/*++
Copyright (c) 2011 Microsoft Corporation

Module Name:

    elim_uncnstr_vars.cpp

Abstract:

    Eliminated unconstrained variables.

Author:

    Leonardo (leonardo) 2011-10-22

Notes:

--*/
#include "tactic/tactical.h"
#include "tactic/generic_model_converter.h"
#include "ast/rewriter/rewriter_def.h"
#include "ast/arith_decl_plugin.h"
#include "ast/bv_decl_plugin.h"
#include "ast/recfun_decl_plugin.h"
#include "ast/array_decl_plugin.h"
#include "ast/datatype_decl_plugin.h"
#include "tactic/core/collect_occs.h"
#include "ast/ast_smt2_pp.h"
#include "ast/ast_ll_pp.h"

namespace {
class elim_uncnstr_tactic : public tactic {

    // unconstrained vars collector

    typedef generic_model_converter mc;

    struct rw_cfg : public default_rewriter_cfg {
        bool                   m_produce_proofs;
        obj_hashtable<expr> &  m_vars;
        obj_hashtable<expr>&   m_nonvars;
        ref<mc>                m_mc;
        arith_util             m_a_util;
        bv_util                m_bv_util;
        array_util             m_ar_util;
        datatype_util          m_dt_util;
        app_ref_vector         m_fresh_vars;
        obj_map<app, app*>     m_cache;
        app_ref_vector         m_cache_domain;
        unsigned long long     m_max_memory;
        unsigned               m_max_steps;
        
        rw_cfg(ast_manager & m, bool produce_proofs, obj_hashtable<expr> & vars, obj_hashtable<expr> & nonvars, mc * _m, 
               unsigned long long max_memory, unsigned max_steps):
            m_produce_proofs(produce_proofs),
            m_vars(vars),
            m_nonvars(nonvars),
            m_mc(_m),
            m_a_util(m),
            m_bv_util(m),
            m_ar_util(m),
            m_dt_util(m),
            m_fresh_vars(m),
            m_cache_domain(m),
            m_max_memory(max_memory),
            m_max_steps(max_steps) {
        }
        
        ast_manager & m() const { return m_a_util.get_manager(); }
        
        bool max_steps_exceeded(unsigned num_steps) const { 
            if (memory::get_allocation_size() > m_max_memory)
                throw tactic_exception(TACTIC_MAX_MEMORY_MSG);
            return num_steps > m_max_steps;
        }
        
        bool uncnstr(expr * arg) const {
            return m_vars.contains(arg) && !m_nonvars.contains(arg);
        }
        
        bool uncnstr(unsigned num, expr * const * args) const {
            for (unsigned i = 0; i < num; i++)
                if (!uncnstr(args[i]))
                    return false;
            return true;
        }
        
        /**
             \brief Create a fresh variable for abstracting (f args[0] ... args[num-1])
            Return true if it a new variable was created, and false if the variable already existed for this
            application. Store the variable in v
        */
        bool mk_fresh_uncnstr_var_for(app * t, app * & v) {
            if (m_cache.find(t, v)) {
                return false; // variable already existed for this application
            }
            
            v = m().mk_fresh_const(nullptr, t->get_sort());
            TRACE("elim_uncnstr_bug", tout << "eliminating:\n" << mk_ismt2_pp(t, m()) << "\n";);
            TRACE("elim_uncnstr_bug_ll", tout << "eliminating:\n" << mk_bounded_pp(t, m()) << "\n";);
            m_fresh_vars.push_back(v);
            if (m_mc) m_mc->hide(v);
            m_cache_domain.push_back(t);
            m_cache.insert(t, v);
            return true;
        }

        bool mk_fresh_uncnstr_var_for(func_decl * f, unsigned num, expr * const * args, app * & v) {
            return mk_fresh_uncnstr_var_for(m().mk_app(f, num, args), v);
        }
        
        bool mk_fresh_uncnstr_var_for(func_decl * f, expr * arg1, expr * arg2, app * & v) {
            return mk_fresh_uncnstr_var_for(m().mk_app(f, arg1, arg2), v);
        }
        
        bool mk_fresh_uncnstr_var_for(func_decl * f, expr * arg, app * & v) {
            return mk_fresh_uncnstr_var_for(m().mk_app(f, arg), v);
        }
        
        void add_def(expr * v, expr * def) {
            SASSERT(uncnstr(v));
            SASSERT(to_app(v)->get_num_args() == 0);
            if (m_mc)
                m_mc->add(to_app(v)->get_decl(), def);
        }
        
        void add_defs(unsigned num, expr * const * args, expr * u, expr * identity) {
            if (m_mc) {
                add_def(args[0], u);
                for (unsigned i = 1; i < num; i++)
                    add_def(args[i], identity);
            }
        }
        
        // return a term that is different from t.
        bool mk_diff(expr * t, expr_ref & r) {
            sort * s = t->get_sort();
            if (m().is_bool(s)) {
                r = m().mk_not(t);
                return true;
            }
            family_id fid = s->get_family_id();
            if (fid == m_a_util.get_family_id()) {
                r = m_a_util.mk_add(t, m_a_util.mk_numeral(rational(1), s));
                return true;
            }
            if (fid == m_bv_util.get_family_id()) {
                r = m().mk_app(m_bv_util.get_family_id(), OP_BNOT, t);
                return true;
            }
            if (fid == m_ar_util.get_family_id()) {
                if (m().is_uninterp(get_array_range(s)))
                    return false;
                unsigned arity = get_array_arity(s);
                for (unsigned i = 0; i < arity; i++)
                    if (m().is_uninterp(get_array_domain(s, i)))
                        return false;
                // building 
                // r = (store t i1 ... in d)
                // where i1 ... in are arbitrary values
                // and d is a term different from (select t i1 ... in)
                ptr_buffer<expr> new_args;
                new_args.push_back(t);
                for (unsigned i = 0; i < arity; i++)
                    new_args.push_back(m().get_some_value(get_array_domain(s, i)));
                expr_ref sel(m());
                sel = m().mk_app(fid, OP_SELECT, new_args.size(), new_args.data());
                expr_ref diff_sel(m());
                if (!mk_diff(sel, diff_sel))
                    return false;
                new_args.push_back(diff_sel);
                r = m().mk_app(fid, OP_STORE, new_args.size(), new_args.data());
                return true;
            }
            if (fid == m_dt_util.get_family_id()) {
                // In the current implementation, I only handle the case where
                // the datatype has a recursive constructor.
                ptr_vector<func_decl> const & constructors = *m_dt_util.get_datatype_constructors(s);
                for (func_decl * constructor : constructors) {
                    unsigned num    = constructor->get_arity();
                    unsigned target = UINT_MAX;
                    for (unsigned i = 0; i < num; i++) {
                        sort * s_arg = constructor->get_domain(i);
                        if (s == s_arg) {
                            target = i;
                            continue;
                        }
                        if (m().is_uninterp(s_arg))
                            break;
                    }
                    if (target == UINT_MAX)
                        continue;
                    // use the constructor the distinct term constructor(...,t,...)
                    ptr_buffer<expr> new_args;
                    for (unsigned i = 0; i < num; i++) {
                        if (i == target) {
                            new_args.push_back(t);
                        }
                        else {
                            new_args.push_back(m().get_some_value(constructor->get_domain(i)));
                        }
                    }
                    r = m().mk_app(constructor, new_args.size(), new_args.data());
                    return true;
                }
                // TODO: handle more cases.
                return false;
            }
            return false;
        }

        app * process_eq(func_decl * f, expr * arg1, expr * arg2) {
            expr * v;
            expr * t;
            if (uncnstr(arg1)) {
                v = arg1;
                t = arg2;
            }
            else if (uncnstr(arg2)) {
                v = arg2;
                t = arg1;
            }
            else {
                return nullptr;
            }
            
            sort * s = arg1->get_sort();
            
            // Remark:
            // I currently do not support unconstrained vars that have
            // uninterpreted sorts, for the following reasons:
            // - Soundness
            //     (forall ((x S) (y S)) (= x y))
            //     (not (= c1 c2))
            //
            //   The constants c1 and c2 have only one occurrence in
            //   the formula above, but they are not really unconstrained.
            //   The quantifier forces S to have interpretations of size 1.
            //   If we replace (= c1 c2) with fresh k. The formula will
            //   become satisfiable. 
            //
            // - Even if the formula is quantifier free, I would still
            //   have to build an interpretation for the eliminated 
            //   variables.
            //
            if (!m().is_fully_interp(s))
                return nullptr;
            
            // If the interpreted sort has only one element,
            // then it is unsound to eliminate the unconstrained variable in the equality
            sort_size sz = s->get_num_elements();
            
            if (sz.is_finite() && sz.size() <= 1)
                return nullptr;
            
            if (!m_mc) {
                // easy case, model generation is disabled.
                app * u;
                mk_fresh_uncnstr_var_for(f, arg1, arg2, u);
                return u;
            }
            
            expr_ref d(m());
            if (mk_diff(t, d)) {
                app * u;
                if (!mk_fresh_uncnstr_var_for(f, arg1, arg2, u))
                    return u;
                add_def(v, m().mk_ite(u, t, d));
                return u;
            }
            return nullptr;
        }
        
        app * process_basic_app(func_decl * f, unsigned num, expr * const * args) {
            SASSERT(f->get_family_id() == m().get_basic_family_id());
            switch (f->get_decl_kind()) {
            case OP_ITE:
                SASSERT(num == 3);
                if (uncnstr(args[1]) && uncnstr(args[2])) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    add_def(args[1], r);
                    add_def(args[2], r);
                    return r;
                }
                if (uncnstr(args[0]) && uncnstr(args[1])) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    add_def(args[0], m().mk_true());
                    add_def(args[1], r);
                    return r;
                }
                if (uncnstr(args[0]) && uncnstr(args[2])) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    add_def(args[0], m().mk_false());
                    add_def(args[2], r);
                    return r;
                }
                return nullptr;
            case OP_NOT:
                SASSERT(num == 1);
                if (uncnstr(args[0])) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    if (m_mc)
                        add_def(args[0], m().mk_not(r));
                    return r;
                }
                return nullptr;
            case OP_AND:
                if (num > 0 && uncnstr(num, args)) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    if (m_mc)
                        add_defs(num, args, r, m().mk_true());
                    return r;
                }
                return nullptr;
            case OP_OR:
                if (num > 0 && uncnstr(num, args)) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    if (m_mc)
                        add_defs(num, args, r, m().mk_false());
                    return r;
                }
                return nullptr;
            case OP_EQ:
                SASSERT(num == 2);
                return process_eq(f, args[0], args[1]);
            default:
                return nullptr;
            }
        }

        app * process_le_ge(func_decl * f, expr * arg1, expr * arg2, bool le) {
            expr * v;
            expr * t;
            if (uncnstr(arg1)) {
                v = arg1;
                t = arg2;
            }
            else if (uncnstr(arg2)) {
                v = arg2;
                t = arg1;
                le = !le;
            }
            else {
                return nullptr;
            }
            app * u;
            if (!mk_fresh_uncnstr_var_for(f, arg1, arg2, u))
                return u;
            if (!m_mc)
                return u;
            // v = ite(u, t, t + 1) if le
            // v = ite(u, t, t - 1) if !le
            add_def(v, m().mk_ite(u, t, m_a_util.mk_add(t, m_a_util.mk_numeral(rational(le ? 1 : -1), arg1->get_sort()))));
            return u;
        }

        app * process_add(family_id fid, decl_kind add_k, decl_kind sub_k, unsigned num, expr * const * args) {
            if (num == 0)
                return nullptr;
            unsigned i;
            expr * v = nullptr;
            for (i = 0; i < num; i++) {
                expr * arg = args[i];
                if (uncnstr(arg)) {
                    v = arg;
                    break;
                }
            }
            if (v == nullptr)
                return nullptr;
            app  * u;
            if (!mk_fresh_uncnstr_var_for(m().mk_app(fid, add_k, num, args), u))
                return u;
            if (!m_mc) 
                return u;
            ptr_buffer<expr> new_args;
            for (unsigned j = 0; j < num; j++) {
                if (j == i)
                    continue;
                new_args.push_back(args[j]);
            }
            if (new_args.empty()) {
                add_def(v, u);
            }
            else {
                expr * rest;
                if (new_args.size() == 1)
                    rest = new_args[0];
                else
                    rest = m().mk_app(fid, add_k, new_args.size(), new_args.data());
                add_def(v, m().mk_app(fid, sub_k, u, rest));
            }
            return u;
        }
        
        app * process_arith_mul(func_decl * f, unsigned num, expr * const * args) {
            if (num == 0)
                return nullptr;
            sort * s = args[0]->get_sort();
            if (uncnstr(num, args)) {
                app * r;
                if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                    return r;
                if (m_mc)
                    add_defs(num, args, r, m_a_util.mk_numeral(rational(1), s));
                return r;
            }
            // c * v case for reals
            bool is_int;
            rational val;
            if (num == 2 && uncnstr(args[1]) && m_a_util.is_numeral(args[0], val, is_int) && !is_int) {
                if (val.is_zero())
                    return nullptr;
                app * r;
                if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                    return r;
                if (m_mc) {
                    val = rational(1) / val;
                    add_def(args[1], m_a_util.mk_mul(m_a_util.mk_numeral(val, false), r));
                }
                return r;
            }
            return nullptr;
        }
        
        app * process_arith_app(func_decl * f, unsigned num, expr * const * args) {
            
            SASSERT(f->get_family_id() == m_a_util.get_family_id());
            switch (f->get_decl_kind()) {
            case OP_ADD:
                return process_add(f->get_family_id(), OP_ADD, OP_SUB, num, args);
            case OP_MUL:
                return process_arith_mul(f, num, args);
            case OP_LE:
                SASSERT(num == 2);
                return process_le_ge(f, args[0], args[1], true);
            case OP_GE:
                SASSERT(num == 2);
                return process_le_ge(f, args[0], args[1], false);
            default:
                return nullptr;
            }
        }
        
        app * process_bv_mul(func_decl * f, unsigned num, expr * const * args) {
            if (num == 0)
                return nullptr;
            if (uncnstr(num, args)) {
                sort * s = args[0]->get_sort();
                app * r;
                if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                    return r;
                if (m_mc)
                    add_defs(num, args, r, m_bv_util.mk_numeral(rational(1), s));
                return r;
            }
            // c * v (c is even) case
            unsigned bv_size;
            rational val;
            rational inv;
            if (num == 2 && 
                uncnstr(args[1]) && 
                m_bv_util.is_numeral(args[0], val, bv_size) &&
                val.mult_inverse(bv_size, inv)) {
                app * r;
                if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                    return r;
                sort * s = args[1]->get_sort();
                if (m_mc)
                    add_def(args[1], m_bv_util.mk_bv_mul(m_bv_util.mk_numeral(inv, s), r));
                return r;
            }
            return nullptr;
        }
        
        app * process_extract(func_decl * f, expr * arg) {
            if (!uncnstr(arg))
                return nullptr;
            app * r;
            if (!mk_fresh_uncnstr_var_for(f, arg, r))
                return r;
            if (!m_mc)
                return r;
            unsigned high    = m_bv_util.get_extract_high(f);
            unsigned low     = m_bv_util.get_extract_low(f);
            unsigned bv_size = m_bv_util.get_bv_size(arg->get_sort());
            if (bv_size == high - low + 1) {
                add_def(arg, r);
            }
            else {
                ptr_buffer<expr> args;
                if (high < bv_size - 1)
                    args.push_back(m_bv_util.mk_numeral(rational(0), bv_size - high - 1));
                args.push_back(r);
                if (low > 0)
                    args.push_back(m_bv_util.mk_numeral(rational(0), low));
                add_def(arg, m_bv_util.mk_concat(args.size(), args.data()));
            }
            return r;
        }
        
        app * process_bv_div(func_decl * f, expr * arg1, expr * arg2) {
            if (uncnstr(arg1) && uncnstr(arg2)) {
                sort * s = arg1->get_sort();
                app * r;
                if (!mk_fresh_uncnstr_var_for(f, arg1, arg2, r))
                    return r;
                if (!m_mc)
                    return r;
                add_def(arg1, r);
                add_def(arg2, m_bv_util.mk_numeral(rational(1), s));
                return r;
            }
            return nullptr;
        }
        
        app * process_concat(func_decl * f, unsigned num, expr * const * args) {
            if (num == 0)
                return nullptr;
            if (!uncnstr(num, args))
                return nullptr;
            app * r;
            if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                return r;
            if (m_mc) {
                unsigned i = num;
                unsigned low = 0;
                while (i > 0) {
                    --i;
                    expr * arg  = args[i];
                    unsigned sz = m_bv_util.get_bv_size(arg);
                    add_def(arg, m_bv_util.mk_extract(low + sz - 1, low, r));
                    low += sz;
                }
            }
            return r;
        }
        
        app * process_bv_le(func_decl * f, expr * arg1, expr * arg2, bool is_signed) {
            if (m_produce_proofs) {
                // The result of bv_le is not just introducing a new fresh name,
                // we need a side condition.
                // TODO: the correct proof step
                return nullptr;
            }
            if (uncnstr(arg1)) {
                // v <= t
                expr * v = arg1;
                expr * t = arg2;
                // v <= t --->  (u or t == MAX)   u is fresh
                //     add definition v = ite(u or t == MAX, t, t+1)
                unsigned bv_sz = m_bv_util.get_bv_size(arg1);
                rational MAX;
                if (is_signed)
                    MAX = rational::power_of_two(bv_sz - 1) - rational(1);
                else
                    MAX = rational::power_of_two(bv_sz) - rational(1);
                app * u;
                bool is_new = mk_fresh_uncnstr_var_for(f, arg1, arg2, u);
                app * r = m().mk_or(u, m().mk_eq(t, m_bv_util.mk_numeral(MAX, bv_sz)));
                if (m_mc && is_new)
                    add_def(v, m().mk_ite(r, t, m_bv_util.mk_bv_add(t, m_bv_util.mk_numeral(rational(1), bv_sz))));
                return r;
            }
            if (uncnstr(arg2)) {
                // v >= t
                expr * v = arg2;
                expr * t = arg1;
                // v >= t --->  (u ot t == MIN)  u is fresh
                //    add definition v = ite(u or t == MIN, t, t-1)
                unsigned bv_sz = m_bv_util.get_bv_size(arg1);
                rational MIN;
                if (is_signed)
                    MIN = -rational::power_of_two(bv_sz - 1);
                else
                    MIN = rational(0);
                app * u;
                bool is_new = mk_fresh_uncnstr_var_for(f, arg1, arg2, u);
                app * r = m().mk_or(u, m().mk_eq(t, m_bv_util.mk_numeral(MIN, bv_sz)));
                if (m_mc && is_new)
                    add_def(v, m().mk_ite(r, t, m_bv_util.mk_bv_sub(t, m_bv_util.mk_numeral(rational(1), bv_sz))));
                return r;
            }
            return nullptr;
        }
        
        app * process_bv_app(func_decl * f, unsigned num, expr * const * args) {
            SASSERT(f->get_family_id() == m_bv_util.get_family_id());
            switch (f->get_decl_kind()) {
            case OP_BADD:
                return process_add(f->get_family_id(), OP_BADD, OP_BSUB, num, args);
            case OP_BMUL:
                return process_bv_mul(f, num, args);
            case OP_BSDIV:
            case OP_BUDIV:
            case OP_BSDIV_I:
            case OP_BUDIV_I:
                SASSERT(num == 2);
                return process_bv_div(f, args[0], args[1]);
            case OP_SLEQ:
                SASSERT(num == 2);
                return process_bv_le(f, args[0], args[1], true);
            case OP_ULEQ:
                SASSERT(num == 2);
                return process_bv_le(f, args[0], args[1], false);
            case OP_CONCAT:
                return process_concat(f, num, args);
            case OP_EXTRACT:
                SASSERT(num == 1);
                return process_extract(f, args[0]);
            case OP_BNOT:
                SASSERT(num == 1);
                if (uncnstr(args[0])) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    if (m_mc)
                        add_def(args[0], m().mk_app(f, r));
                    return r;
                }
                return nullptr;
            case OP_BOR:
                if (num > 0 && uncnstr(num, args)) {
                    sort * s = args[0]->get_sort();
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    if (m_mc)
                        add_defs(num, args, r, m_bv_util.mk_numeral(rational(0), s));
                    return r;
                }
                return nullptr;
            default:
                return nullptr;
            }
        }
        
        app * process_array_app(func_decl * f, unsigned num, expr * const * args) {
            SASSERT(f->get_family_id() == m_ar_util.get_family_id());
            switch (f->get_decl_kind()) {
            case OP_SELECT:
                if (uncnstr(args[0])) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    sort * s = args[0]->get_sort();
                    if (m_mc)
                        add_def(args[0], m_ar_util.mk_const_array(s, r));
                    return r;
                }
                return nullptr;
            case OP_STORE:
                if (uncnstr(args[0]) && uncnstr(args[num-1])) {
                    app * r;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, r))
                        return r;
                    if (m_mc) {
                        add_def(args[num-1], m().mk_app(m_ar_util.get_family_id(), OP_SELECT, num-1, args));
                        add_def(args[0], r);
                    }
                    return r;
                }
            default:
                return nullptr;
            }
        }
        
        app * process_datatype_app(func_decl * f, unsigned num, expr * const * args) {
            if (m_dt_util.is_accessor(f)) {
                SASSERT(num == 1);
                if (uncnstr(args[0])) {
                    if (!m_mc) {
                        app * r;
                        mk_fresh_uncnstr_var_for(f, num, args, r);
                        return r;
                    }
                    func_decl * c = m_dt_util.get_accessor_constructor(f);
                    for (unsigned i = 0; i < c->get_arity(); i++) 
                        if (!m().is_fully_interp(c->get_domain(i)))
                            return nullptr;
                    app * u;
                    if (!mk_fresh_uncnstr_var_for(f, num, args, u))
                        return u;
                    ptr_vector<func_decl> const & accs = *m_dt_util.get_constructor_accessors(c);
                    ptr_buffer<expr> new_args;
                    for (unsigned i = 0; i < accs.size(); i++) {
                        if (accs[i] == f) 
                            new_args.push_back(u);
                        else
                            new_args.push_back(m().get_some_value(c->get_domain(i)));
                    }
                    add_def(args[0], m().mk_app(c, new_args.size(), new_args.data()));
                    return u;
                }
            }
            return nullptr;
        }
        
        br_status reduce_app(func_decl * f, unsigned num, expr * const * args, expr_ref & result, proof_ref & result_pr) {
            if (num == 0)
                return BR_FAILED;
            family_id fid = f->get_family_id();
            if (fid == null_family_id)
                return BR_FAILED;
            
            for (unsigned i = 0; i < num; i++) {
                if (!is_ground(args[i]))
                    return BR_FAILED; // non-ground terms are not handled.
            }
            
            app * u = nullptr;
            
            if (fid == m().get_basic_family_id())
                u = process_basic_app(f, num, args);
            else if (fid == m_a_util.get_family_id())
                u = process_arith_app(f, num, args);
            else if (fid == m_bv_util.get_family_id())
                u = process_bv_app(f, num, args);
            else if (fid == m_ar_util.get_family_id())
                u = process_array_app(f, num, args);
            else if (fid == m_dt_util.get_family_id())
                u = process_datatype_app(f, num, args);
            
            if (u == nullptr)
                return BR_FAILED;
            
            result = u;
            if (m_produce_proofs) {
                expr * s    = m().mk_app(f, num, args);
                expr * eq   = m().mk_eq(s, u);
                proof * pr1 = m().mk_def_intro(eq);
                result_pr   = m().mk_apply_def(s, u, pr1);
            }
            
            return BR_DONE;
        }
    };
    
    class rw : public rewriter_tpl<rw_cfg> {
        rw_cfg m_cfg;
    public:
        rw(ast_manager & m, bool produce_proofs, obj_hashtable<expr> & vars, obj_hashtable<expr>& nonvars, mc * _m,
            unsigned long long max_memory, unsigned max_steps):
            rewriter_tpl<rw_cfg>(m, produce_proofs, m_cfg),
            m_cfg(m, produce_proofs, vars, nonvars, _m, max_memory, max_steps) {
        }
    };
    
    ast_manager &                    m_manager;
    ref<mc>                          m_mc;
    obj_hashtable<expr>              m_vars;
    obj_hashtable<expr>              m_nonvars;
    scoped_ptr<rw>                   m_rw;
    unsigned                         m_num_elim_apps = 0;
    unsigned long long               m_max_memory;
    unsigned                         m_max_steps;
    
    ast_manager & m() { return m_manager; }
    
    void init_mc(bool produce_models) {
        m_mc = nullptr;
        if (produce_models) {
            m_mc = alloc(mc, m(), "elim_uncstr");
        }
    }
    
    void init_rw(bool produce_proofs) {
        m_rw = alloc(rw, m(), produce_proofs, m_vars, m_nonvars, m_mc.get(), m_max_memory, m_max_steps);            
    }

    void run(goal_ref const & g, goal_ref_buffer & result) {
        bool produce_proofs = g->proofs_enabled();
        TRACE("goal", g->display(tout););
        tactic_report report("elim-uncnstr", *g);
        m_vars.reset();
        collect_occs p;
        p(*g, m_vars);
        if (m_vars.empty() || recfun::util(m()).has_defs()) {
            result.push_back(g.get());
            // did not increase depth since it didn't do anything.
            return;
        }
        bool modified = true;
        TRACE("elim_uncnstr", tout << "unconstrained variables...\n";
                for (expr * v : m_vars) tout << mk_ismt2_pp(v, m()) << " "; 
                tout << "\n";);
        init_mc(g->models_enabled());
        init_rw(produce_proofs);
        
        expr_ref   new_f(m());
        proof_ref  new_pr(m());
        unsigned round = 0;
        unsigned size  = g->size();
        unsigned idx   = 0;
        while (true) {
            for (; idx < size; idx++) {
                expr * f = g->form(idx);
                m_rw->operator()(f, new_f, new_pr);
                if (f == new_f)
                    continue;
                modified = true;
                if (produce_proofs) {
                    proof * pr = g->pr(idx);
                    new_pr     = m().mk_modus_ponens(pr, new_pr);
                }
                g->update(idx, new_f, new_pr, g->dep(idx));
            }
            if (!modified) {
                if (round == 0) {                        
                }
                else {
                    m_num_elim_apps = m_rw->cfg().m_fresh_vars.size();
                    g->add(m_mc.get());                        
                }
                TRACE("elim_uncnstr", if (m_mc) m_mc->display(tout); else tout << "no mc\n";);
                m_mc = nullptr;
                m_rw = nullptr;                    
                result.push_back(g.get());
                g->inc_depth();
                TRACE("goal", g->display(tout););
                return;
            }
            modified = false;
            round ++;
            size       = g->size();
            m_rw->reset(); // reset cache
            m_vars.reset(); 
            {
                collect_occs p;
                p(*g, m_vars);
            }
            if (m_vars.empty()) 
                idx = size; // force to finish 
            else
                idx = 0;
        }
    }
    
    params_ref m_params;
public:
   elim_uncnstr_tactic(ast_manager & m, params_ref const & p):
        m_manager(m), m_params(p) {
        updt_params(p);
    }

   char const* name() const override { return "elim_uncstr"; }

    tactic * translate(ast_manager & m) override {
        return alloc(elim_uncnstr_tactic, m, m_params);
    }

    void updt_params(params_ref const & p) override {
        m_params.append(p);
        m_max_memory = megabytes_to_bytes(m_params.get_uint("max_memory", UINT_MAX));
        m_max_steps  = m_params.get_uint("max_steps", UINT_MAX);
    }
    
    void collect_param_descrs(param_descrs & r) override {
        insert_max_memory(r);
        insert_max_steps(r);
    }

    void operator()(goal_ref const & g, 
                    goal_ref_buffer & result) override {
        run(g, result);
        report_tactic_progress(":num-elim-apps", m_num_elim_apps);
    }
    
    void cleanup() override {
        m_mc = nullptr;
        m_rw = nullptr;
        m_vars.reset();
    }

    void collect_statistics(statistics & st) const override {
        st.update("eliminated applications", m_num_elim_apps);
    }
    
    void reset_statistics() override {
        m_num_elim_apps = 0;
    }

    void user_propagate_register_expr(expr* e) override { 
        m_nonvars.insert(e);
    }

    void user_propagate_clear() override {
        m_nonvars.reset();
    }


};
}

tactic * mk_elim_uncnstr_tactic(ast_manager & m, params_ref const & p) {
    return clean(alloc(elim_uncnstr_tactic, m, p));
}
