/*++
Copyright (c) 2006 Microsoft Corporation

Module Name:

    theory_datatype.cpp

Abstract:

    <abstract>

Author:

    Leonardo de Moura (leonardo) 2008-10-31.

Revision History:

--*/

#include "util/stats.h"
#include "ast/ast_pp.h"
#include "ast/ast_ll_pp.h"
#include "ast/ast_smt2_pp.h"
#include "smt/smt_context.h"
#include "smt/theory_datatype.h"
#include "smt/theory_array.h"
#include "smt/smt_model_generator.h"
#include <iostream>

namespace smt {
    
    class dt_eq_justification : public ext_theory_eq_propagation_justification {
    public:
        dt_eq_justification(family_id fid, context& ctx, literal antecedent, enode * lhs, enode * rhs):
            ext_theory_eq_propagation_justification(fid, ctx, 1, &antecedent, 0, nullptr, lhs, rhs) {
        }
        // Remark: the assignment must be propagated back to the datatype theory.
        theory_id get_from_theory() const override { return null_theory_id; }
    };

    theory_datatype::final_check_st::final_check_st(theory_datatype * th) : th(th) {
        SASSERT(th->m_to_unmark.empty());
        SASSERT(th->m_to_unmark2.empty());
        th->m_used_eqs.reset();
        th->m_stack.reset();
        th->m_parent.reset();
    }

    theory_datatype::final_check_st::~final_check_st() {
        th->clear_mark();
    }

    void theory_datatype::clear_mark() {
        unmark_enodes(m_to_unmark.size(), m_to_unmark.data());
        unmark_enodes2(m_to_unmark2.size(), m_to_unmark2.data());
        m_to_unmark.reset();
        m_to_unmark2.reset();
    }
    
    void theory_datatype::oc_mark_on_stack(enode * n) {
        n = n->get_root();
        n->set_mark();
        m_to_unmark.push_back(n); 
    }

    void theory_datatype::oc_mark_cycle_free(enode * n) {
        n = n->get_root();
        n->set_mark2();
        m_to_unmark2.push_back(n); 
    }

    void theory_datatype::oc_push_stack(enode * n) {
        m_stack.push_back(std::make_pair(EXIT, n));
        m_stack.push_back(std::make_pair(ENTER, n));
    }


    theory* theory_datatype::mk_fresh(context* new_ctx) { 
        return alloc(theory_datatype, *new_ctx);
    }

    /**
       \brief Assert the axiom (antecedent => lhs = rhs)
       antecedent may be null_literal
    */
    void theory_datatype::assert_eq_axiom(enode * lhs, expr * rhs, literal antecedent) {

        if (antecedent != null_literal) {
            std::function<void(void)> fn = [&]() {
                app_ref body(m);
                body = m.mk_eq(lhs->get_expr(), rhs);
                body = m.mk_implies(ctx.bool_var2expr(antecedent.var()), body);                
                log_axiom_instantiation(body, 1, &lhs);
            };
            scoped_trace_stream _st(m, fn);
        }

        if (m.proofs_enabled()) {
            literal l(mk_eq(lhs->get_expr(), rhs, true));
            ctx.mark_as_relevant(l);
            if (antecedent != null_literal) {
                literal lits[2] = {l, ~antecedent};
                ctx.mk_th_axiom(get_id(), 2, lits);
            }
            else {
                literal lits[1] = {l};
                ctx.mk_th_axiom(get_id(), 1, lits);
            }
        }
        else {
            ctx.internalize(rhs, false);
            TRACE(datatype, tout << "adding axiom:\n" << pp(lhs, m) << "\n=\n" << mk_pp(rhs, m) << "\n";);
            if (antecedent == null_literal) {
                ctx.assign_eq(lhs, ctx.get_enode(rhs), eq_justification::mk_axiom());
            }
            else if (ctx.get_assignment(antecedent) != l_true) {
                literal l(mk_eq(lhs->get_expr(), rhs, true));
                ctx.mark_as_relevant(l);
                ctx.mark_as_relevant(antecedent);
                literal lits[2] = {l, ~antecedent};
                ctx.mk_th_axiom(get_id(), 2, lits);
            }
            else {
                SASSERT(ctx.get_assignment(antecedent) == l_true);
                enode * _rhs = ctx.get_enode(rhs);
                justification * js = ctx.mk_justification(dt_eq_justification(get_id(), ctx, antecedent, lhs, _rhs));
                TRACE(datatype, tout << "assigning... #" << lhs->get_owner_id() << " #" << _rhs->get_owner_id() << "\n";
                      tout << "v" << lhs->get_th_var(get_id()) << " v" << _rhs->get_th_var(get_id()) << "\n";);
                TRACE(datatype_detail, display(tout););
                ctx.assign_eq(lhs, _rhs, eq_justification(js));
            }
        }
    }

    /**
       \brief Assert the equality (= n (c (acc_1 n) ... (acc_m n))) where 
       where acc_i are the accessors of constructor c.
    */
    void theory_datatype::assert_is_constructor_axiom(enode * n, func_decl * c, literal antecedent) {
        app* e = n->get_expr();
        TRACE(datatype_bug, tout << "creating axiom (= n (c (acc_1 n) ... (acc_m n))) for\n" 
            << mk_pp(c, m) << " " << mk_pp(e, m) << "\n";);
        m_stats.m_assert_cnstr++;
        SASSERT(m_util.is_constructor(c));
        SASSERT(m_util.is_datatype(e->get_sort()));

        SASSERT(c->get_range() == e->get_sort());
        ptr_vector<expr> args;
        ptr_vector<func_decl> const & accessors = *m_util.get_constructor_accessors(c);
        SASSERT(c->get_arity() == accessors.size());
        for (func_decl * d : accessors) {
            SASSERT(d->get_arity() == 1);
            args.push_back(m.mk_app(d, e));
        }
        expr_ref mk(m.mk_app(c, args), m);

        assert_eq_axiom(n, mk, antecedent);
    }

    /**
       \brief Given a constructor n := (c a_1 ... a_m) assert the axioms
       (= (acc_1 n) a_1)
       ...
       (= (acc_m n) a_m)
    */
    void theory_datatype::assert_accessor_axioms(enode * n) {
        m_stats.m_assert_accessor++;

        SASSERT(is_constructor(n));
        func_decl * d     = n->get_decl();
        ptr_vector<func_decl> const & accessors   = *m_util.get_constructor_accessors(d);
        SASSERT(n->get_num_args() == accessors.size());
        app_ref_vector bindings(m);
        vector<std::tuple<enode *, enode *>> used_enodes;
        used_enodes.push_back(std::make_tuple(nullptr, n));
        for (unsigned i = 0; i < n->get_num_args(); ++i) {
            bindings.push_back(n->get_arg(i)->get_expr());
        }
        unsigned base_id = m.has_trace_stream() && accessors.size() > 0 ? m_util.plugin().get_axiom_base_id(d->get_name()) : 0;
        unsigned i = 0;
        for (func_decl * acc : accessors) {
            app_ref acc_app(m.mk_app(acc, n->get_expr()), m);
            enode * arg       = n->get_arg(i);

            std::function<void(void)> fn = [&]() {
                app_ref body(m);
                body = m.mk_eq(arg->get_expr(), acc_app);
                log_axiom_instantiation(body, base_id + 3*i, bindings.size(), bindings.data(), base_id - 3, used_enodes);
            };
            scoped_trace_stream _st(m, fn);
            assert_eq_axiom(arg, acc_app, null_literal);
            ++i;
        }
    }

    /**
       \brief Sign a conflict for r := is_mk(a), c := mk(...), not(r),  and c == a.
    */
    void theory_datatype::sign_recognizer_conflict(enode * c, enode * r) {
        SASSERT(is_constructor(c));
        SASSERT(is_recognizer(r));
        SASSERT(m_util.get_recognizer_constructor(r->get_decl()) == c->get_decl());
        SASSERT(c->get_root() == r->get_arg(0)->get_root());
        TRACE(recognizer_conflict,
              tout << mk_ismt2_pp(c->get_expr(), m) << "\n" << mk_ismt2_pp(r->get_expr(), m) << "\n";);
        literal l(ctx.enode2bool_var(r));
        SASSERT(ctx.get_assignment(l) == l_false);
        l.neg();
        SASSERT(ctx.get_assignment(l) == l_true);
        enode_pair p(c, r->get_arg(0));
        clear_mark();
        ctx.set_conflict(ctx.mk_justification(ext_theory_conflict_justification(get_id(), ctx, 1, &l, 1, &p)));
    }

    /**
       \brief Given a field update n := { r with field := v } for constructor C, assert the axioms:
       (=> (is-C r) (= (acc_j n) (acc_j r))) for acc_j != field
       (=> (is-C r) (= (field n) v))         for acc_j != field
       (=> (not (is-C r)) (= n r))
       (=> (is-C r) (is-C n))
    */
    void theory_datatype::assert_update_field_axioms(enode * n) {
        m_stats.m_assert_update_field++;
        SASSERT(is_update_field(n));
        app*        own  = n->get_expr();
        expr*       arg1 = own->get_arg(0);
        func_decl * upd  = n->get_decl();
        func_decl * acc  = to_func_decl(upd->get_parameter(0).get_ast());
        func_decl * con  = m_util.get_accessor_constructor(acc);
        func_decl * rec  = m_util.get_constructor_is(con);
        ptr_vector<func_decl> const & accessors = *m_util.get_constructor_accessors(con);
        app_ref rec_app(m.mk_app(rec, arg1), m);
        app_ref acc_app(m);
        ctx.internalize(rec_app, false);
        literal is_con(ctx.get_bool_var(rec_app));        
        for (func_decl* acc1 : accessors) {
            enode* arg;
            if (acc1 == acc) {
                arg = n->get_arg(1);
            }
            else {
                acc_app = m.mk_app(acc1, arg1);
                ctx.internalize(acc_app, false);
                arg = ctx.get_enode(acc_app);
            }
            app_ref acc_own(m.mk_app(acc1, own), m);
            assert_eq_axiom(arg, acc_own, is_con); 
        }
        // update_field is identity if 'n' is not created by a matching constructor.        
        app_ref imp(m.mk_implies(m.mk_not(rec_app), m.mk_eq(n->get_expr(), arg1)), m);
        assert_eq_axiom(n, arg1, ~is_con);

        app_ref n_is_con(m.mk_app(rec, own), m);
        ctx.internalize(n_is_con, false);
        literal lits[2] = { ~is_con, literal(ctx.get_bool_var(n_is_con)) };
        ctx.mark_as_relevant(lits[0]);
        ctx.mark_as_relevant(lits[1]);
        std::function<literal_vector(void)> fn = [&]() { return literal_vector(2, lits); };
        scoped_trace_stream _st(*this, fn);
        ctx.mk_th_axiom(get_id(), 2, lits);
    }

    theory_var theory_datatype::mk_var(enode * n) {
        theory_var r  = theory::mk_var(n);
        VERIFY(r == static_cast<theory_var>(m_find.mk_var()));
        SASSERT(r == static_cast<int>(m_var_data.size()));
        m_var_data.push_back(alloc(var_data));
        var_data * d  = m_var_data[r];
        ctx.attach_th_var(n, this, r);
        if (is_constructor(n)) {
            d->m_constructor = n;
            assert_accessor_axioms(n);
        }
        else if (is_update_field(n)) {
            assert_update_field_axioms(n);
        }
        else if (m_util.is_datatype(n->get_sort())) {
            sort * s      = n->get_sort();
            if (m_util.get_datatype_num_constructors(s) == 1) {
                func_decl * c = m_util.get_datatype_constructors(s)->get(0);
                assert_is_constructor_axiom(n, c, null_literal);
            }
            else {
                if (params().m_dt_lazy_splits == 0 || (params().m_dt_lazy_splits == 1 && !s->is_infinite()))
                    mk_split(r);
            }
        }
        return r;
    }

    bool theory_datatype::internalize_atom(app * atom, bool gate_ctx) {
        TRACE(datatype, tout << "internalizing atom:\n" << mk_pp(atom, m) << "\n";);
        return internalize_term(atom);
    }

    bool theory_datatype::internalize_term(app * term) {
        force_push();
        TRACE(datatype, tout << "internalizing term:\n" << mk_pp(term, m) << "\n";);
        unsigned num_args = term->get_num_args();
        for (unsigned i = 0; i < num_args; i++)
            ctx.internalize(term->get_arg(i), m.is_bool(term) && has_quantifiers(term));
        // the internalization of the arguments may trigger the internalization of term.
        if (ctx.e_internalized(term))
            return true;
        enode * e       = ctx.mk_enode(term, false, m.is_bool(term), true); // possible optimization, the third argument may be set to false, if the term (actually, atom) is not in the context of a gate.
        if (m.is_bool(term)) {
            bool_var bv = ctx.mk_bool_var(term);
            ctx.set_var_theory(bv, get_id());
            ctx.set_enode_flag(bv, true);
        }
        if (is_constructor(term) || is_update_field(term)) {
            SASSERT(!is_attached_to_var(e));
            // *** We must create a theory variable for each argument that has sort datatype ***
            //
            // The apply_sort_cnstr method will not create a theory
            // variable for an expression N when sort of N has an
            // infinite number of elements.
            //
            // This may create problems during model construction.
            // For example, suppose we have
            //    x1 = cons(v1, x2)
            // and x1 and x2 are lists of integers. 
            // This sort has an infinite number of elements. So, in principle,
            // we do not need a theory variable for x2. 
            // Recall that if an expression is not associated with a
            // theory variable, then a fresh value is associated with
            // it.
            // Moreover, fresh variables of sort S can only be created after the
            // interpretation for each (relevant) expression of sort S in the
            // logical context is created.  Returning to the example,
            // to create the interpretation of x1 we need the
            // interpretation for x2.  So, x2 cannot be a fresh value,
            // since it would have to be created after x1.
            //
            for (unsigned i = 0; i < num_args; i++) {
                enode * arg = e->get_arg(i);
                sort * s    = arg->get_sort();
                if (m_autil.is_array(s) && m_util.is_datatype(get_array_range(s))) {
                    app_ref def(m_autil.mk_default(arg->get_expr()), m);
                    if (!ctx.e_internalized(def)) {
                        ctx.internalize(def, false);
                    }
                    arg = ctx.get_enode(def);       
                }
                if (!m_util.is_datatype(s) && !m_sutil.is_seq(s))
                    continue;
                if (is_attached_to_var(arg))
                    continue;
                mk_var(arg);
            }
            mk_var(e);
        }
        else {
            SASSERT(is_accessor(term) || is_recognizer(term));
            SASSERT(term->get_num_args() == 1);
            enode * arg = e->get_arg(0);
            if (!is_attached_to_var(arg))
                mk_var(arg);
            SASSERT(is_attached_to_var(arg));
        }
        if (is_recognizer(term)) {
            enode * arg = e->get_arg(0);
            theory_var v = arg->get_th_var(get_id());
            SASSERT(v != null_theory_var);
            // When relevancy propagation is enabled, the recognizer is only added when it is marked as relevant.
            if (!ctx.relevancy()) 
                add_recognizer(v, e);
        }
        return true;
    }

    void theory_datatype::apply_sort_cnstr(enode * n, sort * s) {
        force_push();
        // Remark: If s is an infinite sort, then it is not necessary to create
        // a theory variable. 
        // 
        // Actually, when the logical context has quantifiers, it is better to 
        // disable this optimization.
        // Example:
        // 
        //   (forall (l list) (a Int) (= (len (cons a l)) (+ (len l) 1)))
        //   (assert (> (len a) 1)
        //   
        // If the theory variable is not created for 'a', then a wrong model will be generated.
        TRACE(datatype, tout << "apply_sort_cnstr: #" << n->get_owner_id() << " " << pp(n, m) << "\n";);
        TRACE(datatype_bug, 
              tout << "apply_sort_cnstr:\n" << pp(n, m) << " ";
              tout << m_util.is_datatype(s) << " ";
              if (m_util.is_datatype(s)) tout << "is-infinite: " << s->is_infinite() << " "; 
              if (m_util.is_datatype(s)) tout << "attached: " << is_attached_to_var(n) << " ";
              tout << "\n";);

        if (!is_attached_to_var(n) &&
            (ctx.has_quantifiers() || 
             (m_util.is_datatype(s) && m_util.has_nested_rec()) || 
             (m_util.is_datatype(s) && !s->is_infinite()))) {
            mk_var(n);
        }
    }

    void theory_datatype::new_eq_eh(theory_var v1, theory_var v2) {
        force_push();
        m_find.merge(v1, v2);
    }

    bool theory_datatype::use_diseqs() const {
        return false;
    }

    void theory_datatype::new_diseq_eh(theory_var v1, theory_var v2) {
        UNREACHABLE();
    }

    void theory_datatype::assign_eh(bool_var v, bool is_true) {
        force_push();
        enode * n     = ctx.bool_var2enode(v);
        if (!is_recognizer(n))
            return;
        TRACE(datatype, tout << "assigning recognizer: #" << n->get_owner_id() << " is_true: " << is_true << "\n" 
              << enode_pp(n, ctx) << "\n";);
        SASSERT(n->get_num_args() == 1);
        enode * arg   = n->get_arg(0);
        theory_var tv = arg->get_th_var(get_id());
        tv = m_find.find(tv);
        var_data * d  = m_var_data[tv];
        func_decl * r = n->get_decl();
        func_decl * c = m_util.get_recognizer_constructor(r);
        if (is_true) {
            SASSERT(tv != null_theory_var);
            if (d->m_constructor != nullptr && d->m_constructor->get_decl() == c)
                return; // do nothing
            assert_is_constructor_axiom(arg, c, literal(v));
        }
        else {
            if (d->m_constructor != nullptr) {
                if (d->m_constructor->get_decl() == c) {
                    // conflict
                    sign_recognizer_conflict(d->m_constructor, n);
                }
            }
            else {
                propagate_recognizer(tv, n);
            }
        }
    }

    void theory_datatype::relevant_eh(app * n) {
        force_push();
        TRACE(datatype, tout << "relevant_eh: " << mk_pp(n, m) << "\n";);
        SASSERT(ctx.relevancy());
        if (is_recognizer(n)) {
            SASSERT(ctx.e_internalized(n));
            enode * e = ctx.get_enode(n);
            theory_var v = e->get_arg(0)->get_th_var(get_id());
            SASSERT(v != null_theory_var);
            add_recognizer(v, e);
        }
    }

    void theory_datatype::push_scope_eh() {
        if (lazy_push())
            return;
        theory::push_scope_eh();
        m_trail_stack.push_scope();
    }

    void theory_datatype::pop_scope_eh(unsigned num_scopes) {
        if (lazy_pop(num_scopes))
            return;
        m_trail_stack.pop_scope(num_scopes);
        unsigned num_old_vars = get_old_num_vars(num_scopes);
        std::for_each(m_var_data.begin() + num_old_vars, m_var_data.end(), delete_proc<var_data>());
        m_var_data.shrink(num_old_vars);
        theory::pop_scope_eh(num_scopes);
        SASSERT(m_find.get_num_vars() == m_var_data.size());
        SASSERT(m_find.get_num_vars() == get_num_vars());
    }

    final_check_status theory_datatype::final_check_eh() {
        force_push();
        int num_vars = get_num_vars();
        final_check_status r = FC_DONE;
        final_check_st _guard(this); 
        for (int v = 0; v < num_vars; v++) {
            if (v == static_cast<int>(m_find.find(v))) {
                enode * node = get_enode(v);
                sort* s = node->get_sort();
                if (!m_util.is_datatype(s))
                    continue;
                if (m_util.is_recursive(s) && !oc_cycle_free(node) && occurs_check(node)) {
                    // conflict was detected... 
                    // return...
                    return FC_CONTINUE;
                }
                if (params().m_dt_lazy_splits > 0) {
                    // using lazy case splits...
                    var_data * d = m_var_data[v];
                    if (d->m_constructor == nullptr) {
                        clear_mark();
                        mk_split(v);
                        r = FC_CONTINUE;
                    }
                }
            }
        }
        return r;
    }

    // Assuming `app` is equal to a constructor term, return the constructor enode
    inline enode * theory_datatype::oc_get_cstor(enode * app) {
        theory_var v = app->get_root()->get_th_var(get_id());
        if (v == null_theory_var)
            return nullptr;
        v = m_find.find(v);
        var_data * d = m_var_data[v];
        return d->m_constructor;
    }

    void theory_datatype::explain_is_child(enode* parent, enode* child) {
        enode * parentc = oc_get_cstor(parent);        
        if (parent != parentc) 
            m_used_eqs.push_back(enode_pair(parent, parentc));

        // collect equalities on all children that may have been used.
        bool found = false;
        for (enode * arg : enode::args(parentc)) {
            // found an argument which is equal to root
            if (arg->get_root() == child->get_root()) {
                if (arg != child) {
                    m_used_eqs.push_back(enode_pair(arg, child));
                }
                found = true;
            }
            sort * s = arg->get_sort();
            if (m_autil.is_array(s) && m_util.is_datatype(get_array_range(s))) {
                for (enode* aarg : get_array_args(arg)) {
                    if (aarg->get_root() == child->get_root()) {
                        if (aarg != child) {
                            m_used_eqs.push_back(enode_pair(aarg, child));
                        }
                        found = true;
                    }
                }
            }
            sort* se = nullptr;
            if (m_sutil.is_seq(s, se) && m_util.is_datatype(se)) {
                enode* sibling;
                for (enode* aarg : get_seq_args(arg, sibling)) {
                    if (aarg->get_root() == child->get_root()) {
                        if (aarg != child) 
                            m_used_eqs.push_back(enode_pair(aarg, child));
                        found = true;
                    }
                }
                if (sibling && sibling != arg)
                    m_used_eqs.push_back(enode_pair(arg, sibling));

            }
        }
        VERIFY(found);
    }

    // explain the cycle root -> ... -> app -> root
    void theory_datatype::occurs_check_explain(enode * app, enode * root) {
        TRACE(datatype, tout << "occurs_check_explain " << mk_bounded_pp(app->get_expr(), m) << " <-> " << mk_bounded_pp(root->get_expr(), m) << "\n";);

        // first: explain that root=v, given that app=cstor(...,v,...)

        explain_is_child(app, root);

        // now explain app=cstor(..,v,..) where v=root, and recurse with parent of app
        while (app->get_root() != root->get_root()) {
            enode* parent_app = m_parent[app->get_root()];
            explain_is_child(parent_app, app);
            SASSERT(is_constructor(parent_app));
            app = parent_app;            
        }
        
        SASSERT(app->get_root() == root->get_root());
        if (app != root) {
            m_used_eqs.push_back(enode_pair(app, root));
        }

        TRACE(datatype,
              tout << "occurs_check\n";
              for (enode_pair const& p : m_used_eqs) 
                  tout << enode_eq_pp(p, ctx);
              for (auto const& [a,b] : m_used_eqs)
                  tout << mk_pp(a->get_expr(), m) << " = " << mk_pp(b->get_expr(), m) << "\n";
              );
    }

    // start exploring subgraph below `app`
    bool theory_datatype::occurs_check_enter(enode * app) {
        app = app->get_root();
        theory_var v = app->get_th_var(get_id());
        if (v == null_theory_var) {
            return false;
        }
        v = m_find.find(v);
        var_data * d = m_var_data[v];

        if (!d->m_constructor) 
            return false;
        enode * parent = d->m_constructor;
        oc_mark_on_stack(parent);
        auto process_arg = [&](enode* aarg) {
            if (oc_cycle_free(aarg)) 
                return false;                
            if (oc_on_stack(aarg)) {
                occurs_check_explain(parent, aarg);
                return true;
            }
            if (m_util.is_datatype(aarg->get_sort())) {
                m_parent.insert(aarg->get_root(), parent);
                oc_push_stack(aarg);
            }
            return false;            
        };
        
        for (enode * arg : enode::args(parent)) {
            if (oc_cycle_free(arg)) 
                continue;
            if (oc_on_stack(arg)) {
                // arg was explored before app, and is still on the stack: cycle
                occurs_check_explain(parent, arg);
                return true;
            }
            // explore `arg` (with parent)
            expr* earg = arg->get_expr();
            sort* s = earg->get_sort(), *se = nullptr;
            if (m_util.is_datatype(s)) {
                m_parent.insert(arg->get_root(), parent);
                oc_push_stack(arg);
            }
            else if (m_sutil.is_seq(s, se) && m_util.is_datatype(se)) {
                enode* sibling;
                for (enode* sarg : get_seq_args(arg, sibling)) {
                    if (process_arg(sarg)) 
                        return true;
                }
            }
            else if (m_autil.is_array(s) && m_util.is_datatype(get_array_range(s))) {
                for (enode* aarg : get_array_args(arg)) 
                    if (process_arg(aarg))
                        return true;
            }
        }
        return false;
    }

    ptr_vector<enode> const& theory_datatype::get_seq_args(enode* n, enode*& sibling) {
        m_args.reset();
        m_todo.reset();
        auto add_todo = [&](enode* n) {
            if (!n->is_marked()) {
                n->set_mark();
                m_todo.push_back(n);
            }
        };
        
        for (enode* sib : *n) {
            if (m_sutil.str.is_concat_of_units(sib->get_expr())) {
                add_todo(sib);
                sibling = sib;
                break;
            }
        }
            
        for (unsigned i = 0; i < m_todo.size(); ++i) {
            enode* n = m_todo[i];
            expr* e = n->get_expr();
            if (m_sutil.str.is_unit(e))
                m_args.push_back(n->get_arg(0));
            else if (m_sutil.str.is_concat(e)) 
                for (expr* arg : *to_app(e)) 
                    add_todo(ctx.get_enode(arg));
        }
        for (enode* n : m_todo)
            n->unset_mark();

        return m_args;
    }
    
    ptr_vector<enode> const& theory_datatype::get_array_args(enode* n) {
        m_args.reset();
        theory_array* th = dynamic_cast<theory_array*>(ctx.get_theory(m_autil.get_family_id()));
        for (enode* p : th->parent_selects(n)) 
            m_args.push_back(p);            
        app_ref def(m_autil.mk_default(n->get_expr()), m);
        m_args.push_back(ctx.get_enode(def));
        return m_args;
    }

    /**
       \brief Check if n can be reached starting from n and following equalities and constructors.
       For example, occur_check(a1) returns true in the following set of equalities:
       a1 = cons(v1, a2)
       a2 = cons(v2, a3)
       a3 = cons(v3, a1)
    */
    bool theory_datatype::occurs_check(enode * n) {
        TRACE(datatype_verbose, tout << "occurs check: " << enode_pp(n, ctx) << "\n";);
        m_stats.m_occurs_check++;

        bool res = false;
        oc_push_stack(n);

        // DFS traversal from `n`. Look at top element and explore it.
        while (!res && !m_stack.empty()) {
            stack_op op = m_stack.back().first;
            enode * app = m_stack.back().second;
            m_stack.pop_back();

            if (oc_cycle_free(app))
                continue;

            TRACE(datatype_verbose, tout << "occurs check loop: " << enode_pp(app, ctx) << (op==ENTER?" enter":" exit")<< "\n";);

            switch (op) {
            case ENTER:
                res = occurs_check_enter(app);
                break;

            case EXIT:
                oc_mark_cycle_free(app);
                break;
            }
        }

        if (res) {
            // m_used_eqs should contain conflict
            clear_mark();
            ctx.set_conflict(ctx.mk_justification(ext_theory_conflict_justification(get_id(), ctx, 0, nullptr, m_used_eqs.size(), m_used_eqs.data())));
        }
        return res;
    }
        
    void theory_datatype::reset_eh() {
        m_trail_stack.reset();
        std::for_each(m_var_data.begin(), m_var_data.end(), delete_proc<var_data>());
        m_var_data.reset();
        theory::reset_eh();
        m_util.reset();
        m_stats.reset();
    }

    bool theory_datatype::is_shared(theory_var v) const {
        // In principle, parametric theories such as Array Theory and
        // Datatype Theory need to implement this method. However, the datatype theory
        // propagates all implied equalities. And, the is_shared method is essentially used
        // to create interface equalities. So, it is safe to return false.
        return false;
    }

    theory_datatype_params const& theory_datatype::params() const { 
        return ctx.get_fparams(); 
    }

    theory_datatype::theory_datatype(context& ctx):
        theory(ctx, ctx.get_manager().mk_family_id("datatype")),
        m_util(m),
        m_autil(m),
        m_sutil(m),
        m_find(*this) {
    }

    theory_datatype::~theory_datatype() {
        std::for_each(m_var_data.begin(), m_var_data.end(), delete_proc<var_data>());
        m_var_data.reset();
    }

    void theory_datatype::display(std::ostream & out) const {
        unsigned num_vars = get_num_vars();
        if (num_vars == 0) return;
        out << "Theory datatype:\n";
        for (unsigned v = 0; v < num_vars; v++) 
            display_var(out, v);
    }

    void theory_datatype::collect_statistics(::statistics & st) const {
        st.update("datatype occurs check", m_stats.m_occurs_check);
        st.update("datatype splits", m_stats.m_splits);
        st.update("datatype constructor ax", m_stats.m_assert_cnstr);
        st.update("datatype accessor ax", m_stats.m_assert_accessor);
        st.update("datatype update ax", m_stats.m_assert_update_field);
    }
    
    void theory_datatype::display_var(std::ostream & out, theory_var v) const {
        var_data * d = m_var_data[v];
        out << "v" << v << " #" << get_enode(v)->get_owner_id() << " -> v" << m_find.find(v) << " ";
        if (d->m_constructor)
            out << enode_pp(d->m_constructor, ctx);
        else
            out << "(null)";
        out << "\n";
    }

    bool theory_datatype::include_func_interp(func_decl* f) {
        if (!m_util.is_accessor(f))
            return false;
        func_decl* con = m_util.get_accessor_constructor(f);
        for (enode* app : ctx.enodes_of(f)) {
            enode* arg = app->get_arg(0);
            enode* arg_con = oc_get_cstor(arg);
            if (arg_con && is_constructor(arg_con) && arg_con->get_decl() != con) 
                return true;
        }
        return false; 
    }

    void theory_datatype::init_model(model_generator & mg) {
        m_factory = alloc(datatype_factory, m, mg.get_model());
        mg.register_factory(m_factory);
    }

    class datatype_value_proc : public model_value_proc {
        func_decl *                     m_constructor;
        svector<model_value_dependency> m_dependencies;
    public:
        datatype_value_proc(func_decl * d):m_constructor(d) {}
        void add_dependency(enode * n) { m_dependencies.push_back(model_value_dependency(n)); }
        void get_dependencies(buffer<model_value_dependency> & result) override {
            result.append(m_dependencies.size(), m_dependencies.data());
        }
        app * mk_value(model_generator & mg, expr_ref_vector const & values) override {
            SASSERT(values.size() == m_dependencies.size());
            return mg.get_manager().mk_app(m_constructor, values.size(), values.data());
        }
    };

    model_value_proc * theory_datatype::mk_value(enode * n, model_generator & mg) {
        theory_var v = n->get_th_var(get_id());
        v            = m_find.find(v);
        SASSERT(v != null_theory_var);
        var_data * d = m_var_data[v];
        SASSERT(d->m_constructor);
        func_decl * c_decl = d->m_constructor->get_decl();
        datatype_value_proc * result = alloc(datatype_value_proc, c_decl);
        for (enode* arg : enode::args(d->m_constructor)) 
            result->add_dependency(arg);
        TRACE(datatype, 
              tout << pp(n, m) << "\n";
              tout << "depends on\n";
              for (enode* arg : enode::args(d->m_constructor)) 
                  tout << " " << pp(arg, m) << "\n";
              );
        return result;
    }

    void theory_datatype::merge_eh(theory_var v1, theory_var v2, theory_var, theory_var) {
        // v1 is the new root
        TRACE(datatype, tout << "merging v" << v1 << " v" << v2 << "\n";);
        SASSERT(v1 == static_cast<int>(m_find.find(v1)));
        var_data * d1 = m_var_data[v1];
        var_data * d2 = m_var_data[v2];
        if (d2->m_constructor != nullptr) {
            if (d1->m_constructor != nullptr && d1->m_constructor->get_decl() != d2->m_constructor->get_decl()) {
                enode_pair p(d1->m_constructor, d2->m_constructor);
                SASSERT(d1->m_constructor->get_root() == d2->m_constructor->get_root());
                ctx.set_conflict(ctx.mk_justification(ext_theory_conflict_justification(get_id(), ctx, 0, nullptr, 1, &p)));
            }
            if (d1->m_constructor == nullptr) {
                m_trail_stack.push(set_ptr_trail<enode>(d1->m_constructor)); 
                // check whether there is a recognizer in d1 that conflicts with d2->m_constructor;
                if (!d1->m_recognizers.empty()) {
                    unsigned c_idx = m_util.get_constructor_idx(d2->m_constructor->get_decl());
                    enode * recognizer = d1->m_recognizers[c_idx];
                    if (recognizer != nullptr && ctx.get_assignment(recognizer) == l_false) {
                        sign_recognizer_conflict(d2->m_constructor, recognizer);
                        return;
                    }
                }
                d1->m_constructor = d2->m_constructor;
            }
        }
        for (enode* e : d2->m_recognizers) 
            if (e)
                add_recognizer(v1, e);
    }

    void theory_datatype::unmerge_eh(theory_var v1, theory_var v2) {
        // do nothing
    }

    void theory_datatype::add_recognizer(theory_var v, enode * recognizer) {
        SASSERT(is_recognizer(recognizer));
        v = m_find.find(v);
        var_data * d = m_var_data[v];
        sort * s     = recognizer->get_decl()->get_domain(0);
        if (d->m_recognizers.empty()) {
            SASSERT(m_util.is_datatype(s));
            d->m_recognizers.resize(m_util.get_datatype_num_constructors(s), nullptr);
        }
        SASSERT(d->m_recognizers.size() == m_util.get_datatype_num_constructors(s));
        unsigned c_idx = m_util.get_recognizer_constructor_idx(recognizer->get_decl());
        if (d->m_recognizers[c_idx] == nullptr) {
            lbool val = ctx.get_assignment(recognizer);
            TRACE(datatype, tout << "adding recognizer to v" << v << " rec: #" << recognizer->get_owner_id() << " val: " << val << "\n";);
            if (val == l_true) {
                // do nothing... 
                // If recognizer assignment was already processed, then
                // d->m_constructor is already set.
                // Otherwise, it will be set when assign_eh is invoked.
                return; 
            }
            if (val == l_false && d->m_constructor != nullptr) {
                func_decl * c_decl = m_util.get_recognizer_constructor(recognizer->get_decl());
                if (d->m_constructor->get_decl() == c_decl) {
                    // conflict
                    sign_recognizer_conflict(d->m_constructor, recognizer);
                }
                return;
            }
            SASSERT(val == l_undef || (val == l_false && d->m_constructor == nullptr));
            d->m_recognizers[c_idx] = recognizer;
            m_trail_stack.push(set_vector_idx_trail(d->m_recognizers, c_idx));
            if (val == l_false) {
                propagate_recognizer(v, recognizer);
            }
        }
    }

    /**
       \brief Propagate a recognizer assigned to false.
    */
    void theory_datatype::propagate_recognizer(theory_var v, enode * recognizer) {
        SASSERT(is_recognizer(recognizer));
        SASSERT(static_cast<int>(m_find.find(v)) == v);
        SASSERT(ctx.get_assignment(recognizer) == l_false);
        unsigned num_unassigned  = 0;
        unsigned unassigned_idx  = UINT_MAX;
        enode * n       = get_enode(v);
        sort * dt       = n->get_sort();
        var_data * d    = m_var_data[v];
        if (d->m_recognizers.empty()) {
            theory_var w = recognizer->get_arg(0)->get_th_var(get_id());
            SASSERT(w != null_theory_var);
            add_recognizer(w, recognizer);
        }
        CTRACE(datatype, d->m_recognizers.empty(), ctx.display(tout););
        SASSERT(!d->m_recognizers.empty());
        literal_vector lits;
        enode_pair_vector eqs;
        ptr_vector<enode>::const_iterator it  = d->m_recognizers.begin();
        ptr_vector<enode>::const_iterator end = d->m_recognizers.end();
        for (unsigned idx = 0; it != end; ++it, ++idx) {
            enode * r = *it;
            if (r && ctx.get_assignment(r) == l_true)
                return; // nothing to be propagated
            if (r && ctx.get_assignment(r) == l_false) {
                SASSERT(r->get_num_args() == 1);
                lits.push_back(literal(ctx.enode2bool_var(r), true));
                if (n != r->get_arg(0)) {
                    // Argument of the current recognizer is not necessarily equal to n.
                    // This can happen when n and r->get_arg(0) are in the same equivalence class.
                    // We must add equality as an assumption to the conflict or propagation
                    SASSERT(n->get_root() == r->get_arg(0)->get_root());
                    eqs.push_back(enode_pair(n, r->get_arg(0)));
                }
                continue;
            }
            if (num_unassigned == 0)
                unassigned_idx = idx;
            num_unassigned++;
        }
        TRACE(datatype, tout << "propagate " << num_unassigned << " eqs: " << eqs.size() << "\n";);
        if (num_unassigned == 0) {
            // conflict
            SASSERT(!lits.empty());
            TRACE(datatype_conflict, tout << mk_ismt2_pp(recognizer->get_expr(), m) << "\n";
                  for (literal l : lits) 
                      ctx.display_detailed_literal(tout, l) << "\n";
                  for (auto const& p : eqs) 
                      tout << enode_eq_pp(p, ctx);
                  );
            ctx.set_conflict(ctx.mk_justification(ext_theory_conflict_justification(get_id(), ctx, lits.size(), lits.data(), eqs.size(), eqs.data())));
        }
        else if (num_unassigned == 1) {
            // propagate remaining recognizer
            SASSERT(!lits.empty());
            enode * r = d->m_recognizers[unassigned_idx];
            literal consequent;
            if (!r) {
                ptr_vector<func_decl> const & constructors = *m_util.get_datatype_constructors(dt);
                func_decl * rec = m_util.get_constructor_is(constructors[unassigned_idx]);
                app_ref rec_app(m.mk_app(rec, n->get_expr()), m);
                ctx.internalize(rec_app, false);
                consequent = literal(ctx.get_bool_var(rec_app));
            }
            else {
                consequent = literal(ctx.enode2bool_var(r));
            }
            ctx.mark_as_relevant(consequent);
            ctx.assign(consequent, 
                       ctx.mk_justification(ext_theory_propagation_justification(get_id(), ctx, lits.size(), lits.data(), 
                                                                                 eqs.size(), eqs.data(), consequent)));
        }
        else {
            // there are more than 2 unassigned recognizers...
            // if eager splits are enabled... create new case split
            if (params().m_dt_lazy_splits == 0 || (!dt->is_infinite() && params().m_dt_lazy_splits == 1))
                mk_split(v);
        }
    }

    /**
       \brief Create a new case split for v. That is, create the atom (is_mk v) and mark it as relevant.
       If first is true, it means that v does not have recognizer yet.
    */
    void theory_datatype::mk_split(theory_var v) {
        v                     = m_find.find(v);
        enode * n             = get_enode(v);
        sort * s              = n->get_sort();
        func_decl * non_rec_c = m_util.get_non_rec_constructor(s); 
        unsigned non_rec_idx  = m_util.get_constructor_idx(non_rec_c);
        var_data * d          = m_var_data[v];
        SASSERT(d->m_constructor == nullptr);
        func_decl * r         = nullptr;
        m_stats.m_splits++;

        TRACE(datatype_bug, tout << "non_rec_c: " << non_rec_c->get_name() << " #rec: " << d->m_recognizers.size() << "\n";);

        if (d->m_recognizers.empty()) {
            r = m_util.get_constructor_is(non_rec_c);
        }
        else {
            enode * recognizer    = d->m_recognizers[non_rec_idx];
            if (recognizer == nullptr) {
                r = m_util.get_constructor_is(non_rec_c);
            }
            else if (!ctx.is_relevant(recognizer)) {
                ctx.mark_as_relevant(recognizer);
                return;
            }
            else if (ctx.get_assignment(recognizer) != l_false) {
                // if is l_true, then we are done
                // otherwise wait recognizer to be assigned.
                return;
            }
            else {
                // look for a slot of d->m_recognizers that is 0, or it is not marked as relevant and is unassigned.
                ptr_vector<enode>::const_iterator it  = d->m_recognizers.begin();
                ptr_vector<enode>::const_iterator end = d->m_recognizers.end();
                for (unsigned idx = 0; it != end; ++it, ++idx) {
                    enode * curr = *it;
                    if (curr == nullptr) {
                        ptr_vector<func_decl> const & constructors = *m_util.get_datatype_constructors(s);
                        // found empty slot...
                        r = m_util.get_constructor_is(constructors[idx]);
                        break;
                    }
                    else if (!ctx.is_relevant(curr)) { 
                        ctx.mark_as_relevant(curr);
                        return;
                    }
                    else if (ctx.get_assignment(curr) != l_false) 
                        return;
                }
                if (r == nullptr)
                    return; // all recognizers are asserted to false... conflict will be detected...
            }
        }
        SASSERT(r != nullptr);
        app_ref r_app(m.mk_app(r, n->get_expr()), m);
        TRACE(datatype, tout << "creating split: " << mk_pp(r_app, m) << "\n";);
        ctx.internalize(r_app, false);
        bool_var bv     = ctx.get_bool_var(r_app);
        ctx.set_true_first_flag(bv);
        ctx.mark_as_relevant(bv);
    }


};
