and the ν-indexed explored
    // subautomaton Q_ν — nothing is materialized as a regex, which keeps the
    // reachable state space finite (termination).
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_cycle_decomposition(nielsen_node* node) {
        // Look for a str_mem with a variable-headed string
        for (unsigned mi = 0; mi < node->str_mems().size(); ++mi) {
            str_mem const& mem = node->str_mems()[mi];
            SASSERT(mem.well_formed());
            if (!mem.is_plain() || mem.is_primitive())
                continue;
            euf::snode const* first = mem.m_str->first();
            SASSERT(first);
            if (!first->is_var())
                continue;

            euf::snode const* x = first;
            euf::snode const* R = mem.m_regex;

            // Lazily explore R's full automaton (once, cached) so that
            // collect_scc_for_projection sees the complete SCC of R and the
            // stabilizer view / cycle guard gate on a complete Q.
            ensure_automaton_explored(R);

            // R must sit on a cycle (an SCC of the partial DFA).
            uint_set scc;
            if (!collect_scc_for_projection(R, scc))
                continue;
            // Mark the SCC edges; this gives a ν identifying the current Q_SCC.
            // (We trigger on absence of a matching guard, NOT on novelty.)
            mark_scc_projection_edges(scc);
            const unsigned nu = m_projection_extract_idx;
            if (nu == 0)
                continue;

            // Trigger condition: x must not already carry a cycle guard for the
            // current SCC snapshot ν.  All states of one SCC share a single ν, so
            // the guard is keyed on ν rather than on the (changing) head R: as the
            // derivation walks the SCC the head moves, but the lineage is already
            // guarded against re-traversing the cycle.  An already-decomposed
            // variable whose guard refers to a strictly smaller, stale ν is
            // re-decomposed to adopt the enlarged SCC.
            bool already_guarded = false;
            for (str_mem const& g : node->str_mems()) {
                if (g.is_guard() && g.m_nu >= nu
                    && g.m_str && g.m_str->first() == x) {
                    already_guarded = true;
                    break;
                }
            }
            if (already_guarded)
                continue;

            sort* seq_sort = x->get_expr()->get_sort();

            // Construct the replacement x = x' x''
            euf::snode const* xp      = m_sg.mk(m_sk.mk("cycle", x->get_expr(), R->get_expr(), seq_sort));
            euf::snode const* xpp     = get_tail(x, compute_length_expr(xp).get());
            euf::snode const* xp_xpp  = m_sg.mk_concat(xp, xpp);

            nielsen_node* child = mk_child(node);
            SASSERT(child->m_str_mem[mi] == mem);
            nielsen_edge* e     = mk_edge(node, child, "cycle decomp", false);
            const nielsen_subst s(x, xp_xpp, mem.m_dep);
            e->add_subst(s);
            child->apply_subst(m_sg, s);

            // remove from mi^th element of the child the leading token, as it is immediately subsumed
            SASSERT(child->m_str_mem[mi].m_str->first() == xp);
            child->m_str_mem[mi].m_str = dir_drop(m_sg, child->m_str_mem[mi].m_str, 1, true);

            // x' ∈ stab(R, Q_ν)   (stabilizer view, F = {R}, current state = R)
            child->add_str_mem(str_mem::mk_view(xp, R, R, nu, mem.m_dep));

            // noloop(x'', R, Q_ν)  (cycle guard, two-mode monitor, state = R)
            child->add_str_mem(str_mem::mk_guard(xpp, R, R, nu, mem.m_dep));

            TRACE(seq, tout << "cycle_decomp: x=" << mk_pp(x->get_expr(), m)
                            << " R=" << mk_pp(R->get_expr(), m) << " nu=" << nu << "\n");

#ifdef Z3DEBUG
            std::string dot = partial_dfa_to_dot(R, false);
#endif
            return true;
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_gpower_intr
    // Generalized power introduction: for an equation where one side's head
    // is a variable v and the other side has a ground prefix followed by a
    // variable x that forms a dependency cycle back to v, introduce
    // v = base^n · suffix where base is the ground prefix.
    // Generates side constraints n >= 0 and 0 <= len(suffix) < len(base).
    // mirrors ZIPT's GPowerIntrModifier (SplitGroundPower + TryGetPowerSplitBase)
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_gpower_intr(nielsen_node* node) {
        for (str_eq const& eq : node->str_eqs()) {
            SASSERT(eq.well_formed());
            if (eq.is_trivial())
                continue;

            // Try both directions (ZIPT's ExtendDir(fwd=true/false)).
            for (unsigned od = 0; od < 2; ++od) {
                const bool fwd = (od == 0);
                euf::snode const* lhead = dir_token(eq.m_lhs, fwd);
                euf::snode const* rhead = dir_token(eq.m_rhs, fwd);
                if (!lhead || !rhead)
                    continue;

                // Orientation 1: RHS directional head is var, scan LHS in that
                // direction for ground prefix + matching cycle var.
                if (rhead->is_var() && !lhead->is_var()) {
                    euf::snode_vector toks;
                    collect_tokens_dir(eq.m_lhs, fwd, toks);
                    euf::snode_vector ground_prefix;
                    euf::snode const* target_var = nullptr;
                    for (unsigned i = 0; i < toks.size(); ++i) {
                        if (toks[i]->is_var()) {
                            target_var = toks[i];
                            break;
                        }
                        ground_prefix.push_back(toks[i]);
                    }
                    if (target_var && !ground_prefix.empty() && target_var->id() == rhead->id()) {
                        if (fire_gpower_intro(node, eq, rhead, ground_prefix, fwd))
                            return true;
                    }
                }

                // Orientation 2: LHS directional head is var, scan RHS analogously.
                if (lhead->is_var() && !rhead->is_var()) {
                    euf::snode_vector toks;
                    collect_tokens_dir(eq.m_rhs, fwd, toks);
                    euf::snode_vector ground_prefix;
                    euf::snode const* target_var = nullptr;
                    for (unsigned i = 0; i < toks.size(); ++i) {
                        if (toks[i]->is_var()) {
                            target_var = toks[i];
                            break;
                        }
                        ground_prefix.push_back(toks[i]);
                    }
                    if (target_var && !ground_prefix.empty() && target_var->id() == lhead->id()) {
                        if (fire_gpower_intro(node, eq, lhead, ground_prefix, fwd))
                            return true;
                    }
                }
            }
            // TODO: Extend to transitive cycles across multiple equations
            // (ZIPT's varDep + HasDepCycle). Currently only self-cycles are detected.
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_regex_factorization (Boolean Closure)
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_regex_factorization(nielsen_node* node) {
        if (m_regex_factorization_threshold == 0)
            return false;

        struct rf_split {
            euf::snode const* m_p;
            euf::snode const* m_q;
            dep_tracker m_dep;
        };

        for (str_mem const& mem : node->str_mems()) {
            SASSERT(mem.well_formed());

            // split() handles all regex forms (incl. complement / intersection),
            // so the classical restriction is no longer needed.
            if (mem.m_str->is_empty() || mem.is_primitive())
                continue;

            // View / guard memberships (Section 3.3) are handled by the
            // cycle machinery and the synchronous product, not by factorization.
            if (!mem.is_plain())
                continue;

            split_set pairs;
            auto [head, tail] = split_membership(mem.m_str, mem.m_regex, sg(), m_regex_factorization_threshold, pairs);
            if (!head) {
                SASSERT(!tail);
                continue;
            }
            SASSERT(tail);

            euf::snode const* const first = mem.m_str->first();

            vector<rf_split> feasible;
            dep_tracker eliminated_dep = mem.m_dep;

            for (auto const &[tp, tq] : pairs) {
                euf::snode const* sn_p = m_sg.mk(tp);
                euf::snode const* sn_q = m_sg.mk(tq);

                // Also check intersection with other primitive constraints on `head`.
                // Only valid when head is the single token `first`; for a multi-token
                // head Δ constrains the whole prefix, so we only check Δ ≠ ∅.
                euf::snode_vector regexes_p;
                regexes_p.push_back(sn_p);
                dep_tracker first_filter_dep = nullptr;
                if (head == first) {
                    for (auto const& prev_mem : node->str_mems()) {
                        if (prev_mem.m_str == first) {
                            regexes_p.push_back(prev_mem.m_regex);
                            first_filter_dep = m_dep_mgr.mk_join(first_filter_dep, prev_mem.m_dep);
                        }
                    }
                }
                if (m_seq_regex->check_intersection_emptiness(regexes_p, 100) == l_true) {
                    eliminated_dep = m_dep_mgr.mk_join(eliminated_dep, first_filter_dep);
                    continue;
                }

                feasible.push_back({ sn_p, sn_q, m_dep_mgr.mk_join(mem.m_dep, first_filter_dep) });
                if (feasible.size() > m_regex_factorization_threshold)
                    break;
            }

            if (feasible.empty()) {
                node->set_general_conflict();
                node->set_conflict(backtrack_reason::regex, eliminated_dep);
                return true;
            }

            if (feasible.size() > m_regex_factorization_threshold)
                continue;

            for (auto& [m_p, m_q, m_dep] : feasible) {
                nielsen_node* child = mk_child(node);
                mk_edge(node, child, "regex fact", true);

                // remove the original mem from child
                auto& child_mems = child->str_mems();
                for (unsigned k = 0; k < child_mems.size(); ++k) {
                    if (child_mems[k] == mem) {
                        child_mems[k] = child_mems.back();
                        child_mems.pop_back();
                        break;
                    }
                }

                child->add_str_mem(str_mem(head, m_p, m_dep));
                child->add_str_mem(str_mem(tail, m_q, m_dep));
            }
            return true;
        }
        return false;
    }

    bool nielsen_graph::fire_gpower_intro(
        nielsen_node* node, str_eq const& eq,
        euf::snode const* var, euf::snode_vector const& ground_prefix_orig, const bool fwd) {

        // Compress repeated patterns in the ground prefix (mirrors ZIPT's LcpCompressionFull).
        // E.g., [a,b,a,b] has minimal period 2 → use [a,b] as the power base.
        // This ensures we use the minimal repeating unit: x = (ab)^n · suffix
        // instead of x = (abab)^n · suffix.
        // (mirrors ZIPT: if b.Length == 1 && b is PowerToken pt => b = pt.Base)
        euf::snode_vector ground_prefix;
        const unsigned n = ground_prefix_orig.size();
        unsigned period = n;
        for (unsigned p = 1; p <= n / 2; ++p) {
            if (n % p != 0)
                continue;
            bool match = true;
            for (unsigned i = p; i < n && match; ++i)
                match = ground_prefix_orig[i]->id() == ground_prefix_orig[i % p]->id();
            if (match) {
                period = p;
                break;
            }
        }
        for (unsigned i = 0; i < period; ++i) {
            ground_prefix.push_back(ground_prefix_orig[i]);
        }

        // If the compressed prefix is a single power snode, unwrap it to use
        // its base tokens, avoiding nested powers.
        // E.g., [(ab)^3] → [a, b] so we get (ab)^n instead of ((ab)^3)^n.
        // (mirrors ZIPT: if b.Length == 1 && b is PowerToken pt => b = pt.Base)
        if (ground_prefix.size() == 1 && ground_prefix[0]->is_power()) {
            expr* base_e = get_power_base_expr(ground_prefix[0], m_seq);
            if (base_e) {
                euf::snode const* base_sn = m_sg.mk(base_e);
                if (base_sn) {
                    euf::snode_vector base_toks;
                    collect_tokens_dir(base_sn, fwd, base_toks);
                    if (!base_toks.empty()) {
                        ground_prefix.reset();
                        ground_prefix.append(base_toks);
                    }
                }
            }
        }

        const unsigned base_len = ground_prefix.size();

        // Build base string expression from ground prefix tokens.
        // Each s_char snode's get_expr() is already seq.unit(ch) (a string).
        expr_ref base_str(m);
        for (unsigned i = 0; i < base_len; ++i) {
            expr* tok_expr = ground_prefix[i]->get_expr();
            if (!tok_expr) return false;
            if (i == 0)
                base_str = tok_expr;
            else if (fwd)
                base_str = m_seq.str.mk_concat(base_str, tok_expr);
            else
                base_str = m_seq.str.mk_concat(tok_expr, base_str);
        }

        // Create fresh exponent variable and power expression: base^n
        const expr_ref fresh_n = get_or_create_gpower_n_var(var);
        const expr_ref power_expr(m_seq.str.mk_power(base_str, fresh_n), m);
        euf::snode const* power_snode = m_sg.mk(power_expr);
        if (!power_snode)
            return false;

        const expr_ref zero(a.mk_int(0), m);

        // Generate children mirroring ZIPT's GetDecompose:
        // P(t0 · t1 · ... · t_{k-1}) = P(t0) | t0·P(t1) | ... | t0·...·t_{k-2}·P(t_{k-1})
        // For char tokens P(c) = {ε}, for power tokens P(u^m) = {u^m', 0 ≤ m' ≤ m}.
        // Child at position i substitutes var → base^n · t0·...·t_{i-1} · P(t_i).
        for (unsigned i = 0; i < base_len; ++i) {
            euf::snode const* tok = ground_prefix[i];

            // Skip char position when preceding token is a power:
            // The power case at i-1 with 0 ≤ m' ≤ exp already covers m' = exp,
            // which produces the same result. Using the original exponent here
            // creates a rigid coupling that causes cycling.
            if (!tok->is_power() && i > 0 && ground_prefix[i - 1]->is_power())
                continue;

            // Build full-token prefix: ground_prefix[0..i-1]
            euf::snode const* prefix_sn = nullptr;
            for (unsigned j = 0; j < i; ++j)
                prefix_sn = (j == 0) ? ground_prefix[0] : dir_concat(m_sg, prefix_sn, ground_prefix[j], fwd);

            euf::snode const* replacement;
            expr_ref fresh_m(m);

            if (tok->is_power()) {
                // Token is a power u^exp: use fresh m' with 0 ≤ m' ≤ exp
                const expr * inner_exp = get_power_exponent(tok);
                expr* inner_base = get_power_base_expr(tok, m_seq);
                if (inner_exp && inner_base) {
                    fresh_m = get_or_create_gpower_m_var(var);
                    expr_ref partial_pow(m_seq.str.mk_power(inner_base, fresh_m), m);
                    euf::snode const* partial_sn = m_sg.mk(partial_pow);
                    euf::snode const* suffix_sn = prefix_sn ? dir_concat(m_sg, prefix_sn, partial_sn, fwd) : partial_sn;
                    replacement = dir_concat(m_sg, power_snode, suffix_sn, fwd);
                }
                else {
                    // Fallback: use full token (shouldn't normally happen)
                    euf::snode const* suffix_sn = prefix_sn ? dir_concat(m_sg, prefix_sn, tok, fwd) : tok;
                    replacement = dir_concat(m_sg, power_snode, suffix_sn, fwd);
                }
            }
            else
                // Token is a char: P(char) = ε, suffix = just the prefix
                replacement = prefix_sn ? dir_concat(m_sg, power_snode, prefix_sn, fwd) : power_snode;

            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "power intr", true);
            nielsen_subst s(var, replacement, eq.m_dep); // TODO review - ensure var does not occur in replacement.
            e->add_subst(s);
            child->apply_subst(m_sg, s);

            // Side constraint: n >= 0
            e->add_side_constraint(mk_constraint(a.mk_ge(fresh_n, zero), eq.m_dep));

            // Side constraints for fresh partial exponent
            if (fresh_m.get()) {
                expr* inner_exp = get_power_exponent(tok);
                // m' >= 0
                e->add_side_constraint(mk_constraint(a.mk_ge(fresh_m, zero), eq.m_dep));
                // m' <= inner_exp
                e->add_side_constraint(mk_constraint(a.mk_ge(inner_exp, fresh_m), eq.m_dep));
            }
        }

        return true;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_signature_split
    // Heuristic equation split based on a shortest prefix signature (i, j):
    // prefixes u[0..i-1], v[0..j-1] must contain at least one variable and
    // every variable in one prefix must occur in the other prefix.
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_signature_split(nielsen_node* node) {
        auto first_var_pos = [](euf::snode_vector const& toks) {
            for (unsigned k = 0; k < toks.size(); ++k)
                if (toks[k]->is_var())
                    return k;
            return toks.size();
        };

        auto const& eqs = node->str_eqs();
        for (unsigned eq_idx = 0; eq_idx < eqs.size(); ++eq_idx) {
            str_eq const& eq = eqs[eq_idx];
            SASSERT(eq.well_formed());
            if (eq.is_trivial())
                continue;

            euf::snode_vector lhs_toks, rhs_toks;
            eq.m_lhs->collect_tokens(lhs_toks);
            eq.m_rhs->collect_tokens(rhs_toks);

            // Start from the first variable on each side; if one side has no
            // variable, this equation has no usable signature.
            const unsigned i0 = first_var_pos(lhs_toks);
            const unsigned j0 = first_var_pos(rhs_toks);
            if (i0 == lhs_toks.size() || j0 == rhs_toks.size())
                continue;

            std::unordered_map<expr*, unsigned> lhs_first, rhs_first;
            lhs_first.reserve(lhs_toks.size());
            rhs_first.reserve(rhs_toks.size());

            for (unsigned k = 0; k < lhs_toks.size(); ++k) {
                if (!lhs_toks[k]->is_var())
                    continue;
                expr* x = lhs_toks[k]->get_expr();
                if (!lhs_first.contains(x))
                    lhs_first.emplace(x, k);
            }
            for (unsigned k = 0; k < rhs_toks.size(); ++k) {
                if (!rhs_toks[k]->is_var())
                    continue;
                expr* x = rhs_toks[k]->get_expr();
                if (!rhs_first.contains(x))
                    rhs_first.emplace(x, k);
            }

            svector<unsigned> lhs_need_j(lhs_toks.size(), UINT_MAX);
            svector<unsigned> rhs_need_i(rhs_toks.size(), UINT_MAX);

            // Prefix summary arrays:
            // lhs_need_j[k] = maximum first-occurrence index in rhs for any
            // variable seen in lhs[0..k]. Symmetric for rhs_need_i.
            // A value of UINT_MAX means "no variable requirement yet".
            // A value of UINT_MAX-1 means "fail: some prefix variable does not
            // occur on the opposite side at all".
            constexpr unsigned FAIL_MARK = UINT_MAX - 1;
            unsigned need = UINT_MAX;
            for (unsigned k = 0; k < lhs_toks.size(); ++k) {
                if (lhs_toks[k]->is_var()) {
                    auto it = rhs_first.find(lhs_toks[k]->get_expr());
                    if (it == rhs_first.end())
                        need = FAIL_MARK;
                    else if (need != FAIL_MARK)
                        need = (need == UINT_MAX) ? it->second : std::max(need, it->second);
                }
                lhs_need_j[k] = need;
            }

            need = UINT_MAX;
            for (unsigned k = 0; k < rhs_toks.size(); ++k) {
                if (rhs_toks[k]->is_var()) {
                    auto it = lhs_first.find(rhs_toks[k]->get_expr());
                    if (it == lhs_first.end())
                        need = FAIL_MARK;
                    else if (need != FAIL_MARK)
                        need = (need == UINT_MAX) ? it->second : std::max(need, it->second);
                }
                rhs_need_i[k] = need;
            }

            unsigned i = i0 + 1;
            unsigned j = j0 + 1;

            // Compute least fixpoint for (i, j): grow one side only when the
            // current prefix on the other side requires it.
            bool changed = true;
            while (changed) {
                changed = false;

                const unsigned req_j = lhs_need_j[i - 1];
                if (req_j == FAIL_MARK) {
                    i = lhs_toks.size();
                    break;
                }
                if (req_j != UINT_MAX && req_j + 1 > j) {
                    j = req_j + 1;
                    changed = true;
                }

                const unsigned req_i = rhs_need_i[j - 1];
                if (req_i == FAIL_MARK) {
                    j = rhs_toks.size();
                    break;
                }
                if (req_i != UINT_MAX && req_i + 1 > i) {
                    i = req_i + 1;
                    changed = true;
                }
            }

            if (i >= lhs_toks.size() || j >= rhs_toks.size())
                continue;

            // Decompose u = u1·u2 and v = v1·v2 at signature indices.
            euf::snode const* u1 = m_sg.drop_right(eq.m_lhs, lhs_toks.size() - i);
            euf::snode const* u2 = m_sg.drop_left(eq.m_lhs, i);
            euf::snode const* v1 = m_sg.drop_right(eq.m_rhs, rhs_toks.size() - j);
            euf::snode const* v2 = m_sg.drop_left(eq.m_rhs, j);
            // NSB review: if we keep this skolem function it should include arguments
            // to not clash with other values of i, j
            // Why not use
            // x := str.substr(u2, 0, str.len(u2) - str.len(v1)),
            const auto x_e = m_sk.mk("signature-split", eq.m_lhs->get_expr(), eq.m_rhs->get_expr(), eq.m_lhs->get_sort());
            euf::snode const* x = m_sg.mk(x_e);

            for (unsigned branch = 0; branch < 2; ++branch) {
                nielsen_node* child = mk_child(node);
                mk_edge(node, child, "signature split", true);

                auto& child_eqs = child->str_eqs();
                child_eqs[eq_idx] = child_eqs.back();
                child_eqs.pop_back();

                // Two-way split:
                // (1) u1·x = v1   and   u2 = x·v2
                // (2) u1 = v1·x   and   x·u2 = v2
                if (branch == 0) {
                    child_eqs.push_back(str_eq(m_sg.mk_concat(u1, x), v1, eq.m_dep));
                    child_eqs.push_back(str_eq(u2, m_sg.mk_concat(x, v2), eq.m_dep));
                }
                else {
                    child_eqs.push_back(str_eq(u1, m_sg.mk_concat(v1, x), eq.m_dep));
                    child_eqs.push_back(str_eq(m_sg.mk_concat(x, u2), v2, eq.m_dep));
                }
            }
            return true;
        }
        return false;
    }

    bool nielsen_graph::apply_regex_if_split(nielsen_node *node) {
        for (str_mem const &mem : node->str_mems()) {
            SASSERT(mem.well_formed());

            expr *r_expr = mem.m_regex->get_expr();
            expr_ref c(m), th(m), el(m);
            if (!bool_rewriter(m).decompose_ite(r_expr, c, th, el))
                continue;

            bool created = false;

            // Worklist: (regex_expr, accumulated_conditions).
            // Call decompose_ite in a loop until no more ite sub-expressions,
            // branching on c=true and c=false and accumulating conditions.
            vector<std::pair<expr_ref, expr_ref_vector>> worklist;
            worklist.push_back({expr_ref(r_expr, m), expr_ref_vector(m)});

            while (!worklist.empty()) {
                auto [r, cs] = std::move(worklist.back());
                worklist.pop_back();

                if (m_seq.re.is_empty(r))
                    continue;

                expr_ref c2(m), th2(m), el2(m);
                if (!bool_rewriter(m).decompose_ite(r, c2, th2, el2)) {
                    // No ite remaining: leaf → create child node with regex updated to r.
                    // Canonicalize with th_rewriter so that the resolved leaf shares
                    // its snode id with the corresponding partial-DFA state (which is
                    // built by brzozowski_deriv); otherwise un-simplified residuals
                    // like (a|∅)·R≠a·R break view/guard Q-membership and lap checks.
                    euf::snode const* new_regex_snode = mk_rewrite(r);
                    nielsen_node *child = mk_child(node);
                    nielsen_edge* e = mk_edge(node, child, "regex if", true);
                    for (const auto f : cs) {
                        e->add_side_constraint(constraint(f, mem.m_dep, m));
                    }
                    for (str_mem &cm : child->str_mems()) {
                        if (cm == mem) {
                            cm.m_regex = new_regex_snode;
                            // A guard whose symbolic step lands back on the cycle
                            // head closed a lap within Q → this branch is dead.
                            if (cm.is_guard() && new_regex_snode == cm.m_root) {
                                child->set_general_conflict();
                                child->set_conflict(backtrack_reason::regex, cm.m_dep);
                            }
                            break;
                        }
                    }
                    created = true;
                    continue;
                }

                th_rewriter tr(m);
                expr_ref c_simp(c2, m);
                tr(c_simp);

                if (m.is_true(c_simp)) {
                    if (!m_seq.re.is_empty(th2))
                        worklist.push_back({th2, std::move(cs)});
                }
                else if (m.is_false(c_simp)) {
                    if (!m_seq.re.is_empty(el2))
                        worklist.push_back({el2, std::move(cs)});
                }
                else {
                    if (!m_seq.re.is_empty(th2)) {
                        expr_ref_vector cs_true(cs);
                        cs_true.push_back(c2);
                        worklist.push_back({th2, std::move(cs_true)});
                    }
                    if (!m_seq.re.is_empty(el2)) {
                        expr_ref_vector cs_false(cs);
                        cs_false.push_back(mk_not(c2));
                        worklist.push_back({el2, std::move(cs_false)});
                    }
                }
            }

            if (created)
                return true;
        }
        return false;
    }
    // -----------------------------------------------------------------------
    // Modifier: apply_regex_var_split
    // For str_mem x·s ∈ R where x is a variable, split using minterms:
    //   (1) x → ε (empty)
    //   (2) x → c · x' for each minterm character class c
    // More general than regex_char_split; uses minterm partitioning rather
    // than just extracting concrete characters.
    // mirrors ZIPT's RegexVarSplitModifier
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_regex_var_split(nielsen_node* node) {
        for (str_mem const& mem : node->str_mems()) {
            SASSERT(mem.well_formed());
            if (mem.is_primitive())
                continue;
            euf::snode const* first = mem.m_str->first();
            SASSERT(first);

            // std::cout << "Considering regex: " << spp(mem.m_regex, m) << std::endl;

            // Branch 1: x → ε (progress)
            {
                nielsen_node* child = mk_child(node);
                nielsen_edge* e = mk_edge(node, child, "regex var split", true);
                const nielsen_subst s(first, m_sg.mk_empty_seq(first->get_sort()), mem.m_dep);
                e->add_subst(s);
                child->apply_subst(m_sg, s);
            }

            // Branch 2: x → ?c · x, where ?c is a fresh symbolic char.
            // for variables at mod_count 0 and other terms, use symbolic (str.len expr)
            // NSB review:
            // it really is seq.nth (length-of-prefix that was chopped of for first)
            // assume len(x) contains the expression for the current length of x,
            // then the suffix where the current x is located is at str.len(x) - len(x)
            // (seq.nth x (- (str.len x) len(x))
            //
            euf::snode const* fresh_char = m_sg.mk(get_or_create_char_var(first));

            euf::snode const* tail = get_tail(first, 1, true);
            euf::snode const* replacement = m_sg.mk_concat(fresh_char, tail);
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "regex var split", false);

            const nielsen_subst s(first, replacement, mem.m_dep);
            e->add_subst(s);
            child->apply_subst(m_sg, s);

            return true;
        }
        return false;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_power_split
    // For a variable x facing a power token u^n, branch:
    //   (1) x = u^m · prefix(u) with m < n (bounded power prefix)
    //   (2) x = u^n · x' (the variable extends past the full power)
    // Generates integer side constraints for the fresh exponent variables.
    // mirrors ZIPT's PowerSplitModifier
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_power_split(nielsen_node* node) {

        euf::snode const* power = nullptr;
        euf::snode const* var_head = nullptr;
        str_eq const* eq = nullptr;
        bool fwd = true;
        if (!find_power_vs_var(node, power, var_head, eq, fwd))
            return false;

        SASSERT(power->is_power() && power->num_args() >= 1);
        euf::snode const* base = power->arg0();
        const expr_ref zero(a.mk_int(0), m);

        // Branch 1: enumerate all decompositions of the base.
        // x = base^m · prefix_i(base) where 0 <= m < n
        // Uses the same GetDecompose pattern as fire_gpower_intro.
        {
            euf::snode_vector base_toks;
            collect_tokens_dir(base, fwd, base_toks);
            const unsigned base_len = base_toks.size();
            expr* base_expr = get_power_base_expr(power, m_seq);
            if (!base_expr || base_len == 0)
                return false;

            const expr_ref fresh_m = get_or_create_gpower_n_var(var_head);
            const expr_ref power_m_expr(m_seq.str.mk_power(base_expr, fresh_m), m);
            euf::snode const* power_m_sn = m_sg.mk(power_m_expr);
            if (!power_m_sn)
                return false;

            for (unsigned i = 0; i < base_len; ++i) {
                euf::snode const* tok = base_toks[i];

                // Skip char position when preceding token is a power:
                // the power case at i-1 with 0 <= m' <= exp already covers m' = exp.
                if (!tok->is_power() && i > 0 && base_toks[i - 1]->is_power())
                    continue;

                // Build full-token prefix: base_toks[0..i-1]
                euf::snode const* prefix_sn = nullptr;
                for (unsigned j = 0; j < i; ++j)
                    prefix_sn = (j == 0) ? base_toks[0] : dir_concat(m_sg, prefix_sn, base_toks[j], fwd);

                euf::snode const* replacement;
                expr_ref fresh_inner_m(m);

                if (tok->is_power()) {
                    // Token is a power u^exp: decompose with fresh m', 0 <= m' <= exp
                    const expr * inner_exp = get_power_exponent(tok);
                    expr* inner_base = get_power_base_expr(tok, m_seq);
                    if (inner_exp && inner_base) {
                        fresh_inner_m = get_or_create_gpower_m_var(var_head);
                        expr_ref partial_pow(m_seq.str.mk_power(inner_base, fresh_inner_m), m);
                        euf::snode const* partial_sn = m_sg.mk(partial_pow);
                        euf::snode const* suffix_sn = prefix_sn ? dir_concat(m_sg, prefix_sn, partial_sn, fwd) : partial_sn;
                        replacement = dir_concat(m_sg, power_m_sn, suffix_sn, fwd);
                    }
                    else {
                        euf::snode const* suffix_sn = prefix_sn ? dir_concat(m_sg, prefix_sn, tok, fwd) : tok;
                        replacement = dir_concat(m_sg, power_m_sn, suffix_sn, fwd);
                    }
                }
                else
                    // Token is a char: P(char) = ε, suffix is just the prefix
                    replacement = prefix_sn ? dir_concat(m_sg, power_m_sn, prefix_sn, fwd) : power_m_sn;

                nielsen_node* child = mk_child(node);
                nielsen_edge* e = mk_edge(node, child, "power split", true);
                nielsen_subst s(var_head, replacement, eq->m_dep); // TODO review - ensure var does not occur in replacement.
                e->add_subst(s);
                child->apply_subst(m_sg, s);

                // Side constraint: n >= 0
                e->add_side_constraint(mk_constraint(a.mk_ge(fresh_m, zero), eq->m_dep));

                // Side constraints for fresh partial exponent
                if (fresh_inner_m.get()) {
                    expr* inner_exp = get_power_exponent(tok);
                    // m' >= 0
                    e->add_side_constraint(mk_constraint(a.mk_ge(fresh_inner_m, zero), eq->m_dep));
                    // m' <= inner_exp
                    e->add_side_constraint(mk_constraint(a.mk_ge(inner_exp, fresh_inner_m), eq->m_dep));
                }
            }
        }

        // Branch 2: x = u^n · x' (variable extends past full power, non-progress)
        // so replace x -> u^n · x
        {
            euf::snode const* replacement = dir_concat(m_sg, power, var_head, fwd);
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "power split", false);
            const nielsen_subst s(var_head, replacement, eq->m_dep);
            e->add_subst(s);
            child->apply_subst(m_sg, s);
        }

        return true;
    }

    // -----------------------------------------------------------------------
    // Modifier: apply_var_num_unwinding
    // For a power token u^n facing a variable, branch:
    //   (1) n = 0 → u^n = ε (replace power with empty)
    //   (2) n >= 1 → peel one u from the power
    // Generates integer side constraints for each branch.
    // mirrors ZIPT's VarNumUnwindingModifier
    // -----------------------------------------------------------------------

    bool nielsen_graph::apply_var_num_unwinding_eq(nielsen_node* node) {

        euf::snode const* power = nullptr;
        euf::snode const* var_head = nullptr;
        str_eq const* eq = nullptr;
        bool fwd = true;
        if (!find_power_vs_var(node, power, var_head, eq, fwd))
            return false;

        SASSERT(power->is_power() && power->num_args() >= 1);
        euf::snode const* base = power->arg0();
        expr* exp_n = get_power_exponent(power);
        SASSERT(exp_n);
        const expr_ref zero(a.mk_int(0), m);
        const expr_ref one(a.mk_int(1), m);

        // Branch 1: n = 0 → replace u^n with ε (progress)
        // Side constraint: n = 0
        {
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "unwinding eq 0", true);
            const nielsen_subst s(power, m_sg.mk_empty_seq(power->get_sort()), eq->m_dep);
            e->add_subst(s);
            child->apply_subst(m_sg, s);
            e->add_side_constraint(mk_constraint(a.mk_ge(exp_n, zero), eq->m_dep));
            e->add_side_constraint(mk_constraint(a.mk_le(exp_n, zero), eq->m_dep));
        }

        // Branch 2: n >= 1 → peel one u: u^n → u · u^(n-1)
        // Side constraint: n >= 1
        {
            const expr_ref power_expr(m_seq.str.mk_power(base->get_expr(), a.mk_sub(exp_n, a.mk_int(1))), m);
            euf::snode const* power_snode = m_sg.mk(power_expr);
            euf::snode const* replacement = dir_concat(m_sg, base, power_snode, fwd);
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "unwinding eq &gt;", false);
            const nielsen_subst s(power, replacement, eq->m_dep); // TODO review - ensure var does not occur in replacement.
            e->add_subst(s);
            child->apply_subst(m_sg, s);
            e->add_side_constraint(mk_constraint(a.mk_ge(exp_n, one), eq->m_dep));
        }

        return true;
    }

    bool nielsen_graph::apply_var_num_unwinding_mem(nielsen_node* node) {

        euf::snode const* power = nullptr;
        str_mem const* mem = nullptr;
        bool fwd = true;
        if (!find_power_vs_var(node, power, mem, fwd))
            return false;

        SASSERT(power->is_power() && power->num_args() >= 1);
        euf::snode const* base = power->arg0();
        expr* exp_n = get_power_exponent(power);
        SASSERT(exp_n);
        const expr_ref zero(a.mk_int(0), m);
        const expr_ref one(a.mk_int(1), m);

        // Branch 1: n = 0 → replace u^n with ε (progress)
        // Side constraint: n = 0
        {
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "unwinding mem 0", true);
            const nielsen_subst s(power, m_sg.mk_empty_seq(power->get_sort()), mem->m_dep);
            e->add_subst(s);
            child->apply_subst(m_sg, s);
            e->add_side_constraint(mk_constraint(a.mk_eq(exp_n, zero), mem->m_dep));
        }

        // Branch 2: n >= 1 → peel one u: u^n → u · u^(n-1)
        // Side constraint: n >= 1
        {
            const expr_ref power_expr(m_seq.str.mk_power(base->get_expr(), a.mk_sub(exp_n, a.mk_int(1))), m);
            euf::snode const* power_snode = m_sg.mk(power_expr);

            euf::snode const* replacement = dir_concat(m_sg, base, power_snode, fwd);
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "unwinding mem &gt;", false);
            const nielsen_subst s(power, replacement, mem->m_dep); // TODO review - ensure var does not occur in replacement.
            e->add_subst(s);
            child->apply_subst(m_sg, s);
            e->add_side_constraint(mk_constraint(a.mk_ge(exp_n, one), mem->m_dep));
        }

        return true;
    }

    // Cf. axioms::diseq_axiom
    bool nielsen_graph::axiomatize_diseq(nielsen_node* node) {
        SASSERT(node->m_str_eq.empty() &&
                    std::ranges::all_of(node->m_str_mem, [](str_mem const& mem){ return mem.is_primitive(); }));

        if (node->m_str_deq.empty())
            return false;

        const str_deq& first = node->m_str_deq.back();
        euf::snode const* u = first.m_lhs;
        euf::snode const* v = first.m_rhs;

        const expr_ref u_len(compute_length_expr(u), m);
        const expr_ref v_len(compute_length_expr(v), m);
        expr_ref len_eq(m.mk_eq(u_len, v_len), m);
        str_eq eq_uv(u, v, first.m_dep);
        sort *char_sort = nullptr;
        VERIFY(seq().is_seq(u->get_sort(), char_sort));
        euf::snode const* a = m_sg.mk(seq().str.mk_unit(m_sk.mk("diseq.a", u->get_expr(), v->get_expr(), char_sort).get()));
        euf::snode const* b = m_sg.mk(seq().str.mk_unit(m_sk.mk("diseq.b", u->get_expr(), v->get_expr(), char_sort).get()));
        euf::snode const* w = m_sg.mk(m_sk.mk("diseq.w", u->get_expr(), v->get_expr()).get());
        euf::snode const* up = m_sg.mk(m_sk.mk("diseq.u'", u->get_expr(), v->get_expr()).get());
        euf::snode const* vp = m_sg.mk(m_sk.mk("diseq.v'", u->get_expr(), v->get_expr()).get());
        const expr_ref up_len(compute_length_expr(up), m);
        const expr_ref vp_len(compute_length_expr(vp), m);
        euf::snode const* wau = dir_concat(m_sg, dir_concat(m_sg, w, a, true), up, true);
        euf::snode const* wbv = dir_concat(m_sg, dir_concat(m_sg, w, b, true), vp, true);
        const str_eq u_eq(u, wau, first.m_dep);
        const str_eq v_eq(v, wbv, first.m_dep);

        // Branch 1: |u| < |v|
        {
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "diseq I", true);
            child->m_str_deq.pop_back();
            expr_ref cmp(this->a.mk_lt(u_len, v_len), m);
            m_rw(cmp);
            e->add_side_constraint(constraint(cmp, first.m_dep, m));
        }
        // Branch 2: |v| < |u|
        {
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "diseq II", true);
            child->m_str_deq.pop_back();
            expr_ref cmp(this->a.mk_lt(v_len, u_len), m);
            m_rw(cmp);
            e->add_side_constraint(constraint(cmp, first.m_dep, m));
        }
        // Branch 3: u = wau' && v = wbv' && |u'| = |v'| && a != b
        {
            nielsen_node* child = mk_child(node);
            nielsen_edge* e = mk_edge(node, child, "diseq III", true);
            child->m_str_deq.pop_back();
            child->add_str_eq(u_eq);
            child->add_str_eq(v_eq);
            child->add_constraint(constraint(m.mk_eq(up_len, vp_len), first.m_dep, m));
            e->add_side_constraint(constraint(m.mk_not(m.mk_eq(a->get_expr(), b->get_expr())), first.m_dep, m));
        }

        return true;
    }

    dep_tracker nielsen_graph::collect_conflict_deps() const {
        dep_tracker deps = nullptr;
        // todo: Add visit set if the graph could contain cycles in the future
        // enumerating all nodes would not work due to hot-restarts having created
        // children that are currently not relevant
        vector<nielsen_node const*> to_visit;
        to_visit.push_back(m_root);
        while (!to_visit.empty()) {
            nielsen_node const* n = to_visit.back();
            to_visit.pop_back();
            if (n->reason() == backtrack_reason::children_failed) {
                SASSERT(n->m_conflict_external_literal == sat::null_literal);
                SASSERT(!n->m_conflict_internal);
                for (unsigned i = n->outgoing().size(); i > 0; i--) {
                    nielsen_edge const* e = n->outgoing()[i - 1];
                    to_visit.push_back(e->tgt());
                }
                continue;
            }
            // not true anymore since we might have done a hot-restart where we previously created the child:
            //SASSERT(n->outgoing().empty());
            SASSERT(n->is_currently_conflict());
            if (n->m_conflict_external_literal != sat::null_literal)
                // We know from the outer solver that this literal is assigned true and contradicts node constraint
                deps = m_dep_mgr.mk_join(deps, m_dep_mgr.mk_leaf(n->m_conflict_external_literal));

            if (n->m_conflict_internal)
                deps = m_dep_mgr.mk_join(deps, n->m_conflict_internal);
        }
        return deps;
    }


    // NSB review: this is one of several methods exposed for testing
    void nielsen_graph::test_aux_explain_conflict(svector<enode_pair>& eqs,
        svector<sat::literal>& mem_literals) const {
        SASSERT(m_root);
        const auto deps = collect_conflict_deps();
        vector<dep_source> vs;
        m_dep_mgr.linearize(deps, vs);
        for (dep_source const& d : vs) {
            if (std::holds_alternative<enode_pair>(d))
                eqs.push_back(std::get<enode_pair>(d));
            else if (std::holds_alternative<sat::literal>(d))
                mem_literals.push_back(std::get<sat::literal>(d));
            else
                UNREACHABLE();
        }
    }


    // -----------------------------------------------------------------------
    // nielsen_graph: length constraint generation
    // -----------------------------------------------------------------------

    expr_ref nielsen_graph::compute_length_expr(euf::snode const* n) {
        if (n->is_empty())
            return expr_ref(a.mk_int(0), m);

        if (n->is_char_or_unit())
            return expr_ref(a.mk_int(1), m);

        if (n->is_concat()) {
            const expr_ref left = compute_length_expr(n->arg0());
            const expr_ref right = compute_length_expr(n->arg(1));
            return expr_ref(a.mk_add(left, right), m);
        }

        //euf::snode const* length_term = nullptr;
        //if (m_length_info.find(n->id(), length_term) && length_term)
        //    return expr_ref(length_term->get_expr(), m);

        return expr_ref(m_seq.str.mk_length(n->get_expr()), m);
    }

    void nielsen_graph::generate_length_constraints(vector<length_constraint>& constraints) {
        if (!m_root)
            return;
        uint_set seen_vars;

        TRACE(seq, display(tout, m_root));

        const seq_util & seq = m_sg.get_seq_util();
        for (str_eq const& eq : m_root->str_eqs()) {
            if (eq.is_trivial())
                continue;

            expr_ref len_lhs = compute_length_expr(eq.m_lhs);
            expr_ref len_rhs = compute_length_expr(eq.m_rhs);
            TRACE(seq,
                tout << "Length constraint from LHS " << snode_label_html(eq.m_lhs, m, true) << " to " << len_lhs << ":\n";
                tout << "Length constraint from RHS " << snode_label_html(eq.m_rhs, m, true) << " to " << len_rhs << "\n");
            expr_ref len_eq(m.mk_eq(len_lhs, len_rhs), m);
            constraints.push_back(length_constraint(len_eq, eq.m_dep, length_kind::eq, true, m));

            // collect variables for non-negativity constraints
            euf::snode_vector tokens;
            eq.m_lhs->collect_tokens(tokens);
            eq.m_rhs->collect_tokens(tokens);
            for (euf::snode const* tok : tokens) {
                if (tok->is_var() && !seen_vars.contains(tok->id())) {
                    seen_vars.insert(tok->id());
                    expr_ref len_var(seq.str.mk_length(tok->get_expr()), m);
                    expr_ref ge_zero(a.mk_ge(len_var, a.mk_int(0)), m);
                    TRACE(seq, tout << "non-negative length " << ge_zero << "\n");
                    constraints.push_back(length_constraint(ge_zero, eq.m_dep, length_kind::nonneg, true, m));
                }
            }
        }

        // Parikh interval reasoning for regex memberships
        for (str_mem const& mem : m_root->str_mems()) {
            expr* re_expr = mem.m_regex->get_expr();
            SASSERT(re_expr && seq.is_re(re_expr));

            unsigned min_len = 0, max_len = UINT_MAX;
            compute_regex_length_interval(mem.m_regex, min_len, max_len);

            expr_ref len_str = compute_length_expr(mem.m_str);

            // generate len(str) >= min_len when min_len > 0
            if (min_len > 0) {
                expr_ref bound(a.mk_ge(len_str, a.mk_int(min_len)), m);
                TRACE(seq, tout << "Parikh " << mk_pp(re_expr, m) << " bound: " << bound << "\n");
                constraints.push_back(length_constraint(bound, mem.m_dep, length_kind::bound, false, m));
            }

            // generate len(str) <= max_len when bounded
            if (max_len < UINT_MAX) {
                expr_ref bound(a.mk_le(len_str, a.mk_int(max_len)), m);
                TRACE(seq, tout << "Parikh " << mk_pp(re_expr, m) << " bound: " << bound << "\n");
                constraints.push_back(length_constraint(bound, mem.m_dep, length_kind::bound, false, m));
            }
        }
    }

    void nielsen_graph::compute_regex_length_interval(euf::snode const* regex, unsigned& min_len, unsigned& max_len) const {
        const seq_util & seq = m_sg.get_seq_util();
        expr* e = regex->get_expr();
        SASSERT(e && seq.is_re(e));
        min_len = seq.re.min_length(e);
        max_len = seq.re.max_length(e);
        // for empty language, min_length may be UINT_MAX (vacuously true);
        // normalize to avoid generating bogus constraints
        if (min_len > max_len) {
            min_len = 0;
            max_len = 0;
        }
    }

    // -----------------------------------------------------------------------
    // int_constraint display
    // -----------------------------------------------------------------------

    std::ostream& constraint::display(std::ostream& out) const {
        return out << fml;
    }

    // -----------------------------------------------------------------------
    // Integer feasibility subsolver implementation
    // Uses the injected simple_solver (default: lp_simple_solver).
    // -----------------------------------------------------------------------

    // -----------------------------------------------------------------------
    // Modification counter: substitution length tracking
    // mirrors ZIPT's LocalInfo.CurrentModificationCnt + NielsenEdge.IncModCount/DecModCount
    // + NielsenNode constructor length assertion logic
    // -----------------------------------------------------------------------

    expr_ref nielsen_graph::get_or_create_char_var(euf::snode const* var) {
        SASSERT(var && var->is_var());
        const expr_ref idx(a.mk_sub(seq().str.mk_length(var->get_expr()), compute_length_expr(var)), m);
        const auto e = seq().str.mk_nth_u(var->get_expr(), idx);
        return expr_ref(m_seq.str.mk_unit(expr_ref(e, m)), m);
    }

    expr_ref nielsen_graph::get_or_create_gpower_n_var(euf::snode const* var) {
        SASSERT(var && var->is_var());
        //unsigned mc = 0;
        //m_mod_cnt.find(var->id(), mc);
        th_rewriter rw(m);
        return m_sk.mk("gpn!", var->get_expr()/*, a.mk_int(mc)*/, a.mk_int());
    }

    expr_ref nielsen_graph::get_or_create_gpower_m_var(euf::snode const* var) {
        SASSERT(var && var->is_var());
        //unsigned mc = 0;
        //m_mod_cnt.find(var->id(), mc);
        th_rewriter rw(m);
        return m_sk.mk("gpm!", var->get_expr()/*, a.mk_int(mc)*/, a.mk_int());
    }

    void nielsen_graph::add_subst_length_constraints(nielsen_edge* e) {
        auto const& substs = e->subst();
        // Compute LHS (|x|) for each non-eliminating substitution
        vector<std::pair<unsigned, expr_ref>> lhs_exprs;
        for (unsigned i = 0; i < substs.size(); ++i) {
            auto const& s = substs[i];
            if (!s.m_var->is_var())
                continue;
            if (!m_seq.is_seq(s.m_var->get_expr()))
                continue;
            expr_ref lhs = compute_length_expr(s.m_var);
            expr_ref rhs = compute_length_expr(s.m_replacement);
            e->add_side_constraint(mk_constraint(a.mk_eq(lhs, rhs), s.m_dep));
        }
    }

    void nielsen_graph::assert_to_subsolver(const constraint& c) const {
        m_length_solver.assert_expr(c.fml, c.dep);
    }

    void nielsen_graph::assert_to_subsolver(expr* e) const {
        m_length_solver.assert_expr(e);
    }

    void nielsen_graph::assert_node_side_constraints(nielsen_node* node) const {
        // Assert only the constraints that are new to this node (beyond those
        // inherited from its parent via clone_from).  The parent's constraints are
        // already present in the enclosing solver scope; asserting them again would
        // be redundant (though harmless).  This is called by search_dfs right after
        // simplify_and_init, which is where new constraints are produced.
        for (unsigned i = node->m_parent_ic_count; i < node->constraints().size(); ++i) {
            auto& c = node->constraints()[i];
            auto lit = m_context_solver.literal_if_false(c.fml);
            // std::cout << "Internalizing literal " << mk_pp(c.fml, m) << " [" << (lit == sat::null_literal) << "]" <<
            // std::endl;
            if (lit != sat::null_literal) {
                node->set_external_conflict(lit, c.dep);
                return;
            }
            assert_to_subsolver(c);
        }
    }

    void nielsen_graph::generate_node_length_constraints(nielsen_node* node) {
        if (node->m_node_len_constraints_generated)
            return;
        node->m_node_len_constraints_generated = true;

        // Skip the root node — its length constraints are already asserted
        // at the base solver level by assert_root_constraints_to_solver().
        if (node == m_root)
            return;

        // TODO: Do we really need this?
        uint_set seen_vars;

        for (str_eq const& eq : node->str_eqs()) {
            if (eq.is_trivial())
                continue;

            expr_ref len_lhs = compute_length_expr(eq.m_lhs);
            expr_ref len_rhs = compute_length_expr(eq.m_rhs);
            //node->add_constraint(mk_constraint(m.mk_eq(len_lhs, len_rhs), eq.m_dep));
            node->add_constraint(mk_constraint(a.mk_eq(len_lhs, len_rhs), eq.m_dep));

            // non-negativity for each variable (mod-count-aware)
            euf::snode_vector tokens;
            eq.m_lhs->collect_tokens(tokens);
            eq.m_rhs->collect_tokens(tokens);
            for (euf::snode const* tok : tokens) {
                if (tok->is_var() && !seen_vars.contains(tok->id())) {
                    seen_vars.insert(tok->id());
                    expr_ref len_var = compute_length_expr(tok);
                    node->add_constraint(mk_constraint(a.mk_ge(len_var, a.mk_int(0)), eq.m_dep));
                }
            }
        }

        // Parikh interval bounds for regex memberships at this node
        for (str_mem const& mem : node->str_mems()) {
            expr* re_expr = mem.m_regex->get_expr();
            SASSERT(re_expr && m_seq.is_re(re_expr));

            unsigned min_len = 0, max_len = UINT_MAX;
            compute_regex_length_interval(mem.m_regex, min_len, max_len);

            expr_ref len_str = compute_length_expr(mem.m_str);

            if (min_len > 0)
                node->add_constraint(mk_constraint(a.mk_ge(len_str, a.mk_int(min_len)), mem.m_dep));
            if (max_len < UINT_MAX)
                node->add_constraint(mk_constraint(a.mk_le(len_str, a.mk_int(max_len)), mem.m_dep));
        }
    }

    bool nielsen_graph::check_int_feasibility() const {
        // In incremental mode the solver already holds all path constraints
        // (root length constraints at the base level, edge side_constraints and node
        // constraints pushed/popped as the DFS descends and backtracks).
        // A plain check() is therefore sufficient.
        return m_length_solver.check() != l_false;
    }

    dep_tracker nielsen_graph::get_subsolver_dependency(nielsen_node* /*n*/) const {
        // check_int_feasibility() already called m_solver.check() which computed
        // the UNSAT core in terms of tracked assumption literals and their deps.
        return m_length_solver.core();
    }

    bool nielsen_graph::check_lp_le(expr* lhs, expr* rhs, nielsen_node* n, dep_tracker& dep) {
        dep = nullptr;

        rational lhs_lo, rhs_up;
        literal_vector lits;
        enode_pair_vector eqs;
        if (m_context_solver.lower_bound(lhs, lhs_lo, lits, eqs) &&
            m_context_solver.upper_bound(rhs, rhs_up, lits, eqs) && lhs_lo > rhs_up)
            return false;

        // lhs <= lhs_up <= rhs_lo <= rhs
        // => lhs <= rhs is entailed

        lits.reset();
        eqs.reset();
        rational rhs_lo, lhs_up;
        if (m_context_solver.upper_bound(lhs, lhs_up, lits, eqs) &&
            m_context_solver.lower_bound(rhs, rhs_lo, lits, eqs) &&
            lhs_up <= rhs_lo) {
            for (auto lit : lits)
                dep = m_dep_mgr.mk_join(dep, m_dep_mgr.mk_leaf(lit));
            for (enode_pair eq : eqs)
                dep = m_dep_mgr.mk_join(dep, m_dep_mgr.mk_leaf(eq));
            return true;
        }

        // fall through - ask the solver [expensive]

        // TODO: Maybe cache the result?

        // The solver already holds all path constraints incrementally.
        // Temporarily add NOT(lhs <= rhs), i.e. lhs >= rhs + 1.
        // If that is unsat, then lhs <= rhs is entailed.
        const expr_ref one(a.mk_int(1), m);
        const expr_ref rhs_plus_one(a.mk_add(rhs, one), m);

        m_length_solver.push();
        assert_to_subsolver(a.mk_ge(lhs, rhs_plus_one));
        const lbool result = m_length_solver.check();
        if (result == l_false)
            dep = m_length_solver.core();
        m_length_solver.pop(1);
        if (result == l_false) {
            n->add_constraint(constraint(a.mk_le(lhs, rhs), dep, m));
            return true;
        }
        return false;
    }

    constraint nielsen_graph::mk_constraint(expr *fml, dep_tracker const &dep) const {
        return constraint(fml, dep, m);
    }

    expr* nielsen_graph::get_power_exponent(euf::snode const* power) {
        SASSERT(power);
        if (!power->is_power())
            return nullptr;
        SASSERT(power->num_args() == 2);
        euf::snode const* exp_snode = power->arg(1);
        return exp_snode ? exp_snode->get_expr() : nullptr;
    }

    // -----------------------------------------------------------------------
    // Regex widening: overapproximate string and check intersection emptiness
    // Mirrors ZIPT NielsenNode.CheckRegexWidening (NielsenNode.cs:1350-1380)
    // -----------------------------------------------------------------------

    bool nielsen_graph::check_regex_widening(nielsen_node const& node, str_mem const& mem, dep_tracker& dep) {
        const auto str = mem.m_str;
        const auto regex = mem.m_regex;
        SASSERT(m_seq_regex);
        // Only apply to ground regexes with non-trivial strings
        if (!regex->is_ground())
            return false;


        // Build the overapproximation regex for the string.
        // Variables → intersection of their primitive regex constraints (or Σ*)
        // Symbolic chars → re.range from char_ranges (or full_char)
        // Concrete chars → to_re(unit(c))
        euf::snode_vector tokens;
        str->collect_tokens(tokens);
        if (tokens.empty())
            return false;

        SASSERT(dep);

        euf::snode const* approx = nullptr;
        for (euf::snode const* tok : tokens) {
            euf::snode const* tok_re = nullptr;

            if (tok->is_char()) {
                // Concrete character → to_re(unit(c))
                expr* te = tok->get_expr();
                SASSERT(te);
                tok_re = m_sg.mk(m_seq.re.mk_to_re(te));
            }
            else if (tok->is_var()) {
                // Variable → intersection of primitive regex constraints, or Σ*
                euf::snode const* x_range = m_seq_regex->collect_primitive_regex_intersection(tok, node, m_dep_mgr, dep);
                if (x_range)
                    tok_re = x_range;
                else {
                    // Unconstrained variable: approximate as Σ*
                    sort* str_sort = m_seq.str.mk_string_sort();
                    expr_ref all_re(m_seq.re.mk_full_seq(m_seq.re.mk_re(str_sort)), m);
                    tok_re = m_sg.mk(all_re);
                }
                TRACE(seq, tout << "intersection-collection\n" << spp(tok, m)
                                << "\n" <<  spp(tok_re, m) << "\n");
            }
            else if (tok->is_unit()) {
                // Symbolic char — try to get char_range
                if (node.char_ranges().contains(tok->id())) {
                    auto& cs = node.char_ranges()[tok->id()];
                    if (!cs.first.is_empty()) {
                        // Build union of re.range for each interval
                        euf::snode const* range_re = nullptr;
                        for (auto const& r : cs.first.ranges()) {
                            expr_ref rng(m_seq.re.mk_range(
                                m_seq.str.mk_string(zstring(r.m_lo)),
                                m_seq.str.mk_string(zstring(r.m_hi - 1))), m);
                            euf::snode const* rng_sn = m_sg.mk(rng);
                            if (!range_re)
                                range_re = rng_sn;
                            else {
                                expr_ref u(m_seq.re.mk_union(range_re->get_expr(), rng_sn->get_expr()), m);
                                range_re = m_sg.mk(u);
                            }
                        }
                        dep = dep_mgr().mk_join(dep, cs.second);
                        tok_re = range_re;
                    }
                }
                if (!tok_re) {
                    // Unconstrained symbolic char: approximate as full_char (single char, any value)
                    sort* str_sort = m_seq.str.mk_string_sort();
                    expr_ref fc(m_seq.re.mk_full_char(m_seq.re.mk_re(str_sort)), m);
                    tok_re = m_sg.mk(fc);
                }
            }
            else {
                // Unknown token type — approximate as Σ*
                sort* str_sort = m_seq.str.mk_string_sort();
                expr_ref all_re(m_seq.re.mk_full_seq(m_seq.re.mk_re(str_sort)), m);
                tok_re = m_sg.mk(all_re);
            }

            SASSERT(tok_re);

            if (!approx)
                approx = tok_re;
            else {
                expr* ae = approx->get_expr();
                expr* te = tok_re->get_expr();
                SASSERT(ae && te);
                approx = m_sg.mk(m_seq.re.mk_concat(ae, te));
            }
        }

        if (!approx)
            return false;

        // Check intersection(approx, regex) for emptiness
        // Build intersection regex
        expr* ae = approx->get_expr();
        expr* re = regex->get_expr();
        SASSERT(ae && re);
        const expr_ref inter(m_seq.re.mk_inter(ae, re), m);
        euf::snode const* inter_sn = m_sg.mk(inter);
        SASSERT(inter_sn);
        // TODO: Minimize the conflict here
        const lbool result = m_seq_regex->is_empty_bfs(inter_sn, 5000);
        TRACE(seq, tout << "widen empty-intersect: " << result << " " << mk_pp(re, m)
            << " <= " << mk_pp(ae, m) << "\n" << mem_pp(mem, m) << "\n";
        display(tout, &node) << "\n");
        return result == l_true;
    }

    // -----------------------------------------------------------------------
    // Leaf-node regex intersection emptiness check
    // Mirrors ZIPT NielsenNode.CheckRegex (NielsenNode.cs:1311-1329)
    // -----------------------------------------------------------------------

    // -----------------------------------------------------------------------
    // Synchronous product over plain / view / guard / co-view components.
    // -----------------------------------------------------------------------

    lbool nielsen_graph::comp_accepting(prod_comp const& c) const {
        if (c.m_dead)
            return l_false;
        switch (c.m_kind) {
        case mem_kind::plain:
            return m_sg.re_nullable(c.m_state);
        case mem_kind::stab_view:
            if (c.m_complemented)
                return (c.m_sink || c.m_state != c.m_root) ? l_true : l_false;
            return (c.m_state == c.m_root) ? l_true : l_false;
        case mem_kind::no_loop:
            return l_true; // guard accepts on every prefix it has not failed on
        }
        return l_undef;
    }

    nielsen_graph::prod_comp nielsen_graph::comp_step(prod_comp const& c, euf::snode const* mt) {
        prod_comp r = c;
        if (c.m_dead)
            return r;
        switch (c.m_kind) {
        case mem_kind::plain: {
            euf::snode const* d = m_sg.brzozowski_deriv(c.m_state, mt);
            if (!d || d->is_fail()) r.m_dead = true; else r.m_state = d;
            return r;
        }
        case mem_kind::stab_view: {
            if (c.m_complemented) {
                if (c.m_sink) return r;                    // Σ*
                if (!projection_state_in_Q(c.m_state->get_expr(), c.m_nu)) { r.m_sink = true; return r; }
                euf::snode const* d = m_sg.brzozowski_deriv(c.m_state, mt);
                if (!d || d->is_fail()) { r.m_sink = true; return r; } // ~∅ = Σ*
                r.m_state = d;
                return r;
            }
            if (!projection_state_in_Q(c.m_state->get_expr(), c.m_nu)) { r.m_dead = true; return r; }
            euf::snode const* d = m_sg.brzozowski_deriv(c.m_state, mt);
            if (!d || d->is_fail()) { r.m_dead = true; return r; }
            r.m_state = d;
            return r;
        }
        case mem_kind::no_loop: {
            if (c.m_sink) return r;                        // discharged: Σ*
            if (!projection_state_in_Q(c.m_state->get_expr(), c.m_nu)) { r.m_sink = true; return r; }
            euf::snode const* d = m_sg.brzozowski_deriv(c.m_state, mt);
            if (!d || d->is_fail()) { r.m_sink = true; return r; } // lap cannot close through ∅
            if (d == c.m_root) { r.m_dead = true; return r; }      // lap completed → forbidden
            r.m_state = d;
            return r;
        }
        }
        return r;
    }

    lbool nielsen_graph::check_product_emptiness(vector<prod_comp> const& comps0, unsigned max_states) {
        if (comps0.empty())
            return l_false; // empty intersection = Σ* (non-empty)

        auto encode = [](vector<prod_comp> const& cs) {
            std::vector<unsigned> key;
            key.reserve(cs.size() * 5);
            for (auto const& c : cs) {
                key.push_back(static_cast<unsigned>(c.m_kind));
                key.push_back((c.m_complemented ? 1u : 0u) | (c.m_sink ? 2u : 0u) | (c.m_dead ? 4u : 0u));
                key.push_back(c.m_state ? c.m_state->id() : UINT_MAX);
            }
            return key;
        };

        std::set<std::vector<unsigned>> visited;
        vector<vector<prod_comp>> work;
        work.push_back(comps0);
        visited.insert(encode(comps0));
        unsigned explored = 0;

        while (!work.empty()) {
            if (!m.inc())
                return l_undef;
            if (explored >= max_states)
                return l_undef;
            vector<prod_comp> cur = work.back();
            work.pop_back();
            ++explored;

            bool any_dead = false;
            for (auto const& c : cur) if (c.m_dead) { any_dead = true; break; }
            if (any_dead)
                continue;

            // simultaneously accepting?
            bool all_acc = true, any_undef = false;
            for (auto const& c : cur) {
                const lbool a = comp_accepting(c);
                if (a == l_false) { all_acc = false; break; }
                if (a == l_undef) any_undef = true;
            }
            if (all_acc && !any_undef)
                return l_false; // found a common word

            // joint first-character partition = minterms of the intersection of
            // all still-discriminating component states.
            expr* combined = nullptr;
            for (auto const& c : cur) {
                if (c.m_sink || c.m_dead) continue;
                combined = combined ? m_seq.re.mk_inter(combined, c.m_state->get_expr())
                                    : c.m_state->get_expr();
            }
            if (!combined)
                continue; // no discriminating state and not accepting: dead end
            euf::snode_vector mts;
            m_sg.compute_minterms(m_sg.mk(combined), mts);

            for (euf::snode const* mt : mts) {
                vector<prod_comp> nxt;
                bool dead = false;
                for (auto const& c : cur) {
                    prod_comp d = comp_step(c, mt);
                    if (d.m_dead) { dead = true; break; }
                    nxt.push_back(d);
                }
                if (dead)
                    continue;
                if (visited.insert(encode(nxt)).second)
                    work.push_back(nxt);
            }
        }
        return l_true; // exhausted with no accepting tuple → empty
    }

    bool nielsen_graph::collect_var_components(euf::snode const* var, nielsen_node const& node,
                                               vector<prod_comp>& out, dep_tracker& dep) {
        bool found = false;
        for (auto const& mem : node.str_mems()) {
            if (!mem.is_primitive())
                continue;
            if (mem.m_str->first() != var)
                continue;
            switch (mem.m_kind) {
            case mem_kind::plain:
                out.push_back(prod_comp::mk_plain(mem.m_regex));
                break;
            case mem_kind::stab_view:
                out.push_back(prod_comp::mk_view(mem.m_regex, mem.m_root, mem.m_nu, false));
                break;
            case mem_kind::no_loop:
                out.push_back(prod_comp::mk_guard(mem.m_regex, mem.m_root, mem.m_nu, mem.m_discharged));
                break;
            }
            dep = m_dep_mgr.mk_join(dep, mem.m_dep);
            found = true;
        }
        return found;
    }

    bool nielsen_graph::check_leaf_regex(nielsen_node const& node, dep_tracker& dep) {
        SASSERT(m_seq_regex);

        // distinct variables carrying a primitive constraint
        uint_set seen;
        for (auto const& mem : node.str_mems()) {
            SASSERT(mem.is_primitive());
            euf::snode const* const first = mem.m_str->first();
            SASSERT(first && first->is_var());
            if (seen.contains(first->id()))
                continue;
            seen.insert(first->id());

            vector<prod_comp> comps;
            dep_tracker d = nullptr;
            collect_var_components(first, node, comps, d);
            const lbool result = check_product_emptiness(comps, 5000);
            if (result == l_true) {
                TRACE(seq, tout << "empty intersection\n");
                dep = d;
                return false;
            }
        }
        return true;
    }

    // -----------------------------------------------------------------------
    // Statistics collection
    // -----------------------------------------------------------------------

    void nielsen_graph::collect_statistics(::statistics& st) const {
        st.update("nseq solve calls",     m_stats.m_num_solve_calls);
        st.update("nseq dfs nodes",       m_stats.m_num_dfs_nodes);
        st.update("nseq sat",             m_stats.m_num_sat);
        st.update("nseq unsat",           m_stats.m_num_unsat);
        st.update("nseq unknown",         m_stats.m_num_unknown);
        st.update("nseq simplify clash",  m_stats.m_num_simplify_conflict);
        st.update("nseq extensions",      m_stats.m_num_extensions);
        st.update("nseq fresh vars",      m_stats.m_num_fresh_vars);
        st.update("nseq max depth",       m_stats.m_max_depth);

        // modifier breakdown
        st.update("nseq mod det",              m_stats.m_mod_det);
        st.update("nseq mod power epsilon",    m_stats.m_mod_power_epsilon);
        st.update("nseq mod num cmp",          m_stats.m_mod_num_cmp);
        st.update("nseq mod const num unwind", m_stats.m_mod_const_num_unwinding);
        st.update("nseq mod eq split",         m_stats.m_mod_eq_split);
        st.update("nseq mod star intr",        m_stats.m_mod_star_intr);
        st.update("nseq mod cycle subsump",    m_stats.m_mod_cycle_subsumption);
        st.update("nseq mod gpower intr",      m_stats.m_mod_gpower_intr);
        st.update("nseq mod regex fact",       m_stats.m_mod_regex_factorization);
        st.update("nseq mod const nielsen",    m_stats.m_mod_const_nielsen);
        st.update("nseq mod signature split",  m_stats.m_mod_signature_split);
        st.update("nseq mod regex var",        m_stats.m_mod_regex_var_split);
        st.update("nseq mod regex if",         m_stats.m_mod_regex_if_split);
        st.update("nseq mod power split",      m_stats.m_mod_power_split);
        st.update("nseq mod var nielsen",      m_stats.m_mod_var_nielsen);
        st.update("nseq mod var num unwind (eq)",   m_stats.m_mod_var_num_unwinding_eq);
        st.update("nseq mod var num unwind (mem)",   m_stats.m_mod_var_num_unwinding_mem);
        st.update("nseq mod axiomatized disequalities",   m_stats.m_ax_diseq);
        st.update("nseq unsat-cache size",                (unsigned) m_unsat_node_cache.size());
        st.update("nseq unsat-cache hits",                m_num_cache_hits);
    }

}