Reorganizing the code

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

src/spc/README

@@ -0,0 +1,2 @@
+Superposition Calculus.
+This module is currently disabled.

src/spc/expr_offset.h

@@ -0,0 +1,53 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    expr_offset.h
+
+Abstract:
+
+    Expressions + Offsets. 
+    In order to avoid creating variants of terms, we use a pair (expression, offset),
+    where offset is just a small integer.
+    Non ground terms with different offsets are always considered
+    disequal. For example, (f x):1 is different from (f x):2, and 
+    (f x):1 is unifiable with x:2.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-28.
+
+Revision History:
+
+--*/
+#ifndef _EXPR_OFFSET_H_
+#define _EXPR_OFFSET_H_
+
+#include"ast.h"
+
+class expr_offset {
+    expr *    m_expr;
+    unsigned  m_offset;
+public:
+    expr_offset():m_expr(0), m_offset(0) {}
+    expr_offset(expr * e, unsigned o):m_expr(e), m_offset(o) {}
+    
+    expr * get_expr() const { return m_expr; }
+    unsigned get_offset() const { return m_offset; }
+    bool operator==(expr_offset const & other) const { return m_expr == other.m_expr && m_offset == other.m_offset; }
+    bool operator!=(expr_offset const & other) const { return !operator==(other); }
+
+    unsigned hash() const {
+        unsigned a = m_expr->get_id();
+        unsigned b = m_offset;
+        unsigned c = 17;
+        mix(a, b, c);
+        return c;
+    }
+};
+
+typedef std::pair<expr_offset, expr_offset> expr_offset_pair;
+typedef pair_hash<obj_hash<expr_offset>, obj_hash<expr_offset> > expr_offset_pair_hash;
+
+#endif /* _EXPR_OFFSET_H_ */

src/spc/expr_offset_map.h

@@ -0,0 +1,94 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    expr_offset_map.h
+
+Abstract:
+
+    A generic mapping from (expression, offset) to a value T.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-01.
+
+Revision History:
+
+--*/
+#ifndef _EXPR_OFFSET_MAP_H_
+#define _EXPR_OFFSET_MAP_H_
+
+#include"expr_offset.h"
+#include"vector.h"
+
+/**
+   \brief A mapping from expr_offset to some value of type T.
+*/
+template<typename T>
+class expr_offset_map {
+    struct data {
+        T        m_data;
+        unsigned m_timestamp;
+        data():m_timestamp(0) {}
+    };
+    vector<svector<data> > m_map;
+    unsigned               m_timestamp;
+public:
+    expr_offset_map():
+        m_timestamp(1) {}
+    
+    bool contains(expr_offset const & n) const {
+        unsigned off = n.get_offset();
+        if (off < m_map.size()) {
+            svector<data> const & v = m_map[off];
+            unsigned id             = n.get_expr()->get_id();
+            if (id < v.size()) 
+                return v[id].m_timestamp == m_timestamp;
+        }
+        return false;
+    }
+
+    bool find(expr_offset const & n, T & r) const {
+        unsigned off = n.get_offset();
+        if (off < m_map.size()) {
+            svector<data> const & v = m_map[off];
+            unsigned id             = n.get_expr()->get_id();
+            if (id < v.size() && v[id].m_timestamp == m_timestamp) {
+                r = v[id].m_data;
+                return true;
+            }
+        }
+        return false;
+    }
+
+    void insert(expr_offset const & n, T const & r) {
+        unsigned off = n.get_offset();
+        if (off >= m_map.size())
+            m_map.resize(off+1, svector<data>());
+        svector<data> & v = m_map[off];
+        unsigned id       = n.get_expr()->get_id();
+        if (id >= v.size())
+            v.resize(id+1);
+        v[id].m_data      = r;
+        v[id].m_timestamp = m_timestamp;
+    }
+
+    void reset() {
+        m_timestamp++;
+        if (m_timestamp == UINT_MAX) {
+            typename vector<svector<data> >::iterator it  = m_map.begin();
+            typename vector<svector<data> >::iterator end = m_map.end();
+            for (; it != end; ++it) {
+                svector<data> & v = *it;
+                typename svector<data>::iterator it2  = v.begin();
+                typename svector<data>::iterator end2 = v.end();
+                for (; it2 != end2; ++it2)
+                    it2->m_timestamp = 0;
+            }
+            m_timestamp = 1;
+        }
+    }
+};
+
+#endif /* _EXPR_OFFSET_MAP_H_ */

src/spc/expr_stat.cpp

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+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    expr_stat.cpp
+
+Abstract:
+
+    Expression statistics (symbol count, var count, depth, ...)
+    
+    All functions in these module assume expressions do not contain
+    nested quantifiers.
+    
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-05.
+
+Revision History:
+
+--*/
+#include"for_each_expr.h"
+#include"expr_stat.h"
+
+void get_expr_stat(expr * n, expr_stat & r) {
+    typedef std::pair<expr *, unsigned> pair;
+    buffer<pair> todo;
+    todo.push_back(pair(n, 0));
+    while (!todo.empty()) {
+        pair & p       = todo.back();
+        n              = p.first;
+        unsigned depth = p.second;
+        unsigned j;
+        todo.pop_back();
+        r.m_sym_count++;
+        if (depth > r.m_depth)
+            r.m_depth = depth;
+        switch (n->get_kind()) {
+        case AST_APP:
+            j = to_app(n)->get_num_args();
+            if (j == 0)
+                r.m_const_count++;
+            while (j > 0) {
+                --j;
+                todo.push_back(pair(to_app(n)->get_arg(j), depth + 1));
+            }
+            break;
+        case AST_VAR:
+            if (to_var(n)->get_idx() > r.m_max_var_idx) 
+                r.m_max_var_idx = to_var(n)->get_idx();
+            r.m_ground = false;
+            break;
+        case AST_QUANTIFIER:
+            todo.push_back(pair(to_quantifier(n)->get_expr(), depth+1));
+            break;
+        default:
+            UNREACHABLE();
+        }
+    }
+}
+
+unsigned get_symbol_count(expr * n) {
+    unsigned r = 0;
+    ptr_buffer<expr> todo;
+    todo.push_back(n);
+    while (!todo.empty()) {
+        n = todo.back();
+        unsigned j;
+        todo.pop_back();
+        r++;
+        switch (n->get_kind()) {
+        case AST_APP:
+            j = to_app(n)->get_num_args();
+            while (j > 0) {
+                --j;
+                todo.push_back(to_app(n)->get_arg(j));
+            }
+            break;
+        case AST_QUANTIFIER:
+            todo.push_back(to_quantifier(n)->get_expr());
+            break;
+        default:
+            break;
+        }
+    }
+    return r;
+}
+

src/spc/expr_stat.h

@@ -0,0 +1,50 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    expr_stat.h
+
+Abstract:
+
+    Expression statistics (symbol count, var count, depth, ...)
+    
+    All functions in these module assume expressions do not contain
+    nested quantifiers.
+    
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-05.
+
+Revision History:
+
+--*/
+#ifndef _EXPR_STAT_H_
+#define _EXPR_STAT_H_
+
+class expr;
+
+struct expr_stat {
+    unsigned m_sym_count; // symbol count
+    unsigned m_depth;     // depth
+    unsigned m_const_count; // constant count
+    unsigned m_max_var_idx;  
+    bool     m_ground;
+    expr_stat():m_sym_count(0), m_depth(0), m_const_count(0), m_max_var_idx(0), m_ground(true) {}
+};
+
+/**
+   \brief Collect statistics regarding the given expression. 
+
+   \warning This function traverses the dag as a tree.
+*/
+void get_expr_stat(expr * n, expr_stat & r);
+
+/**
+   \brief Return the number of symbols in \c n.
+
+   \warning This function traverses the dag as a tree.
+*/
+unsigned get_symbol_count(expr * n);
+
+#endif /* _EXPR_STAT_H_ */

src/spc/kbo.cpp

@@ -0,0 +1,276 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    kbo.cpp
+
+Abstract:
+
+    Knuth-Bendix ordering.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-28.
+
+Revision History:
+
+--*/
+#include"kbo.h"
+#include"ast_pp.h"
+
+inline unsigned kbo::f_weight(func_decl * f) const {
+    // TODO
+    return 1;
+}
+
+inline unsigned kbo::var_weight() const {
+    return m_var_weight;
+}
+
+inline void kbo::reset() {
+    m_weight_balance = 0;
+    m_deltas.reset();
+    m_num_pos = 0;
+    m_num_neg = 0;
+}
+
+/**
+   \brief Increase the balance of the given variable.
+*/
+inline void kbo::inc(expr_offset v) {
+    SASSERT(is_var(v.get_expr()));
+    int val;
+    unsigned v_idx  = to_var(v.get_expr())->get_idx();
+    unsigned offset = v.get_offset();
+    if (m_deltas.find(v_idx, offset, val)) {
+        if (val == -1) 
+            m_num_neg--;
+        else if (val == 0)
+            m_num_pos++;
+        m_deltas.insert(v_idx, offset, val + 1);
+    }
+    else {
+        m_deltas.insert(v_idx, offset, 1);
+        m_num_pos ++;
+    }
+}
+
+/**
+   \brief Decreate the balance of the given variable.
+*/
+inline void kbo::dec(expr_offset v) {
+    int val;
+    unsigned v_idx  = to_var(v.get_expr())->get_idx();
+    unsigned offset = v.get_offset();
+    if (m_deltas.find(v_idx, offset, val)) {
+        if (val == 0) 
+            m_num_neg++;
+        else if (val == 1)
+            m_num_pos--;
+        m_deltas.insert(v_idx, offset, val - 1);
+    }
+    else {
+        m_deltas.insert(v_idx, offset, -1);
+        m_num_neg ++;
+    }
+}
+
+/**
+   \brief Accumulate the variables and weight balance of t. Return
+   true if t contains target_var.
+*/
+template<bool pos>
+bool kbo::VWBc(expr_offset t, expr_offset target_var) {
+    SASSERT(target_var.get_expr() == 0 || is_var(target_var.get_expr()));
+    svector<expr_offset> & todo = m_vwbc_todo;
+    expr_offset s;
+    bool found = false;
+    unsigned j;
+    SASSERT(todo.empty());
+    todo.push_back(t);
+    while (!todo.empty()) {
+        t        = todo.back();
+        if (t == target_var)
+            found = true;
+        expr * n        = t.get_expr();
+        unsigned offset = t.get_offset();
+        todo.pop_back();
+        switch (n->get_kind()) {
+        case AST_VAR:
+            if (m_subst && m_subst->find(to_var(n), offset, s))
+                todo.push_back(s);
+            else if (pos) {
+                inc(t);
+                m_weight_balance += var_weight();
+            }
+            else {
+                dec(t);
+                m_weight_balance -= var_weight();
+            }
+            break;
+        case AST_APP:
+            if (pos)
+                m_weight_balance += f_weight(to_app(n)->get_decl());
+            else
+                m_weight_balance -= f_weight(to_app(n)->get_decl());
+            j = to_app(n)->get_num_args();
+            while (j > 0) {
+                --j;
+                todo.push_back(expr_offset(to_app(n)->get_arg(j), offset));
+            }
+            break;
+        default:
+            UNREACHABLE();
+            break;
+        }
+    }
+    return found;
+}
+
+template<bool pos>
+inline void kbo::VWB(expr_offset t, unsigned idx) {
+    expr_offset null(0, 0);
+    app * n = to_app(t.get_expr());
+    unsigned num = n->get_num_args();
+    for (; idx < num; idx++)
+        VWBc<pos>(expr_offset(n->get_arg(idx), t.get_offset()), null);
+}
+
+inline bool is_unary_app(expr * n) {
+    return is_app(n) && to_app(n)->get_num_args() == 1;
+}
+
+inline kbo::result kbo::no_neg() const {
+    return m_num_neg == 0 ? GREATER : UNCOMPARABLE;
+}
+
+inline kbo::result kbo::no_pos() const {
+    return m_num_pos == 0 ? LESSER : UNCOMPARABLE;
+}
+
+order::result kbo::compare(expr_offset const & t1, expr_offset const & t2, substitution * s) {
+    reset();
+    m_subst = s;
+
+    if (t1 == t2) 
+        return EQUAL;
+
+    expr * n1 = t1.get_expr();
+    expr * n2 = t2.get_expr();
+
+    // f(s) >_{kbo} f(t) iff s >_{kbo} t
+    while (is_unary_app(n1) && is_unary_app(n2) && to_app(n1)->get_decl() == to_app(n2)->get_decl()) {
+        n1 = to_app(n1)->get_arg(0);
+        n2 = to_app(n2)->get_arg(0);
+    }
+
+    svector<entry> & todo = m_compare_todo;
+    SASSERT(todo.empty());
+    todo.push_back(entry(find(expr_offset(n1, t1.get_offset())),
+                         find(expr_offset(n2, t2.get_offset())),
+                         0));
+
+    result res = UNKNOWN;
+
+    while (!todo.empty()) {
+        entry & e = todo.back();
+        expr_offset t1 = e.m_t1;
+        expr_offset t2 = e.m_t2;
+        expr * n1 = t1.get_expr();
+        expr * n2 = t2.get_expr();
+        TRACE("kbo", tout << "processing with idx: " << e.m_idx << "\n" << 
+              mk_pp(n1, m_manager) << "\n" << mk_pp(n2, m_manager) << "\n";
+              tout << "wb : " << m_weight_balance << "\n";);
+        SASSERT(!is_quantifier(n1) && !is_quantifier(n2));
+        bool v1 = is_var(n1);
+        bool v2 = is_var(n2);
+        if (v1 && v2) {
+            todo.pop_back();
+            inc(t1);
+            dec(t2);
+            res = t1 == t2 ? EQUAL : UNCOMPARABLE;
+        }
+        else if (v1) {
+            todo.pop_back();
+            res = VWBc<false>(t2, t1) ? LESSER : UNCOMPARABLE;
+            inc(t1);
+            m_weight_balance += var_weight();
+        }
+        else if (v2) {
+            todo.pop_back();
+            res = VWBc<true>(t1, t2) ? GREATER : UNCOMPARABLE;
+            dec(t2);
+            m_weight_balance -= var_weight();
+        }
+        else {
+            func_decl * f = to_app(n1)->get_decl();
+            func_decl * g = to_app(n2)->get_decl();
+            result lex;
+            if (f != g || to_app(n1)->get_num_args() != to_app(n2)->get_num_args()) {
+                VWB<true>(t1, 0);
+                VWB<false>(t2, 0);
+                lex = UNCOMPARABLE;
+            }
+            else {
+                unsigned & idx = e.m_idx;
+                // when idx > 0, res contains the result for child (idx - 1)
+                if (idx > 0 && res != EQUAL) {
+                    VWB<true>(t1, idx);
+                    VWB<false>(t2, idx);
+                    lex = res;
+                }
+                else if (idx == to_app(n1)->get_num_args()) {
+                    // all children were visited
+                    lex = EQUAL;
+                }
+                else if (idx < to_app(n1)->get_num_args()) {
+                    expr_offset c1 = find(expr_offset(to_app(n1)->get_arg(idx), t1.get_offset()));
+                    expr_offset c2 = find(expr_offset(to_app(n2)->get_arg(idx), t2.get_offset()));
+                    idx++; // move curr entry child idx
+                    entry new_entry(c1, c2, 0);
+                    todo.push_back(new_entry);
+                    continue; // process child before continuing
+                }
+            }
+            
+            todo.pop_back();
+            m_weight_balance += f_weight(f);
+            m_weight_balance -= f_weight(g);
+
+            if (m_weight_balance > 0)
+                res = no_neg();
+            else if (m_weight_balance < 0)
+                res = no_pos();
+            else if (f_greater(f, g))
+                res = no_neg();
+            else if (f_greater(g, f))
+                res = no_pos();
+            else if (f != g)
+                res = UNCOMPARABLE;
+            else if (lex == EQUAL)
+                res = EQUAL;
+            else if (lex == GREATER)
+                res = no_neg();
+            else if (lex == LESSER)
+                res = no_pos();
+            else
+                res = UNCOMPARABLE;
+        }
+        TRACE("kbo", tout << "result: " << res << "\n";);
+    }
+
+    return res;
+}
+
+bool kbo::greater(expr_offset const & t1, expr_offset const & t2, substitution * s) {
+    return compare(t1, t2, s) == GREATER;
+}
+
+int kbo::compare_ge(expr_offset const & t1, expr_offset const & t2, substitution * s) {
+    switch (compare(t1, t2, s)) {
+    case GREATER: return 1;
+    case EQUAL:   return 0;
+    default: return -1;
+    }
+}

src/spc/kbo.h

@@ -0,0 +1,70 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    kbo.h
+
+Abstract:
+
+    Knuth-Bendix ordering.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-28.
+
+Revision History:
+
+--*/
+#ifndef _KBO_H_
+#define _KBO_H_
+
+#include"order.h"
+
+class kbo : public order {
+    struct entry {
+        expr_offset m_t1;
+        expr_offset m_t2;
+        unsigned    m_idx;
+        entry():m_idx(UINT_MAX) {}
+        entry(expr_offset const & t1, expr_offset const & t2, unsigned idx):
+            m_t1(t1), m_t2(t2), m_idx(idx) {}
+    };
+
+    unsigned              m_var_weight;
+    int                   m_weight_balance;
+    var_offset_map<int>   m_deltas;
+    unsigned              m_num_pos;
+    unsigned              m_num_neg;
+    svector<expr_offset>  m_vwbc_todo;
+    svector<entry>        m_compare_todo;
+
+    unsigned f_weight(func_decl * f) const;
+    unsigned var_weight() const;
+    
+    void reset();
+    void inc(expr_offset v);
+    void dec(expr_offset v);
+
+    template<bool pos>
+    bool VWBc(expr_offset t, expr_offset target_var);
+
+    template<bool pos>
+    void VWB(expr_offset t, unsigned idx);
+    
+    result no_neg() const;
+    result no_pos() const;
+
+public:
+    kbo(ast_manager & m, precedence * p, unsigned var_weight = 1):order(m, p), m_var_weight(var_weight) {}
+    virtual ~kbo() {}
+    virtual void reserve(unsigned num_offsets, unsigned num_vars) { m_deltas.reserve(num_offsets, num_vars); }
+    virtual void reserve_offsets(unsigned num_offsets) { m_deltas.reserve_offsets(num_offsets); }
+    virtual void reserve_vars(unsigned num_vars) { m_deltas.reserve_vars(num_vars); }
+    virtual result compare(expr_offset const & t1, expr_offset const & t2, substitution * s);
+    result compare(expr * t1, expr * t2) { return compare(expr_offset(t1, 0), expr_offset(t2, 0), 0); }
+    virtual bool greater(expr_offset const & t1, expr_offset const & t2, substitution * s);
+    virtual int compare_ge(expr_offset const & t1, expr_offset const & t2, substitution * s);
+};
+
+#endif /* _KBO_H_ */

src/spc/lpo.cpp

@@ -0,0 +1,184 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    lpo.h
+
+Abstract:
+
+    Lexicographical Path Ordering
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-01.
+
+Revision History:
+
+--*/
+#include"lpo.h"
+
+/**
+   \brief Check whether the variable in t1 occurs in t2.
+*/
+bool lpo::occurs(expr_offset const & t1, expr_offset const & t2) {
+    SASSERT(is_var(t1.get_expr()));
+    if (is_ground(t2.get_expr()))
+        return false;
+    m_todo.reset();
+    m_todo.push_back(t2);
+    while (!m_todo.empty()) {
+        expr_offset t = m_todo.back();
+        m_todo.pop_back();
+        t = find(t);
+        expr * n        = t.get_expr();
+        if (is_ground(n))
+            continue;
+        unsigned offset = t.get_offset();
+        unsigned j;
+        switch (n->get_kind()) {
+        case AST_VAR:
+            if (t == t1)
+                return true;
+            break;
+        case AST_APP:
+            j = to_app(n)->get_num_args();
+            while (j > 0) {
+                --j;
+                expr * arg = to_app(n)->get_arg(j);
+                if (!is_ground(arg)) 
+                    m_todo.push_back(expr_offset(arg, offset));
+            }
+            break;
+        default:
+            UNREACHABLE();
+        }
+    }
+    return false;
+}
+
+inline bool lpo::greater(expr_offset s, expr_offset t, unsigned depth) {
+    return lpo::compare(s, t, depth) == GREATER;
+}
+
+/**
+   \brief Return true if s >_{lpo} t_i forall children t_i of t.
+*/
+bool lpo::dominates_args(expr_offset s, expr_offset t, unsigned depth) {
+    SASSERT(is_app(t.get_expr()));
+    unsigned num_args = to_app(t.get_expr())->get_num_args();
+    unsigned off      = t.get_offset();
+    for (unsigned i = 0; i < num_args; i++) {
+        expr * t_i = to_app(t.get_expr())->get_arg(i);
+        if (!greater(s, expr_offset(t_i, off), depth+1))
+            return false;
+    }
+    return true;
+}
+
+/**
+   \brief Return true if s_i >=_{lpo} t for some arg s_i of s.
+ */
+bool lpo::arg_dominates_expr(expr_offset s, expr_offset t, unsigned depth) {
+    SASSERT(is_app(s.get_expr()));
+    unsigned num_args = to_app(s.get_expr())->get_num_args();
+    unsigned off      = s.get_offset();
+    for (unsigned i = 0; i < num_args; i++) {
+        expr * s_i = to_app(s.get_expr())->get_arg(i);
+        result r   = compare(expr_offset(s_i, off), t, depth+1);
+        if (r == EQUAL || r == GREATER)
+            return true;
+    }
+    return false;
+}
+
+order::result lpo::lex_compare(expr_offset s, expr_offset t, unsigned depth) {
+    SASSERT(is_app(s.get_expr()));
+    SASSERT(is_app(t.get_expr()));
+    app * _s = to_app(s.get_expr());
+    app * _t = to_app(t.get_expr());
+    unsigned num_args1 = _s->get_num_args();
+    unsigned num_args2 = _t->get_num_args();
+    unsigned num_args  = std::min(num_args1, num_args2);
+    unsigned off1      = s.get_offset();
+    unsigned off2      = t.get_offset();
+    result r = EQUAL;
+    for (unsigned i = 0; i < num_args; i++) {
+        r = compare(expr_offset(_s->get_arg(i), off1), expr_offset(_t->get_arg(i), off2), depth+1);
+        if (r != EQUAL)
+            break;
+    }
+    if (r == EQUAL) {
+        if (num_args1 > num_args2)
+            return GREATER;
+        if (num_args1 < num_args2)
+            return NOT_GTEQ;
+    }
+    return r;
+}
+
+inline order::result lpo::compare_core(expr_offset s, expr_offset t, unsigned depth) {
+    s = find(s);
+    t = find(t);
+    
+    if (max_depth(depth))
+        return UNKNOWN;
+
+    if (is_var(s.get_expr()))
+        return s == t ? EQUAL : UNCOMPARABLE;
+    else if (is_var(t.get_expr()))
+        return occurs(t, s) ? GREATER : UNCOMPARABLE;
+    else {
+        func_decl * f = to_app(s.get_expr())->get_decl();
+        func_decl * g = to_app(t.get_expr())->get_decl();
+        if (f_greater(f, g))
+            return dominates_args(s, t, depth) ? GREATER : NOT_GTEQ;
+        else if (f != g)
+            return arg_dominates_expr(s, t, depth) ? GREATER : NOT_GTEQ;
+        else {
+            result r = lex_compare(s, t, depth);
+            if (r == GREATER) {
+                if (dominates_args(s, t, depth))
+                    return GREATER;
+            }
+            else if (r == EQUAL)
+                return EQUAL;
+            return to_app(s.get_expr())->get_num_args() > 1 && arg_dominates_expr(s, t, depth) ? GREATER : NOT_GTEQ;
+        }
+    }
+}
+
+order::result lpo::compare(expr_offset s, expr_offset t, unsigned depth) {
+    TRACE("lpo", tout << "comparing:\n" << mk_pp(s.get_expr(), m_manager) << "\n" << mk_pp(t.get_expr(), m_manager) << "\n";);
+    result r = compare_core(s, t, depth);
+    TRACE("lpo", tout << "result of comparing:\n" << mk_pp(s.get_expr(), m_manager) << "\n" << mk_pp(t.get_expr(), m_manager) << "\nresult: " << r << "\n";);
+    return r;
+}
+
+bool lpo::greater(expr_offset const & t1, expr_offset const & t2, substitution * s) {
+    m_subst = s;
+    return greater(t1, t2, static_cast<unsigned>(0));
+}
+
+order::result lpo::compare(expr_offset const & t1, expr_offset const & t2, substitution * s) {
+    m_subst = s;
+    result r = compare(t1, t2, static_cast<unsigned>(0));
+    if (r != NOT_GTEQ) 
+        return r;
+    r = compare(t2, t1, static_cast<unsigned>(0));
+    if (r == GREATER)
+        return LESSER;
+    if (r == UNKNOWN)
+        return UNKNOWN;
+    return UNCOMPARABLE;
+}
+
+int lpo::compare_ge(expr_offset const & t1, expr_offset const & t2, substitution * s) {
+    m_subst = s;
+    result r = compare(t1, t2, static_cast<unsigned>(0));
+    switch (r) {
+    case GREATER: return 1;
+    case EQUAL:   return 0;
+    default:      return -1;
+    }
+}

src/spc/lpo.h

@@ -0,0 +1,49 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    lpo.h
+
+Abstract:
+
+    Lexicographical Path Ordering
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-01.
+
+Revision History:
+
+--*/
+#ifndef _LPO_H_
+#define _LPO_H_
+
+#include"order.h"
+#include"vector.h"
+#include"map.h"
+
+class lpo : public order {
+    svector<expr_offset>  m_todo;
+
+    bool occurs(expr_offset const & t1, expr_offset const & t2);
+    bool greater(expr_offset s, expr_offset t, unsigned depth);
+    bool dominates_args(expr_offset s, expr_offset t, unsigned depth);
+    bool arg_dominates_expr(expr_offset s, expr_offset t, unsigned depth);
+    result lex_compare(expr_offset s, expr_offset t, unsigned depth);
+    result compare_core(expr_offset s, expr_offset t, unsigned depth);
+    result compare(expr_offset s, expr_offset t, unsigned depth);
+    bool max_depth(unsigned d) { /* TODO */ return false; }
+public:
+    lpo(ast_manager & m, precedence * p):order(m, p) {}
+    virtual ~lpo() {}
+    
+    virtual result compare(expr_offset const & t1, expr_offset const & t2, substitution * s);
+    result compare(expr * t1, expr * t2) { return compare(expr_offset(t1, 0), expr_offset(t2, 0), static_cast<substitution*>(0)); }
+
+    virtual bool greater(expr_offset const & t1, expr_offset const & t2, substitution * s);
+    bool greater(expr_offset const & t1, expr_offset const & t2) { return greater(t1, t2); }
+    virtual int compare_ge(expr_offset const & t1, expr_offset const & t2, substitution * s);
+};
+
+#endif /* _LPO_H_ */

src/spc/marker.h

@@ -0,0 +1,56 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    marker.h
+
+Abstract:
+
+    Auxiliary object for managing markings
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-07.
+
+Revision History:
+
+--*/
+#ifndef _MARKER_H_
+#define _MARKER_H_
+
+#include"vector.h"
+
+/**
+   \brief Keep track of all marked objects. Unmark them when the method
+   unmark or destructor is invoked.
+*/
+template<typename T>
+class marker {
+    ptr_vector<T> m_to_unmark;
+public:
+    ~marker() {
+        unmark();
+    }
+
+    void mark(T * obj) { 
+        obj->set_mark(true);
+        m_to_unmark.push_back(obj);
+    }
+    
+    bool is_marked(T * obj) const {
+        return obj->is_marked();
+    }
+
+    void unmark() {
+        typename ptr_vector<T>::iterator it  = m_to_unmark.begin();
+        typename ptr_vector<T>::iterator end = m_to_unmark.end();
+        for (; it != end; ++it) {
+            T * obj = *it;
+            obj->set_mark(false);
+        }
+        m_to_unmark.reset();
+    }
+};
+
+#endif /* _MARKER_H_ */

src/spc/matcher.cpp

@@ -0,0 +1,81 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    matcher.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#include"matcher.h"
+
+matcher::matcher(ast_manager & m):
+    m_manager(m) {
+}
+
+bool matcher::operator()(expr * e1, expr * e2, substitution & s) {
+    reset();
+    m_subst = &s;
+    m_todo.push_back(expr_pair(e1, e2));
+    while (!m_todo.empty()) {
+        expr_pair const & p = m_todo.back();
+        // if (m_cache.contains(p)) {
+        //    m_todo.pop_back();
+        //    continue;
+        // }
+
+        if (is_var(p.first)) {
+            expr_offset r;
+            if (m_subst->find(to_var(p.first), 0, r)) {
+                if (r.get_expr() != p.second)
+                    return false;
+            }
+            else {
+                m_subst->insert(to_var(p.first), 0, expr_offset(p.second, 1));
+            }
+            m_todo.pop_back();
+            continue;
+        }
+
+
+        if (is_var(p.second))
+            return false;
+
+        app * n1 = to_app(p.first);
+        app * n2 = to_app(p.second);
+
+        if (n1->get_decl() != n2->get_decl())
+            return false;
+
+        unsigned num_args1 = n1->get_num_args();
+        if (num_args1 != n2->get_num_args())
+            return false;
+
+        m_todo.pop_back();
+
+        if (num_args1 == 0)
+            continue;
+        
+        // m_cache.insert(p);
+        unsigned j = num_args1;
+        while (j > 0) {
+            --j;
+            m_todo.push_back(expr_pair(n1->get_arg(j), n2->get_arg(j)));
+        }
+    }
+    return true;
+}
+
+void matcher::reset() {
+    // m_cache.reset();
+    m_todo.reset();
+}

src/spc/matcher.h

@@ -0,0 +1,64 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    matcher.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#ifndef _MATCHER_H_
+#define _MATCHER_H_
+
+#include"substitution.h"
+#include"hashtable.h"
+
+/**
+   \brief Functor for matching expressions.
+*/
+class matcher {
+    typedef std::pair<expr *, expr *> expr_pair;
+    typedef pair_hash<obj_ptr_hash<expr>, obj_ptr_hash<expr> > expr_pair_hash;
+    typedef hashtable<expr_pair, expr_pair_hash, default_eq<expr_pair> > cache;
+
+    ast_manager &         m_manager;
+    substitution *        m_subst;
+    cache                 m_cache;
+    svector<expr_pair>    m_todo;
+
+    void reset();
+
+public:
+    matcher(ast_manager & m);
+
+    /**
+       \brief Return true if e2 is an instance of e1.
+       In case of success (result is true), it will store the substitution that makes e1 equals to e2 into s.
+       
+       For example:
+       1) e1 = f(g(x), x), e2 = f(g(h(a)), h(a))
+       The result is true, and s will contain x -> h(a)
+
+       2) e1 = f(a, x)     e2 = f(x, a)
+       The result is false.
+
+       3) e1 = f(x, x)     e2 = f(y, a)
+       The result is false
+
+       4) e1 = f(x, y)     e2 = f(h(z), a)
+       The result is true, and s contains x->h(z) and y->a
+    */
+    bool operator()(expr * e1, expr * e2, substitution & s);
+};
+
+#endif /* _MATCHER_H_ */
+

src/spc/normalize_vars.cpp

@@ -0,0 +1,88 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    normalize_vars.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-16.
+
+Revision History:
+
+--*/
+#include"normalize_vars.h"
+
+expr * normalize_vars::operator()(expr * n) {
+    SASSERT(m_todo.empty());
+    m_todo.push_back(n);
+    while (!m_todo.empty()) {
+        n = m_todo.back();
+        if (m_cache.contains(n)) {
+            m_todo.pop_back();
+            continue;
+        }
+        if (is_var(n)) {
+            m_todo.pop_back();
+            unsigned idx = to_var(n)->get_idx();
+            var * new_var = m_map.get(idx, 0);
+            if (new_var == 0) {
+                new_var = m_manager.mk_var(m_next_var, to_var(n)->get_sort());
+                m_next_var++;
+                m_new_vars.push_back(new_var);
+                m_map.setx(idx, new_var, 0);
+            }
+            SASSERT(new_var->get_sort() == to_var(n)->get_sort());
+            m_cache.insert(n, new_var);
+        }
+        else {
+            SASSERT(is_app(n));
+            bool visited = true;
+            unsigned num_args = to_app(n)->get_num_args();
+            unsigned j = num_args;
+            while (j > 0) {
+                --j;
+                expr * child = to_app(n)->get_arg(j);
+                if (!m_cache.contains(child)) {
+                    m_todo.push_back(child);
+                    visited = false;
+                }
+            }
+            if (visited) {
+                m_todo.pop_back();
+                m_new_children.reset();
+                bool modified = false;
+                for (unsigned i = 0; i < num_args; i++) {
+                    expr * child = to_app(n)->get_arg(i);
+                    expr * new_child = 0;
+                    m_cache.find(child, new_child);
+                    SASSERT(new_child);
+                    if (child != new_child)
+                        modified = true;
+                    m_new_children.push_back(new_child);
+                }
+                if (!modified) 
+                    m_cache.insert(n, n);
+                else
+                    m_cache.insert(n, m_manager.mk_app(to_app(n)->get_decl(), m_new_children.size(), m_new_children.c_ptr()));
+            }
+        }
+    }
+    expr * r = 0;
+    m_cache.find(n, r);
+    SASSERT(r);
+    return r;
+}
+ 
+void normalize_vars::reset() {
+    m_cache.reset();
+    m_map.reset();
+    m_new_vars.reset();
+    m_next_var = 0;
+}
+

src/spc/normalize_vars.h

@@ -0,0 +1,47 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    normalize_vars.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-16.
+
+Revision History:
+
+--*/
+#ifndef _NORMALIZE_VARS_H_
+#define _NORMALIZE_VARS_H_
+
+#include"ast.h"
+#include"obj_hashtable.h"
+
+class normalize_vars {
+    ast_manager &    m_manager;
+    var_ref_vector   m_new_vars;
+    unsigned         m_next_var;
+    ptr_vector<var>  m_map;
+    typedef obj_map<expr, expr *> cache;
+    cache            m_cache;
+    ptr_vector<expr> m_todo;
+    ptr_vector<expr> m_new_children;
+public:
+    normalize_vars(ast_manager & m):
+        m_manager(m),
+        m_new_vars(m) {
+    }
+
+    expr * operator()(expr * n);
+    void reset();
+    unsigned get_num_vars() const { return m_new_vars.size(); }
+    var * const * get_vars() const { return m_new_vars.c_ptr(); }
+};
+
+#endif /* _NORMALIZE_VARS_H_ */
+

src/spc/order.cpp

@@ -0,0 +1,51 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    order.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-15.
+
+Revision History:
+
+--*/
+
+#include"order.h"
+
+bool order::equal(expr_offset const & _t1, expr_offset const & _t2, substitution * s) {
+    if (_t1 == _t2)
+        return true;
+    if (s == 0)
+        return false;
+    m_eq_todo.reset();
+    m_eq_todo.push_back(expr_offset_pair(_t1, _t2));
+    while (!m_eq_todo.empty()) {
+        expr_offset_pair const & p = m_eq_todo.back();
+        expr_offset t1 = find(p.first);
+        expr_offset t2 = find(p.second);
+        m_eq_todo.pop_back();
+        if (t1 == t2)
+            continue;
+        expr * n1 = t1.get_expr();
+        expr * n2 = t2.get_expr();
+        if (!is_app(n1) || !is_app(n2))
+            return false;
+        if (to_app(n1)->get_decl() != to_app(n2)->get_decl())
+            return false;
+        if (to_app(n1)->get_num_args() != to_app(n2)->get_num_args())
+            return false;
+        unsigned num = to_app(n1)->get_num_args();
+        for (unsigned i = 0; i < num; i++) 
+            m_eq_todo.push_back(expr_offset_pair(expr_offset(to_app(n1)->get_arg(i), t1.get_offset()),
+                                                 expr_offset(to_app(n2)->get_arg(i), t2.get_offset())));
+    }
+    return true;
+}
+

src/spc/order.h

@@ -0,0 +1,87 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    order.h
+
+Abstract:
+
+    Abstract class for term orderings.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-28.
+
+Revision History:
+
+--*/
+#ifndef _ORDER_H_
+#define _ORDER_H_
+
+#include"substitution.h"
+#include"precedence.h"
+#include"trace.h"
+
+class order {
+protected:
+    ast_manager &  m_manager;
+    precedence *   m_precedence;
+    substitution * m_subst;
+
+    typedef std::pair<expr_offset, expr_offset> expr_offset_pair;
+    svector<expr_offset_pair> m_eq_todo;
+
+    expr_offset find(expr_offset t) {
+        while (m_subst && is_var(t.get_expr()) && m_subst->find(to_var(t.get_expr()), t.get_offset(), t))
+            ;
+        return t;
+    }
+
+    bool f_greater(func_decl * f, func_decl * g) const {
+        bool r = m_precedence->compare(f, g) > 0;
+        TRACE("order", tout << f->get_name() << " greater than " << g->get_name() << " == " << r << "\n";);
+        return r;
+    }
+public:
+    enum result {
+        UNKNOWN,
+        UNCOMPARABLE,
+        EQUAL,
+        GREATER,
+        LESSER,
+        NOT_GTEQ
+    };
+    static bool ok(result r) { return r == EQUAL || r == GREATER || r == LESSER; }
+    order(ast_manager & m, precedence * p):m_manager(m), m_precedence(p) { SASSERT(p); }
+    virtual ~order() { dealloc(m_precedence); }
+    virtual void reserve(unsigned num_offsets, unsigned num_vars) {}
+    virtual void reserve_offsets(unsigned num_offsets) {}
+    virtual void reserve_vars(unsigned num_vars) {}
+    ast_manager & get_manager() { return m_manager; }
+
+    virtual result compare(expr_offset const & t1, expr_offset const & t2, substitution * s) = 0;
+    result compare(expr * t1, expr * t2, unsigned offset, substitution * s) { return compare(expr_offset(t1, offset), expr_offset(t2, offset), s); }
+    result compare(expr * t1, expr * t2) { return compare(expr_offset(t1, 0), expr_offset(t2, 0), 0); }
+    virtual bool greater(expr_offset const & t1, expr_offset const & t2, substitution * s) = 0;
+    bool greater(expr * t1, expr * t2) { return greater(expr_offset(t1,0), expr_offset(t2,0), 0); }
+    bool greater(expr * t1, expr * t2, substitution * s) { return greater(expr_offset(t1,0), expr_offset(t2,0), s); }
+    bool greater(expr * t1, expr * t2, unsigned offset, substitution * s) { 
+        return greater(expr_offset(t1, offset), expr_offset(t2, offset), s); 
+    }
+    /**
+       \brief Return a value > 0 if t1 is greater than t2, 0 if t1 == t2, and < 0 otherwise (uncomparable, unknown, lesser).
+    */
+    virtual int compare_ge(expr_offset const & t1, expr_offset const & t2, substitution * s) = 0;
+
+    /**
+       \brief Return true if the given terms are equal modulo the given substitution
+    */
+    bool equal(expr_offset const & t1, expr_offset const & t2, substitution * s);
+
+    bool equal(expr * t1, expr * t2, unsigned offset = 0, substitution * s = 0) {
+        return equal(expr_offset(t1, offset), expr_offset(t2, offset), s);
+    }
+};
+
+#endif /* _ORDER_H_ */

src/spc/precedence.cpp

@@ -0,0 +1,191 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    precedence.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-08.
+
+Revision History:
+
+--*/
+#include"precedence.h"
+#include"warning.h"
+
+lex_precedence::lex_precedence(unsigned n, precedence ** ps):
+    m_precedences(n, ps) {
+}
+
+lex_precedence::~lex_precedence() {
+    std::for_each(m_precedences.begin(), m_precedences.end(), delete_proc<precedence>());
+}
+
+int lex_precedence::compare(func_decl * f, func_decl * g) {
+    int r = 0;
+    ptr_vector<precedence>::iterator it  = m_precedences.begin();
+    ptr_vector<precedence>::iterator end = m_precedences.end();
+    for (; it != end; ++it) {
+        r = (*it)->compare(f, g);
+        if (r != 0)
+            return r;
+    }
+    return r;
+}
+
+inv_precedence::inv_precedence(precedence * p):
+    m_precedence(p) {
+    SASSERT(p);
+}
+
+inv_precedence::~inv_precedence() {
+    dealloc(m_precedence);
+}
+
+int inv_precedence::compare(func_decl * f, func_decl * g) {
+    return m_precedence->compare(g, f);
+}
+
+int arbitrary_precedence::compare(func_decl * f, func_decl * g) { 
+    return static_cast<int>(f->get_decl_id()) - static_cast<int>(g->get_decl_id()); 
+}
+
+int arity_precedence::compare(func_decl * f, func_decl * g) { 
+    return static_cast<int>(f->get_arity()) - static_cast<int>(g->get_arity());
+}
+
+int interpreted_precedence::compare(func_decl * f, func_decl * g) {
+    return static_cast<int>(f->get_family_id() == null_family_id) - static_cast<int>(g->get_family_id() == null_family_id);
+}
+
+inline int ext_precedence::get_func_pos(func_decl * f) {
+    unsigned id = f->get_decl_id();
+    return m_cached.get(id, m_undefined);
+}
+
+int ext_precedence::compare(func_decl * f, func_decl * g) {
+    return get_func_pos(f) - get_func_pos(g);
+}
+
+ext_precedence::ext_precedence(ast_manager & m, unsigned num_decls, func_decl ** decls):
+    m_undefined(num_decls),
+    m_cached_domain(m) {
+    for (unsigned i = 0; i < num_decls; i++) {
+        m_cached.setx(decls[i]->get_decl_id(), i, m_undefined);
+        m_cached_domain.push_back(decls[i]);
+    }
+}
+
+ext_precedence::~ext_precedence() {
+}
+
+int abstract_user_precedence::get_decl_pos(decl * d) {
+    unsigned id = d->get_decl_id();
+    int pos = m_cached.get(id, -1);
+    if (pos == -1) {
+        if (!m_symbol2pos.find(d->get_name(), pos))
+            pos = m_undefined;
+        m_cached.setx(id, pos, -1);
+        SASSERT(pos != -1);
+    }
+    return pos;
+}
+
+abstract_user_precedence::abstract_user_precedence(ast_manager & m, unsigned num_syms, symbol * syms):
+    m_undefined(num_syms),
+    m_cached_domain(m) {
+    for (unsigned i = 0; i < num_syms; i++)
+        m_symbol2pos.insert(syms[i], i);
+}
+
+abstract_user_precedence::~abstract_user_precedence() {
+}
+
+int user_precedence::compare(func_decl * f, func_decl * g) {
+    return get_decl_pos(f) - get_decl_pos(g);
+}
+
+int user_sort_precedence::compare(func_decl * f, func_decl * g) {
+    return get_decl_pos(f->get_range()) - get_decl_pos(g->get_range());
+}
+
+static precedence * mk_default_precedence(ast_manager & m, order_params const & params) {
+    ptr_buffer<precedence> ps;
+    if (!params.m_order_precedence.empty())
+        ps.push_back(alloc(user_precedence, m, params.m_order_precedence.size(), params.m_order_precedence.c_ptr()));
+    ps.push_back(alloc(interpreted_precedence));
+    ps.push_back(alloc(arity_precedence));
+    ps.push_back(alloc(arbitrary_precedence));
+    return alloc(lex_precedence, ps.size(), ps.c_ptr());
+}
+
+static precedence * mk_inv_precedence(bool inv, precedence * p) {
+    return inv ? alloc(inv_precedence,p) : p;
+}
+
+static precedence * mk_lex_precedence(ptr_buffer<precedence> const & ps) {
+    unsigned sz = ps.size();
+    if (sz == 0)
+        return alloc(arbitrary_precedence);
+    else if (sz == 1)
+        return ps[0];
+    else 
+        return alloc(lex_precedence, sz, ps.c_ptr());
+}
+
+precedence * mk_precedence(ast_manager & m, order_params const & params) {
+    if (params.m_order_precedence_gen.empty())
+        return mk_default_precedence(m, params);
+
+    symbol user("user");
+    symbol definition("definition");
+    symbol interpreted("interpreted");
+    symbol frequency("frequency");
+    symbol arity("arity");
+    symbol arbitrary("arbitrary");
+    symbol inv("-");
+
+    ptr_buffer<precedence> ps;
+
+    svector<symbol>::const_iterator it  =  params.m_order_precedence_gen.begin();
+    svector<symbol>::const_iterator end =  params.m_order_precedence_gen.end();
+    bool prev_inv = false;
+    for (; it != end; ++it) {
+        symbol curr = *it;
+        if (curr == user) {
+            if (params.m_order_precedence.empty())
+                ps.push_back(mk_inv_precedence(prev_inv, alloc(user_precedence, m, params.m_order_precedence.size(), params.m_order_precedence.c_ptr())));
+        }
+        else if (curr == definition) {
+            warning_msg("definition precedence was not implement yet.");
+        }
+        else if (curr == interpreted) {
+            ps.push_back(mk_inv_precedence(prev_inv, alloc(interpreted_precedence)));
+        }
+        else if (curr == frequency) {
+            warning_msg("frequency precedence was not implement yet.");
+        }
+        else if (curr == arity) {
+            ps.push_back(mk_inv_precedence(prev_inv, alloc(arity_precedence)));
+        }
+        else if (curr == arbitrary) {
+            ps.push_back(mk_inv_precedence(prev_inv, alloc(arbitrary_precedence)));
+            // it is pointless to continue, arbitrary_precedence is a total order
+            return mk_lex_precedence(ps);
+        }
+        else if (curr == inv) {
+            prev_inv = true;
+        }
+        else {
+            warning_msg("invalid precedence generator: ignoring atom '%s'.", curr.bare_str());
+        }
+    }
+    
+    return mk_lex_precedence(ps);
+}

src/spc/precedence.h

@@ -0,0 +1,142 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    precedence.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-08.
+
+Revision History:
+
+--*/
+#ifndef _PRECEDENCE_H_
+#define _PRECEDENCE_H_
+
+#include"ast.h"
+#include"order_params.h"
+
+/**
+   \brief Abstract functor used to implement an order on function symbols.
+*/
+class precedence {
+public:
+    virtual ~precedence() {}
+    virtual int compare(func_decl * f, func_decl * g) = 0;
+    bool operator()(func_decl * f, func_decl * g) { return compare(f, g) < 0; }
+};
+
+/**
+   \brief Compose different precedence functors using lexicographical order.
+*/
+class lex_precedence : public precedence {
+    ptr_vector<precedence> m_precedences;
+public:
+    lex_precedence(unsigned n, precedence ** ps);
+    virtual ~lex_precedence();
+    virtual int compare(func_decl * f, func_decl * g);
+};
+
+/**
+   \brief Invert functor
+ */
+class inv_precedence : public precedence {
+    precedence * m_precedence;
+public:
+    inv_precedence(precedence * p);
+    virtual ~inv_precedence();
+    virtual int compare(func_decl * f, func_decl * g);
+};
+
+/**
+   \brief An arbitrary total order based on the func_decl ids.
+*/
+class arbitrary_precedence : public precedence {
+public:
+    virtual int compare(func_decl * f, func_decl * g);
+};
+
+/**
+   \brief Precedence based on the arity. 
+
+   \remark This is not a total order, so it must be combined
+   with other precedence functors (e.g., arbitrary_precedence).
+*/
+class arity_precedence : public precedence {
+public:
+    virtual int compare(func_decl * f, func_decl * g);
+};
+
+/**
+   \brief Interpreted function symbols are smaller.
+ */
+class interpreted_precedence : public precedence {
+public:
+    virtual int compare(func_decl * f, func_decl * g);
+};
+
+/**
+   \brief A precedence given as a sequence of func_decls.
+   This functor is used to encapsulate automatically/externally generated
+   precedences.
+*/
+class ext_precedence : public precedence {
+    unsigned          m_undefined;  // position for func_decl's not specified by the user.
+    int_vector        m_cached;     // mapping: decl -> int 
+    decl_ref_vector   m_cached_domain;
+    
+    int get_func_pos(func_decl * f);
+public:
+    ext_precedence(ast_manager & m, unsigned num_decls, func_decl ** decls);
+    virtual ~ext_precedence();
+    virtual int compare(func_decl * f, func_decl * g);
+};
+
+/**
+   \brief Abstract class for user precedences based on
+   function or sort symbols.
+*/
+class abstract_user_precedence : public precedence {
+protected:
+    symbol_table<int> m_symbol2pos;
+    unsigned          m_undefined;  // position for symbols not specified by the user.
+    int_vector        m_cached;     // mapping: decl -> int 
+    decl_ref_vector   m_cached_domain;
+
+    int get_decl_pos(decl * d);
+public:
+    abstract_user_precedence(ast_manager & m, unsigned num_syms, symbol * syms);
+    virtual ~abstract_user_precedence();
+};
+
+/**
+   \brief A user defined precedence given as a sequence of symbols.
+
+   \remark User provided precedences are usually not total.
+*/
+class user_precedence : public abstract_user_precedence {
+public:
+    user_precedence(ast_manager & m, unsigned num_syms, symbol * syms):abstract_user_precedence(m, num_syms, syms) {}
+    virtual int compare(func_decl * f, func_decl * g);
+};
+
+/**
+   \brief A user defined precedence given as a sequence of sort symbols.
+   The functions are ordered based on their range sort.
+*/
+class user_sort_precedence : public abstract_user_precedence {
+public:
+    user_sort_precedence(ast_manager & m, unsigned num_syms, symbol * syms):abstract_user_precedence(m, num_syms, syms) {}
+    virtual int compare(func_decl * f, func_decl * g);
+};
+
+precedence * mk_precedence(ast_manager & m, order_params const & params);
+
+#endif /* _PRECEDENCE_H_ */
+

src/spc/preprocessor.cpp

@@ -0,0 +1,155 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    preprocessor.cpp
+
+Abstract:
+
+    Preprocessor
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-17.
+
+Revision History:
+
+--*/
+#include"ast_pp.h"
+#include"ast_ll_pp.h"
+#include"preprocessor.h"
+#include"for_each_expr.h"
+#include"num_occurs.h"
+
+preprocessor::preprocessor(ast_manager & m, defined_names & d, simplifier & s, preprocessor_params & p):
+    m_params(p),
+    m_manager(m),
+    m_simp(s),
+    m_nnf(m, d, p),
+    m_cnf(m, d, p),
+    m_der(m),
+    m_push_app_ite(s, p.m_lift_ite == LI_CONSERVATIVE),
+    m_cnf_todo(m),
+    m_cnf_todo_prs(m),
+    m_push_todo(m),
+    m_push_todo_prs(m) {
+    switch (m_params.m_cnf_mode) {
+    case CNF_QUANT: 
+        if (m_params.m_nnf_mode == NNF_SKOLEM)
+            m_params.m_nnf_mode = NNF_QUANT;
+        break;
+    case CNF_OPPORTUNISTIC:
+        if (m_params.m_nnf_mode == NNF_SKOLEM)
+            m_params.m_nnf_mode = NNF_QUANT;
+        break;
+    case CNF_FULL:
+        m_params.m_nnf_mode = NNF_FULL;
+        break;
+    default:
+        break;
+    }
+}
+
+#ifdef _TRACE
+struct num_occurs_pp {
+    
+    ast_manager &  m_manager;
+    std::ostream & m_out;
+    num_occurs     m_occurs;
+
+    num_occurs_pp(ast_manager & m, std::ostream & out, expr * root):
+        m_manager(m),
+        m_out(out) {
+        m_occurs(root);
+    }
+    void operator()(var * n) {}
+    void operator()(app * n) { 
+        unsigned val = m_occurs.get_num_occs(n);
+        if (val > 1 && m_manager.is_bool(n))
+            m_out << "#" << n->get_id() << " -> " << val << " " << n->get_ref_count() << "\n";
+    }
+    void operator()(quantifier * n) {}
+};
+#endif
+
+void preprocessor::operator()(expr * e, proof * in_pr, expr_ref_vector & result, proof_ref_vector & result_prs) {
+    m_cnf_todo.reset();
+    m_cnf_todo_prs.reset();
+    
+    expr_ref   r1(m_manager);
+    proof_ref  pr1(m_manager);
+    m_simp(e, r1, pr1);
+    in_pr = m_manager.mk_modus_ponens(in_pr, pr1);
+    
+    expr_ref   r2(m_manager);
+    proof_ref  pr2(m_manager);
+    m_nnf(r1, m_cnf_todo, m_cnf_todo_prs, r2, pr2);
+    in_pr = m_manager.mk_modus_ponens(in_pr, pr2);
+
+    TRACE("preprocessor", tout << mk_ll_pp(r2, m_manager);
+          num_occurs_pp proc(m_manager, tout, r2);
+          for_each_expr(proc, r2););
+    
+    m_cnf_todo.push_back(r2);
+    m_cnf_todo_prs.push_back(in_pr);
+
+    unsigned sz = m_cnf_todo.size();
+    for (unsigned i = 0; i < sz; i++) {
+        m_push_todo.reset();
+        m_push_todo_prs.reset();
+
+        expr * e = m_cnf_todo.get(i);
+        if (m_params.m_lift_ite != LI_NONE) {
+            m_push_app_ite(e, r1, pr1);
+        }
+        else {
+            r1  = e;
+            pr1 = 0;
+        }
+
+        TRACE("preprocessor", tout << mk_ll_pp(r1, m_manager););
+        
+        expr_ref aux(r1, m_manager);
+        m_simp(aux, r1, pr2);
+        pr1 = m_manager.mk_transitivity(pr1, pr2);
+
+        TRACE("preprocessor", tout << mk_ll_pp(r1, m_manager););
+
+        aux = r1;
+        m_der(aux, r1, pr2);
+        pr1 = m_manager.mk_transitivity(pr1, pr2);
+
+        TRACE("preprocessor", tout << mk_ll_pp(r1, m_manager););
+
+        if (m_manager.proofs_enabled())
+            in_pr = m_manager.mk_modus_ponens(m_cnf_todo_prs.get(i), pr1);
+        else
+            in_pr = 0;
+
+        aux = r1;
+        m_cnf(aux, m_push_todo, m_push_todo_prs, r1, pr1);
+        m_push_todo.push_back(r1);
+
+        TRACE("preprocessor", tout << mk_ll_pp(r1, m_manager););
+
+        if (m_manager.proofs_enabled()) {
+            in_pr = m_manager.mk_modus_ponens(in_pr, pr1);
+            m_push_todo_prs.push_back(in_pr);
+        }
+
+        unsigned sz2 = m_push_todo.size();
+        for (unsigned j = 0; j < sz2; j++) {
+            expr * e = m_push_todo.get(j);
+            m_simp(e, r1, pr1);
+            
+            TRACE("preprocessor", tout << mk_ll_pp(r1, m_manager););
+
+            if (m_manager.proofs_enabled())
+                in_pr = m_manager.mk_modus_ponens(m_push_todo_prs.get(j), pr1);
+            else
+                in_pr = 0;
+            push_assertion(m_manager, r1, in_pr, result, result_prs);
+        }
+    }
+}

src/spc/preprocessor.h

@@ -0,0 +1,51 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    preprocessor.h
+
+Abstract:
+
+    Preprocess AST before adding them to the logical context
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-17.
+
+Revision History:
+
+--*/
+#ifndef _PREPROCESSOR_H_
+#define _PREPROCESSOR_H_
+
+#include"preprocessor_params.h"
+#include"simplifier.h"
+#include"pattern_inference.h"
+#include"nnf.h"
+#include"cnf.h"
+#include"der.h"
+#include"push_app_ite.h"
+
+/**
+   \brief Functor used to preprocess expressions before adding them to
+   the logical context.
+*/
+class preprocessor {
+    preprocessor_params & m_params;
+    ast_manager &         m_manager;
+    simplifier &          m_simp;
+    nnf                   m_nnf;
+    cnf                   m_cnf;
+    der_star              m_der;                  
+    push_app_ite          m_push_app_ite;
+    expr_ref_vector       m_cnf_todo;
+    proof_ref_vector      m_cnf_todo_prs;
+    expr_ref_vector       m_push_todo;
+    proof_ref_vector      m_push_todo_prs;
+public:
+    preprocessor(ast_manager & m, defined_names & d, simplifier & s, preprocessor_params & p);
+    void operator()(expr * e, proof * in_pr, expr_ref_vector & result, proof_ref_vector & result_prs);
+};
+
+#endif /* _PREPROCESSOR_H_ */

src/spc/sparse_use_list.h

@@ -0,0 +1,107 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    sparse_use_list.h
+
+Abstract:
+
+    Sparse use list index.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-13.
+
+Revision History:
+
+--*/
+#ifndef _SPARSE_USE_LIST_H_
+#define _SPARSE_USE_LIST_H_
+
+#include"ast.h"
+#include"obj_hashtable.h"
+
+/**
+   \brief (Generic sparse) use-list data-structure.
+*/
+template<typename T, typename Set>
+class sparse_use_list {
+    typedef obj_map<T, Set *> use_list;
+    use_list        m_use_list;
+
+public:
+    typedef typename Set::iterator iterator;
+    sparse_use_list() {}
+    ~sparse_use_list() { 
+        reset();
+    }
+
+    void insert(typename Set::data const & parent, T * child) {
+        Set * parents = 0;
+        if (!m_use_list.find(child, parents)) {
+            parents = alloc(Set);
+            m_use_list.insert(child, parents);
+        }
+        SASSERT(parents);
+        parents->insert(parent);
+    }
+
+    /**
+       \brief Return 0 if child did not contain any parents.
+       Return 1, if child does not have more parents after 
+       removing parent.
+       Return 2 otherwise.
+    */
+    unsigned erase(typename Set::data const & parent, T * child) {
+        Set * parents = 0;
+        if (m_use_list.find(child, parents)) {
+            parents->erase(parent);
+            if (parents->empty()) {
+                dealloc(parents);
+                m_use_list.erase(child);
+                return 1;
+            }
+            return 2;
+        }
+        return 0;
+    }
+
+    void reset() {
+        typename use_list::iterator it  = m_use_list.begin();
+        typename use_list::iterator end = m_use_list.end();
+        for (; it != end; ++it)
+            dealloc(it->m_value);
+        m_use_list.reset();
+    }
+
+    Set * get_parents(T * e) {
+        Set * parents = 0;
+        m_use_list.find(e, parents);
+        return parents;
+    }
+    
+    iterator begin(T * e) {
+        Set * parents = 0;
+        m_use_list.find(e, parents);
+        SASSERT(parents);
+        return parents->begin();
+    }
+
+    iterator end(T * e) {
+        Set * parents = 0;
+        m_use_list.find(e, parents);
+        SASSERT(parents);
+        return parents->end();
+    }
+
+    bool empty(T * e) const {
+        Set * parents = 0;
+        if (m_use_list.find(e, parents))
+            return parents->empty();
+        return true;
+    }
+};
+
+#endif /* _SPARSE_USE_LIST_H_ */
+

src/spc/spc_asserted_literals.cpp

@@ -0,0 +1,170 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_asserted_literals.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-14.
+
+Revision History:
+
+--*/
+
+#include"spc_asserted_literals.h"
+#include"ast_pp.h"
+
+namespace spc {
+
+    asserted_literals::asserted_literals(ast_manager & m):
+        m_manager(m),
+        m_subst(m),
+        m_tmp_eq1(2),
+        m_tmp_eq2(2) {
+        for (unsigned i = 0; i < 2; i++) {
+            m_st[i]           = alloc(substitution_tree, m_manager);
+            m_expr2clause[i]  = alloc(expr2clause);
+        }
+        m_subst.reserve_offsets(3);
+    }
+
+    asserted_literals::~asserted_literals() {
+        for (unsigned i = 0; i < 2; i++) {
+            dealloc(m_st[i]);
+            dealloc(m_expr2clause[i]);
+        }
+    }
+        
+    void asserted_literals::insert(clause * cls) {
+        if (cls->get_num_literals() == 1) {
+            TRACE("asserted_literals", tout << "inserting clause into asserted_literals index:\n";
+                  cls->display(tout, m_manager); tout << "\n";);
+            literal const & l = cls->get_literal(0);
+            unsigned neg = static_cast<unsigned>(l.sign());
+            expr * atom  = l.atom();
+            m_st[neg]->insert(to_app(atom));
+            m_expr2clause[neg]->insert(atom, cls);
+            m_subst.reserve_vars(m_st[neg]->get_approx_num_regs());
+        }
+    }
+
+    void asserted_literals::erase(clause * cls) {
+        if (cls->get_num_literals() == 1) {
+            literal const & l = cls->get_literal(0);
+            unsigned neg = static_cast<unsigned>(l.sign());
+            expr * atom  = l.atom();
+            m_expr2clause[neg]->erase(atom);
+            m_st[neg]->erase(to_app(atom));
+        }
+    }
+    
+    void asserted_literals::reset() {
+        for (unsigned i = 0; i < 2; i++) {
+            m_st[i]->reset();
+            m_expr2clause[i]->reset();
+        }
+    }
+
+    struct asserted_literals_visitor : public st_visitor {
+        expr * m_target;
+        asserted_literals_visitor(substitution & s):st_visitor(s), m_target(0) {}
+        virtual bool operator()(expr * e) {
+            m_target = e;
+            return false; // stop
+        }
+    };
+
+    /**
+       \brief Return an unit clause that is a generalization
+       of the given literal.
+       Return 0 if such clause does not exist.
+    */
+    clause * asserted_literals::gen(expr * atom, bool n) {
+        if (is_app(atom)) {
+            TRACE("asserted_literals", tout << "checking if there is generalizer for: " << n << "\n" << 
+                  mk_pp(atom, m_manager) << "\n";);
+            unsigned neg = static_cast<unsigned>(n);
+            m_subst.reset_subst();
+            asserted_literals_visitor visitor(m_subst); 
+            TRACE("asserted_literals_bug", tout << "query: " << mk_pp(atom, m_manager) << "\n"; m_st[neg]->display(tout);
+                  m_subst.display(tout););
+            m_st[neg]->gen(to_app(atom), visitor);
+            if (visitor.m_target != 0) {
+                clause * cls = 0;
+                m_expr2clause[neg]->find(visitor.m_target, cls);
+                SASSERT(cls);
+                return cls;
+            }
+            if (m_manager.is_eq(atom)) {
+                m_subst.reset();
+                m_tmp_eq1.copy_swapping_args(to_app(atom));
+                m_st[neg]->gen(m_tmp_eq1.get_app(), visitor);
+                if (visitor.m_target != 0) {
+                    clause * cls = 0;
+                    m_expr2clause[neg]->find(visitor.m_target, cls);
+                    SASSERT(cls);
+                    return cls;
+                }
+            }
+        }
+        return 0;
+    }
+
+    /**
+       \brief Return an unit clause that is a generalization
+       of the equality (= lhs rhs)
+       Return 0 if such clause does not exist.
+    */
+    clause * asserted_literals::gen_eq(expr * lhs, expr * rhs) {
+        expr * args[2] = { lhs, rhs };
+        func_decl_ref eq_decl(m_manager.mk_func_decl(m_manager.get_basic_family_id(), OP_EQ, 0, 0, 2, args), m_manager);
+        m_tmp_eq2.set_decl(eq_decl);
+        m_tmp_eq2.set_arg(0, lhs);
+        m_tmp_eq2.set_arg(1, rhs);
+        return gen(m_tmp_eq2.get_app(), false);
+    }
+
+    /**
+       \brief Return a unit equality clause (= s t) that (eq) subsumes (= lhs rhs).
+       That is, lhs and rhs have the form u[s'] and u[t'] and there is 
+       a substitution sigma s.t. sigma(s) = s' and sigma(t) = t'.
+       Return 0 if such clause does not exist.
+    */
+    clause * asserted_literals::subsumes(expr * lhs, expr * rhs) {
+        while (true) {
+            TRACE("eq_subsumption", tout << "eq_subsumption loop:\n" << mk_pp(lhs, m_manager) << "\n" << 
+                  mk_pp(rhs, m_manager) << "\n";);
+            clause * subsumer = gen_eq(lhs, rhs);
+            if (subsumer)
+                return subsumer;
+            if (!is_app(lhs) || !is_app(rhs) ||
+                to_app(lhs)->get_decl() != to_app(rhs)->get_decl() || 
+                to_app(lhs)->get_num_args() != to_app(rhs)->get_num_args())
+                return 0;
+            expr * d1 = 0;
+            expr * d2 = 0;
+            unsigned num_args = to_app(lhs)->get_num_args();
+            for (unsigned i = 0; i < num_args; i++) {
+                expr * c1 = to_app(lhs)->get_arg(i);
+                expr * c2 = to_app(rhs)->get_arg(i);
+                if (c1 != c2) {
+                    if (d1) 
+                        return 0;
+                    d1 = c1;
+                    d2 = c2;
+                }
+            }
+            SASSERT(d1);
+            lhs = d1;
+            rhs = d2;
+        }
+        return 0;
+    }
+
+};

src/spc/spc_asserted_literals.h

@@ -0,0 +1,69 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_asserted_literals.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-14.
+
+Revision History:
+
+--*/
+#ifndef _SPC_ASSERTED_LITERALS_H_
+#define _SPC_ASSERTED_LITERALS_H_
+
+#include"spc_clause.h"
+#include"substitution_tree.h"
+#include"obj_hashtable.h"
+
+namespace spc {
+    
+    /**
+       \brief Index for the asserted literals in the logical context.
+       
+       This index is used to implement forward unit subsumption, 
+       equality subsumption, positive simplify-reflect, and
+       negative simplify-reflect.
+    */
+    class asserted_literals {
+    protected:
+        typedef obj_map<expr, clause*> expr2clause;
+        ast_manager &       m_manager;
+        substitution_tree * m_st[2];
+        expr2clause *       m_expr2clause[2];
+        substitution        m_subst;
+        tmp_app             m_tmp_eq1;
+        tmp_app             m_tmp_eq2;
+    public:
+        asserted_literals(ast_manager & m);
+        ~asserted_literals();
+
+        void insert(clause * cls);
+        void erase(clause * cls);
+        void reset();
+        void reserve_vars(unsigned num_vars) { m_subst.reserve_vars(num_vars); }
+
+        clause * gen(literal const & l) {
+            return gen(l.atom(), l.sign());
+        }
+
+        clause * gen(expr * atom, bool neg);
+        clause * gen_eq(expr * lhs, expr * rhs);
+        clause * subsumes(expr * lhs, expr * rhs);
+
+        bool has_pos_literals() const { return !m_st[0]->empty(); }
+        bool has_neg_literals() const { return !m_st[1]->empty(); }
+        bool has_literals() const { return has_pos_literals() || has_neg_literals(); }
+    };
+    
+};
+
+#endif /* _SPC_ASSERTED_LITERALS_H_ */
+

src/spc/spc_clause.cpp

@@ -0,0 +1,287 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_clause.cpp
+
+Abstract:
+
+    Superposition Calculus Clause
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#include"spc_clause.h"
+#include"splay_tree_def.h"
+
+template class splay_tree<spc::clause *, spc::clause::compare>;
+
+namespace spc {
+
+    clause::clause(ast_manager & m, unsigned num_lits, literal * lits, justification * p, unsigned scope_lvl):
+        m_id(UINT_MAX),
+        m_time(UINT_MAX),
+        m_scope_lvl(scope_lvl),
+        m_bidx(UINT_MAX),
+        m_processed(false),
+        m_indexed(false), 
+        m_has_sel_lit(false), 
+        m_justification(p) {
+        
+        set_fields(num_lits, lits);
+
+        m_num_lits_capacity = m_num_lits[0] + m_num_lits[1];
+        
+        memcpy(m_lits, lits, sizeof(literal) * get_num_literals());
+
+        for (unsigned i = 0; i < num_lits; i++)
+            m.inc_ref(m_lits[i].atom());
+        m_justification->inc_ref();
+        m_justification->set_owner(this);
+        
+        sort_literals();
+    }
+
+    clause * clause::mk(ast_manager & m, unsigned num_lits, literal * lits, justification * p, unsigned scope_lvl) {
+        void * mem   = m.get_allocator().allocate(sizeof(clause) + num_lits * sizeof(literal));
+        return new (mem) clause(m, num_lits, lits, p, scope_lvl);
+    }
+
+    void clause::init(unsigned id, unsigned time) {
+        SASSERT(m_id == UINT_MAX);
+        SASSERT(m_time == UINT_MAX);
+
+        m_id          = id;
+        m_time        = time;
+        m_proof_depth = 0;
+
+        justification_stat j_stat;
+        get_justification_stat(m_justification, j_stat);
+
+        m_proof_depth = j_stat.m_proof_depth;
+        
+        if (j_stat.m_max_scope_lvl > m_scope_lvl) 
+            m_scope_lvl = j_stat.m_max_scope_lvl;
+        
+        update_parents(j_stat.m_parent_clauses);
+    }
+
+    void clause::update_parents(ptr_buffer<clause> & parents) {
+        ptr_buffer<clause>::iterator it  = parents.begin();
+        ptr_buffer<clause>::iterator end = parents.end();
+        for (; it != end; ++it) {
+            clause * parent = *it;
+            parent->add_child(this);
+        }
+    }
+
+    void clause::deallocate(ast_manager & m) {
+
+        justification_stat j_stat;
+        get_justification_stat(get_justification(), j_stat);
+
+        ptr_buffer<clause>::iterator it  = j_stat.m_parent_clauses.begin();
+        ptr_buffer<clause>::iterator end = j_stat.m_parent_clauses.end();
+        for (; it != end; ++it) {
+            clause * parent = *it;
+            parent->del_child(this);
+        }
+
+        dec_ref(get_justification(), m);
+        
+        unsigned num_lits = get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++)
+            m.dec_ref(get_literal(i).atom());
+
+        unsigned capacity = get_num_literals_capacity();
+        this->~clause();
+        m.get_allocator().deallocate(sizeof(clause) +  capacity * sizeof(literal), this);
+    }
+
+    void clause::select_literal(unsigned idx) {
+        SASSERT(idx < get_num_literals());
+        m_lits[idx].set_selected(true);
+        m_has_sel_lit = true;
+    }
+
+    /**
+       \brief Return true if l is maximal in the clause, given a substitution s.
+       
+       s(l) is considered maximal if there is no literal l' in the clause such s(l') is greater
+       than s(l).
+    */
+    bool clause::is_maximal(order & o, literal const & l, unsigned offset, substitution * s) const {
+        unsigned num_lits = get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l_prime = m_lits[i];
+            if (l != l_prime && greater(o, l_prime, l, offset, s))
+                return false;
+        }
+        return true;
+    }
+
+    /**
+       \brief Return true if l is a maximal selected literal in the clause, given a substitution s.
+       
+       s(l) is considered maximal selected literal if there is no
+       selected literal l' in the clause such s(l') is greater than s(l).
+    */
+    bool clause::is_sel_maximal(order & o, literal const & l, unsigned offset, substitution * s) const {
+        if (!l.is_selected())
+            return false;
+        unsigned num_lits = get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l_prime = m_lits[i];
+            if (l != l_prime && l_prime.is_selected() && greater(o, l_prime, l, offset, s))
+                return false;
+        }
+        return true;
+    }
+
+    /**
+       \brief Return true if l is eligible for resolution.
+    */
+    bool clause::is_eligible_for_resolution(order & o, literal const & l, unsigned offset, substitution * s) const {
+        if (has_sel_lit()) 
+            return is_sel_maximal(o, l, offset, s);
+        else
+            return is_maximal(o, l, offset, s);
+    }
+
+    /**
+       \brief Return true if l is eligible for paramodulation.
+    */
+    bool clause::is_eligible_for_paramodulation(order & o, literal const & l, unsigned offset, substitution * s) const {
+        return !has_sel_lit() && is_maximal(o, l, offset, s);
+    }
+
+    /**
+       \brief Try to orient literals.
+    */
+    void clause::try_to_orient_literals(order & o) {
+        o.reserve_vars(get_num_vars());
+        unsigned num_lits = get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal & l = m_lits[i];
+            l.try_to_orient(o);
+        }
+    }
+
+    void clause::set_fields(unsigned num_lits, literal * lits) {
+        clause_stat c_stat;
+        get_clause_stat(num_lits, lits, c_stat);
+
+        m_num_vars          = c_stat.m_max_var_idx + 1;
+        m_sym_count         = c_stat.m_sym_count;
+        m_const_count       = c_stat.m_const_count;
+        m_depth             = c_stat.m_depth;
+        m_num_lits[0]       = c_stat.m_num_lits[0];
+        m_num_lits[1]       = c_stat.m_num_lits[1];
+        m_ground            = c_stat.m_ground;
+    }
+
+    struct lit_lt {
+        bool operator()(literal const & l1, literal const & l2) const {
+            if (l1.is_ground() > l2.is_ground())
+                return true;
+            if (l1.is_ground() != l2.is_ground())
+                return false;
+            if (l1.get_approx_depth() > l2.get_approx_depth())
+                return true;
+            if (l1.get_approx_depth() != l2.get_approx_depth())
+                return false;
+            if (l1.get_approx_sym_count() > l2.get_approx_sym_count())
+                return true;
+            if (l1.get_approx_sym_count() != l2.get_approx_sym_count())
+                return false;
+            if (l1.get_approx_const_count() > l2.get_approx_const_count())
+                return true;
+            if (l1.get_approx_const_count() != l2.get_approx_const_count())
+                return false;
+            return l1.get_id() < l2.get_id();
+        }
+    };
+
+    /**
+       \brief Sort literals to improve the performance of subsumption tests.
+    */
+    void clause::sort_literals() {
+        DEBUG_CODE({
+            unsigned num_lits = get_num_literals();
+            for (unsigned i = 0; i < num_lits; i++) {
+                SASSERT(m_lits[i].has_stats());
+            }
+        });
+        std::sort(m_lits, m_lits + get_num_literals(), lit_lt());
+    }
+
+    /**
+       \brief Replace clause literal with the given literals.
+       Use the given justification to justify the new clause.
+    */
+    void clause::update_lits(ast_manager & m, unsigned num_lits, literal * lits, justification * j) {
+        unsigned old_num_lits = get_num_literals();
+        SASSERT(num_lits <= old_num_lits);
+     
+        for (unsigned i = 0; i < num_lits; i++)
+            m.inc_ref(lits[i].atom());
+
+        for (unsigned i = 0; i < old_num_lits; i++)
+            m.dec_ref(m_lits[i].atom());
+        
+        for (unsigned i = 0; i < num_lits; i++) 
+            m_lits[i] = lits[i];
+        
+        set_fields(num_lits, m_lits);
+
+        SASSERT(get_num_literals() == num_lits);
+
+        j->inc_ref();
+        m_justification->set_owner(0); // release ownership
+        dec_ref(m_justification, m);
+        m_justification = j;
+        m_justification->set_owner(this);
+
+        sort_literals();
+
+        justification_stat j_stat;
+        get_justification_stat(m_justification, j_stat);
+
+        m_proof_depth = j_stat.m_proof_depth;
+        
+        SASSERT(m_scope_lvl == j_stat.m_max_scope_lvl);
+        
+        update_parents(j_stat.m_parent_clauses);
+    }
+
+    void clause::display(std::ostream & out, ast_manager & m, bool detailed) {
+        if (get_num_literals() == 0) {
+            out << "empty-clause";
+            return;
+        }
+        out << "#" << m_id << ": (clause ";
+        spc::display(out, get_num_literals(), m_lits, m, detailed);
+        out << ")";
+        if (m_processed)
+            out << "*";
+    }
+
+    void get_clause_stat(unsigned num_lits, literal * lits, clause_stat & stat) {
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal_stat c;
+            lits[i].get_stat(c);
+            stat.m_sym_count   += c.m_sym_count;
+            stat.m_depth        = std::max(stat.m_depth, c.m_depth);
+            stat.m_max_var_idx  = std::max(stat.m_max_var_idx, c.m_max_var_idx);
+            stat.m_const_count += c.m_const_count;
+            stat.m_ground      &= c.m_ground;
+            stat.m_num_lits[static_cast<unsigned>(lits[i].sign())]++;
+        }
+    }
+
+};

src/spc/spc_clause.h

@@ -0,0 +1,152 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_clause.h
+
+Abstract:
+
+    Superposition Calculus Clause
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#ifndef _SPC_CLAUSE_H_
+#define _SPC_CLAUSE_H_
+
+#include"ast.h"
+#include"splay_tree.h"
+#include"use_list.h"
+#include"spc_literal.h"
+#include"spc_justification.h"
+#include"use_list.h"
+
+namespace spc {
+
+    class context;
+
+    /**
+       \brief Superposition Calculus clause.
+    */
+    class clause {
+        struct compare {
+            // ignoring potential overflow/underflow
+            int operator()(clause * c1, clause * c2) const {
+                return static_cast<int>(c1->get_id()) - static_cast<int>(c2->get_id());
+            }
+        };
+    public:
+        typedef splay_tree<clause *, compare> set;
+    private:
+        unsigned            m_id;           // clause unique id
+        unsigned            m_time;         // how old is the clause. 
+        unsigned            m_num_vars;     // approx. number of variables (i.e., max_var_id + 1)
+        unsigned            m_sym_count;    // number of symbols
+        unsigned            m_const_count;  // number of constants
+        unsigned            m_depth;        // depth (i.e., max depth of a literal)
+        unsigned            m_proof_depth;  
+        unsigned            m_scope_lvl;    // which scope level owns the clause
+        unsigned            m_num_lits[2];  // number of positive [0] and negative [1] literals. 
+        unsigned            m_num_lits_capacity; // some of the clause literals can be simplified and removed, this field contains the original number of literals (used for GC).
+        unsigned            m_bidx;         // position on the backtracking stack
+        bool                m_ground:1;
+        bool                m_processed:1;  
+        bool                m_indexed:1;
+        bool                m_has_sel_lit:1;
+        justification *     m_justification;
+        set                 m_children;
+        literal             m_lits[0];
+        friend class context;
+
+        void set_fields(unsigned num_lits, literal * lits);
+        unsigned get_bidx() const { return m_bidx; }
+        void init(unsigned idx, unsigned time);
+        void update_parents(ptr_buffer<clause> & parents);
+        void set_bidx(unsigned idx) { SASSERT(m_bidx == UINT_MAX); m_bidx = idx; }
+        void add_child(clause * c) { m_children.insert(c); }
+        void del_child(clause * c) { m_children.erase(c); }
+        void set_processed(bool f) { m_processed = f; }
+        void set_indexed(bool f) { m_indexed = f; }
+        void sort_literals();
+        /**
+           \brief Release ownership of the justification.
+        */
+        justification * release_justification() { justification * r = m_justification; m_justification = 0; return r; }
+
+        clause(ast_manager & m, unsigned num_lits, literal * lits, justification * p, unsigned scope_lvl);
+
+    public:
+        static clause * mk(ast_manager & m, unsigned num_lits, literal * lits, justification * p, unsigned scope_lvl);
+        void deallocate(ast_manager & m);
+
+        unsigned get_id() const { SASSERT(m_id != UINT_MAX); return m_id; }
+        unsigned get_time() const { return m_time; }
+        unsigned get_symbol_count() const { return m_sym_count; }
+        unsigned get_proof_depth() const { return m_proof_depth; }
+        unsigned get_num_literals() const { return m_num_lits[0] + m_num_lits[1]; }
+        unsigned get_num_literals_capacity() const { return m_num_lits_capacity; }
+        unsigned get_num_pos_literals() const { return m_num_lits[0]; }
+        unsigned get_num_neg_literals() const { return m_num_lits[1]; }
+        unsigned get_depth() const { return m_depth; }
+        unsigned get_const_count() const { return m_const_count; }
+        unsigned get_scope_lvl() const { return m_scope_lvl; }
+        unsigned get_num_vars() const { return m_num_vars; }
+        bool empty() const { return m_num_lits[0] == 0 && m_num_lits[1] == 0; }
+        literal const & get_literal(unsigned idx) const { return m_lits[idx]; }
+        literal & get_literal(unsigned idx) { return m_lits[idx]; }
+        literal * get_literals() const { return const_cast<literal*>(m_lits); }
+        justification * get_justification() const { return m_justification; }
+        bool is_processed() const { return m_processed; }
+        bool is_indexed() const { return m_indexed; }
+        bool is_ground() const { return m_ground; }
+        void select_literal(unsigned idx);
+        bool is_maximal(order & o, literal const & l, unsigned offset = 0, substitution * s = 0) const;
+        bool is_sel_maximal(order & o, literal const & l, unsigned offset = 0, substitution * s = 0) const ;
+        bool is_eligible_for_resolution(order & o, literal const & l, unsigned offset = 0, substitution * s = 0) const;
+        bool is_eligible_for_paramodulation(order & o, literal const & l, unsigned offset = 0, substitution * s = 0) const;
+        bool has_sel_lit() const { return m_has_sel_lit; }
+        void try_to_orient_literals(order & o);
+        void update_lits(ast_manager & m, unsigned num_lits, literal * lits, justification * j);
+
+        void display(std::ostream & out, ast_manager & m, bool detailed = false);
+        unsigned hash() const { return m_id; }
+    };
+
+    typedef ptr_vector<clause> clause_vector;
+
+    /**
+       \brief Clause Statistics (used to build clauses, subsumption, etc).
+    */
+    struct clause_stat : public expr_stat {
+        unsigned m_num_lits[2];
+        clause_stat() {
+            m_num_lits[0] = 0;
+            m_num_lits[1] = 0;
+        }
+    };
+
+    /**
+       \brief Compute the statistics for a clause with num_lits
+       literals lits, and store the results in stat.
+    */
+    void get_clause_stat(unsigned num_lits, literal * lits, clause_stat & stat);
+
+    /**
+       \brief A mapping from clause-id's to clauses
+    */
+    class id2clause {
+        ptr_vector<clause> m_clauses;
+    public:
+        void insert(clause * c) { return m_clauses.setx(c->get_id(), c, 0); }
+        void erase(clause * c) { unsigned id = c->get_id(); if (id < m_clauses.size()) m_clauses[id] = 0; }
+        clause * operator()(unsigned id) const { return m_clauses.get(id, 0); }
+    };
+};
+
+#endif /* _SPC_CLAUSE_H_ */
+

src/spc/spc_clause_pos_set.h

@@ -0,0 +1,58 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_clause_pos_set.h
+
+Abstract:
+
+    A set of pairs (clause, index).
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-16.
+
+Revision History:
+
+--*/
+#ifndef _SPC_CLAUSE_POS_SET_H_
+#define _SPC_CLAUSE_POS_SET_H_
+
+#include"hashtable.h"
+
+namespace spc {
+
+    typedef std::pair<clause *, unsigned> clause_pos_pair;
+
+    class clause_pos_entry {
+        clause_pos_pair m_data;
+    public:
+        typedef clause_pos_pair data;
+        clause_pos_entry() { m_data.first = 0; }
+        unsigned get_hash() const { return m_data.first->get_id(); }
+        bool is_free() const { return m_data.first == 0; }
+        bool is_deleted() const { return m_data.first == reinterpret_cast<clause *>(1); }
+        bool is_used() const { 
+            return m_data.first != reinterpret_cast<clause *>(0) && m_data.first != reinterpret_cast<clause *>(1);
+        }
+        clause_pos_pair const & get_data() const { return m_data; }
+        clause_pos_pair & get_data() { return m_data; }
+        void set_data(clause_pos_pair const & d) { 
+            SASSERT(d.first != 0 && d.first != reinterpret_cast<clause*>(1));
+            m_data = d; 
+        }
+        void set_hash(unsigned h) { SASSERT(m_data.first->get_id() == h); }
+        void mark_as_deleted() { m_data.first = reinterpret_cast<clause *>(1); }
+        void mark_as_free() { m_data.first = 0; }
+    };
+    
+    struct clause_pos_pair_hash {
+        unsigned operator()(clause_pos_pair const & p) const { return p.first->get_id(); }
+    };
+
+    typedef core_hashtable<clause_pos_entry, clause_pos_pair_hash, default_eq<clause_pos_pair> > clause_pos_set;
+};
+
+#endif /* _SPC_CLAUSE_POS_SET_H_ */
+

src/spc/spc_clause_selection.cpp

@@ -0,0 +1,121 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_clause_selection.cpp
+
+Abstract:
+
+    Superposition Calculus Clause Selection
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#include"spc_clause_selection.h"
+
+namespace spc {
+  
+    const unsigned default_heap_size = 1024;
+
+    clause_selection::clause_selection(unsigned num_heaps, clause_eval * const * fs, unsigned * slot_size):
+        m_curr_slot(0),
+        m_counter(0),
+        m_fs(num_heaps, fs) {
+        SASSERT(num_heaps > 0);
+        for (unsigned i = 0; i < num_heaps; i++) {
+            m_heaps.push_back(alloc(heap<lt>, default_heap_size, lt(m_id2clause, *(fs[i]))));
+            SASSERT(slot_size[i] > 0);
+            m_slot_size.push_back(slot_size[i]);
+        }
+    }
+
+    clause_selection::~clause_selection() {
+        std::for_each(m_heaps.begin(), m_heaps.end(), delete_proc<heap<lt> >());
+        std::for_each(m_fs.begin(), m_fs.end(), delete_proc<clause_eval>());
+    }
+
+    void clause_selection::reserve(unsigned cid) {
+        unsigned capacity = m_heaps[0]->get_bounds();
+        if (cid >= capacity) {
+            unsigned new_capacity = 2 * cid + 1;
+            SASSERT(cid < new_capacity);
+            ptr_vector<heap<lt> >::iterator it  = m_heaps.begin();
+            ptr_vector<heap<lt> >::iterator end = m_heaps.end();
+            for (; it != end; ++it) {
+                heap<lt> * h = *it;
+                h->reserve(new_capacity);;
+            }
+        }
+    }
+
+    void clause_selection::reset() {
+        ptr_vector<heap<lt> >::iterator it  = m_heaps.begin();
+        ptr_vector<heap<lt> >::iterator end = m_heaps.end();
+        for (; it != end; ++it) {
+            heap<lt> * h = *it;
+            h->reset();
+        }
+    }
+    
+    void clause_selection::insert(clause * c) {
+        reserve(c->get_id());
+        m_id2clause.insert(c);
+        ptr_vector<heap<lt> >::iterator it  = m_heaps.begin();
+        ptr_vector<heap<lt> >::iterator end = m_heaps.end();
+        for (; it != end; ++it) {
+            heap<lt> * h = *it;
+            h->insert(c->get_id());
+        }
+    }
+    
+    void clause_selection::erase(clause * c) {
+        // remark: it is not necessary to remove c from m_id2clause
+        ptr_vector<heap<lt> >::iterator it  = m_heaps.begin();
+        ptr_vector<heap<lt> >::iterator end = m_heaps.end();
+        SASSERT(it != end);
+        if (!(*it)->contains(c->get_id()))
+            return;
+        for (; it != end; ++it) {
+            heap<lt> * h = *it;
+            h->erase(c->get_id());
+        }
+    }
+
+    bool clause_selection::empty() const {
+        ptr_vector<heap<lt> >::const_iterator it  = m_heaps.begin();
+        ptr_vector<heap<lt> >::const_iterator end = m_heaps.end();
+        for (; it != end; ++it)
+            if (!(*it)->empty())
+                return false;
+        return true;
+    }
+    
+    clause * clause_selection::get_best() {
+        heap<lt> * h = m_heaps[m_curr_slot];
+        if (h->empty())
+            return 0;
+        unsigned cid = m_heaps[m_curr_slot]->erase_min();
+        clause * c = m_id2clause(cid);
+        SASSERT(c);
+        // remove clause from the other heaps
+        unsigned num_heaps = m_heaps.size();
+        for (unsigned i = 0; i < num_heaps; i++) {
+            if (m_curr_slot != i)
+                m_heaps[i]->erase(cid);
+        }
+        // remark: it is not necessary to remove c from m_id2clause
+        m_counter++;
+        if (m_counter >= m_slot_size[m_curr_slot]) {
+            m_counter = 0;
+            m_curr_slot++;
+            if (m_curr_slot >= m_slot_size.size()) 
+                m_curr_slot = 0;
+        }
+        return c;
+    }
+};

src/spc/spc_clause_selection.h

@@ -0,0 +1,85 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_clause_selection.h
+
+Abstract:
+
+    Superposition Calculus Clause Selection
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#ifndef _SPC_CLAUSE_SELECTION_H_
+#define _SPC_CLAUSE_SELECTION_H_
+
+#include"spc_clause.h"
+#include"heap.h"
+
+namespace spc {
+    
+    /**
+       \brief Abstract functor for evaluating how 'good' a clause is.
+       Smaller values mean better clauses.
+    */
+    struct clause_eval {
+        virtual ~clause_eval() {}
+        virtual unsigned operator()(clause * c) const = 0;
+    };
+
+    /**
+       \brief Clause selection heuristic. It supports different priority queues.
+    */
+    class clause_selection {
+        class lt {
+            id2clause &   m_id2clause;
+            clause_eval & m_func;
+        public:
+            lt(id2clause & m, clause_eval & f):
+                m_id2clause(m), m_func(f) {}
+            bool operator()(int cidx1, int cidx2) const {
+                return m_func(m_id2clause(cidx1)) < m_func(m_id2clause(cidx2));
+            }
+        };
+
+        id2clause                m_id2clause;
+        ptr_vector<heap<lt> >    m_heaps;
+        unsigned_vector          m_slot_size;
+        unsigned                 m_curr_slot;
+        unsigned                 m_counter;
+        ptr_vector<clause_eval>  m_fs;
+        void reserve(unsigned cid);
+    public:
+        clause_selection(unsigned num_heaps, clause_eval * const * fs, unsigned * slots);
+        ~clause_selection();
+        void insert(clause * c);
+        void erase(clause * c);
+        bool empty() const;
+        void reset();
+        clause * get_best();
+    };
+
+    struct symbol_count_clause_eval : public clause_eval {
+        virtual ~symbol_count_clause_eval() {}
+        virtual unsigned operator()(clause * c) const { return c->get_symbol_count(); }
+    };
+     
+    struct time_clause_eval : public clause_eval {
+        virtual ~time_clause_eval() {}
+        virtual unsigned operator()(clause * c) const { return c->get_time(); }
+    };
+    
+    struct proof_depth_clause_eval : public clause_eval {
+        virtual ~proof_depth_clause_eval() {}
+        virtual unsigned operator()(clause * c) const { return c->get_proof_depth(); }
+    };
+};
+
+#endif /* _SPC_CLAUSE_SELECTION_H_ */
+

src/spc/spc_context.cpp

@@ -0,0 +1,504 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_context.cpp
+
+Abstract:
+
+    Superposition Calculus Engine
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#include"spc_context.h"
+#include"buffer.h"
+#include"ast_pp.h"
+#include"ast_ll_pp.h"
+#include"ast_smt2_pp.h"
+#include"warning.h"
+
+namespace spc {
+
+    context::context(ast_manager & m, order & o, clause_selection & cs, literal_selection & ls, simplifier & s, spc_params & params):
+        m_manager(m),
+        m_params(params),
+        m_alloc(m.get_allocator()),
+        m_order(o),
+        m_cls_sel(cs),
+        m_lit_sel(ls),
+        m_simplifier(s),
+        m_time(0),
+        m_scope_lvl(0),
+        m_sem_taut(m),
+        m_asserted_literals(m),
+        m_rewriter(m, s, m_order, m_asserted_literals),
+        m_der(m), 
+        m_subsumption(m, m_asserted_literals, params),
+        m_eq_resolution(m, m_order, m_stats),
+        m_factoring(m, m_order, m_stats),
+        m_superposition(m, m_order, m_stats),
+        m_unsat(0) {
+        m_order.reserve_offsets(3);
+    }
+
+    context::~context() {
+        reset();
+    }
+
+    void context::reset() {
+        m_cls_sel.reset();
+        m_time      = 0;
+        m_scope_lvl = 0;
+
+        if (m_unsat)
+            m_unsat = 0;
+        for (unsigned i = 0; i <= m_scope_lvl; i++) {
+            del_clauses(i);
+            if (i < m_clauses_to_unfreeze.size())
+                m_clauses_to_unfreeze[i].reset();
+        }
+
+        m_asserted_literals.reset();
+        m_rewriter.reset();
+        m_subsumption.reset();
+        m_superposition.reset();
+    }
+
+    /**
+       \brief Insert the given clause into the indexes of processed clauses.
+    */
+    void context::insert_index(clause * cls) {
+        TRACE("insert_index", tout << "indexing clause, num_vars: " << cls->get_num_vars() << "\n"; 
+              cls->display(tout, m_manager); tout << "\n";);
+        m_order.reserve_vars(cls->get_num_vars());
+        m_lit_sel(cls);
+        m_asserted_literals.insert(cls);
+        m_rewriter.insert(cls);
+        m_subsumption.insert(cls);
+        m_superposition.insert(cls);
+        cls->set_indexed(true);
+    }
+
+    void context::erase_index(clause * cls) {
+        if (cls->is_indexed()) {
+            m_asserted_literals.erase(cls);
+            m_rewriter.erase(cls);
+            m_subsumption.erase(cls);
+            m_superposition.erase(cls);
+            cls->set_indexed(false);
+        }
+    }
+
+    void context::set_conflict(clause * cls) {
+        SASSERT(cls->get_num_literals() == 0);
+        m_unsat = cls;
+        if (m_params.m_spc_trace) {
+            cls->display(std::cout, m_manager); std::cout << " "; 
+            cls->get_justification()->display(std::cout);
+            std::cout << "\n";
+            std::cout.flush();
+        }
+    }
+
+    void context::del_clause(clause * cls) {
+        TRACE("context", tout << "deleting clause:\n"; cls->display(tout, m_manager); tout << "\n";);
+        m_stats.m_num_del_clause++;
+
+        erase_index(cls);
+        if (!cls->is_processed())
+            m_cls_sel.erase(cls);
+
+        unsigned scope_lvl = cls->get_scope_lvl();
+        unsigned bidx      = cls->get_bidx();
+        m_clauses_to_delete[scope_lvl][bidx] = 0;
+
+        cls->deallocate(m_manager);
+    }
+
+    void context::freeze_clause_until(clause * cls, unsigned scope_lvl) {
+        if (cls->get_scope_lvl() >= scope_lvl) {
+            del_clause(cls);
+            return;
+        }
+        TRACE("context", tout << "freezing clause until: " << scope_lvl << ":\n"; cls->display(tout, m_manager); tout << "\n";);
+        if (scope_lvl >= m_clauses_to_unfreeze.size())
+            m_clauses_to_unfreeze.resize(scope_lvl+1, clause_vector());
+
+        erase_index(cls);
+        cls->set_processed(false);
+
+        m_clauses_to_unfreeze[scope_lvl].push_back(cls);
+    }
+
+    void context::unfreeze_clause(clause * cls) {
+        TRACE("context", tout << "unfreezing clausel: "; cls->display(tout, m_manager); tout << "\n";);
+        SASSERT(!cls->is_processed());
+        m_cls_sel.insert(cls);
+    }
+
+    void context::init_clause(clause * cls) {
+        m_stats.m_num_mk_clause++;
+
+        cls->init(m_cls_id_gen.mk(), m_time);
+        m_time++;
+        unsigned scope_lvl = cls->get_scope_lvl();
+
+        if (scope_lvl >= m_clauses_to_delete.size())
+            m_clauses_to_delete.resize(scope_lvl+1, clause_vector());
+        
+        clause_vector & cv = m_clauses_to_delete[scope_lvl];
+        unsigned bidx = cv.size();
+        cv.push_back(cls);
+        cls->set_bidx(bidx);
+    }
+
+    clause * context::mk_clause(unsigned num_lits, literal * lits, justification * p, unsigned scope_lvl) {
+        clause * cls = clause::mk(m_manager, num_lits, lits, p, scope_lvl);
+        init_clause(cls);
+        return cls;
+    }
+
+    void context::assert_expr(expr * n, proof * p, unsigned scope_lvl) {
+        TRACE("spc_assert_expr", tout << mk_ismt2_pp(n, m_manager) << "\n";);
+        SASSERT(scope_lvl <= m_scope_lvl);
+        justification_ref ref(m_manager);
+        ref = justification_proof_wrapper::mk(p, m_manager);
+        assert_expr(n, ref, scope_lvl);
+    }
+
+    void invalid_clause(expr * n) {
+        warning_msg("ignoring formula containing an universally quantified boolean variable.");
+    }
+
+    void context::assert_expr(expr * n, justification * p, unsigned scope_lvl) {
+        SASSERT(scope_lvl <= m_scope_lvl);
+        buffer<literal> lits;
+        if (is_forall(n))
+            n = to_quantifier(n)->get_expr();
+        if (m_manager.is_or(n)) {
+            unsigned num = to_app(n)->get_num_args();
+            for (unsigned i = 0; i < num; i++) {
+                expr * c    = to_app(n)->get_arg(i);
+                bool is_neg = m_manager.is_not(c);
+                if (is_var(c) || (is_neg && is_var(to_app(c)->get_arg(0)))) {
+                    invalid_clause(n);
+                    return;
+                }
+                if (is_neg)
+                    lits.push_back(literal(to_app(c)->get_arg(0), true));
+                else 
+                    lits.push_back(literal(c, false));
+            }
+        }
+        else if (m_manager.is_false(n)) {
+            // skip
+        }
+        else if (m_manager.is_not(n)) {
+            if (is_var(to_app(n)->get_arg(0))) {
+                invalid_clause(n);
+                return;
+            }
+            lits.push_back(literal(to_app(n)->get_arg(0), true));
+        }
+        else {
+            if (is_var(n)) {
+                invalid_clause(n);
+                return;
+            }
+            lits.push_back(literal(n, false));
+        }
+        
+        if (trivial(lits.size(), lits.c_ptr()))
+            return;
+
+        clause * cls = mk_clause(lits.size(), lits.c_ptr(), p, scope_lvl);
+        m_cls_sel.insert(cls);
+        if (cls->get_num_literals() == 0)
+            set_conflict(cls);
+    }
+
+    /**
+       \brief Return true if the given clause (set of literals) is trivial.
+       That is, it contains the literal s = s or complementary literals.
+    */
+    bool context::trivial(unsigned num_lits, literal * lits) {
+        SASSERT(m_found_literals.empty());
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal l       = lits[i];
+            if (m_found_literals.contains_neg(l) || l.is_true(m_manager)) {
+                m_found_literals.reset();
+                m_stats.m_num_trivial++;
+                return true;
+            }
+            m_found_literals.insert(l);
+        }
+        m_found_literals.reset();
+        return false;
+    }
+
+    bool context::trivial(clause * cls) {
+        return trivial(cls->get_num_literals(), cls->get_literals());
+    }
+    
+    /**
+       \brief Simplify the given clause using the set of processed clauses.
+       Return the simplified clause.
+    */
+    clause * context::simplify(clause * cls) {
+        clause * old_cls = cls;
+        m_der(cls);
+        cls = m_rewriter(old_cls);
+        if (cls != old_cls) {
+            // freeze old clause until simplified clause is deleted.
+            freeze_clause_until(old_cls, cls->get_scope_lvl());
+            init_clause(cls);
+            m_stats.m_num_simplified++;
+        }
+        m_der(cls);
+        return cls;
+    }
+    
+    /**
+       \brief Use the given clause to simplify the set of processed clauses.
+       
+       \remark: processed clauses that can be simplified, are moved to the
+       set of unprocessed clauses.
+    */
+    void context::simplify_processed(clause * cls) {
+        // TODO
+    }
+
+    /**
+       \brief Return true if the clause is redundant. 
+    */
+    bool context::redundant(clause * cls) {
+        int r_scope_lvl = -1;
+        if (trivial(cls)) {
+            TRACE("redundant", tout << "clause is trivial:\n"; cls->display(tout, m_manager); tout << "\n";);
+            r_scope_lvl = 0;
+        }
+        else if (m_sem_taut(cls->get_num_literals(), cls->get_literals())) {
+            TRACE("redundant", tout << "clause is a semantic tautology:\n"; cls->display(tout, m_manager); tout << "\n";);
+            r_scope_lvl = 0;
+        }
+        else {
+            clause * subsumer = m_subsumption.forward(cls);
+            if (subsumer != 0) {
+                TRACE("redundant", tout << "clause was subsumed: "; cls->display(tout, m_manager);
+                      tout << "\nsubsumer:\n"; subsumer->display(tout, m_manager); tout << "\n";);
+                r_scope_lvl = subsumer->get_scope_lvl();
+                m_stats.m_num_subsumed++;
+            }
+        }
+
+        if (r_scope_lvl >= 0) {
+            m_stats.m_num_redundant++;
+            TRACE("spc_saturate", tout << "clause is redundant until level: " << r_scope_lvl << " ...\n";);
+            freeze_clause_until(cls, r_scope_lvl);
+            return true;
+        }
+
+        return false;
+    }
+
+    /**
+       \brief Process a newly generated clause.
+    */
+    void context::process_new_clause(clause * cls) {
+        if (cls) {
+            SASSERT(cls->get_justification() != 0);
+            init_clause(cls);
+            if (trivial(cls)) {
+                del_clause(cls);
+                return;
+            }
+            cls = simplify(cls);
+            if (trivial(cls)) {
+                del_clause(cls);
+                return;
+            }
+            // if (!redundant(cls)) {
+            m_cls_sel.insert(cls);
+            if (cls->get_num_literals() == 0)
+                set_conflict(cls);
+            // }
+        }
+    }
+
+    /**
+       \brief Apply superposition (left&right), resolution, (equality) factoring, and equality resolution
+       with the given clause and the set of processed clauses.
+    */
+    void context::generate(clause * cls) {
+        m_new_clauses.reset();
+        m_eq_resolution(cls, m_new_clauses);
+        m_factoring(cls, m_new_clauses);
+        m_superposition(cls, m_new_clauses);
+
+        ptr_vector<clause>::iterator it  = m_new_clauses.begin();
+        ptr_vector<clause>::iterator end = m_new_clauses.end();
+        for (; it != end; ++it) {
+            TRACE("spc_generate", tout << "new generated clause:\n"; (*it)->display(tout, m_manager); tout << "\n";);
+            process_new_clause(*it);
+        }
+    }
+
+    void context::saturate(unsigned threshold) {
+        if (inconsistent())
+            return;
+        TRACE("spc_saturate", tout << "initial state:\n"; display(tout););
+        unsigned i = 0;
+        ptr_buffer<clause> to_simplify;
+        while (i < threshold && !processed_all()) {
+            i++;
+            m_stats.m_num_processed++;
+            clause * cls = m_cls_sel.get_best();
+            if (m_params.m_spc_trace) {
+                cls->display(std::cout, m_manager); std::cout << " "; 
+                cls->get_justification()->display(std::cout);
+                std::cout << "\n";
+                std::cout.flush();
+            }
+            cls->set_processed(true);
+            TRACE("spc_saturate", tout << "get best: "; cls->display(tout, m_manager); tout << "\n";);
+            cls = simplify(cls);
+            
+            TRACE("spc_saturate", tout << "clause after simplification: "; cls->display(tout, m_manager); tout << "\n";);
+            if (redundant(cls))
+                continue;
+            if (cls->empty()) {
+                set_conflict(cls);
+                break;
+            }
+            cls->try_to_orient_literals(m_order);
+            simplify_processed(cls);
+            insert_index(cls);
+            generate(cls);
+            if (inconsistent())
+                break;
+        }
+
+        TRACE("spc_saturate", tout << "final state:\n"; display(tout););
+
+#if 0
+        IF_VERBOSE(10000, 
+                   display(std::cout););
+        display_statistics(std::cout);
+        if (m_unsat && m_manager.fine_grain_proofs()) {
+            std::cout << mk_ll_pp(m_unsat->get_justification()->get_proof(), m_manager);
+        }
+#endif 
+    }
+
+    void context::push_scope() {
+        m_scope_lvl++;
+        m_time_trail.push_back(m_time);
+    }
+
+    void context::del_clauses(unsigned scope_lvl) {
+        if (scope_lvl < m_clauses_to_delete.size()) {
+            clause_vector & cv = m_clauses_to_delete[m_scope_lvl];
+            clause_vector::iterator it  = cv.begin();
+            clause_vector::iterator end = cv.end();
+            for (; it != end; ++it) {
+                clause * cls = *it;
+                if (cls)
+                    del_clause(cls);
+            }
+            cv.reset();
+        }
+    }
+
+    void context::unfreeze_clauses(unsigned scope_lvl) {
+        if (scope_lvl < m_clauses_to_unfreeze.size()) {
+            clause_vector & cv = m_clauses_to_unfreeze[m_scope_lvl];
+            clause_vector::iterator it  = cv.begin();
+            clause_vector::iterator end = cv.end();
+            for (; it != end; ++it)
+                unfreeze_clause(*it);
+            cv.reset();
+        }
+    }
+
+    void context::pop_scope(unsigned num_scopes) {
+        SASSERT(num_scopes >= m_scope_lvl);
+        unsigned new_lvl = m_scope_lvl - num_scopes;
+        m_time = m_time_trail[new_lvl];
+        m_time_trail.shrink(new_lvl);
+
+        if (m_unsat && new_lvl < m_unsat->get_scope_lvl()) 
+            m_unsat = 0;
+        
+        while (m_scope_lvl > new_lvl) {
+            del_clauses(m_scope_lvl);
+            unfreeze_clauses(m_scope_lvl);
+            m_scope_lvl --;
+        }
+    }
+
+    void context::display(std::ostream & out, vector<clause_vector> const & cvs, unsigned scope_lvl, bool frozen) const {
+        if (scope_lvl < cvs.size()) {
+            bool first = true;
+            clause_vector const & cv = cvs[scope_lvl];
+            clause_vector::const_iterator it  = cv.begin();
+            clause_vector::const_iterator end = cv.end();
+            for (; it != end; ++it) {
+                clause * cls = *it;
+                if (cls) {
+                    if (first) {
+                        out << "level " << scope_lvl << ":\n";
+                        first = false;
+                    }
+                    cls->display(out, m_manager);
+                    if (frozen)
+                        out << " [frozen]";
+                    out << "\n";
+                }
+            }
+        }
+    }
+    
+    void context::display(std::ostream & out) const {
+        for (unsigned i = 0; i <= m_scope_lvl; i++) {
+            display(out, m_clauses_to_delete, i, false);
+            display(out, m_clauses_to_unfreeze, i, true);
+        }
+    }
+
+    void context::display_statistics(std::ostream & out) const {
+        m_stats.display(out);
+    }
+    
+
+/**
+   Generate new clauses
+
+   5) Object equality resolution 1
+   
+   (R or X = i)  
+   ==>
+   sigma(R)
+
+   sigma = { X -> j }
+   where i and j are distinct objects
+   sigma(X = i) is not smaller or equal than any other literal in the clause
+
+   6) Object equality resolution 2
+   
+   (R or X = Y)
+   ==>
+   sigma(R)
+
+   sigma = { X -> i, Y -> j }
+   For every pair of distinct objects i and j
+   sigma(X = Y) is not smaller or equal than any other literal in the clause
+   
+*/
+
+};

src/spc/spc_context.h

@@ -0,0 +1,122 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_context.h
+
+Abstract:
+
+    Superposition Calculus Engine
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#ifndef _SPC_CONTEXT_H_
+#define _SPC_CONTEXT_H_
+
+#include"spc_params.h"
+#include"spc_clause.h"
+#include"spc_clause_selection.h"
+#include"spc_literal_selection.h"
+#include"spc_semantic_tautology.h"
+#include"spc_rewriter.h"
+#include"spc_asserted_literals.h"
+#include"spc_subsumption.h"
+#include"spc_eq_resolution.h"
+#include"spc_factoring.h"
+#include"spc_superposition.h"
+#include"spc_statistics.h"
+#include"spc_der.h"
+#include"substitution_tree.h"
+#include"order.h"
+
+namespace spc {
+
+    /**
+       \brief Logical context of the superposition calculus engine.
+    */
+    class context {
+    public:
+        statistics                  m_stats;
+    protected:
+        typedef clause::set clause_set;
+        
+        ast_manager &               m_manager;
+        spc_params &                m_params;
+        small_object_allocator &    m_alloc;
+        order &                     m_order;
+        clause_selection &          m_cls_sel;
+        literal_selection &         m_lit_sel;
+        simplifier &                m_simplifier;
+        unsigned                    m_time;
+        unsigned                    m_scope_lvl;
+        id_gen                      m_cls_id_gen;
+        found_literals              m_found_literals;
+        semantic_tautology          m_sem_taut;
+        asserted_literals           m_asserted_literals;
+        rewriter                    m_rewriter;
+        der                         m_der;
+        subsumption                 m_subsumption;
+        eq_resolution               m_eq_resolution;
+        factoring                   m_factoring;
+        superposition               m_superposition;
+        vector<clause_vector>       m_clauses_to_unfreeze;
+        vector<clause_vector>       m_clauses_to_delete;
+        unsigned_vector             m_time_trail;
+        clause *                    m_unsat;
+        ptr_vector<clause>          m_new_clauses;
+
+        void insert_index(clause * cls);
+        void erase_index(clause * cls);
+
+        void init_clause(clause * cls);
+        clause * mk_clause(unsigned num_lits, literal * lits, justification * p, unsigned scope_lvl);
+
+        void del_clause(clause * cls);
+        void del_clauses(unsigned scope_lvl);
+
+        void freeze_clause_until(clause * cls, unsigned scope_lvl);
+        void unfreeze_clause(clause * cls);
+        void unfreeze_clauses(unsigned scope_lvl);
+
+        bool trivial(unsigned num_lits, literal * lits);
+        bool trivial(clause * cls);
+        clause * simplify(clause * cls);
+        void simplify_processed(clause * cls);
+        bool redundant(clause * cls);
+        void generate(clause * cls);
+        void process_new_clause(clause * cls);
+
+        void display(std::ostream & out, vector<clause_vector> const & cvs, unsigned scope_lvl, bool frozen) const;
+        
+        void set_conflict(clause * cls);
+
+    public:
+        context(ast_manager & m, order & o, clause_selection & cs, literal_selection & ls, simplifier & s, spc_params & params);
+        ~context();
+        
+        simplifier & get_simplifier() { return m_simplifier; }
+        order & get_order() { return m_order; }
+        ast_manager & get_manager() { return m_manager; }
+
+        unsigned get_scope_lvl() const { return m_scope_lvl; }
+
+        void assert_expr(expr * n, proof * p, unsigned scope_lvl = 0);
+        void assert_expr(expr * n, justification * p, unsigned scope_lvl = 0);
+        void saturate(unsigned threshold);
+        bool inconsistent() const { return m_unsat != 0; }
+        bool processed_all() const { return m_cls_sel.empty(); }
+        void push_scope();
+        void pop_scope(unsigned num_scopes);
+        void reset();
+        void display(std::ostream & out) const;
+        void display_statistics(std::ostream & out) const;
+    };
+};
+
+#endif /* _SPC_CONTEXT_H_ */

src/spc/spc_decl_plugin.cpp

@@ -0,0 +1,135 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_decl_plugin.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-12.
+
+Revision History:
+
+--*/
+#include"spc_decl_plugin.h"
+
+std::ostream & operator<<(std::ostream & out, spc_op_kind k) {
+    switch (k) {
+    case PR_DEMODULATION:              out << "demod"; break;
+    case PR_SPC_REWRITE:               out << "rewrite"; break;
+    case PR_SPC_RESOLUTION:            out << "res"; break;
+    case PR_SUPERPOSITION:             out << "sup"; break;
+    case PR_EQUALITY_RESOLUTION:       out << "eq_res"; break;
+    case PR_FACTORING:                 out << "fact"; break;
+    case PR_SPC_DER:                   out << "der"; break;
+    case PR_SPC_ASSERTED:              out << "asserted"; break;
+    default:                           out << "unknown"; break;
+    }
+    return out;
+}
+
+spc_decl_plugin::spc_decl_plugin() :
+    m_demodulation("demod"),
+    m_spc_rewrite("sp-rw"),
+    m_spc_resolution("sp-res"),
+    m_superposition("sp"),
+    m_equality_resolution("eq-res"),
+    m_factoring("fact"),
+    m_spc_der("spc-der") {
+}
+    
+spc_decl_plugin::~spc_decl_plugin() {
+}
+
+sort * spc_decl_plugin::mk_sort(decl_kind k, unsigned num_parameters, parameter const* parameters) {
+    UNREACHABLE();
+    return 0;
+}
+
+func_decl * spc_decl_plugin::mk_func_decl(decl_kind k, unsigned num_parameters, parameter const * parameters, 
+                                          unsigned arity, sort * const * domain, sort * range) {
+    
+#define MK_PROOF(SYM) m_manager->mk_func_decl(SYM, arity, domain, m_manager->mk_proof_sort(), func_decl_info(m_family_id, k))
+    
+    SASSERT(num_parameters == 0);
+    switch (k) {
+        /*
+          #1: (forall (x) (= t[x] s[x]))
+          [demod #1] (= t[a] s[a])
+        */
+    case PR_DEMODULATION:           return MK_PROOF(m_demodulation);
+        /*
+          Justifies a rewriting (simplification step) in the superposition engine.
+          It has n+1 antecedents. The first antecedent is the clause being simplified.
+          The other antecedents are demodulators.
+          The consequent is the simplied clause.
+        */
+    case PR_SPC_REWRITE:            return MK_PROOF(m_spc_rewrite);
+        /*
+          Resolution proof:
+
+          #1: (or C l)
+          #2: (or D (not l'))
+          [sp-res #1 #2]: sigma(or C D)
+
+          where sigma is the mgu of l and l'
+
+        */
+    case PR_SPC_RESOLUTION:         return MK_PROOF(m_spc_resolution);
+        /*
+          Superposition proof:
+
+          #1: (or (= s t) R)
+          #2: D[u]
+          [sp #1 #2]: sigma(or R D[t])
+
+          where sigma is the mgu(u, s)
+        */
+    case PR_SUPERPOSITION:          return MK_PROOF(m_superposition);
+        /*
+          Equality resolution proof:
+
+          #1: (or (not (= s t)) R)
+          [eq-res #1]: sigma R
+
+          where sigma is the mgu of s and t.
+        */
+    case PR_EQUALITY_RESOLUTION:    return MK_PROOF(m_equality_resolution);
+        /*
+          Proof object for factoring and equality-factoring:
+
+          #1: (or P[t] P[s] R)
+          [fact #1]: sigma(or P[t] R)
+
+          where sigma is the mgu(t,s)
+
+          #1: (or (= s t) (= u v) R)
+          [fact #1]: sigma(or (not (= t v)) (= u v) R)
+
+          where sigma = mgu(s, u)
+        */
+    case PR_FACTORING:              return MK_PROOF(m_factoring);
+        /*
+          Proof object for destructive equality resolution:
+
+          #1: (or (not (= x t)) C[x])
+          [spc-der #1]: C[t]
+
+          t does not contain x.
+          
+          Several variables may be eliminated simultaneously.
+        */
+    case PR_SPC_DER:                return MK_PROOF(m_spc_der);
+    default:
+        UNREACHABLE();
+        return 0;
+    }
+
+}
+
+

src/spc/spc_decl_plugin.h

@@ -0,0 +1,61 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_decl_plugin.h
+
+Abstract:
+
+    Proof declarations for Superposition Calculus Engine.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-12.
+
+Revision History:
+
+--*/
+#ifndef _SPC_DECL_PLUGIN_H_
+#define _SPC_DECL_PLUGIN_H_
+
+#include"ast.h"
+
+enum spc_op_kind {
+    PR_DEMODULATION,
+    PR_SPC_REWRITE,
+    PR_SPC_RESOLUTION,
+    PR_SUPERPOSITION,
+    PR_EQUALITY_RESOLUTION,
+    PR_FACTORING,
+    PR_SPC_DER,
+    PR_SPC_ASSERTED,
+    PR_SPC_LAST_ID
+};
+
+std::ostream & operator<<(std::ostream & out, spc_op_kind k);
+  
+class spc_decl_plugin : public decl_plugin {
+    symbol m_demodulation;
+    symbol m_spc_rewrite;
+    symbol m_spc_resolution;
+    symbol m_superposition;
+    symbol m_equality_resolution;
+    symbol m_factoring;
+    symbol m_spc_der;
+
+public:
+    spc_decl_plugin();
+    
+    virtual ~spc_decl_plugin();
+
+    virtual decl_plugin * mk_fresh() { return alloc(spc_decl_plugin); }
+    
+    virtual sort * mk_sort(decl_kind k, unsigned num_parameters, parameter const* parameters);
+    
+    virtual func_decl * mk_func_decl(decl_kind k, unsigned num_parameters, parameter const * parameters, 
+                                     unsigned arity, sort * const * domain, sort * range);
+};
+
+#endif /* _SPC_DECL_PLUGIN_H_ */
+

src/spc/spc_der.cpp

@@ -0,0 +1,80 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_der.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-17.
+
+Revision History:
+
+--*/
+#include"spc_der.h"
+#include"occurs.h"
+
+namespace spc {
+
+    der::der(ast_manager & m):
+        m_manager(m),
+        m_subst(m),
+        m_spc_fid(m.get_family_id("spc")) {
+        m_subst.reserve_offsets(1);
+    }
+
+    void der::apply(clause * cls, unsigned j, expr * lhs, expr * rhs) {
+        TRACE("der", tout << "applying der at: " << j << "\n"; cls->display(tout, m_manager); tout << "\n";);
+        m_subst.reserve_vars(cls->get_num_vars());
+        m_subst.reset();
+        m_subst.insert(expr_offset(lhs, 0), expr_offset(rhs, 0));
+        literal_buffer new_lits(m_manager);
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            if (i != j) {
+                literal const & l = cls->get_literal(i);
+                expr_ref new_atom(m_manager);
+                m_subst.apply(l.atom(), new_atom);
+                new_lits.push_back(literal(new_atom, l.sign()));
+            }
+        }
+        justification * js = mk_der_justification(m_manager, m_spc_fid, cls->get_justification(), new_lits.size(), new_lits.c_ptr());
+        cls->update_lits(m_manager, new_lits.size(), new_lits.c_ptr(), js);
+    }
+
+    bool der::apply(clause * cls) {
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l = cls->get_literal(i);
+            if (l.sign() && m_manager.is_eq(l.atom())) {
+                expr * lhs = l.lhs();
+                expr * rhs = l.rhs();
+                if (is_var(lhs) && !occurs(lhs, rhs)) {
+                    apply(cls, i, lhs, rhs);
+                    return true;
+                }
+                else if (is_var(rhs) && !occurs(rhs, lhs)) {
+                    apply(cls, i, rhs, lhs);
+                    return true;
+                }
+            }
+        }
+        return false;
+    }
+    
+    /**
+       \brief Clause cls is destructively updated.
+    */
+    void der::operator()(clause * cls) {
+        while(apply(cls))
+            ;
+    }
+
+
+};
+

src/spc/spc_der.h

@@ -0,0 +1,52 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_der.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-17.
+
+Revision History:
+
+--*/
+#ifndef _SPC_DER_H_
+#define _SPC_DER_H_
+
+#include"spc_clause.h"
+
+namespace spc {
+
+    /**
+       \brief Functor for applying destructive equality resolution.
+       This is similar to the Functor in der.h, but this one applies
+       the simplification on clauses instead of ast's.
+
+       x != s or R
+       ==>
+       sigma(R)
+       
+       where
+       sigma = mgu(x, s)
+    */
+    class der { 
+        ast_manager &   m_manager;
+        substitution    m_subst;
+        unsigned_vector m_to_keep;
+        family_id       m_spc_fid;
+        void apply(clause * cls, unsigned j, expr * lhs, expr * rhs);
+        bool apply(clause * cls);
+    public:
+        der(ast_manager & m);
+        void operator()(clause * cls);
+    };
+};
+
+#endif /* _SPC_DER_H_ */
+

src/spc/spc_eq_resolution.cpp

@@ -0,0 +1,44 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_eq_resolution.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-14.
+
+Revision History:
+
+--*/
+#include"spc_eq_resolution.h"
+
+namespace spc {
+
+    /**
+       \brief Apply equality resolution rule on the given clause.
+       Store the produced clauses in new_clauses.
+    */
+    void eq_resolution::operator()(clause * cls, ptr_vector<clause> & new_clauses) {
+        m_subst.reserve_vars(cls->get_num_vars());
+        unsigned num = cls->get_num_literals();
+        for (unsigned i = 0; i < num; i++) {
+            literal const & l = cls->get_literal(i);
+            expr * atom = l.atom();
+            if (l.sign() && m_manager.is_eq(atom)) {
+                expr * lhs = to_app(atom)->get_arg(0);
+                expr * rhs = to_app(atom)->get_arg(1);
+                m_subst.reset();
+                if (m_unifier(lhs, rhs, m_subst, false) && cls->is_eligible_for_resolution(m_order, l, 0, &m_subst)) {
+                    m_stats.m_num_eq_resolution++;
+                    new_clauses.push_back(mk_result(cls, i));
+                }
+            }
+        }
+    }
+};

src/spc/spc_eq_resolution.h

@@ -0,0 +1,50 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_eq_resolution.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-14.
+
+Revision History:
+
+--*/
+#ifndef _SPC_EQ_RESOLUTION_H_
+#define _SPC_EQ_RESOLUTION_H_
+
+#include"spc_unary_inference.h"
+#include"spc_statistics.h"
+
+namespace spc {
+
+    /**
+       \brief Functor for applying equality resolution.
+
+       s != t or R
+       ==>
+       sigma(R)
+    */
+    class eq_resolution : public unary_inference { 
+    protected:
+        statistics & m_stats;
+        family_id    m_spc_fid;
+        virtual justification * mk_justification(justification * parent, unsigned num_lits, literal * new_lits) {
+            return mk_eq_res_justification(m_manager, m_spc_fid, parent, num_lits, new_lits);
+        }
+    public:
+        eq_resolution(ast_manager & m, order & ord, statistics & stats):unary_inference(m, ord), m_stats(stats), m_spc_fid(m.get_family_id("spc")) {}
+        virtual ~eq_resolution() {}
+        void operator()(clause * cls, ptr_vector<clause> & new_clauses);
+    };
+};
+
+
+#endif /* _SPC_EQ_RESOLUTION_H_ */
+

src/spc/spc_factoring.cpp

@@ -0,0 +1,156 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_factoring.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-14.
+
+Revision History:
+
+--*/
+#include"spc_factoring.h"
+
+namespace spc {
+
+    /**
+       \brief Create a new clause by removing literal at position j, apply substitution m_subst,
+       and adding a disequality lhs != rhs.
+    */
+    clause * factoring::mk_eq_fact_result(clause * cls, unsigned j, expr * lhs, expr * rhs) {
+        sbuffer<literal> new_literals;
+
+        expr_ref new_eq(m_manager.mk_eq(lhs, rhs), m_manager);
+        expr_ref new_eq_after_subst(m_manager);
+        m_subst.apply(new_eq, new_eq_after_subst);
+        new_literals.push_back(literal(new_eq_after_subst, true));
+
+        unsigned num = cls->get_num_literals();
+        for (unsigned i = 0; i < num; i++) {
+            if (i != j) {
+                literal const & l = cls->get_literal(i);
+                expr_ref new_atom(m_manager);
+                m_subst.apply(l.atom(), new_atom);
+                new_literals.push_back(literal(new_atom, l.sign()));
+            }
+        }
+        
+        justification * js = mk_factoring_justification(m_manager, m_spc_fid, cls->get_justification(), new_literals.size(), 
+                                                        new_literals.c_ptr());
+        clause * new_cls = clause::mk(m_manager, new_literals.size(), new_literals.c_ptr(), js, cls->get_scope_lvl());
+        m_stats.m_num_eq_factoring++;
+        return new_cls;
+    }
+
+    /**
+       \brief Try to apply equality factoring using the eq literal stored at position j.
+       Assume lhs and rhs are the left hand side of this equality (they may be swapped).
+    */
+    void factoring::try_eq_factoring(clause * cls, unsigned j, expr * lhs, expr * rhs, ptr_vector<clause> & new_clauses) {
+        literal const & l1 = cls->get_literal(j);
+        sort * s = m_manager.get_sort(lhs);
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l2 = cls->get_literal(i);
+            if (i == j)
+                continue;
+            if (l2.sign())
+                continue;
+            expr * atom = l2.atom();
+            if (!m_manager.is_eq(atom))
+                continue;
+            expr * lhs2 = to_app(atom)->get_arg(0);
+            if (m_manager.get_sort(lhs2) != s)
+                continue;
+            expr * rhs2 = to_app(atom)->get_arg(1);
+            m_subst.reset();
+            if (m_unifier(lhs, lhs2, m_subst, false) && 
+                (l1.is_oriented() || !m_order.greater(rhs, lhs, &m_subst)) && 
+                cls->is_eligible_for_paramodulation(m_order, l1, 0, &m_subst)) {
+                new_clauses.push_back(mk_eq_fact_result(cls, j, rhs, rhs2));
+            }
+            m_subst.reset();
+            if (m_unifier(lhs, rhs2, m_subst, false) &&
+                (l1.is_oriented() || !m_order.greater(rhs, lhs, &m_subst)) && 
+                cls->is_eligible_for_paramodulation(m_order, l1, 0, &m_subst)) {
+                new_clauses.push_back(mk_eq_fact_result(cls, j, rhs, lhs2));
+            }
+        }
+    }
+
+    /**
+       \brief Try to apply equality factoring using the eq literal stored at position i.
+    */
+    void factoring::try_eq_factoring(clause * cls, unsigned i, ptr_vector<clause> & new_clauses) {
+        if (cls->get_num_pos_literals() <= 1)
+            return;
+        literal const & l = cls->get_literal(i);
+        app * eq   = to_app(l.atom());
+        expr * lhs = eq->get_arg(0);
+        expr * rhs = eq->get_arg(1);
+        if (l.is_oriented()) {
+            if (!l.is_left()) 
+                std::swap(lhs, rhs);
+            try_eq_factoring(cls, i, lhs, rhs, new_clauses);
+        }
+        else {
+            try_eq_factoring(cls, i, lhs, rhs, new_clauses);
+            try_eq_factoring(cls, i, rhs, lhs, new_clauses);
+        }
+    }
+
+    /**
+       \brief Try to apply (ordering) factoring rule.
+    */
+    void factoring::try_factoring(clause * cls, unsigned j, ptr_vector<clause> & new_clauses) {
+        literal const & l1 = cls->get_literal(j);
+        if (l1.sign() && cls->get_num_neg_literals() <= 1)
+            return;
+        if (!l1.sign() && cls->get_num_pos_literals() <= 1)
+            return;
+        unsigned num = cls->get_num_literals();
+        for (unsigned i = 0; i < num; i++) {
+            if (i == j)
+                continue;
+            literal const & l2 = cls->get_literal(i);
+            if (l1.sign() != l2.sign())
+                continue;
+            m_subst.reset();
+            if (m_unifier(l1.atom(), l2.atom(), m_subst, false) &&
+                cls->is_eligible_for_resolution(m_order, l1, 0, &m_subst)) {
+                new_clauses.push_back(mk_result(cls, i));
+                m_stats.m_num_factoring++;
+            }
+        }
+    }
+
+    /**
+       \brief Apply factoring rule on the given clause.
+       Store the produced clauses into new_clauses.
+    */
+    void factoring::operator()(clause * cls, ptr_vector<clause> & new_clauses) {
+        if (cls->get_num_pos_literals() <= 1 && cls->get_num_neg_literals() <= 1)
+            return;
+
+        m_subst.reserve_vars(cls->get_num_vars());
+        unsigned num = cls->get_num_literals();
+        for (unsigned i = 0; i < num; i++) {
+            literal const & l = cls->get_literal(i);
+            expr * atom = l.atom();
+            // remark: if the clause has selected literals then the literal will not be eligible
+            // for paramodulation and eq_resolution will not be applied.
+            if (!l.sign() && m_manager.is_eq(atom) && !cls->has_sel_lit()) 
+                try_eq_factoring(cls, i, new_clauses);
+            if (l.is_selected() || !cls->has_sel_lit())
+                try_factoring(cls, i, new_clauses);
+        }
+    }
+  
+};

src/spc/spc_factoring.h

@@ -0,0 +1,66 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_factoring.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-14.
+
+Revision History:
+
+--*/
+#ifndef _SPC_FACTORING_H_
+#define _SPC_FACTORING_H_
+
+#include"spc_unary_inference.h"
+#include"spc_statistics.h"
+
+namespace spc {
+
+    /**
+       \brief Functor for applying factoring.
+
+       - Equality Factoring
+       s = t or u = v or R
+       ==>
+       sigma(t != v or u = v or R)
+       
+       sigma = mgu(s, u)
+       sigma(s) not greater than sigma(t)
+       sigma(s = t) is eligible for paramodulation.
+       
+       - Factoring
+       P(t) or P(s) or R
+       ==>
+       sigma(P(t) or R)
+
+       sigma = mgu(t,s)
+       sigma(P(t)) is eligible for resolution.
+    */
+    class factoring : public unary_inference {
+    protected:
+        statistics & m_stats;
+        family_id    m_spc_fid;
+        virtual justification * mk_justification(justification * parent, unsigned num_lits, literal * new_lits) {
+            return mk_factoring_justification(m_manager, m_spc_fid, parent, num_lits, new_lits);
+        }
+        clause * mk_eq_fact_result(clause * cls, unsigned j, expr * lhs, expr * rhs);
+        void try_eq_factoring(clause * cls, unsigned j, expr * lhs, expr * rhs, ptr_vector<clause> & new_clauses);
+        void try_eq_factoring(clause * cls, unsigned i, ptr_vector<clause> & new_clauses);
+        void try_factoring(clause * cls, unsigned j, ptr_vector<clause> & new_clauses);
+    public:
+        factoring(ast_manager & m, order & ord, statistics & stats):unary_inference(m, ord), m_stats(stats), m_spc_fid(m.get_family_id("spc")) {}
+        virtual ~factoring() {}
+        void operator()(clause * cls, ptr_vector<clause> & new_clauses);
+    };
+};
+
+#endif /* _SPC_FACTORING_H_ */
+

src/spc/spc_justification.cpp

@@ -0,0 +1,184 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_justification.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-08.
+
+Revision History:
+
+--*/
+#include"spc_justification.h"
+#include"spc_clause.h"
+#include"marker.h"
+
+namespace spc {
+
+    void get_justification_stat(justification * p, justification_stat & stat) {
+        // Remark: justification objects that are not associated
+        // with clauses may be shared. That is, they may be parent of
+        // several different justification objects.
+        marker<justification> m;
+        ptr_buffer<justification> todo;
+        todo.push_back(p);
+        while (!todo.empty()) {
+            justification * p = todo.back();
+            todo.pop_back();
+            if (!m.is_marked(p)) {
+                m.mark(p);
+                clause * cls = p->get_clause();
+                if (cls) {
+                    if (cls->get_proof_depth() > stat.m_proof_depth)
+                        stat.m_proof_depth = cls->get_proof_depth();
+                    if (cls->get_scope_lvl() > stat.m_max_scope_lvl)
+                        stat.m_max_scope_lvl = cls->get_scope_lvl();
+                    stat.m_parent_clauses.push_back(cls);
+                }
+                else {
+                    p->get_parents(todo);
+                }
+            }
+        }
+    }
+
+    void justification::display(std::ostream & out) {
+        out << "[" << get_rule_id();
+        ptr_buffer<justification> ps;
+        get_parents(ps);
+        unsigned sz = ps.size();
+        for (unsigned i = 0; i < sz; i++) {
+            out << " ";
+            justification * js = ps[i];
+            clause * cls = js->get_clause();
+            if (cls)
+                out << "#" << cls->get_id();
+            else
+                js->display(out);
+        }
+        out << "]";
+    }
+
+    justification * justification_proof_wrapper::mk(proof * p, ast_manager & m) {
+        void * mem = m.get_allocator().allocate(sizeof(justification_proof_wrapper));
+        return new (mem) justification_proof_wrapper(p, m);
+    }
+
+    proof * justification_proof_wrapper::get_proof() const { 
+        return m_proof; 
+    }
+
+    unsigned justification_proof_wrapper::del_eh(ast_manager & m) { 
+        m.dec_ref(m_proof); 
+        return sizeof(justification_proof_wrapper);
+    }
+
+    void dec_ref(justification * p, ast_manager & m) {
+        if (p->dec_ref()) {
+            ptr_buffer<justification> to_delete;
+            ptr_buffer<justification> parents;
+            to_delete.push_back(p);
+            while (!to_delete.empty()) {
+                justification * p = to_delete.back();
+                to_delete.pop_back();
+                SASSERT(p->get_ref_count() == 0);
+                parents.reset();
+                p->get_parents(parents);
+                ptr_buffer<justification>::iterator it  = parents.begin();
+                ptr_buffer<justification>::iterator end = parents.end();
+                for (; it != end; ++it) {
+                    justification * parent = *it;
+                    if (parent->dec_ref())
+                        to_delete.push_back(parent);
+                }
+                unsigned sz = p->del_eh(m);
+                p->~justification();
+                m.get_allocator().deallocate(sz, p);
+            }
+        }
+    }
+
+    /**
+       \brief Return a proof for a new clause formed by the literals lits[0] ... lits[num_lits - 1].
+       This clause was produced using a main clause C, where the proof of C is \c main_pr,
+       and the auxiliary proofs auxs[0] ... aux[num_auxs-1].
+       
+       \remark If fine_grain_proofs() is false, then 0 is returned.
+    */
+    proof * mk_proof(ast_manager & m, family_id spc_fid, spc_op_kind pid, unsigned num_lits, literal * lits, proof * main_pr, 
+                     unsigned num_auxs, proof * const * auxs) {
+        if (m.fine_grain_proofs()) {
+            expr * new_fact_body = mk_or(m, num_lits, lits);
+            
+            SASSERT(main_pr);
+            SASSERT(m.has_fact(main_pr));
+            expr * fact     = m.get_fact(main_pr);
+            expr * new_fact = 0;
+            if (is_quantifier(fact))
+                new_fact = m.update_quantifier(to_quantifier(fact), new_fact_body);
+            else 
+                new_fact = new_fact_body;
+
+            ptr_buffer<expr> args;
+            args.push_back(main_pr);
+            args.append(num_auxs, (expr**) auxs);
+            args.push_back(new_fact);
+            
+            return m.mk_app(spc_fid, pid, args.size(), args.c_ptr());
+        }
+        return 0;
+    }
+
+    justification * rewrite_justification::mk(ast_manager & m, justification * head, 
+                                              unsigned num_demodulators, justification * const * demodulators, proof * pr) {
+        void * mem = m.get_allocator().allocate(get_obj_size(num_demodulators, m.fine_grain_proofs()));
+        return new (mem) rewrite_justification(m, head, num_demodulators, demodulators, pr);
+    }
+
+    rewrite_justification::rewrite_justification(ast_manager & m, justification * head, 
+                                                 unsigned num_demodulators, justification * const * demodulators, proof * pr):
+        m_num_demodulators(num_demodulators) {
+        SASSERT(m.fine_grain_proofs() == (pr != 0));
+        m_fields[0] = head;
+        head->inc_ref();
+        for (unsigned i = 0; i < num_demodulators; i++) {
+            m_fields[i+1] = demodulators[i];
+            demodulators[i]->inc_ref();
+        }
+        if (m.fine_grain_proofs()) {
+            SASSERT(pr);
+            m_fields[num_demodulators+1] = pr;
+            m.inc_ref(pr);
+        }
+    }
+            
+    void rewrite_justification::get_parents(ptr_buffer<justification> & parents) {
+        unsigned num_parents = m_num_demodulators+1;
+        for (unsigned i = 0; i < num_parents; i++)
+            parents.push_back(reinterpret_cast<justification*>(m_fields[i]));
+    }
+
+    proof * rewrite_justification::get_proof() const {
+        return reinterpret_cast<proof*>(m_fields[m_num_demodulators+1]);
+    }
+
+    unsigned rewrite_justification::del_eh(ast_manager & m) {
+        if (m.fine_grain_proofs()) {
+            m.dec_ref(reinterpret_cast<proof*>(m_fields[m_num_demodulators+1]));
+            return get_obj_size(m_num_demodulators, true);
+        }
+        return get_obj_size(m_num_demodulators, false);
+    }
+
+    proof * mk_rewrite_proof(ast_manager & m, family_id spc_fid, unsigned num_lits, literal * lits, proof * main_pr, 
+                             unsigned num_auxs, proof * const * auxs) {
+        return mk_proof(m, spc_fid, PR_SPC_REWRITE, num_lits, lits, main_pr, num_auxs, auxs);
+    }
+};

src/spc/spc_justification.h

@@ -0,0 +1,337 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_justification.h
+
+Abstract:
+
+    Proof-like objects for tracking dependencies in the superposition
+    calculus engine, and generating proofs.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#ifndef _SPC_JUSTIFICATION_H_
+#define _SPC_JUSTIFICATION_H_
+
+#include"ast.h"
+#include"spc_literal.h"
+#include"spc_decl_plugin.h"
+
+namespace spc {
+
+    class clause;
+    
+    /**
+       \brief Proof-like object use to track dependencies and produce
+       proofs.
+
+       \remark All justification objects must be allocated using the
+       small_object_allocator in ast_manager.
+    */
+    class justification {
+        clause * m_owner;
+        unsigned m_ref_count:30;
+        unsigned m_mark:1;
+        unsigned m_assumption:1;
+        
+        friend class clause;
+        void set_owner(clause * cls) { m_owner = cls; }
+
+    public:
+        justification(bool assumption = false):
+            m_owner(0),
+            m_ref_count(0),
+            m_mark(false), 
+            m_assumption(assumption) {
+        }
+        
+        virtual ~justification() {}
+
+        void inc_ref() { 
+            m_ref_count++;
+        }
+
+        bool dec_ref() {
+            SASSERT(m_ref_count > 0);
+            m_ref_count--;
+            return m_ref_count == 0;
+        }
+            
+        unsigned get_ref_count() {
+            return m_ref_count;
+        }
+
+        void set_mark(bool f) { m_mark = f; }
+     
+        bool is_marked() const { return m_mark; }
+
+        /**
+           \brief Return the clause justified by this object. 
+           
+           \remark for some justification objects that clause is
+           supressed. Example: intermediate steps.
+        */
+        clause * get_clause() { return m_owner; }
+
+        /**
+           \brief Return the expr justified by this object. 
+           This method returns a non null value only when
+           proof generation is enabled. 
+        */
+        virtual expr * get_expr(ast_manager & m) { return 0; }
+
+        /**
+           \brief Return a non-zero value if the justification
+           is wrapping a proof object.
+        */
+        virtual proof * get_proof() const { return 0; }
+
+        /**
+           \brief Return the parent justifications.
+        */
+        virtual void get_parents(ptr_buffer<justification> & parents) {}
+
+        /**
+           \brief Return the name of the rule used.
+        */
+        virtual spc_op_kind get_rule_id() = 0;
+        
+        /**
+           \brief Return true if the justification is an external assumption.
+        */
+        bool assumption() const { return m_assumption; }
+
+        void display(std::ostream & out);
+
+        /**
+           \brief This method is invoked before the object is deleted.
+           Return the amount of memory consumed by this object.
+        */
+        virtual unsigned del_eh(ast_manager & m) = 0;
+    };
+
+    struct justification_stat {
+        unsigned           m_proof_depth;
+        unsigned           m_max_scope_lvl;
+        ptr_buffer<clause> m_parent_clauses;
+        justification_stat():
+            m_proof_depth(0),
+            m_max_scope_lvl(0) {
+        }
+    };
+
+    void get_justification_stat(justification * p, justification_stat & stat);
+
+    void dec_ref(justification * p, ast_manager & m);
+
+    /**
+       \brief Smart pointer for justification objects.
+    */
+    class justification_ref {
+        justification * m_obj;
+        ast_manager &   m_manager;
+        void inc_ref() { if (m_obj) m_obj->inc_ref(); }
+        void dec_ref() { if (m_obj) spc::dec_ref(m_obj, m_manager); }
+    public:
+        justification_ref(ast_manager & m):m_obj(0), m_manager(m) {}
+        justification_ref(justification * j, ast_manager & m):
+            m_obj(j), m_manager(m) {
+            inc_ref();
+        }
+        ~justification_ref() {
+            dec_ref();
+        }
+        operator justification*() const { return m_obj; }
+        operator bool() const { return m_obj != 0; }
+        bool operator!() const { return m_obj == 0; }
+        justification * operator->() const { return m_obj; }
+        justification const & operator*() const { return *m_obj; }
+        justification_ref & operator=(justification * n) {
+            if (n)
+                n->inc_ref();
+            dec_ref();
+            m_obj = n;
+            return *this;
+        }
+        justification_ref & operator=(justification_ref & n) {
+            SASSERT(&m_manager == &n.m_manager);
+            n.inc_ref();
+            dec_ref();
+            m_obj = n.m_obj;
+            return *this;
+        }
+    };
+    
+    class justification_proof_wrapper : public justification {
+        proof * m_proof;
+        justification_proof_wrapper(proof * p, ast_manager & m):m_proof(p) { m.inc_ref(m_proof); }
+    public:
+        static justification * mk(proof * p, ast_manager & m);
+        virtual ~justification_proof_wrapper() {}
+        virtual proof * get_proof() const;
+        virtual spc_op_kind get_rule_id() { return PR_SPC_ASSERTED; };
+        virtual unsigned del_eh(ast_manager & m);
+    };
+
+    proof * mk_proof(ast_manager & m, family_id spc_fid, spc_op_kind pid, unsigned num_lits, literal * lits, proof * main_pr, unsigned num_auxs, 
+                     proof * const * auxs);
+
+    /**
+       \brief Justification for rewriting steps: demodulation, duplicate literal deletion, resolved literal deletion.
+    */
+    class rewrite_justification : public justification {
+        unsigned m_num_demodulators;
+        void *   m_fields[0];
+        static unsigned get_obj_size(unsigned num_demodulators, bool fine_grain) { 
+            return sizeof(rewrite_justification) + (num_demodulators + (fine_grain ? 2 : 1)) * sizeof(void *); 
+        }
+        rewrite_justification(ast_manager & m, justification * head, 
+                              unsigned num_demodulators, justification * const * demodulators, proof * pr);
+    public:
+        static justification * mk(ast_manager & m, justification * head, 
+                                  unsigned num_demodulators, justification * const * demodulators, proof * pr = 0);
+        virtual ~rewrite_justification() {}
+        virtual proof * get_proof() const;
+        virtual spc_op_kind get_rule_id() { return PR_SPC_REWRITE; }
+        virtual void get_parents(ptr_buffer<justification> & parents);
+        virtual unsigned del_eh(ast_manager & m);
+    };
+
+    proof * mk_rewrite_proof(ast_manager & m, family_id spc_fid, unsigned num_lits, literal * lits, proof * main_pr, unsigned num_auxs, 
+                             proof * const * auxs);
+
+    template<spc_op_kind Kind> 
+    class unary_justification : public justification {
+    protected:
+        justification * m_parent;
+        proof *         m_proof; 
+
+        unary_justification(ast_manager & m, justification * p, proof * pr):
+            m_parent(p),
+            m_proof(pr) {
+            p->inc_ref();
+            SASSERT(m.fine_grain_proofs() == (pr != 0));
+            if (m.fine_grain_proofs())
+                m.inc_ref(pr);
+        }
+
+    public:
+        virtual proof * get_proof() const {
+            return m_proof;
+        }
+
+        virtual void get_parents(ptr_buffer<justification> & parents) {
+            parents.push_back(m_parent);
+        }
+        
+        virtual unsigned del_eh(ast_manager & m) {
+            if (m.fine_grain_proofs()) 
+                m.dec_ref(m_proof);
+            return sizeof(unary_justification);
+        }
+        
+        virtual spc_op_kind get_rule_id() {
+            return Kind;
+        }
+        
+        static justification * mk(ast_manager & m, family_id spc_fid, justification * p, unsigned num_lits, literal * lits) {
+            proof * pr = 0;
+            if (m.fine_grain_proofs())
+                pr = mk_proof(m, spc_fid, Kind, num_lits, lits, p->get_proof(), 0, 0);
+            void * mem = m.get_allocator().allocate(sizeof(unary_justification));
+            return new (mem) unary_justification(m, p, pr);
+        }
+    };
+
+    inline justification * mk_eq_res_justification(ast_manager & m, family_id spc_fid, justification * p, unsigned num_lits, literal * lits) {
+        return unary_justification<PR_EQUALITY_RESOLUTION>::mk(m, spc_fid, p, num_lits, lits);
+    }
+
+    inline justification * mk_factoring_justification(ast_manager & m, family_id spc_fid, justification * p, unsigned num_lits, literal * lits) {
+        return unary_justification<PR_FACTORING>::mk(m, spc_fid, p, num_lits, lits);
+    }
+
+    inline justification * mk_der_justification(ast_manager & m, family_id spc_fid, justification * p, unsigned num_lits, literal * lits) {
+        return unary_justification<PR_SPC_DER>::mk(m, spc_fid, p, num_lits, lits);
+    }
+
+    template<spc_op_kind Kind> 
+    class binary_justification : public justification {
+    protected:
+        justification * m_parent1;
+        justification * m_parent2;
+        proof *         m_proof; 
+
+        binary_justification(ast_manager & m, justification * p1, justification * p2, proof * pr):
+            m_parent1(p1),
+            m_parent2(p2),
+            m_proof(pr) {
+            p1->inc_ref();
+            p2->inc_ref();
+            SASSERT(m.fine_grain_proofs() == (pr != 0));
+            if (m.fine_grain_proofs())
+                m.inc_ref(pr);
+        }
+
+    public:
+        virtual proof * get_proof() const {
+            return m_proof;
+        }
+
+        virtual void get_parents(ptr_buffer<justification> & parents) {
+            parents.push_back(m_parent1);
+            parents.push_back(m_parent2);
+        }
+        
+        virtual unsigned del_eh(ast_manager & m) {
+            if (m.fine_grain_proofs()) 
+                m.dec_ref(m_proof);
+            return sizeof(binary_justification);
+        }
+        
+        virtual spc_op_kind get_rule_id() {
+            return Kind;
+        }
+        
+        static justification * mk(ast_manager & m, family_id spc_fid, justification * p1, justification * p2, unsigned num_lits, literal * lits,
+                                  unsigned num_vars, var * const * vars) {
+            proof * pr = 0;
+            if (m.fine_grain_proofs()) {
+                ptr_buffer<sort> sorts;
+                sbuffer<symbol>  names;
+                for (unsigned i = 0; i < num_vars; i++) {
+                    sorts.push_back(vars[num_vars - i - 1]->get_sort());
+                    names.push_back(symbol(num_vars - i - 1));
+                }
+                expr * body = mk_or(m, num_lits, lits);
+                expr * new_fact = 0;
+                if (num_vars == 0)
+                    new_fact = body;
+                else
+                    new_fact = m.mk_forall(sorts.size(), sorts.c_ptr(), names.c_ptr(), body);
+                pr = m.mk_app(spc_fid, Kind, p1->get_proof(), p2->get_proof(), new_fact); 
+            }
+            void * mem = m.get_allocator().allocate(sizeof(binary_justification));
+            return new (mem) binary_justification(m, p1, p2, pr);
+        }
+    };
+
+    inline justification * mk_superposition_justification(ast_manager & m, family_id spc_fid, justification * p1, justification * p2,
+                                                          unsigned num_lits, literal * lits, unsigned num_vars, var * const * vars) {
+        return binary_justification<PR_SUPERPOSITION>::mk(m, spc_fid, p1, p2, num_lits, lits, num_vars, vars);
+    }
+
+    inline justification * mk_resolution_justification(ast_manager & m, family_id spc_fid, justification * p1, justification * p2,
+                                                          unsigned num_lits, literal * lits, unsigned num_vars, var * const * vars) {
+        return binary_justification<PR_SPC_RESOLUTION>::mk(m, spc_fid, p1, p2, num_lits, lits, num_vars, vars);
+    }
+};
+
+#endif /* _SPC_JUSTIFICATION_H_ */

src/spc/spc_literal.cpp

@@ -0,0 +1,432 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_literal.cpp
+
+Abstract:
+
+    Superposition Calculus literal
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#include"spc_literal.h"
+#include"ast_pp.h"
+
+namespace spc {
+
+    void literal::try_to_orient(order & o) {
+        ast_manager & m = o.get_manager();
+        if (!m_sign && m.is_eq(m_atom)) {
+            expr * lhs = to_app(m_atom)->get_arg(0);
+            expr * rhs = to_app(m_atom)->get_arg(1);
+            TRACE("spc_orient", tout << "trying to orient:\n" << mk_pp(lhs, m) << "\n" << mk_pp(rhs, m) << "\n";);
+            switch (o.compare(lhs, rhs)) {
+            case order::GREATER:
+                m_oriented = true;
+                m_left     = true;
+                TRACE("spc_orient", tout << "greater\n";);
+                return;
+            case order::LESSER:
+                m_oriented = true;
+                m_left     = false;
+                TRACE("spc_orient", tout << "smaller\n";);
+                return;
+            default:
+                return;
+            }
+        }
+    }
+        
+    void literal::get_stat(literal_stat & stat) {
+        get_expr_stat(m_atom, stat);
+        m_stats       = true;
+        m_ground      = stat.m_ground;
+        m_sym_count   = stat.m_sym_count   > SYM_COUNT_MAX   ? SYM_COUNT_MAX   : stat.m_sym_count;
+        m_depth       = stat.m_depth       > DEPTH_MAX       ? DEPTH_MAX       : stat.m_depth;
+        m_const_count = stat.m_const_count > CONST_COUNT_MAX ? CONST_COUNT_MAX : stat.m_const_count;
+    }
+    
+    expr * literal::to_expr(ast_manager & m) const {
+        if (is_true(m)) 
+            return m.mk_true();
+        else if (is_false(m))
+            return m.mk_false();
+        else if (m_sign)
+            return m.mk_not(m_atom);
+        else
+            return m_atom;
+    }
+
+    void literal::display(std::ostream & out, ast_manager & m, bool detailed) const {
+        pp_params p;
+        p.m_pp_single_line = true;
+        
+        if (m_sign) 
+            out << "(not ";
+        
+        if (m_oriented) {
+            expr * lhs = to_app(m_atom)->get_arg(0);
+            expr * rhs = to_app(m_atom)->get_arg(1);
+            if (!m_left)
+                std::swap(lhs, rhs);
+            out << "(-> ";
+            ast_pp(out, lhs, m, p);
+            out << " ";
+            ast_pp(out, rhs, m, p);
+            out << ")";
+        }
+        else {
+            ast_pp(out, m_atom, m, p);            
+        }
+
+        if (m_sign) 
+            out << ")";
+    
+        if (detailed && m_stats) {
+            out << "[" << m_ground << ", " << m_depth << ", " << m_sym_count << ", " << m_const_count << "]";
+        }
+    
+        if (m_selected)
+            out << "$";
+        if (m_p_indexed)
+            out << "!";
+        if (m_r_indexed)
+            out << "@";
+    }
+
+    void display(std::ostream & out, unsigned num_lists, literal * lits, ast_manager & m, bool detailed) {
+        for (unsigned i = 0; i < num_lists; i++) {
+            if (i > 0) out << " ";
+            lits[i].display(out, m, detailed);
+        }
+    }
+
+    /**
+       \brief Given an eq literal store in lhs and rhs the left and right hand sides. If they can be oriented
+       given the substitution s, then return true, and make lhs the maximal one.
+    */
+    bool can_orient(order & o, literal const & l, unsigned offset, substitution * s, expr * & lhs, expr * & rhs) {
+        SASSERT(o.get_manager().is_eq(l.atom()));
+        lhs = l.lhs();
+        rhs = l.rhs();
+        if (l.is_oriented()) {
+            if (!l.is_left())
+                std::swap(lhs, rhs);
+            return true;
+        }
+        else {
+            order::result comp = o.compare(lhs, rhs, offset, s);
+            if (comp == order::GREATER)
+                return true;
+            else if (comp == order::LESSER) {
+                std::swap(lhs, rhs);
+                return true;
+            }
+            return false;
+        }
+    }
+
+    /**
+       \brief Compare literal signs. Negative sign is bigger than the positive one.
+    */
+    inline order::result compare_signs(bool sign1, bool sign2) {
+        if (sign1 && !sign2)
+            return order::GREATER;
+        else if (!sign1 && sign2)
+            return order::LESSER;
+        else
+            return order::EQUAL;
+    }
+
+    /**
+       \brief Compare two literals (modulo a substitution) using the given term ordering.
+    */
+    order::result compare(order & o, literal const & l1, literal const & l2, unsigned offset, substitution * s) {
+        ast_manager & m = o.get_manager();
+        expr * n1 = l1.atom();
+        expr * n2 = l2.atom();
+        bool is_eq1 = m.is_eq(n1);
+        bool is_eq2 = m.is_eq(n2);
+        if (is_eq1 && is_eq2) {
+            expr * lhs1 = 0;
+            expr * rhs1 = 0;
+            expr * lhs2 = 0;
+            expr * rhs2 = 0;
+            bool oriented1 = can_orient(o, l1, offset, s, lhs1, rhs1);
+            bool oriented2 = can_orient(o, l2, offset, s, lhs2, rhs2);
+            if (oriented1) {
+                // equation 1 can be oriented
+                if (oriented2) {
+                    // equation 2 can be oriented
+                    // both equations are oriented
+                    SASSERT(oriented1);
+                    SASSERT(oriented2);
+                    order::result r = o.compare(lhs1, lhs2, offset, s);
+                    if (r == order::EQUAL) {
+                        if (l1.pos()) {
+                            if (l2.pos()) 
+                                return o.compare(rhs1, rhs2, offset, s);
+                            else
+                                return order::LESSER;
+                        }
+                        else {
+                            if (l2.pos())
+                                return order::GREATER;
+                            else
+                                return o.compare(rhs1, rhs2, offset, s);
+                        }
+                    }
+                    return r;
+                }
+                else {
+                    // equation 2 cannot be oriented
+                    SASSERT(oriented1);
+                    SASSERT(!oriented2);
+                    SASSERT(o.compare(lhs1, rhs1, offset, s) == order::GREATER);
+                    if (o.equal(lhs1, lhs2, offset, s)) {
+                        order::result r = o.compare(rhs1, rhs2, offset, s);
+                        if (r == order::EQUAL)
+                            return compare_signs(l1.sign(), l2.sign());
+                        return r;
+                    }
+                    if (o.equal(lhs1, rhs2, offset, s)) {
+                        order::result r = o.compare(rhs1, lhs2, offset, s);
+                        if (r == order::EQUAL)
+                            return compare_signs(l1.sign(), l2.sign());
+                        return r;
+                    }
+                    order::result lhs1_lhs2 = o.compare(lhs1, lhs2, offset, s);
+                    order::result lhs1_rhs2 = o.compare(lhs1, rhs2, offset, s);
+                    if (lhs1_lhs2 == lhs1_rhs2) 
+                        return lhs1_lhs2;
+                    order::result rhs1_rhs2 = o.compare(rhs1, rhs2, offset, s);
+                    if (lhs1_lhs2 == rhs1_rhs2)
+                        return lhs1_lhs2;
+                    if (lhs1_rhs2 == order::LESSER && rhs1_rhs2 == order::LESSER)
+                        return order::LESSER;
+                    order::result rhs1_lhs2 = o.compare(rhs1, lhs2, offset, s);
+                    if (lhs1_lhs2 == order::LESSER && rhs1_lhs2 == order::LESSER)
+                        return order::LESSER;
+                    return order::UNCOMPARABLE;
+                }
+            }
+            else {
+                // equation 1 cannot be oriented
+                if (oriented2) {
+                    SASSERT(!oriented1);
+                    SASSERT(oriented2);
+                    // equation 2 can be oriented
+                    if (o.equal(lhs1, lhs2, offset, s)) {
+                        order::result r = o.compare(rhs1, rhs2, offset, s);
+                        if (r == order::EQUAL)
+                            return compare_signs(l1.sign(), l2.sign());
+                        return r;
+                    }
+                    if (o.equal(rhs1, lhs2, offset, s)) {
+                        order::result r = o.compare(lhs1, rhs2, offset, s);
+                        if (r == order::EQUAL)
+                            return compare_signs(l1.sign(), l2.sign());
+                        return r;
+                    }
+                    order::result lhs1_lhs2 = o.compare(lhs1, lhs2, offset, s);
+                    order::result rhs1_lhs2 = o.compare(rhs1, lhs2, offset, s);
+                    if (lhs1_lhs2 == rhs1_lhs2)
+                        return lhs1_lhs2;
+                    order::result rhs1_rhs2 = o.compare(rhs1, rhs2, offset, s);
+                    if (lhs1_lhs2 == rhs1_rhs2)
+                        return lhs1_lhs2;
+                    if (rhs1_lhs2 == order::GREATER && rhs1_rhs2 == order::GREATER)
+                        return order::GREATER;
+                    order::result lhs1_rhs2 = o.compare(lhs1, rhs2, offset, s);
+                    if (lhs1_lhs2 == order::GREATER && lhs1_rhs2 == order::GREATER)
+                        return order::GREATER;
+                    return order::UNCOMPARABLE;
+                }
+                else {
+                    SASSERT(!oriented1);
+                    SASSERT(!oriented2);
+                    if (o.equal(lhs1, lhs2, offset, s)) {
+                        order::result r = o.compare(rhs1, rhs2, offset, s);
+                        if (r == order::EQUAL)
+                            return compare_signs(l1.sign(), l2.sign());
+                        return r;
+                    }
+                    if (o.equal(rhs1, lhs2, offset, s)) {
+                        order::result r = o.compare(lhs1, rhs2, offset, s);
+                        if (r == order::EQUAL)
+                            return compare_signs(l1.sign(), l2.sign());
+                        return r;
+                    }
+                    if (o.equal(lhs1, rhs2, offset, s)) {
+                        order::result r = o.compare(rhs1, lhs2, offset, s);
+                        if (r == order::EQUAL)
+                            return compare_signs(l1.sign(), l2.sign());
+                        return r;
+                    }
+                    if (o.equal(rhs1, rhs2, offset, s)) {
+                        order::result r = o.compare(lhs1, lhs2, offset, s);
+                        if (r == order::EQUAL)
+                            return compare_signs(l1.sign(), l2.sign());
+                        return r;
+                    }
+                    
+                    order::result r;
+                    order::result aux;
+                    switch (o.compare(lhs1, lhs2, offset, s)) {
+                    case order::GREATER:
+                        r   = o.compare(lhs1, rhs2, offset, s);
+                        if (r == order::GREATER)
+                            return order::GREATER;
+                        aux = o.compare(rhs1, rhs2, offset, s);
+                        if (aux == order::GREATER)
+                            return order::GREATER;
+                        if (r == order::LESSER && aux == order::LESSER)
+                            return order::LESSER;
+                        SASSERT(r   != order::EQUAL);
+                        SASSERT(aux != order::EQUAL);
+                        return order::UNCOMPARABLE;
+                    case order::LESSER:
+                        r   = o.compare(rhs1, lhs2, offset, s);
+                        if (r == order::LESSER)
+                            return order::LESSER;
+                        aux = o.compare(rhs1, rhs2, offset, s);
+                        if (aux == order::LESSER)
+                            return order::LESSER;
+                        if (r == order::GREATER && aux == order::GREATER)
+                            return order::GREATER;
+                        SASSERT(r   != order::EQUAL);
+                        SASSERT(aux != order::EQUAL);
+                        return order::UNCOMPARABLE;
+                    case order::EQUAL:
+                        UNREACHABLE();
+                        return order::UNKNOWN;
+                    default:
+                        switch (o.compare(lhs1, rhs2, offset, s)) {
+                        case order::GREATER:
+                            if (o.compare(rhs1, lhs2, offset, s) == order::GREATER)
+                                return order::GREATER;
+                            return order::UNCOMPARABLE;
+                        case order::LESSER:
+                            if (o.compare(rhs1, lhs2, offset, s) == order::LESSER ||
+                                o.compare(rhs1, rhs2, offset, s) == order::LESSER)
+                                return order::LESSER;
+                            return order::UNCOMPARABLE;
+                        case order::EQUAL:
+                            UNREACHABLE();
+                            return order::UNKNOWN;
+                        default:
+                            if (o.compare(rhs1, lhs2, offset, s) == order::GREATER &&
+                                o.compare(rhs1, rhs2, offset, s) == order::GREATER)
+                                return order::GREATER;
+                            return order::UNCOMPARABLE;
+                        }
+                    }
+                }
+            }
+        }
+        else if (is_eq1) {
+            expr * lhs1 = l1.lhs();
+            expr * rhs1 = l1.rhs();
+            if (l1.is_oriented() && !l1.is_left())
+                std::swap(lhs1, rhs1);
+            order::result r = o.compare(lhs1, n2, offset, s);
+            if (!l1.is_oriented() || r != order::GREATER) {
+                order::result r2 = o.compare(rhs1, n2, offset, s);
+                if (r2 == order::GREATER)
+                    return order::GREATER;
+                else if (r != r2)
+                    return order::UNCOMPARABLE;
+            }
+            return r;
+        }
+        else if (is_eq2) {
+            expr * lhs2 = l2.lhs();
+            expr * rhs2 = l2.rhs();
+            if (l2.is_oriented() && !l2.is_left())
+                std::swap(lhs2, rhs2);
+            order::result r = o.compare(n1, lhs2, offset, s);
+            if (!l1.is_oriented() || r != order::LESSER) {
+                order::result r2 = o.compare(n1, rhs2, offset, s);
+                if (r2 == order::LESSER)
+                    return order::LESSER;
+                else if (r != r2)
+                    return order::UNCOMPARABLE;
+            }
+            return r;
+        }
+        else {
+            order::result r = o.compare(n1, n2, offset, s);
+            if (r == order::EQUAL)
+                return compare_signs(l1.sign(), l2.sign());
+            return r;
+        }
+    }
+
+    bool greater(order & o, literal const & l1, literal const & l2, unsigned offset, substitution * s) {
+        order::result r = compare(o, l1, l2, offset, s);
+        TRACE("literal_order", ast_manager & m = o.get_manager();
+              tout << "comparing ";
+              l1.display(tout, m);
+              tout << " ";
+              l2.display(tout, m);
+              tout << " : " << r << "\n";);
+        return r == order::GREATER;
+    }
+
+    void found_literals::insert(literal const & l) {
+        unsigned id = l.get_id();
+        m_marks.reserve(id+1);
+        if (!m_marks.get(id)) {
+            m_marks.set(id);
+            m_lit_ids.push_back(id);
+        }
+    }
+
+    bool found_literals::contains(literal const & l) const {
+        unsigned id = l.get_id();
+        return id < m_marks.size() && m_marks.get(id);
+    }
+
+    bool found_literals::contains_neg(literal const & l) const {
+        unsigned id = l.get_neg_id();
+        return id < m_marks.size() && m_marks.get(id);
+    }
+    
+    void found_literals::reset() {
+        unsigned_vector::iterator it  = m_lit_ids.begin();
+        unsigned_vector::iterator end = m_lit_ids.end();
+        for (; it != end; ++it) 
+            m_marks.unset(*it);
+        m_lit_ids.reset();
+    }
+
+    void literal_buffer::reset() { 
+        buffer<literal>::iterator it  = m_lits.begin();
+        buffer<literal>::iterator end = m_lits.end();
+        for (; it != end; ++it) 
+            m_manager.dec_ref(it->atom());
+        m_lits.reset();
+    }
+
+
+    expr * mk_or(ast_manager & m, unsigned num_lists, literal * lits) {
+        if (num_lists == 0) 
+            return m.mk_false();
+        else if (num_lists == 1)
+            return lits[0].to_expr(m);
+        else {
+            ptr_buffer<expr> new_exprs;
+            for (unsigned i = 0; i < num_lists; i++)
+                new_exprs.push_back(lits[i].to_expr(m));
+            return m.mk_or(new_exprs.size(), new_exprs.c_ptr());
+        }
+    }
+
+};

src/spc/spc_literal.h

@@ -0,0 +1,212 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_literal.h
+
+Abstract:
+
+    Superposition Calculus literal
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#ifndef _SPC_LITERAL_H_
+#define _SPC_LITERAL_H_
+
+#include"ast.h"
+#include"order.h"
+#include"expr_stat.h"
+
+namespace spc {
+    typedef expr_stat literal_stat;
+
+#define DEPTH_NUM_BITS 4
+#define DEPTH_MAX ((1 << DEPTH_NUM_BITS) - 1)
+#define CONST_COUNT_NUM_BITS 4    
+#define CONST_COUNT_MAX ((1 << CONST_COUNT_NUM_BITS) - 1)
+#define SYM_COUNT_NUM_BITS 16
+#define SYM_COUNT_MAX ((1 << SYM_COUNT_NUM_BITS) - 1)
+
+    /**
+       \brief Superposition Calculus literal.
+    */
+    class literal {
+        expr * m_atom;
+        unsigned  m_sign:1;                           // true if a negative literal.
+        unsigned  m_oriented:1;                       // true if it is an oriented equality.
+        unsigned  m_left:1;                           // true if the largest term is on the left-hand-side of the equality (only meaningful if m_oriented == true).
+        unsigned  m_selected:1;                       // true if it is a selected literal.
+        unsigned  m_stats:1;                          // true if the following fields were initialized.
+        unsigned  m_ground:1;                         // true if it is a ground literal
+        unsigned  m_p_indexed:1;                      // true if the literal was inserted into the p (paramodulation) superposition index.
+        unsigned  m_r_indexed:1;                      // true if the literal was inserted into the r (resolution) superposition index.
+        unsigned  m_depth:DEPTH_NUM_BITS;             // approx. depth
+        unsigned  m_const_count:CONST_COUNT_NUM_BITS; // approx. number of constants 
+        unsigned  m_sym_count:SYM_COUNT_NUM_BITS;     // approx. size
+
+        friend class clause;
+        
+        void set_selected(bool f) { 
+            m_selected = f; 
+        }
+        
+    public:
+        literal():
+            m_atom(0), 
+            m_sign(false), 
+            m_oriented(false), 
+            m_left(false), 
+            m_selected(false), 
+            m_stats(false),
+            m_ground(false),
+            m_p_indexed(false),
+            m_r_indexed(false),
+            m_depth(0),
+            m_const_count(0),
+            m_sym_count(0) {
+        }
+
+        literal(expr * atom, bool sign = false):
+            m_atom(atom), 
+            m_sign(sign), 
+            m_oriented(false), 
+            m_left(false), 
+            m_selected(false), 
+            m_stats(false),    
+            m_ground(false),
+            m_p_indexed(false),
+            m_r_indexed(false),
+            m_depth(0),
+            m_const_count(0),
+            m_sym_count(0) {
+        }
+
+        bool sign() const { return m_sign; }
+        bool pos() const { return !m_sign; }
+        bool neg() const { return m_sign; }
+        bool is_oriented() const { return m_oriented; }
+        bool is_left() const { return m_left; }
+        bool is_selected() const { return m_selected; }
+        expr * atom() const { return m_atom; }
+        expr * lhs() const { return to_app(m_atom)->get_arg(0); }
+        expr * rhs() const { return to_app(m_atom)->get_arg(1); }
+        unsigned get_id() const { return m_sign ? (to_app(m_atom)->get_id() << 1) + 1 : (to_app(m_atom)->get_id() << 1); }
+        unsigned get_neg_id() const { return m_sign ? (to_app(m_atom)->get_id() << 1) : (to_app(m_atom)->get_id() << 1) + 1; }
+
+        bool operator==(literal const & other) const { return m_atom == other.m_atom && m_sign == other.m_sign; }
+        bool operator!=(literal const & other) const { return !operator==(other); }
+
+        void set_p_indexed(bool f) { m_p_indexed = f; }
+        void set_r_indexed(bool f) { m_r_indexed = f; }
+        bool is_p_indexed() const { return m_p_indexed; }
+        bool is_r_indexed() const { return m_r_indexed; }
+
+        void try_to_orient(order & o);
+        bool is_true(ast_manager & m) const {
+            return 
+                (!m_sign && m.is_true(m_atom)) || 
+                (!m_sign && m.is_eq(m_atom) && to_app(m_atom)->get_arg(0) == to_app(m_atom)->get_arg(1)) ||
+                (m_sign && m.is_false(m_atom));
+        }
+        bool is_false(ast_manager & m) const {
+            return
+                (m_sign && m.is_true(m_atom)) || 
+                (m_sign && m.is_eq(m_atom) && to_app(m_atom)->get_arg(0) == to_app(m_atom)->get_arg(1)) ||
+                (!m_sign && m.is_false(m_atom));
+        }
+        expr * to_expr(ast_manager & m) const;
+
+        /**
+           \brief Collect literal statistics
+        */
+        void get_stat(literal_stat & stat);
+        void init_stat() { literal_stat st; get_stat(st); }
+        bool has_stats() const { return m_stats; }
+        bool is_ground() const { SASSERT(m_stats); return m_ground; }
+        unsigned get_approx_depth() const { SASSERT(m_stats); return m_depth; }
+        unsigned get_approx_const_count() const { SASSERT(m_stats); return m_const_count; }
+        unsigned get_approx_sym_count() const { SASSERT(m_stats); return m_sym_count; }
+
+        void display(std::ostream & out, ast_manager & m, bool detailed = false) const;
+    };
+
+    COMPILE_TIME_ASSERT(sizeof(expr*) != 4 || sizeof(literal) == sizeof(expr *) + sizeof(unsigned)); // 32 bit machine
+    COMPILE_TIME_ASSERT(sizeof(expr*) != 8 || sizeof(literal) == sizeof(expr *) + sizeof(unsigned) + /* a structure must be aligned */ sizeof(unsigned)); // 64 bit machine
+
+    void display(std::ostream & out, unsigned num_lists, literal * lits, ast_manager & m, bool detailed = false);
+
+    order::result compare(order & o, literal const & l1, literal const & l2, unsigned offset = 0, substitution * s = 0);
+    bool greater(order & o, literal const & l1, literal const & l2, unsigned offset = 0, substitution * s = 0);
+    bool is_maximal(order & o, unsigned num_lists, literal * lits, literal const & l, unsigned offset = 0, substitution * s = 0);
+    bool is_sel_maximal(order & o, unsigned num_lists, literal * lits, literal const & l, unsigned offset = 0, substitution * s = 0);
+
+    /**
+       \brief Set of found literals.
+
+       This object is used to implement duplicate literal elimination, ans syntatic tautology.
+    */
+    class found_literals {
+        bit_vector        m_marks;
+        unsigned_vector   m_lit_ids;
+    public:
+        /**
+           \brief Insert the given literal into the set.
+        */
+        void insert(literal const & l);
+        
+        /**
+           \brief Return true if the set contains \c l.
+        */
+        bool contains(literal const & l) const;
+
+        /**
+           \brief Return true if the set contains the negation of \c l.
+        */
+        bool contains_neg(literal const & l) const;
+
+        bool empty() const { return m_lit_ids.empty(); }
+
+        /**
+           \brief Remove all literals from the set.
+        */
+        void reset();
+    };
+
+    class literal_buffer {
+        ast_manager &   m_manager;
+        buffer<literal> m_lits;
+    public:
+        literal_buffer(ast_manager & m):
+            m_manager(m) {
+        }
+        
+        ~literal_buffer() {
+            reset();
+        }
+        
+        void push_back(literal const & l) { 
+            m_manager.inc_ref(l.atom()); 
+            m_lits.push_back(l); 
+        }
+        
+        void reset();
+
+        unsigned size() const {
+            return m_lits.size();
+        }
+
+        literal * c_ptr() const {
+            return m_lits.c_ptr();
+        }
+    };
+
+    expr * mk_or(ast_manager & m, unsigned num_lists, literal * lits);
+};
+
+#endif /* _SPC_LITERAL_H_ */

src/spc/spc_literal_selection.cpp

@@ -0,0 +1,107 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_literal_selection.cpp
+
+Abstract:
+
+    Superposition Calculus Literal Selection
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-05.
+
+Revision History:
+
+--*/
+#include"spc_literal_selection.h"
+#include"expr_stat.h"
+
+namespace spc {
+
+    void diff_literal_selection::operator()(clause * cls) { 
+        bool found = false;
+        unsigned target = UINT_MAX;
+        unsigned best_count = 0;
+        unsigned num = cls->get_num_literals();
+        for (unsigned i = 0; i < num; i++) {
+            literal & l = cls->get_literal(i);
+            if (l.sign()) {
+                unsigned count;
+                if (m_manager.is_eq(l.atom())) {
+                    unsigned c1 = get_symbol_count(to_app(l.atom())->get_arg(0));
+                    unsigned c2 = get_symbol_count(to_app(l.atom())->get_arg(1));
+                    count = c1 >= c2 ? c1 - c2 : c2 - c1;
+                }
+                else {
+                    count = get_symbol_count(l.atom());
+                }
+                if (count > best_count) {
+                    found      = true;
+                    target     = i;
+                    best_count = count;
+                }
+            }
+        }
+        if (found)
+            cls->select_literal(target);
+    }
+
+    void complex_literal_selection::operator()(clause * cls) { 
+        // look for x != y
+        unsigned num = cls->get_num_literals();
+        for (unsigned i = 0; i < num; i++) {
+            literal & l = cls->get_literal(i);
+            if (l.sign() && m_manager.is_eq(l.atom()) && is_var(to_app(l.atom())->get_arg(0)) && is_var(to_app(l.atom())->get_arg(1))) {
+                cls->select_literal(i);
+                return;
+            }
+        }
+
+        // look for min ground neg literal
+        bool found = false;
+        unsigned target = UINT_MAX;
+        unsigned  best_count = UINT_MAX;
+        for (unsigned i = 0; i < num; i++) {
+            literal & l = cls->get_literal(i);
+            if (l.sign() && is_ground(l.atom())) {
+                unsigned count = get_symbol_count(l.atom());
+                if (count < best_count) {
+                    found      = true;
+                    target     = i;
+                    best_count = count;
+                }
+            }
+        }
+        
+        if (found) {
+            cls->select_literal(target);
+            return;
+        }        
+        
+        diff_literal_selection::operator()(cls);
+    }
+
+    void max_no_selection::operator()(clause * cls) {
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l1 = cls->get_literal(i);
+            if (!l1.sign()) {
+                unsigned j = 0;
+                for (; j < num_lits; j++) {
+                    if (i != j) {
+                        literal const & l2 = cls->get_literal(j);
+                        if (!greater(m_order, l1, l2))
+                            break;
+                    }
+                }
+                if (j == num_lits)
+                    return; // clause has maximal positive literal.
+            }
+        }
+        diff_literal_selection::operator()(cls);
+    }
+
+};

src/spc/spc_literal_selection.h

@@ -0,0 +1,95 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_literal_selection.h
+
+Abstract:
+
+    Superposition Calculus Literal Selection
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-05.
+
+Revision History:
+
+--*/
+#ifndef _SPC_LITERAL_SELECTION_H_
+#define _SPC_LITERAL_SELECTION_H_
+
+#include"spc_clause.h"
+#include"order.h"
+
+namespace spc {
+    
+    /**
+       \brief Abstract functor for literal selection.
+    */
+    class literal_selection {
+    public:
+        virtual ~literal_selection() {}
+        /**
+           \brief Updates the selected status flag of the literals of the given clause.
+        */
+        virtual void operator()(clause * cls) = 0;
+    };
+
+    /**
+       \brief Never selects a literal. This strategy is supposed to be good for planning problems 
+       of TPTP.
+    */
+    class no_literal_selection : public literal_selection {
+    public:
+        virtual void operator()(clause * cls) {}
+    };
+
+    
+    /**
+       \brief Selects a negative literal l with the largest V(l)
+       where V is defined as:
+
+       - difference in symbol count for the left-right hand sides of equalities, .
+
+       - symbol count for other predicates
+       
+    */
+    class diff_literal_selection : public literal_selection {
+    protected:
+        ast_manager & m_manager;
+    public:
+        diff_literal_selection(ast_manager & m):m_manager(m) {}
+        virtual void operator()(clause * cls);
+    };
+
+    /**
+      \brief Selects a negative literal using the following algo:
+
+      - if there is x != y, select it.
+
+      - else if there is negative ground literal, select the smallest one.
+
+      - else if use the approach in diff_literal_selection.
+    */
+    class complex_literal_selection : public diff_literal_selection {
+    public:
+        complex_literal_selection(ast_manager & m):diff_literal_selection(m) {}
+        virtual void operator()(clause * cls);
+    };
+
+    /**
+       \brief Similar to diff_literal_selection, but a literal
+       is not selected if the clause contains a positive literal
+       greater than all other literals.
+    */
+    class max_no_selection : public diff_literal_selection {
+        order & m_order;
+    public:
+        max_no_selection(order & o):diff_literal_selection(o.get_manager()), m_order(o) {}
+        virtual void operator()(clause * cls);
+    };
+
+};
+
+#endif /* _SPC_LITERAL_SELECTION_H_ */

src/spc/spc_prover.cpp

@@ -0,0 +1,132 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_prover.cpp
+
+Abstract:
+
+    Stand-alone SPC prover (it is mainly for debugging purposes).
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-08.
+
+Revision History:
+
+--*/
+#include"spc_prover.h"
+#include"spc_decl_plugin.h"
+#include"for_each_expr.h"
+
+namespace spc {
+    prover::prover(ast_manager & m, front_end_params & params):
+        m_manager(m),
+        m_params(params),
+        m_simplifier(m),
+        m_defined_names(m),
+        m_preprocessor(m, m_defined_names, m_simplifier, params),
+        m_order(0),
+        m_cls_sel(0),
+        m_lit_sel(0),
+        m_context(0),
+        m_exprs(m),
+        m_expr_proofs(m),
+        m_has_theories(false) {
+
+        family_id fid = m_manager.get_family_id("spc");
+        if (!m_manager.has_plugin(fid)) 
+            m_manager.register_plugin(fid, alloc(spc_decl_plugin));
+
+        // This piece of code shows why the old model for passing parameters is broken.
+        // spc::prover must overwrite some parameters, but this modification affects other
+        // components. :-( 
+        // TODO: move everything to the new params_ref object
+        params.m_nnf_mode = NNF_FULL;
+        params.m_cnf_mode = CNF_FULL;
+        params.m_lift_ite = LI_CONSERVATIVE;
+
+        basic_simplifier_plugin * basic = alloc(basic_simplifier_plugin, m_manager);
+        m_simplifier.register_plugin(basic);
+        m_simplifier.register_plugin(alloc(arith_simplifier_plugin, m_manager, *basic, params));
+    }
+
+    prover::~prover() {
+        if (m_context) {
+            dealloc(m_context);
+            dealloc(m_lit_sel);
+            dealloc(m_cls_sel);
+            dealloc(m_order);
+        }
+    }
+
+    void prover::init() {
+        if (m_context)
+            return;
+        precedence * p = mk_precedence(m_manager, m_params);
+
+        // TODO use params to configure the following functors.
+        m_order    = alloc(kbo, m_manager, p); 
+        
+        clause_eval * evals[2]  = { alloc(symbol_count_clause_eval), alloc(time_clause_eval) };
+        unsigned      slots[2] = { 10, 1 };
+        m_cls_sel  = alloc(clause_selection, 2, evals, slots);
+        
+        m_lit_sel  = alloc(max_no_selection, *m_order);
+        // m_lit_sel  = new complex_literal_selection(m_manager);
+        // m_lit_sel  = new diff_literal_selection(m_manager);
+        // m_lit_sel  = new no_literal_selection(); // new diff_literal_selection(m_manager);
+        // END TODO
+        
+        m_context  = alloc(context, m_manager, *m_order, *m_cls_sel, *m_lit_sel, m_simplifier, m_params);
+    }
+
+    struct has_theories_proc {
+        ast_manager & m_manager;
+        has_theories_proc(ast_manager & m):m_manager(m) {}
+        struct found {}; 
+        void operator()(var * n) {}
+        void operator()(app * n) { if (!m_manager.is_builtin_family_id(n->get_family_id())) throw found(); }
+        void operator()(quantifier * n) {}
+    };
+
+    bool has_theories(ast_manager & m, expr * e) {
+        has_theories_proc p(m);
+        try {
+            for_each_expr(p, e);
+        }
+        catch (has_theories_proc::found) {
+            return true;
+        }
+        return false;
+    }
+        
+    void prover::assert_expr(expr * e) {
+        if (!m_has_theories && has_theories(m_manager, e))
+            m_has_theories = true;
+        TRACE("spc_assert", tout << mk_pp(e, m_manager) << "\nhas_theories: " << m_has_theories << "\n";);
+        m_preprocessor(e, m_manager.mk_asserted(e), m_exprs, m_expr_proofs);
+    }
+
+    lbool prover::check() {
+        init();
+        unsigned sz = m_exprs.size();
+        for (unsigned i = 0; i < sz; i++) {
+            expr * curr = m_exprs.get(i);
+            proof * p   = m_manager.proofs_enabled() ? m_expr_proofs.get(i) : m_manager.mk_undef_proof();
+            m_context->assert_expr(curr, p);
+        }
+        m_exprs.reset();
+        m_expr_proofs.reset();
+
+        m_context->saturate(m_params.m_spc_num_iterations);
+        if (m_context->inconsistent())
+            return l_false;
+        else if (m_context->processed_all())
+            return m_has_theories ? l_undef : l_true;
+        else
+            return l_undef;
+    }
+};
+

src/spc/spc_prover.h

@@ -0,0 +1,59 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_prover.h
+
+Abstract:
+
+    Stand-alone SPC prover (it is mainly for debugging purposes).
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-08.
+
+Revision History:
+
+--*/
+#ifndef _SPC_PROVER_H_
+#define _SPC_PROVER_H_
+
+#include"spc_context.h"
+#include"front_end_params.h"
+#include"kbo.h"
+#include"lpo.h"
+#include"basic_simplifier_plugin.h"
+#include"arith_simplifier_plugin.h"
+#include"preprocessor.h"
+#include"defined_names.h"
+#include"lbool.h"
+
+namespace spc {
+    class prover {
+        ast_manager &           m_manager;
+        front_end_params &      m_params;
+        simplifier              m_simplifier;
+        defined_names           m_defined_names;
+        preprocessor            m_preprocessor;
+        order *                 m_order;
+        clause_selection *      m_cls_sel;
+        literal_selection *     m_lit_sel;
+        context *               m_context;
+        expr_ref_vector         m_exprs;
+        proof_ref_vector        m_expr_proofs;
+        bool                    m_has_theories;
+
+        void init();
+
+    public:
+        prover(ast_manager & m, front_end_params & params);
+        ~prover();
+        void assert_expr(expr * e);
+        lbool check();
+        void display_statistics(std::ostream & out) const { if (m_context) m_context->display_statistics(out); }
+    };
+};
+
+#endif /* _SPC_PROVER_H_ */
+

src/spc/spc_rewriter.cpp

@@ -0,0 +1,269 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_rewrite.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-12.
+
+Revision History:
+
+--*/
+#include"spc_rewriter.h"
+#include"spc_decl_plugin.h"
+#include"ast_pp.h"
+
+namespace spc {
+
+    rewriter::rewriter(ast_manager & m, simplifier & simp, order & ord, asserted_literals & al):
+        simplifier(m),
+        m_asserted_literals(al),
+        m_order(ord),
+        m_spc_fid(m.get_family_id("spc")),
+        m_subst(m),
+        m_st(m),
+        m_visitor(m_order, m_subst, m_cls_use_list) {
+
+        reserve_offsets(3);
+
+        borrow_plugins(simp);
+    }
+
+    rewriter::~rewriter() {
+        release_plugins();
+    }
+    
+    inline void rewriter::reserve_vars(unsigned num_vars) {
+        m_subst.reserve_vars(num_vars);
+        m_order.reserve_vars(num_vars);
+        m_asserted_literals.reserve_vars(num_vars);
+    }
+
+    void rewriter::reserve_offsets(unsigned num_offsets) {
+        m_subst.reserve_offsets(num_offsets);
+        m_order.reserve_offsets(num_offsets);
+    }
+
+    inline bool rewriter::demodulator(clause * cls) const {
+        if (cls->get_num_literals() != 1)
+            return false;
+        literal const & l = cls->get_literal(0);
+        return !l.sign() && m_manager.is_eq(l.atom());
+    }
+
+    inline void rewriter::insert(clause * cls, expr * source) {
+        if (!is_var(source)) {
+            TRACE("rewriter_detail", tout << "inserting into rewriter index:\n"; cls->display(tout, m_manager); tout << "\n";);
+            flush_cache();
+            m_st.insert(to_app(source));
+            m_cls_use_list.insert(cls, source);
+        }
+    }
+    
+    void rewriter::insert(clause * cls) {
+        if (demodulator(cls)) {
+            reserve_vars(cls->get_num_vars());
+            literal const & l = cls->get_literal(0);
+            app * eq = to_app(l.atom());
+            if (l.is_oriented()) {
+                expr * source = l.is_left() ? eq->get_arg(0) : eq->get_arg(1);
+                insert(cls, source);
+            }
+            else {
+                insert(cls, eq->get_arg(0));
+                insert(cls, eq->get_arg(1));
+            }
+        }
+    }
+
+    inline void rewriter::erase(clause * cls, expr * source) {
+        if (!is_var(source)) {
+            flush_cache();
+            m_cls_use_list.erase(cls, source);
+            if (m_cls_use_list.empty(source))
+                m_st.erase(to_app(source));
+        }
+    }
+
+    void rewriter::erase(clause * cls) {
+        if (demodulator(cls)) {
+            literal const & l = cls->get_literal(0);
+            app * eq = to_app(l.atom());
+            if (l.is_oriented()) {
+                expr * source = l.is_left() ? eq->get_arg(0) : eq->get_arg(1);
+                erase(cls, source);
+            }
+            else {
+                erase(cls, eq->get_arg(0));
+                erase(cls, eq->get_arg(1));
+            }
+        }
+    }
+
+    bool rewriter::visitor::operator()(expr * e) {
+        if (m_cls_use_list.empty(e)) 
+            return true; // continue;
+        clause_use_list::iterator it  = m_cls_use_list.begin(e);
+        clause_use_list::iterator end = m_cls_use_list.end(e);
+        for (; it != end; ++it) {
+            m_clause = *it;
+            SASSERT(m_clause->get_num_literals() == 1);
+            literal & l = m_clause->get_literal(0);
+            expr * atom = l.atom();
+            SASSERT(!l.sign() && m_manager.is_eq(atom));
+            SASSERT(to_app(atom)->get_arg(0) == e || to_app(atom)->get_arg(1) == e);
+            m_source = to_app(atom)->get_arg(0);
+            m_target = to_app(atom)->get_arg(1);
+            if (m_source != e)
+                std::swap(m_source, m_target);
+            SASSERT(m_source == e);
+            TRACE("rewriter", tout << "found generalization:\n" << mk_pp(m_source, m_manager) << "\n" << 
+                  mk_pp(m_target, m_manager) << "\nsubstitution\n";
+                  m_subst.display(tout); tout << "m_subst: " << &m_subst << "\n";
+                  tout << "checking ordering constraints...\n";);
+            if (l.is_oriented() || m_order.greater(expr_offset(m_source, 1), expr_offset(m_target, 1), &m_subst)) {
+                m_found = true;
+                return false; // stop
+            }
+            TRACE("rewriter", tout << "failed ordering constraints...\n";);
+        }
+        return true; // continue
+    }
+
+    void rewriter::save_justification(justification * j) {
+        if (std::find(m_justifications.begin(), m_justifications.end(), j) == m_justifications.end())
+            m_justifications.push_back(j);
+    }
+
+    proof * rewriter::mk_demodulation_proof(expr * old_expr, expr * new_expr, proof * parent) {
+        if (m_manager.fine_grain_proofs()) {
+            SASSERT(parent);
+            return m_manager.mk_app(m_spc_fid, PR_DEMODULATION, parent, m_manager.mk_eq(old_expr, new_expr));
+        }
+        return 0;
+    }
+
+    void rewriter::reset() {
+        m_st.reset();
+        m_cls_use_list.reset();
+    }
+
+    void rewriter::reduce_literal(literal const & l, literal & l_r, proof * & l_pr) {
+        if (m_st.empty()) {
+            l_r = l;
+            l_pr = 0;
+            return;
+        }
+
+        expr * atom = l.atom();
+        expr *  r;
+        proof * r_pr;
+        m_proofs.reset();
+        while (true) {
+            reduce_core(atom);
+            get_cached(atom, r, r_pr);
+            if (m_manager.fine_grain_proofs() && r_pr)
+                m_proofs.push_back(r_pr);
+            if (atom == r)
+                break;
+            atom = r;
+        }
+        l_r = literal(atom, l.sign());
+        if (m_manager.fine_grain_proofs())
+            l_pr = m_proofs.empty() ? 0 : m_manager.mk_transitivity(m_proofs.size(), m_proofs.c_ptr());
+    }
+
+    clause * rewriter::operator()(clause * cls) {
+        reserve_vars(cls->get_num_vars());
+        SASSERT(m_found_literals.empty());
+        m_justifications.reset();
+        m_max_scope_lvl = cls->get_scope_lvl();
+        
+        literal_buffer    new_literals(m_manager);
+        proof_ref_buffer  new_proofs(m_manager);
+        bool              changed = false;
+
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal & l = cls->get_literal(i);
+            literal l_r;
+            proof * l_pr = 0;
+            reduce_literal(l, l_r, l_pr);
+         
+            if (l != l_r) {
+                changed = true;
+            }
+
+            if (!l_r.is_false(m_manager) && !m_found_literals.contains(l_r)) {
+                m_found_literals.insert(l_r);
+
+                // apply simplify reflect rules
+                expr * atom     = l_r.atom();
+                clause * unit = 0;
+                TRACE("rewriter", tout << "adding literal: " << mk_pp(atom, m_manager) << "\n";);
+                if (l_r.sign()) {
+                    if (m_manager.is_eq(atom))
+                        unit = m_asserted_literals.subsumes(to_app(atom)->get_arg(0), to_app(atom)->get_arg(1));
+                    else
+                        unit = m_asserted_literals.gen(atom, false);
+                }
+                else {
+                    // check if there is a generalization of the negation of the current literal.
+                    unit = m_asserted_literals.gen(atom, true);
+                }
+
+                if (unit) {
+                    // new literal was resolved
+                    justification * j = unit->get_justification();
+                    m_justifications.push_back(j);
+                    changed = true;
+                }
+                else {
+                    // keep new literal
+                    new_literals.push_back(l_r);
+                }
+            }
+            else {
+                // removed duplicate or resolved literal.
+                changed = true;
+            }
+
+            if (m_manager.fine_grain_proofs() && l_pr != 0) {
+                new_proofs.push_back(l_pr);
+            }
+        }
+
+        m_found_literals.reset();
+
+        if (!changed) {
+            m_found_literals.reset();
+            return cls;
+        }
+
+        proof * new_pr = mk_rewrite_proof(m_manager, m_spc_fid, new_literals.size(), new_literals.c_ptr(), cls->get_justification()->get_proof(), 
+                                          new_proofs.size(), new_proofs.c_ptr());
+
+        justification * new_j = rewrite_justification::mk(m_manager, cls->get_justification(), m_justifications.size(), m_justifications.c_ptr(), new_pr);
+        
+        if (m_max_scope_lvl == cls->get_scope_lvl()) {
+            // peform destructive update
+            cls->update_lits(m_manager, new_literals.size(), new_literals.c_ptr(), new_j);
+            return cls;
+        }
+        else {
+            SASSERT(m_max_scope_lvl > cls->get_scope_lvl());
+            // create new clause
+            // the old clause will be frozen
+            return clause::mk(m_manager, new_literals.size(), new_literals.c_ptr(), new_j, m_max_scope_lvl);
+        }
+    }
+
+};
+

src/spc/spc_rewriter.h

@@ -0,0 +1,122 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_rewriter.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-11.
+
+Revision History:
+
+--*/
+#ifndef _SPC_REWRITER_H_
+#define _SPC_REWRITER_H_
+
+#include"simplifier.h"
+#include"order.h"
+#include"substitution_tree.h"
+#include"spc_clause.h"
+#include"spc_asserted_literals.h"
+#include"sparse_use_list.h"
+
+namespace spc {
+    
+    /**
+       \brief Apply rewriting steps using demodulation rule:
+
+       C[s] ==> C[sigma(r)]
+       when 
+       
+       l = r is a known equality (demodulator)
+       sigma(l) = s
+       sigma(l) > sigma(r)
+
+       It also applies the following rules:
+       
+       - Duplicate literal deletion
+
+       - Resolved literal deletion
+
+       - Positive simplify reflect
+       s = t,  (u[p <- sigma(s)] != u[p <- sigma(t)] or R)
+       ==>
+       R
+   
+       - Negative simplify reflect
+       s != t  (sigma(s = t) or R)
+       ===>
+       R
+    */
+    class rewriter : public simplifier {
+    protected:
+        typedef sparse_use_list<expr, ptr_vector<clause> > clause_use_list;
+        asserted_literals &       m_asserted_literals;
+        order &                   m_order;
+        family_id                 m_spc_fid;
+
+        substitution              m_subst;
+        substitution_tree         m_st; // index for potential demodulators left-hand-side
+        clause_use_list           m_cls_use_list; // index for demodulators left-hand-side to equation.
+        found_literals            m_found_literals;
+
+        ptr_vector<justification> m_justifications;
+
+        struct visitor : public st_visitor {
+            ast_manager &     m_manager;
+            order &           m_order;
+            clause_use_list & m_cls_use_list;
+            bool              m_found;
+            clause *          m_clause;
+            expr *            m_source;
+            expr *            m_target;
+
+            visitor(order & ord, substitution & subst, clause_use_list & ul):
+                st_visitor(subst), m_manager(ord.get_manager()), m_order(ord), m_cls_use_list(ul) {
+            }
+            
+            virtual bool operator()(expr * e);
+        };
+
+        unsigned                  m_max_scope_lvl; // maximal scope level used during rewrite.
+        visitor                   m_visitor;
+
+        proof * mk_demodulation_proof(expr * old_expr, expr * new_expr, proof * parent);
+
+        bool demodulator(clause * cls) const;
+        void insert(clause * cls, expr * source);
+        void erase(clause * cls, expr * source);
+        void reserve_vars(unsigned num_vars);
+        void reserve_offsets(unsigned num_offsets);
+        void save_justification(justification * j);
+
+        void reduce_literal(literal const & l, literal & l_r, proof * & l_pr);
+
+    public:
+        rewriter(ast_manager & m, simplifier & s, order & ord, asserted_literals & al);
+        virtual ~rewriter();
+        
+        /**
+           \brief Insert clause into rewriter indexes
+        */
+        void insert(clause * cls);
+
+        /**
+           \brief Remove clause from rewriter indexes
+        */
+        void erase(clause * cls);
+        
+        clause * operator()(clause * cls);
+
+        void reset();
+    };
+};
+
+#endif /* _SPC_REWRITER_H_ */
+

src/spc/spc_semantic_tautology.cpp

@@ -0,0 +1,234 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_semantic_tautology.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-11.
+
+Revision History:
+
+--*/
+#include"spc_semantic_tautology.h"
+#include"ast_pp.h"
+
+namespace spc {
+
+    expr * find(expr2expr & f, expr * n) {
+#ifdef _TRACE
+        expr * _n = n;
+#endif
+        ptr_buffer<expr> path;
+        expr * next;
+        while (f.find(n, next)) {
+            path.push_back(n);
+            n = next;
+        }
+        ptr_buffer<expr>::iterator it  = path.begin();
+        ptr_buffer<expr>::iterator end = path.end();
+        for (; it != end; ++it) {
+            expr * prev = *it;
+            f.insert(prev, n);
+        }
+        SASSERT(n);
+        TRACE("semantic_tautology_detail", tout << "find(#" << _n->get_id() << ") = #" << n->get_id() << "\n";);
+        return n;
+    }
+    
+    semantic_tautology::semantic_tautology(ast_manager & m):
+        m_manager(m),
+        m_cg_table(DEFAULT_HASHTABLE_INITIAL_CAPACITY, cg_hash(m_manager, m_find), cg_eq(m_find)) {
+    }
+
+    unsigned semantic_tautology::cg_hash::operator()(app * n) const { 
+        TRACE("semantic_tautology_detail", tout << "hash code of:\n" << mk_pp(n, m_manager) << "\n";);
+        unsigned r = get_composite_hash<app *, k_hash, c_hash>(n, n->get_num_args(), m_k_hash, m_c_hash); 
+        TRACE("semantic_tautology_detail", tout << "result: " << r << "\n";);
+        return r;
+    }
+
+    bool semantic_tautology::cg_eq::operator()(app * n1, app * n2) const {
+        if (n1->get_decl() != n2->get_decl() || n1->get_num_args() != n2->get_num_args())
+            return false;
+        unsigned num_args = n1->get_num_args();
+        for (unsigned i = 0; i < num_args; i++)
+            if (spc::find(m_find, n1->get_arg(i)) != spc::find(m_find, n2->get_arg(i)))
+                return false;
+        return true;
+    }
+    
+    bool semantic_tautology::is_target(unsigned num_lits, literal * lits) {
+        bool has_diseq     = false;
+        bool has_non_diseq = false;
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l = lits[i];
+            if (l.sign() && m_manager.is_eq(l.atom())) 
+                has_diseq = true;
+            else 
+                has_non_diseq = true;
+        }
+        return has_diseq && has_non_diseq;
+    }
+
+    void semantic_tautology::reset() {
+        m_region.reset();
+        m_init_todo.reset();
+        m_todo.reset();
+        m_already_found.reset();
+        m_use_list.reset();
+        m_find.reset();
+        m_size.reset();
+        m_cg_table.reset();
+    }
+
+    void semantic_tautology::update_use_list(app * parent, expr * child) {
+        list<app*> * use_list = 0;
+        m_use_list.find(child, use_list);
+        use_list = new (m_region) list<app*>(parent, use_list);
+        m_use_list.insert(child, use_list);
+    }
+
+    inline void semantic_tautology::push_init_core(expr * n) {
+        if (is_app(n) && to_app(n)->get_num_args() > 0)
+            m_init_todo.push_back(to_app(n));
+    }
+
+    inline void semantic_tautology::push_init(expr * atom) {
+        if (m_manager.is_eq(atom)) {
+            push_init_core(to_app(atom)->get_arg(0));
+            push_init_core(to_app(atom)->get_arg(1));
+        }
+        else 
+            push_init_core(atom);
+    }
+
+    void semantic_tautology::init_use_list() {
+        while (!m_init_todo.empty()) {
+            app * n = m_init_todo.back();
+            m_init_todo.pop_back();
+            if (!m_already_found.contains(n)) {
+                unsigned num_args = n->get_num_args();
+                SASSERT(num_args > 0);
+                m_cg_table.insert(n);
+                m_already_found.insert(n);
+                for (unsigned i = 0; i < num_args; i++) {
+                    expr * c = n->get_arg(i);
+                    update_use_list(n, c);
+                    push_init_core(c);
+                }
+            }
+        }
+    }
+
+    void semantic_tautology::init(unsigned num_lits, literal * lits) {
+        reset();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l = lits[i];
+            expr * atom = l.atom();
+            push_init(atom);
+            if (l.sign() && m_manager.is_eq(atom))
+                m_todo.push_back(expr_pair(to_app(atom)->get_arg(0), to_app(atom)->get_arg(1)));
+        }
+        init_use_list();
+    }
+
+    void semantic_tautology::remove_parents(expr * n1) {
+        list<app*> * use_list = 0;
+        m_use_list.find(n1, use_list);
+        while (use_list) {
+            TRACE("semantic_tautology", tout << "removing parent from cg_table:\n" << mk_pp(use_list->head(), m_manager) << "\n";);
+            m_cg_table.erase(use_list->head());
+            use_list = use_list->tail();
+        }
+    }
+
+    void semantic_tautology::restore_parents(expr * n1, expr * n2) {
+        list<app*> * use_list = 0;
+        m_use_list.find(n1, use_list);
+        while (use_list) {
+            app * parent = use_list->head();
+            app * other  = 0;
+            if (m_cg_table.find(parent, other)) {
+                TRACE("semantic_tautology", tout << "new congruence:\n" << mk_pp(parent, m_manager) << "\n" << mk_pp(other, m_manager) << "\n";);
+                if (parent != other)
+                    m_todo.push_back(expr_pair(parent, other));
+            }
+            else {
+                TRACE("semantic_tautology", tout << "restoring parent to cg_table:\n" << mk_pp(parent, m_manager) << "\n";);
+                m_cg_table.insert(parent);
+                update_use_list(parent, n2);
+            }
+            use_list = use_list->tail();
+        }
+    }
+
+    void semantic_tautology::assert_eq(expr * n1, expr * n2) {
+        n1 = find(n1);
+        n2 = find(n2);
+        if (n1 == n2)
+            return;
+        TRACE("semantic_tautology", tout << "processing equality:\n" << mk_pp(n1, m_manager) << " " << n1->get_id() << "\n" << 
+              mk_pp(n2, m_manager) << " " << n2->get_id() << "\n";);
+        unsigned sz1 = 1;
+        unsigned sz2 = 1;
+        m_size.find(n1, sz1);
+        m_size.find(n2, sz2);
+        if (sz1 > sz2)
+            std::swap(n1, n2);
+        remove_parents(n1);
+        TRACE("semantic_tautology", tout << "merging equivalence classes\n";);
+        m_find.insert(n1, n2);
+        m_size.insert(n2, sz1 + sz2);
+        restore_parents(n1, n2);
+    }
+
+    void semantic_tautology::process_eqs() {
+        while (!m_todo.empty()) {
+            expr_pair const & p = m_todo.back();
+            expr * lhs = p.first;
+            expr * rhs = p.second;
+            m_todo.pop_back();
+            assert_eq(lhs, rhs);
+        }
+    }
+    
+    bool semantic_tautology::contains_complement(unsigned num_lits, literal * lits, unsigned i, bool sign, expr * atom) {
+        atom = find(atom);
+        for (unsigned j = i + 1; j < num_lits; j++) {
+            literal const & l = lits[j];
+            if (l.sign() != sign && find(l.atom()) == atom)
+                return true;
+        }
+        return false;
+    }
+    
+    bool semantic_tautology::is_tautology(unsigned num_lits, literal * lits) {
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l = lits[i];
+            expr * atom       = l.atom();
+            if (!l.sign() && m_manager.is_eq(atom) && find(to_app(atom)->get_arg(0)) == find(to_app(atom)->get_arg(1))) 
+                return true;
+            if (!m_manager.is_eq(atom) && contains_complement(num_lits, lits, i, l.sign(), atom))
+                return true;
+        }
+        return false;
+    }
+    
+    bool semantic_tautology::operator()(unsigned num_lits, literal * lits) {
+        if (!is_target(num_lits, lits))
+            return false;
+        init(num_lits, lits);
+        process_eqs();
+        bool r = is_tautology(num_lits, lits);
+        TRACE("semantic_tautology", display(tout, num_lits, lits, m_manager); tout << "\nis semantic tautology: " << r << "\n";);
+        return r;
+    }
+
+};

src/spc/spc_semantic_tautology.h

@@ -0,0 +1,114 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_semantic_tautology.h
+
+Abstract:
+
+    Semantic tautology detection
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-11.
+
+Revision History:
+
+--*/
+#ifndef _SPC_SEMANTIC_TAUTOLOGY_H_
+#define _SPC_SEMANTIC_TAUTOLOGY_H_
+
+#include"spc_literal.h"
+#include"list.h"
+#include"obj_hashtable.h"
+#include"map.h"
+
+namespace spc {
+
+    typedef obj_map<expr, expr *> expr2expr;
+
+    expr * find(expr2expr & f, expr * e);
+
+    /**
+       \brief Functor for detecting semantic tautology.
+       A clause C is a semantic tautology if it has the following form:
+
+       s_1 != t_1 or ... or s_n != t_n or s = t or R
+
+       sigma(s_1 = t_1), ..., sigma(s_n t_n) |= sigma(s = t)
+       where sigma maps variables to constants.
+    */
+    class semantic_tautology {
+        typedef std::pair<expr *, expr *> expr_pair;
+        
+        typedef obj_hashtable<expr> already_found;
+        typedef expr2expr find_map;
+        typedef obj_map<expr, list<app*> *> use_list;
+        typedef obj_map<expr, unsigned> size_map;        
+
+        struct k_hash {
+            unsigned operator()(app * n) const { return n->get_decl()->get_id(); }
+        };
+
+        struct c_hash {
+            find_map & m_find;
+            c_hash(find_map & f):m_find(f) {}
+            unsigned operator()(app * n, unsigned i) const { 
+                unsigned id = spc::find(m_find, n->get_arg(i))->get_id();
+                TRACE("semantic_tautology_detail", tout << "child(" << i << ") = #" << id << "\n";);
+                return id;
+            }
+        };
+
+        struct cg_hash {
+            ast_manager & m_manager;
+            k_hash     m_k_hash;
+            c_hash     m_c_hash;
+            cg_hash(ast_manager & m, find_map & f):m_manager(m), m_c_hash(f) {}
+            unsigned operator()(app * n) const;
+        };
+
+        struct cg_eq {
+            find_map & m_find;
+            cg_eq(find_map & f):m_find(f) {}
+            bool operator()(app * n1, app * n2) const;
+        };
+        
+        typedef ptr_hashtable<app, cg_hash, cg_eq> cg_table;
+
+        ast_manager &      m_manager;
+        region             m_region;
+        ptr_vector<app>    m_init_todo;
+        svector<expr_pair> m_todo;
+        already_found      m_already_found;
+        use_list           m_use_list;
+        find_map           m_find;
+        size_map           m_size;
+        cg_table           m_cg_table;
+
+        bool is_target(unsigned num_lits, literal * lits);
+        void reset();
+        void update_use_list(app * parent, expr * child);
+        void push_init_core(expr * n);
+        void push_init(expr * atom);
+        void init_use_list();
+        void init(unsigned num_lits, literal * lits);
+        expr * find(expr * n) { return spc::find(m_find, n); }
+        void remove_parents(expr * n1);
+        void restore_parents(expr * n1, expr * n2);
+        void assert_eq(expr * n1, expr * n2);
+        void process_eqs();
+        bool contains_complement(unsigned num_lits, literal * lits, unsigned i, bool sign, expr * atom);
+        bool is_tautology(unsigned num_lits, literal * lits);
+
+    public:
+        semantic_tautology(ast_manager & m);
+        
+        bool operator()(unsigned num_lits, literal * lits);
+    };
+
+};
+
+#endif /* _SPC_SEMANTIC_TAUTOLOGY_H_ */
+

src/spc/spc_statistics.cpp

@@ -0,0 +1,54 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_statistics.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-17.
+
+Revision History:
+
+--*/
+#include"spc_statistics.h"
+
+namespace spc {
+    
+    void statistics::reset() {
+        m_num_mk_clause = 0;
+        m_num_del_clause = 0;
+        m_num_processed = 0;
+        m_num_superposition = 0;
+        m_num_resolution = 0;
+        m_num_eq_factoring = 0;
+        m_num_factoring = 0;
+        m_num_eq_resolution = 0;
+        m_num_trivial = 0;
+        m_num_simplified = 0;
+        m_num_subsumed = 0;
+        m_num_redundant = 0;
+    }
+    
+    void statistics::display(std::ostream & out) const {
+        out << "num. mk. clause:    " << m_num_mk_clause << "\n";
+        out << "num. del. clause:   " << m_num_del_clause << "\n";
+        out << "num. processed:     " << m_num_processed << "\n";
+        out << "num. superposition: " << m_num_superposition << "\n";
+        out << "num. resolution:    " << m_num_resolution << "\n";
+        out << "num. eq. factoring: " << m_num_eq_factoring << "\n";
+        out << "num. factoring:     " << m_num_factoring << "\n";
+        out << "num. eq. resol.:    " << m_num_eq_resolution << "\n";
+        out << "num. simplified:    " << m_num_simplified << "\n";
+        out << "num. redundant:     " << m_num_redundant << "\n";
+        out << "  num. trivial:     " << m_num_trivial << "\n";
+        out << "  num. subsumed:    " << m_num_subsumed << "\n";
+    }
+
+};
+

src/spc/spc_statistics.h

@@ -0,0 +1,49 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_statistics.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-17.
+
+Revision History:
+
+--*/
+#ifndef _SPC_STATISTICS_H_
+#define _SPC_STATISTICS_H_
+
+#include<iostream>
+
+namespace spc {
+
+    struct statistics {
+        unsigned m_num_mk_clause;
+        unsigned m_num_del_clause;
+        unsigned m_num_processed;
+        unsigned m_num_superposition;
+        unsigned m_num_resolution;
+        unsigned m_num_eq_factoring;
+        unsigned m_num_factoring;
+        unsigned m_num_eq_resolution;
+        unsigned m_num_trivial;
+        unsigned m_num_simplified;
+        unsigned m_num_subsumed;
+        unsigned m_num_redundant;
+        statistics() {
+            reset();
+        }
+        
+        void reset();
+        void display(std::ostream & out) const;
+    };
+};
+
+#endif /* _SPC_STATISTICS_H_ */
+

src/spc/spc_subsumption.cpp

@@ -0,0 +1,698 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_subsumption.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-13.
+
+Revision History:
+
+--*/
+#include"spc_subsumption.h"
+#include"fvi_def.h"
+#include"ast_pp.h"
+
+namespace spc {
+
+    /**
+       \brief Return true if literal l2 is an instance of l1.
+       
+       When ResetSubst == true, m_subst is reset before trying to match l1 and l2.
+
+       When ResetSubst == false, it is assumed that m_subst.push_scope was invoked
+       before invoking match_literal. 
+    */
+    template<bool ResetSubst>
+    bool subsumption::match_literal(literal const & l1, literal const & l2) {
+        if (l1.sign() == l2.sign()) {
+            expr * atom1 = l1.atom();
+            expr * atom2 = l2.atom();
+            bool is_eq1  = m_manager.is_eq(atom1);
+            bool is_eq2  = m_manager.is_eq(atom2);
+            if (is_eq1 && is_eq2) {
+                expr * lhs1 = to_app(atom1)->get_arg(0);
+                expr * rhs1 = to_app(atom1)->get_arg(1);
+                expr * lhs2 = to_app(atom2)->get_arg(0);
+                expr * rhs2 = to_app(atom2)->get_arg(1);
+                if (ResetSubst) 
+                    m_subst.reset_subst();
+                
+                if (m_matcher(lhs1, lhs2, m_subst) && m_matcher(rhs1, rhs2, m_subst))
+                    return true;
+
+                if (ResetSubst)
+                    m_subst.reset_subst();
+                else {
+                    // I'm assuming push_scope was invoked before executing match_literal
+                    // So, pop_scope is equivalent to a local reset.
+                    m_subst.pop_scope();
+                    m_subst.push_scope();
+                }
+
+                return (m_matcher(lhs1, rhs2, m_subst) && m_matcher(rhs1, lhs2, m_subst));
+            }
+            else if (!is_eq1 && !is_eq2) {
+                if (ResetSubst) 
+                    m_subst.reset_subst();
+                return m_matcher(atom1, atom2, m_subst);
+            }
+        }
+        return false;
+    }
+    
+    /**
+       \brief Return true if for every literal l1 in lits1 there is a
+       literal l2 in lits2 such that l2 is an instance of l1.
+    */
+    bool subsumption::can_subsume(unsigned num_lits1, literal * lits1, unsigned num_lits2, literal * lits2) {
+        for (unsigned i = 0; i < num_lits1; i++) {
+            literal const & l1 = lits1[i];
+            unsigned j = 0;
+            for (; j < num_lits2; j++) {
+                literal const & l2 = lits2[j];
+                if (match_literal<true>(l1, l2))
+                    break;
+            }
+            if (j == num_lits2)
+                return false;
+        }
+        return true;
+    }
+
+    /**
+       \brief Return true if cls1 can subsume cls2. It performs a series of quick checks.
+    */
+    bool subsumption::quick_check(clause * cls1, clause * cls2) {
+        return 
+            cls1->get_symbol_count() <= cls2->get_symbol_count() &&
+            cls1->get_const_count() <= cls2->get_const_count() &&
+            cls1->get_depth() <= cls2->get_depth() &&
+            cls1->get_num_pos_literals() <= cls2->get_num_pos_literals() &&
+            cls1->get_num_neg_literals() <= cls2->get_num_neg_literals() &&
+            (!cls1->is_ground() || cls2->is_ground());
+    }
+    
+    /**
+       \brief Return true if the set of literals lits1 subsumes the set of literals lits2.
+    */
+    bool subsumption::subsumes_core(unsigned num_lits1, literal * lits1, unsigned num_lits2, literal * lits2) {
+        enum state {
+            INVOKE, DECIDE, BACKTRACK, RETURN
+        };
+
+        if (num_lits1 == 0)
+            return true;
+        
+        m_stack.reset();
+        m_subst.reset();
+
+        m_stack.push_back(assoc(0, 0));
+        state st = DECIDE;
+
+        unsigned i;
+        assoc * top;
+        unsigned counter = 0;
+#ifdef _TRACE
+        unsigned opt  = 0;
+        unsigned nopt = 0;
+#endif
+        while (true && counter < 5000000) {
+            counter++;
+            switch (st) {
+            case INVOKE:
+                SASSERT(!m_stack.empty());
+                i = m_stack.back().first + 1;
+                if (i >= num_lits1) {
+                    TRACE("subsumption", tout << "subsumption result: YES.\n";);
+                    TRACE_CODE({
+                        if (counter > 10000) {
+                            TRACE("subsumption_perf", 
+                                  tout << "subsumption succeeded: " << counter << " " << opt << " " << nopt << "\n";
+                                  tout << "literals1:\n"; display(tout, num_lits1, lits1, m_manager); tout << "\n";
+                                  tout << "literals2:\n"; display(tout, num_lits2, lits2, m_manager); tout << "\n";);
+                        }
+                    });
+                    return true;
+                }
+                else {
+                    m_stack.push_back(assoc(i, 0));
+                    st = DECIDE;
+                }
+                break;
+            case DECIDE:
+                top = &(m_stack.back());
+                m_subst.push_scope();
+                if (match_literal<false>(lits1[top->first], lits2[top->second]))
+                    st = INVOKE;
+                else
+                    st = BACKTRACK;
+                break;
+            case BACKTRACK: 
+                top = &(m_stack.back());
+                top->second++;
+                m_subst.pop_scope();
+                if (top->second >= num_lits2) 
+                    st = RETURN;
+                else
+                    st = DECIDE;
+                break;
+            case RETURN:
+                top = &(m_stack.back());
+                m_stack.pop_back();
+                if (m_stack.empty()) {
+                    // no more alternatives
+                    TRACE("subsumption", tout << "subsumption result: NO\n";);
+                    TRACE_CODE({
+                        if (counter > 10000) {
+                            TRACE("subsumption_perf",
+                                  tout << "subsumption failed: " << counter << " " << opt << " " << nopt << "\n";
+                                  tout << "literals1:\n"; display(tout, num_lits1, lits1, m_manager); tout << "\n";
+                                  tout << "literals2:\n"; display(tout, num_lits2, lits2, m_manager); tout << "\n";);
+                        }
+                    });
+                    return false;
+                }
+                if (m_subst.top_scope_has_bindings()) {
+                    TRACE_CODE(nopt++;);
+                    st  = BACKTRACK;
+                }
+                else {
+                    TRACE_CODE(opt++;);
+#ifdef Z3DEBUG
+                    unsigned num_bindings = m_subst.get_num_bindings();
+#endif
+                    m_subst.pop_scope();
+                    SASSERT(num_bindings == m_subst.get_num_bindings());
+                    st = RETURN;
+                }
+                break;
+            }
+        }
+        return false;
+    }
+
+    /**
+       \brief Return true if the set of ground literals lits1 subsumes the set of ground literals lits2.
+    */
+    bool subsumption::ground_subsumes_core(unsigned num_lits1, literal * lits1, unsigned num_lits2, literal * lits2) {
+        for (unsigned i = 0; i < num_lits1; i++) {
+            literal const & l1 = lits1[i];
+            unsigned j = 0;
+            for (; j < num_lits2; j++) {
+                literal const & l2 = lits2[j];
+                if (l1 == l2)
+                    break;
+            }
+            if (j == num_lits2)
+                return false;
+        }
+        return true;
+    }
+
+    /**
+       \brief Return true if the literal l1 subsumes the set of literals lits2.
+    */
+    bool subsumption::subsumes_core(literal const & l1, unsigned num_lits2, literal * lits2) {
+        for (unsigned i = 0; i < num_lits2; i++) {
+            if (match_literal<true>(l1, lits2[i]))
+                return true;
+        }
+        return false;
+    }
+
+    subsumption::subsumption(ast_manager & m, asserted_literals & al, spc_params & params):
+        m_manager(m),
+        m_params(params),
+        m_asserted_literals(al),
+        m_subst(m),
+        m_matcher(m),
+        m_found_decls(m), 
+        m_index(0),
+        m_num_processed_clauses(0),
+        m_opt_threshold(params.m_initial_subsumption_index_opt) {
+
+        m_subst.reserve_offsets(1);
+        
+        init_indexes();
+    }
+
+    subsumption::~subsumption() {
+        if (m_index) 
+            dealloc(m_index);
+    }
+
+    /**
+       \brief Return true if cls1 subsumes cls2
+    */
+    bool subsumption::operator()(clause * cls1, clause * cls2) {
+        TRACE("subsumption_detail", tout << "checking if:\n"; cls1->display(tout, m_manager); tout << "\nsubsumes\n";
+              cls2->display(tout, m_manager); tout << "\n";);
+        if (!quick_check(cls1, cls2)) {
+            TRACE("subsumption_detail", tout << "failed quick check\n";);
+            return false;
+        }
+       
+        m_subst.reserve_vars(std::max(cls1->get_num_vars(), cls2->get_num_vars()));
+        unsigned num_lits1 = cls1->get_num_literals();
+        unsigned num_lits2 = cls2->get_num_literals();
+        literal * lits1    = cls1->get_literals();
+        literal * lits2    = cls2->get_literals();
+        if (cls1->is_ground() && cls2->is_ground())
+            return ground_subsumes_core(num_lits1, lits1, num_lits2, lits2);
+        if (num_lits1 == 1)
+            return subsumes_core(lits1[0], num_lits2, lits2);
+        // TODO: REMOVE true below... using it for debugging purposes.
+        if (true || cls1->get_num_neg_literals() >= 3 || cls1->get_num_pos_literals() >= 3)
+            if (!can_subsume(num_lits1, lits1, num_lits2, lits2)) {
+                TRACE("subsumption_detail", tout << "failed can_subsume\n";);
+                return false;
+            }
+        return subsumes_core(num_lits1, lits1, num_lits2, lits2);
+    }
+
+    /**
+       \brief Update the set of function symbols found in the clause being inserted into the index,
+       and the set of clauses found since the index started to be built.
+       
+       Return true if the function symbol should be tracked.
+    */
+    bool subsumption::mark_func_decl(func_decl * f) {
+        if (m_refining_index) {
+            if (!m_cls_found_decl_set.contains(f)) {
+                // update set of func_decls in the curr clause
+                m_cls_found_decl_set.insert(f);
+                m_cls_found_decls.push_back(f);
+                // update global set of founf func_decls
+                unsigned id = f->get_decl_id();
+                m_found_decl_set.reserve(id+1);
+                if (!m_found_decl_set.get(id)) {
+                    m_found_decl_set.set(id);
+                    m_found_decls.push_back(f);
+                }
+            }
+            return true;
+        }
+        else {
+            unsigned id = f->get_decl_id();
+            // if func_decl was not found yet, then ignore it.
+            if (id < m_found_decl_set.size() && m_found_decl_set.get(id)) {
+                if (!m_cls_found_decl_set.contains(f)) {
+                    // update set of func_decls in the curr clause
+                    m_cls_found_decl_set.insert(f);
+                    m_cls_found_decls.push_back(f);
+                }
+                return true;
+            }
+            return false;
+        }
+    }
+
+    /**
+       \brief Increment the number of occurrences of a function symbol in the clause being
+       inserted into the index.
+    */
+    void subsumption::inc_f_count(func_decl * f, bool neg) {
+        decl2nat & f_count = m_f_count[static_cast<unsigned>(neg)];
+        unsigned val;
+        if (f_count.find(f, val)) {
+            f_count.insert(f, val + 1);
+        }
+        else {
+            f_count.insert(f, 1);
+        }
+    }
+
+    /**
+       \brief Update the min/max num. of occurrences of func symbol in a clause.
+    */
+    void subsumption::update_min_max(func_decl * f) {
+        for (unsigned is_neg = 0; is_neg < 1; is_neg++) {
+            decl2nat & f_count = m_f_count[is_neg];
+            decl2nat & f_min   = m_f_min[is_neg];
+            decl2nat & f_max   = m_f_max[is_neg];
+            unsigned count;
+            if (f_count.find(f, count)) {
+                unsigned old_count;
+                if (!f_min.find(f, old_count) || old_count > count) {
+                    f_min.insert(f, count);
+                }
+                if (!f_max.find(f, old_count) || old_count < count) {
+                    f_max.insert(f, count);
+                }
+            }
+        }
+    }
+
+    /**
+       \brief Compute the number of occurences of function symbols in
+       a clause.
+    */
+    void subsumption::update_neg_pos_func_counts(clause * cls) {
+        m_f_count[0].reset();
+        m_f_count[1].reset();
+        m_cls_found_decl_set.reset();
+        m_cls_found_decls.reset();
+        ptr_buffer<expr> todo;
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l = cls->get_literal(i);
+            bool is_neg       = l.sign();
+            expr * n          = l.atom();
+            todo.push_back(n);
+            while (!todo.empty()) {
+                n = todo.back();
+                todo.pop_back();
+                if (is_app(n)) {
+                    func_decl * f = to_app(n)->get_decl();
+                    if (fvi_candidate(f) && mark_func_decl(f)) 
+                        inc_f_count(f, is_neg);
+                    unsigned num = to_app(n)->get_num_args();
+                    for (unsigned i = 0; i < num; i++) 
+                        todo.push_back(to_app(n)->get_arg(i));
+                }
+            }
+        }
+        
+        if (m_refining_index) {
+            ptr_vector<func_decl>::iterator it  = m_cls_found_decls.begin();
+            ptr_vector<func_decl>::iterator end = m_cls_found_decls.end();
+            for (; it != end; ++it) {
+                func_decl * f = *it;
+                update_min_max(f);
+                unsigned val;
+                if (m_f_freq.find(f, val))
+                    m_f_freq.insert(f, val + 1);
+                else
+                    m_f_freq.insert(f, 1);
+            }
+        }
+    }
+
+    /**
+       \brief Store in m_feature_vector the value for the features of cls.
+    */
+    void subsumption::compute_features(clause * cls, unsigned * fvector) {
+        unsigned num = m_features.size();
+        for (unsigned i = 0; i < num; i++) {
+            feature & f = m_features[i];
+            switch (f.m_kind) {
+            case F_GROUND:
+                fvector[i] = cls->is_ground();
+                break;
+            case F_NUM_POS_LITS:
+                fvector[i] = cls->get_num_pos_literals();
+                break;
+            case F_NUM_NEG_LITS:
+                fvector[i] = cls->get_num_neg_literals();
+                break;
+            case F_DEPTH:
+                fvector[i] = cls->get_depth();
+                break;
+            case F_CONST_COUNT:
+                fvector[i] = cls->get_const_count();
+                break;
+            case F_SYM_COUNT:
+                fvector[i] = cls->get_symbol_count();
+                break;
+            case F_NUM_NEG_FUNCS: {
+                unsigned val;
+                if (m_f_count[1].find(f.m_decl, val)) 
+                    fvector[i] = val;
+                else
+                    fvector[i] = 0;
+                break;
+            }
+            case F_NUM_POS_FUNCS: {
+                unsigned val;
+                if (m_f_count[0].find(f.m_decl, val)) 
+                    fvector[i] = val;
+                else
+                    fvector[i] = 0;
+                break;
+            }
+            default:
+                UNREACHABLE();
+            }
+        }
+        TRACE("subsumption_features",
+              tout << "features of: "; cls->display(tout, m_manager); tout << "\n";
+              for (unsigned i = 0; i < num; i++) {
+                  tout << fvector[i] << " ";
+              }
+              tout << "\n";);
+    }
+
+    /**
+       \brief Initialise indexes for forward/backward subsumption.
+    */
+    void subsumption::init_indexes() {
+        // index for forward/backward subsumption
+        // start with simple set of features
+        m_features.push_back(feature(F_GROUND));
+        m_features.push_back(feature(F_NUM_POS_LITS));
+        m_features.push_back(feature(F_NUM_NEG_LITS));
+        m_features.push_back(feature(F_DEPTH));
+        m_features.push_back(feature(F_CONST_COUNT));
+        m_features.push_back(feature(F_SYM_COUNT));
+        m_index = alloc(index, m_features.size(), to_feature_vector(*this));
+    }
+
+    unsigned subsumption::get_value_range(func_decl * f, bool neg) const {
+        unsigned i = static_cast<unsigned>(neg);
+        unsigned min;
+        unsigned max;
+        if (!m_f_min[i].find(f, min))
+            min = 0;
+        if (!m_f_max[i].find(f, max))
+            max = 0;
+        SASSERT(min <= max);
+        return max - min;
+    }
+
+    inline unsigned subsumption::get_value_range(func_decl * f) const {
+        return std::max(get_value_range(f, false), get_value_range(f, true));
+    }
+
+    bool subsumption::f_lt::operator()(func_decl * f1, func_decl * f2) const {
+        unsigned vrange1 = m_owner.get_value_range(f1);
+        unsigned vrange2 = m_owner.get_value_range(f2);
+        if (vrange1 < vrange2)
+            return true;
+        if (vrange1 == vrange2)
+            return f1->get_id() < f2->get_id();
+        return false;
+    }
+
+    /**
+       \brief Optimize the index for (non unit) forward subsumption and 
+       backward subsumption.
+    */
+    void subsumption::optimize_feature_index() {
+        ptr_vector<clause> clauses;
+        m_index->collect(clauses);
+
+        dealloc(m_index);
+        m_features.reset();
+
+        ptr_vector<func_decl> targets;
+        unsigned sz = m_found_decls.size();
+        for (unsigned i = 0; i < sz; i++) {
+            func_decl * f = m_found_decls.get(i);
+            unsigned val;
+            if (m_f_freq.find(f, val) && val > m_params.m_min_func_freq_subsumption_index && get_value_range(f) > 0) 
+                targets.push_back(f);
+        }
+
+        
+        f_lt lt(*this);
+        std::sort(targets.begin(), targets.end(), lt);
+        
+        m_features.push_back(feature(F_GROUND));
+        m_features.push_back(feature(F_NUM_POS_LITS));
+        m_features.push_back(feature(F_NUM_NEG_LITS));
+        m_features.push_back(feature(F_DEPTH));
+       
+        ptr_vector<func_decl>::iterator it  = targets.begin();
+        ptr_vector<func_decl>::iterator end = targets.end();
+        for (; it != end; ++it) {
+            func_decl * f = *it;
+            if (get_value_range(f, false) > 1)
+                m_features.push_back(feature(f, false));
+            if (get_value_range(f, true) > 1)
+                m_features.push_back(feature(f, true));
+            if (m_features.size() > m_params.m_max_subsumption_index_features)
+                break;
+        }
+        
+        m_features.push_back(feature(F_CONST_COUNT));
+        m_features.push_back(feature(F_SYM_COUNT));
+        m_index = alloc(index, m_features.size(), to_feature_vector(*this));
+        m_num_processed_clauses = 0;
+        unsigned new_threshold = static_cast<unsigned>(m_opt_threshold * m_params.m_factor_subsumption_index_opt);
+        if (new_threshold > m_opt_threshold)
+            m_opt_threshold = new_threshold;
+    }
+    
+    /**
+       \brief Insert cls into the indexes used for forward/backward subsumption.
+    */
+    void subsumption::insert(clause * cls) {
+        TRACE("subsumption", tout << "adding clause to subsumption index: " << cls << "\n"; cls->display(tout, m_manager); tout << "\n";);
+        unsigned num_lits = cls->get_num_literals();
+        if (num_lits > 1 || m_params.m_backward_subsumption) {
+            m_index->insert(cls);
+            SASSERT(m_index->contains(cls));
+            m_num_processed_clauses++;
+            if (m_num_processed_clauses > m_opt_threshold)
+                optimize_feature_index();
+        }
+    }
+
+    /**
+       \brief Remove cls from the indexes used for forward/backward subsumption.
+    */
+    void subsumption::erase(clause * cls) {
+        TRACE("subsumption", tout << "removing clause from subsumption index:" << cls << "\n"; cls->display(tout, m_manager); tout << "\n";
+              tout << "num lits.: " << cls->get_num_literals() << ", backward_sub: " << m_params.m_backward_subsumption << "\n";);
+        unsigned num_lits = cls->get_num_literals();
+        if (num_lits > 1 || m_params.m_backward_subsumption)
+            m_index->erase(cls);
+    }
+
+    /**
+       \brief Reset the indexes used for forward/backward subsumption.
+    */
+    void subsumption::reset() {
+        if (m_index)
+            m_index->reset();
+        m_num_processed_clauses = 0;
+        m_opt_threshold = m_params.m_initial_subsumption_index_opt;
+    }
+
+    /**
+       \brief Return an unit clause C in the index that subsumes cls.
+       Return 0 if such clause does not exist.
+    */
+    clause * subsumption::unit_forward(clause * cls) {
+        if (!m_asserted_literals.has_literals())
+            return 0;
+        m_asserted_literals.reserve_vars(cls->get_num_vars());
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l = cls->get_literal(i);
+            clause * subsumer = m_asserted_literals.gen(l);
+            if (subsumer)
+                return subsumer;
+        }
+        return 0;
+    }
+
+    struct non_unit_subsumption_visitor {
+        subsumption & m_owner;
+        clause *      m_new_clause;
+        clause *      m_subsumer;
+        non_unit_subsumption_visitor(subsumption & owner, clause * new_clause):
+            m_owner(owner),
+            m_new_clause(new_clause),
+            m_subsumer(0) {
+        }
+        bool operator()(clause * candidate) {
+            TRACE("subsumption_index", tout << "considering candidate:\n"; candidate->display(tout, m_owner.get_manager()); tout << "\n";);
+            if (candidate->get_num_literals() > 1 && m_owner(candidate, m_new_clause)) {
+                m_subsumer = candidate;
+                return false; // stop subsumer was found
+            }
+            return true; // continue;
+        }
+    };
+
+    /**
+       \brief Return a non unit clause C in the index that subsumes cls.
+       Return 0 if such clause does not exist.
+    */
+    clause * subsumption::non_unit_forward(clause * cls) {
+        non_unit_subsumption_visitor visitor(*this, cls);
+        m_index->visit(cls, visitor, true);
+        return visitor.m_subsumer;
+    }
+
+    /**
+       \brief Return a unit equality clause (= s t) that (eq) subsumes cls.
+       That is, cls contains a literal (= u[s'] u[t']) and there is 
+       a substitution sigma s.t. sigma(s) = s' and sigma(t) = t'.
+       Return 0 if such clause does not exist.
+    */
+    clause * subsumption::eq_subsumption(clause * cls) {
+        if (!m_asserted_literals.has_pos_literals())
+            return 0;
+        m_asserted_literals.reserve_vars(cls->get_num_vars());
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l = cls->get_literal(i);
+            expr * atom = l.atom();
+            if (!l.sign() && m_manager.is_eq(atom)) {
+                expr * lhs = to_app(atom)->get_arg(0);
+                expr * rhs = to_app(atom)->get_arg(1);
+                clause * subsumer = m_asserted_literals.subsumes(lhs, rhs);
+                if (subsumer) {
+                    TRACE("eq_subsumption", tout << "equality subsumption:\n"; cls->display(tout, m_manager); 
+                          tout << "\nis subsumed by:\n"; subsumer->display(tout, m_manager); tout << "\n";);
+                    return subsumer;
+                }
+            }
+        }
+        return 0;
+    }
+
+    /**
+       \brief Return a clause C in the index (i.e., insert(C) was invoked) that subsumes cls.
+       Return 0 if such clause does not exist.
+    */
+    clause * subsumption::forward(clause * cls) {
+        TRACE("subsumption", tout << "trying forward subsumption:\n"; cls->display(tout, m_manager); tout << "\n";);
+        clause * subsumer = unit_forward(cls);
+        if (subsumer)
+            return subsumer;
+        subsumer = non_unit_forward(cls);
+        if (subsumer)
+            return subsumer;
+        if (m_params.m_equality_subsumption) 
+            return eq_subsumption(cls);
+        return 0;
+    }
+
+    struct backward_subsumption_visitor {
+        subsumption &        m_owner;
+        clause *             m_new_clause;
+        ptr_buffer<clause> & m_result;
+        backward_subsumption_visitor(subsumption & owner, clause * new_clause, ptr_buffer<clause> & result):
+            m_owner(owner),
+            m_new_clause(new_clause),
+            m_result(result) {
+        }
+        
+        bool operator()(clause * candidate) {
+            if (m_owner(m_new_clause, candidate))
+                m_result.push_back(candidate); 
+            return true; // always continue in backward subsumption
+        }
+    };
+
+    /**
+       \brief Store in result the set of clauses in the index that are subsumes by cls.
+    */
+    void subsumption::backward(clause * cls, ptr_buffer<clause> & result) {
+        if (m_params.m_backward_subsumption) {
+            backward_subsumption_visitor visitor(*this, cls, result);
+            m_index->visit(cls, visitor, false);
+        }
+    }
+};

src/spc/spc_subsumption.h

@@ -0,0 +1,156 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_subsumption.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-13.
+
+Revision History:
+
+--*/
+#ifndef _SPC_SUBSUMPTION_H_
+#define _SPC_SUBSUMPTION_H_
+
+#include"spc_asserted_literals.h"
+#include"matcher.h"
+#include"fvi.h"
+#include"spc_params.h"
+#include"obj_hashtable.h"
+
+namespace spc {
+
+    class subsumption {
+        ast_manager &         m_manager;
+        spc_params &          m_params;
+        asserted_literals &   m_asserted_literals;
+        substitution          m_subst;
+        matcher               m_matcher;
+
+        // A pair representing the association between l1 and l2 where
+        // first is the position of l1 in lits1 and second the position of l2 in
+        // lits2.
+        typedef std::pair<unsigned, unsigned> assoc; 
+        typedef vector<assoc> stack;
+        stack m_stack;
+
+        template<bool ResetSubst>
+        bool match_literal(literal const & l1, literal const & l2);
+
+        bool can_subsume(unsigned num_lits1, literal * lits1, unsigned num_lits2, literal * lits2);
+        bool quick_check(clause * cls1, clause * cls2);
+        bool subsumes_core(unsigned num_lits1, literal * lits1, unsigned num_lits2, literal * lits2);
+        bool subsumes_core(literal const & l1, unsigned num_lits2, literal * lits2);
+        bool ground_subsumes_core(unsigned num_lits1, literal * lits1, unsigned num_lits2, literal * lits2);
+
+        enum feature_kind {
+            F_GROUND,
+            F_NUM_POS_LITS,
+            F_NUM_NEG_LITS,
+            F_DEPTH,
+            F_CONST_COUNT,
+            F_SYM_COUNT,
+            F_NUM_NEG_FUNCS,
+            F_NUM_POS_FUNCS
+        };
+
+        struct feature {
+            feature_kind m_kind;
+            func_decl *  m_decl;
+            feature(feature_kind k = F_GROUND):m_kind(k) {}
+            feature(func_decl * decl, bool neg):m_kind(neg ? F_NUM_NEG_FUNCS : F_NUM_POS_FUNCS), m_decl(decl) {}
+        };
+
+        vector<feature> m_features;
+
+        bit_vector            m_found_decl_set;
+        func_decl_ref_vector  m_found_decls; // domain of m_found_decl_set;
+
+        /**
+           \brief Return true if the function symbol is considered for feature vector indexing.
+        */
+        bool fvi_candidate(func_decl * f) {
+            return f->get_family_id() == null_family_id || f->get_arity() > 0;
+        }
+        
+        typedef obj_hashtable<func_decl> found_func_decl_set;
+        found_func_decl_set   m_cls_found_decl_set; // temporary set used to track the func_decl's found in a clause
+        ptr_vector<func_decl> m_cls_found_decls;
+
+        bool mark_func_decl(func_decl * f);
+
+        typedef obj_map<func_decl, unsigned> decl2nat;
+        bool        m_refining_index; // if true keep collecting data to refine index.
+        decl2nat    m_f_count[2];     // temporary field used to track the num. of occurs. of function symbols in neg/pos literals.
+        decl2nat    m_f_min[2];
+        decl2nat    m_f_max[2];
+        decl2nat    m_f_freq;
+        
+        void inc_f_count(func_decl * f, bool neg);
+        void update_min_max(func_decl * f);
+        void update_neg_pos_func_counts(clause * cls);
+        
+        void compute_features(clause * cls, unsigned * fvector);
+        
+        struct to_feature_vector;
+        friend struct to_feature_vector;
+
+        struct to_feature_vector {
+            subsumption & m_owner;
+            to_feature_vector(subsumption & o):m_owner(o) {}
+            void operator()(clause * cls, unsigned * fvector) {
+                m_owner.compute_features(cls, fvector);
+            }
+        };
+
+        typedef fvi<clause, to_feature_vector, obj_ptr_hash<clause>, ptr_eq<clause> > index;
+        index *    m_index;
+        unsigned   m_num_processed_clauses;
+        unsigned   m_opt_threshold;
+
+        void init_indexes();
+
+        struct f_lt;
+        friend struct f_lt;
+
+        unsigned get_value_range(func_decl * f, bool neg) const;
+        unsigned get_value_range(func_decl * f) const;
+
+        struct f_lt {
+            subsumption & m_owner;
+            f_lt(subsumption & o):m_owner(o) {}
+            bool operator()(func_decl * f1, func_decl * f2) const;
+        };
+        
+        void optimize_feature_index();
+
+        clause * unit_forward(clause * cls);
+        clause * non_unit_forward(clause * cls);
+        clause * eq_subsumption(expr * lhs, expr * rhs);
+        clause * eq_subsumption(clause * cls);
+
+    public:
+        subsumption(ast_manager & m, asserted_literals & al, spc_params & params);
+        ~subsumption();
+
+        bool operator()(clause * cls1, clause * cls2);
+        void insert(clause * cls);
+        void erase(clause * cls);
+        void reset();
+        clause * forward(clause * cls);
+        void backward(clause * cls, ptr_buffer<clause> & result);
+
+        ast_manager & get_manager() { return m_manager; }
+    };
+
+};
+
+#endif /* _SPC_SUBSUMPTION_H_ */
+

src/spc/spc_superposition.cpp

@@ -0,0 +1,531 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_superposition.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-15.
+
+Revision History:
+
+--*/
+#include"spc_superposition.h"
+#include"ast_pp.h"
+
+namespace spc {
+
+    superposition::superposition(ast_manager & m, order & o, statistics & s):
+        m_manager(m),
+        m_order(o),
+        m_stats(s),
+        m_subst(m),
+        m_p(m),
+        m_r(m),
+        m_normalize_vars(m),
+        m_spc_fid(m.get_family_id("spc")) {
+        m_subst.reserve_offsets(3);
+        m_deltas[0] = 0;
+        m_deltas[1] = 0;
+    }
+
+    superposition::~superposition() {
+    }
+    
+    void superposition::insert_p(clause * cls, expr * lhs, unsigned i) {
+        m_p.insert(lhs);
+        m_subst.reserve_vars(m_p.get_approx_num_regs());
+        m_p2clause_set.insert(clause_pos_pair(cls, i), lhs);
+    }
+
+    void superposition::insert_p(clause * cls, literal & l, unsigned i) {
+        l.set_p_indexed(true);
+        expr * atom = l.atom();
+        if (!m_manager.is_eq(atom))
+            return;
+        if (l.is_oriented())
+            insert_p(cls, l.is_left() ? l.lhs() : l.rhs(), i);
+        else {
+            insert_p(cls, l.lhs(), i);
+            insert_p(cls, l.rhs(), i);
+        }
+    }
+
+    void superposition::insert_r(clause * cls, expr * n, unsigned i, bool lhs) {
+        if (is_app(n)) {
+            unsigned idx = (i << 1) | static_cast<unsigned>(lhs);
+            
+            clause_pos_pair new_pair(cls, idx);
+            SASSERT(m_todo.empty());
+            m_todo.push_back(to_app(n));
+            while (!m_todo.empty()) {
+                app * n = m_todo.back();
+                m_todo.pop_back();
+                clause_pos_set * s = m_r2clause_set.get_parents(n);
+                if (s == 0 || !s->contains(new_pair)) {
+                    m_r.insert(n);
+                    m_r2clause_set.insert(new_pair, n);
+                    unsigned num_args = n->get_num_args();
+                    for (unsigned i = 0; i < num_args; i++) {
+                        expr * c = n->get_arg(i);
+                        if (is_app(c))
+                            m_todo.push_back(to_app(c));
+                    }
+                }
+            }
+        }
+    }
+
+    void superposition::insert_r(clause * cls, literal & l, unsigned i) {
+        l.set_r_indexed(true);
+        expr * atom = l.atom();
+        if (m_manager.is_eq(atom)) {
+            expr * lhs = l.lhs();
+            expr * rhs = l.rhs();
+            if (l.is_oriented()) {
+                bool left = true;
+                if (!l.is_left()) {
+                    left  = false;
+                    std::swap(lhs, rhs);
+                }
+                insert_r(cls, lhs, i, left);
+            }
+            else {
+                insert_r(cls, lhs, i, true);
+                insert_r(cls, rhs, i, false);
+            }
+        }
+        else {
+            insert_r(cls, atom, i, false);
+        }
+        m_subst.reserve_vars(m_r.get_approx_num_regs());
+    }
+
+    void superposition::insert(clause * cls) {
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal & l = cls->get_literal(i);
+            if (l.is_p_indexed() || cls->is_eligible_for_paramodulation(m_order, l)) {
+                if (!l.sign() && m_manager.is_eq(l.atom()))
+                    insert_p(cls, l, i);
+                insert_r(cls, l, i);
+            }
+            else if (l.is_r_indexed() || cls->is_eligible_for_resolution(m_order, l)) {
+                insert_r(cls, l, i);
+            }
+        }
+        TRACE("superposition_detail",
+              tout << "adding clause: "; cls->display(tout, m_manager); tout << "\n";
+              tout << "p index:\n";
+              m_p.display(tout);
+              tout << "r index:\n";
+              m_r.display(tout););
+    }
+
+    void superposition::erase_p(clause * cls, expr * lhs, unsigned i) {
+        m_p2clause_set.erase(clause_pos_pair(cls, i), lhs);
+        if (m_p2clause_set.empty(lhs))
+            m_p.erase(lhs);
+    }
+
+    void superposition::erase_p(clause * cls, literal & l, unsigned i) {
+        expr * atom = l.atom();
+        if (!m_manager.is_eq(atom))
+            return;
+        if (l.is_oriented()) 
+            erase_p(cls, l.is_left() ? l.lhs() : l.rhs(), i);
+        else {
+            erase_p(cls, l.lhs(), i);
+            erase_p(cls, l.rhs(), i);
+        }
+    }
+
+    void superposition::erase_r(clause * cls, literal & l, unsigned i) {
+        clause_pos_pair pair(cls, i);
+
+        expr * atom = l.atom();
+        SASSERT(is_app(atom));
+        SASSERT(m_todo.empty());
+        m_todo.push_back(to_app(atom));
+
+        while (!m_todo.empty()) {
+            app * n = m_todo.back();
+            m_todo.pop_back();
+            switch (m_r2clause_set.erase(pair, n)) {
+            case 0: // pair is not a parent of n
+                break;
+            case 1: // pair is the last parent of n
+                m_r.erase(n);
+            default:
+                unsigned num_args = n->get_num_args();
+                for (unsigned i = 0; i < num_args; i++) {
+                    expr * c = n->get_arg(i);
+                    if (is_app(c))
+                        m_todo.push_back(to_app(c));
+                }
+            }
+        }
+    }
+
+    void superposition::erase(clause * cls) {
+        unsigned num_lits = cls->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal & l = cls->get_literal(i);
+            if (l.is_p_indexed())
+                erase_p(cls, l, i);
+            if (l.is_r_indexed())
+                erase_r(cls, l, i);
+        }
+    }
+
+    void superposition::reset() {
+        m_p.reset();
+        m_p2clause_set.reset();
+        m_r.reset();
+        m_r2clause_set.reset();
+    }
+
+    /**
+       \brief Copy to result the literals of s except literal at position idx. Apply the substitution m_subst,
+       assuming that the variables of s are in the variable bank offset. The deltas for each bank are
+       stored in m_deltas.
+    */
+    void superposition::copy_literals(clause * s, unsigned idx, unsigned offset, literal_buffer & result) {
+        unsigned num_lits = s->get_num_literals();
+        for (unsigned i = 0; i < num_lits; i++)
+            if (i != idx) {
+                literal const & l = s->get_literal(i);
+                expr_ref new_atom(m_manager);
+                m_subst.apply(2, m_deltas, expr_offset(l.atom(), offset), new_atom);
+                TRACE("superposition_copy", tout << "i: " << i << ", idx: " << idx << ", offset: " << offset << "\natom:\n";
+                      tout << mk_pp(l.atom(), m_manager) << "\nnew_atom:\n" << mk_pp(new_atom, m_manager) << "\n";);
+                result.push_back(literal(new_atom, l.sign()));
+            }
+    }
+
+    void superposition::normalize_literals(unsigned num_lits, literal * lits, literal_buffer & result) {
+        m_normalize_vars.reset();
+        for (unsigned i = 0; i < num_lits; i++) {
+            literal const & l = lits[i];
+            result.push_back(literal(m_normalize_vars(l.atom()), l.sign()));
+        }
+    }
+
+    void superposition::mk_sp_clause(unsigned num_lits, literal * lits, justification * p1, justification * p2) {
+        literal_buffer new_literals(m_manager);
+        normalize_literals(num_lits, lits, new_literals);
+        justification * js = mk_superposition_justification(m_manager, m_spc_fid, p1, p2, 
+                                                            new_literals.size(), new_literals.c_ptr(),
+                                                            m_normalize_vars.get_num_vars(), m_normalize_vars.get_vars());
+        clause * new_cls = clause::mk(m_manager, new_literals.size(), new_literals.c_ptr(), js, 0);
+        m_new_clauses->push_back(new_cls);
+        TRACE("superposition", tout << "new superposition clause:\n"; new_cls->display(tout, m_manager); tout << "\n";);
+        m_stats.m_num_superposition++;
+    }
+
+    void superposition::mk_res_clause(unsigned num_lits, literal * lits, justification * p1, justification * p2) {
+        literal_buffer new_literals(m_manager);
+        normalize_literals(num_lits, lits, new_literals);
+        justification * js = mk_resolution_justification(m_manager, m_spc_fid, p1, p2, 
+                                                         new_literals.size(), new_literals.c_ptr(),
+                                                         m_normalize_vars.get_num_vars(), m_normalize_vars.get_vars());
+        clause * new_cls = clause::mk(m_manager, new_literals.size(), new_literals.c_ptr(), js, 0);
+        m_new_clauses->push_back(new_cls);
+        TRACE("superposition", tout << "new resolution clause:\n"; new_cls->display(tout, m_manager); tout << "\n";);
+        m_stats.m_num_resolution++;
+    }
+
+    /**
+       \brief Given the equation (= lhs rhs) of the clause being
+       added, try to apply resolution where the clause being added
+       is the main clause in the superposition rule.
+    */
+    void superposition::try_superposition_main(expr * lhs, expr * rhs) {
+        m_lhs    = lhs;
+        m_rhs    = rhs;
+        m_subst.reset_subst();
+        TRACE("spc_superposition", tout << "try_superposition_main, lhs:\n" << mk_pp(m_lhs, m_manager) << "\nrhs:\n" << mk_pp(m_rhs, m_manager) << "\n";
+              tout << "substitution:\n"; m_subst.display(tout););
+        r_visitor v(*this, m_subst);
+        m_r.unify(lhs, v);
+    }
+
+    /**
+       \brief Try to apply superposition rule using the clause
+       being added (m_clause) as main clause, and its literal m_lit
+       as the equation.
+    */
+    void superposition::try_superposition_main() {
+        expr * lhs = m_lit->lhs();
+        expr * rhs = m_lit->rhs();
+        TRACE("spc_superposition", tout << "trying superposition:\n" << mk_pp(lhs, m_manager) << "\n" << mk_pp(rhs, m_manager) << "\nis_oriented: " << m_lit->is_oriented() << "\n";);
+        if (m_lit->is_oriented()) {
+            if (!m_lit->is_left())
+                std::swap(lhs, rhs);
+            try_superposition_main(lhs, rhs);
+        }
+        else {
+            try_superposition_main(lhs, rhs);
+            try_superposition_main(rhs, lhs);
+        }
+    }
+
+    void superposition::found_r(expr * r) {
+        TRACE("spc_superposition", tout << "found_r:\n" << mk_pp(r, m_manager) << "\n";
+              tout << "substitution:\n"; m_subst.display(tout););
+        if (m_r2clause_set.empty(r))
+            return;
+        TRACE("spc_superposition", tout << "r2clause is not empty.\n";);
+        if (!m_lit->is_oriented() && m_order.greater(m_rhs, m_lhs, &m_subst))
+            return;
+        TRACE("spc_superposition", tout << "order restriction was met.\n";);
+        if (!m_clause->is_eligible_for_paramodulation(m_order, *m_lit, 0, &m_subst))
+            return;
+        TRACE("spc_superposition", tout << "literal is eligible for paramodulation.\n";);
+        r2clause_set::iterator it  = m_r2clause_set.begin(r);
+        r2clause_set::iterator end = m_r2clause_set.end(r);
+        for (; it != end; ++it) {
+            clause_pos_pair & p = *it;
+            clause * aux_cls    = p.first;
+            unsigned aux_idx    = p.second >> 1;
+            //
+            // The following optimization is incorrect (if statement).
+            // For example, it prevents the Z3 from proving the trivial benchmark
+            // c = X, 
+            // a != b
+            // using the order a < b < c
+            //
+            // To prove, this example we need to generate the clause Y = X by applying superposition of c = X on itself.
+            // We can see that by renaming the first clause to c = Y, and then, substituting c in the original by Y.
+            //
+            // Actually, this optimization is correct when the set of variables in m_lhs is a superset of the set of variables in m_rhs, 
+            // because in this case, the new literal will be equivalent to true. In the example above, this is not the case,
+            // since m_lhs does not contain any variable, and m_rhs contains one.
+            // 
+            
+            //
+            // if (r == m_lhs && m_clause == aux_cls && m_idx == aux_idx)
+            //    continue;
+            //
+            bool     in_lhs     = (p.second & 1) != 0;
+            TRACE("spc_superposition", tout << "aux_cls:\n"; aux_cls->display(tout, m_manager); tout << "\naux_idx: " << aux_cls << ", in_lhs: " << in_lhs << "\n";);
+            literal & aux_lit   = aux_cls->get_literal(aux_idx);
+            if (!aux_cls->is_eligible_for_resolution(m_order, aux_lit, 1, &m_subst))
+                continue;
+            literal_buffer new_literals(m_manager);
+            m_subst.reset_cache();
+            if (m_manager.is_eq(aux_lit.atom())) {
+                expr * lhs = aux_lit.lhs();
+                expr * rhs = aux_lit.rhs();
+                TRACE("spc_superposition", tout << "aux_lit lhs:\n" << mk_pp(lhs, m_manager) << "\nrhs:\n" << mk_pp(rhs, m_manager) << "\n";);
+                if (!in_lhs)
+                    std::swap(lhs, rhs);
+                if (!aux_lit.is_oriented() && m_order.greater(rhs, lhs, 1, &m_subst)) {
+                    TRACE("spc_superposition", tout << "failed order constraint.\n";);
+                    continue;
+                }
+                expr_ref new_lhs(m_manager), new_rhs(m_manager);
+                m_subst.apply(2, m_deltas, expr_offset(lhs, 1), expr_offset(r, 1), expr_offset(m_rhs, 0), new_lhs);
+                m_subst.apply(2, m_deltas, expr_offset(rhs, 1), new_rhs);
+                TRACE("spc_superposition", tout << "aux_lit new_lhs:\n" << mk_pp(new_lhs, m_manager) << "\nnew_rhs:\n" << mk_pp(new_rhs, m_manager) << "\n";);
+                expr * new_eq  = m_manager.mk_eq(new_lhs, new_rhs);
+                new_literals.push_back(literal(new_eq, aux_lit.sign()));
+            }
+            else {
+                expr_ref new_atom(m_manager);
+                m_subst.apply(2, m_deltas, expr_offset(aux_lit.atom(), 1), new_atom);
+                new_literals.push_back(literal(new_atom, aux_lit.sign()));
+            }
+            copy_literals(m_clause, m_idx, 0,   new_literals);
+            copy_literals(aux_cls,  aux_idx, 1, new_literals);
+            TRACE("superposition", tout << "found r target: " << mk_pp(r, m_manager) << " for \n" <<
+                  mk_pp(m_lhs, m_manager) << "\nmain clause: "; m_clause->display(tout, m_manager);
+                  tout << "\naux clause: "; aux_cls->display(tout, m_manager); tout << "\nat pos: " <<
+                  aux_idx << "\n";);
+            mk_sp_clause(new_literals.size(), new_literals.c_ptr(), m_clause->get_justification(), aux_cls->get_justification());
+        }
+    }
+
+    /**
+       \brief Try to apply superposition rule using the clause
+       being added (m_clause) as the aux clause, and its literal m_lit
+       as the target.
+    */
+    void superposition::try_superposition_aux() {
+        TRACE("superposition_aux", tout << "superposition aux:\n"; m_clause->display(tout, m_manager); 
+              tout << "\nusing literal: " << m_idx << "\n";);
+        if (m_manager.is_eq(m_lit->atom())) {
+            expr * lhs = m_lit->lhs();
+            expr * rhs = m_lit->rhs();
+            if (m_lit->is_oriented()) {
+                if (!m_lit->is_left())
+                    std::swap(lhs, rhs);
+                try_superposition_aux(lhs, rhs);
+            }
+            else {
+                try_superposition_aux(lhs, rhs);
+                try_superposition_aux(rhs, lhs);
+            }
+        }
+        else {
+            try_superposition_aux(m_lit->atom(), 0);
+        }
+    }
+
+    /**
+       \brief Use the clause being added as the auxiliary clause in the superposition rule.
+    */
+    void superposition::try_superposition_aux(expr * lhs, expr * rhs) {
+        TRACE("superposition_aux", tout << "try_superposition_aux\n" << mk_pp(lhs, m_manager) << "\n" << mk_pp(rhs, m_manager) << "\n";);
+        if (is_var(lhs))
+            return;
+        m_lhs = lhs;
+        m_rhs = rhs;
+        SASSERT(m_todo.empty());
+        m_todo.push_back(to_app(lhs));
+        while (!m_todo.empty()) {
+            m_target = m_todo.back();
+            m_todo.pop_back();
+            m_subst.reset_subst();
+            p_visitor v(*this, m_subst);
+            TRACE("superposition_aux", tout << "trying to find unifier for:\n" << mk_pp(m_target, m_manager) << "\n";);
+            m_p.unify(m_target, v);
+            unsigned j = m_target->get_num_args();
+            while (j > 0) {
+                --j;
+                expr * arg = m_target->get_arg(j);
+                if (is_app(arg))
+                    m_todo.push_back(to_app(arg));
+            }
+        }
+    }
+
+    void superposition::found_p(expr * p) {
+        TRACE("superposition_found_p", tout << "found p:\n" << mk_pp(p, m_manager) << "\n";);
+        if (m_p2clause_set.empty(p)) {
+            TRACE("superposition_found_p", tout << "clause set is empty.\n";);
+            return;
+        }
+        if (m_rhs && !m_lit->is_oriented() && m_order.greater(m_rhs, m_lhs, &m_subst)) {
+            TRACE("superposition_found_p", tout << "aux clause failed not rhs > lhs constraint.\n";);
+            return;
+        }
+        if (!m_clause->is_eligible_for_resolution(m_order, *m_lit, 0, &m_subst)) {
+            TRACE("superposition_found_p", tout << "aux literal is not eligible for resolution.\n";);
+            return;
+        }
+        p2clause_set::iterator it  = m_p2clause_set.begin(p);
+        p2clause_set::iterator end = m_p2clause_set.end(p);
+        for (; it != end; ++it) {
+            clause_pos_pair & pair   = *it;
+            clause * main_cls        = pair.first;
+            TRACE("superposition_found_p", tout << "p clause:\n"; main_cls->display(tout, m_manager); tout << "\n";);
+            unsigned lit_idx         = pair.second;
+            if (p == m_lhs && m_clause == main_cls && m_idx == lit_idx)
+                continue;
+            literal const & main_lit = main_cls->get_literal(lit_idx);
+            SASSERT(m_manager.is_eq(main_lit.atom()));
+            expr * lhs               = main_lit.lhs();
+            expr * rhs               = main_lit.rhs();
+            if (rhs == p)
+                std::swap(lhs, rhs);
+            SASSERT(lhs == p);
+            TRACE("superposition_found_p", tout << "lhs: " << mk_pp(lhs, m_manager) << "\nrhs: " << mk_pp(rhs, m_manager) << "\n";);
+            if (!main_lit.is_oriented() && m_order.greater(rhs, lhs, 1, &m_subst))
+                continue;
+            if (!main_cls->is_eligible_for_paramodulation(m_order, main_lit, 1, &m_subst))
+                continue;
+            literal_buffer new_literals(m_manager);
+            m_subst.reset_cache();
+            TRACE("superposition_found_p", tout << "creating new_lhs\n";);
+            expr_ref new_lhs(m_manager);
+            m_subst.apply(2, m_deltas, expr_offset(m_lhs, 0), expr_offset(m_target, 0), expr_offset(rhs, 1), new_lhs);
+            // FIX: m_subst.reset_cache();
+            TRACE("superposition_found_p", tout << "new_lhs: " << mk_pp(new_lhs, m_manager) << "\n";
+                  m_subst.display(tout););
+            expr * new_atom = 0;
+            if (m_rhs) {
+                TRACE("superposition_found_p", tout << "creating new_rhs\n";);
+                expr_ref new_rhs(m_manager);
+                m_subst.apply(2, m_deltas, expr_offset(m_rhs, 0), new_rhs);
+                TRACE("superposition_found_p", tout << "new_rhs: " << mk_pp(new_rhs, m_manager) << "\n";);
+                new_atom = m_manager.mk_eq(new_lhs, new_rhs);
+            }
+            else
+                new_atom = new_lhs;
+            TRACE("superposition_found_p", tout << "new_atom: " << mk_pp(new_atom, m_manager) << "\n"; m_subst.display(tout););
+            new_literals.push_back(literal(new_atom, m_lit->sign()));
+            TRACE("superposition_found_p", tout << "copying literals\n";);
+            copy_literals(main_cls, lit_idx, 1, new_literals);
+            copy_literals(m_clause, m_idx, 0, new_literals);
+            TRACE("superposition", tout << "found p target: " << mk_pp(p, m_manager) << " for \n" <<
+                  mk_pp(m_lhs, m_manager) << "\nmain clause: "; main_cls->display(tout, m_manager);
+                  tout << "\naux clause: "; m_clause->display(tout, m_manager); tout << "\n";);
+            mk_sp_clause(new_literals.size(), new_literals.c_ptr(), main_cls->get_justification(), m_clause->get_justification());
+        }
+    }
+
+    /**
+       \brief Try to apply resolution rule using the clause being added (m_clause).
+    */
+    void superposition::try_resolution() {
+        m_subst.reset_subst();
+        res_visitor v(*this, m_subst);
+        m_r.unify(m_lit->atom(), v);
+    }
+
+    void superposition::found_res(expr * r) {
+        if (m_r2clause_set.empty(r))
+            return;
+        if (!m_clause->is_eligible_for_resolution(m_order, *m_lit, 0, &m_subst))
+            return;
+        r2clause_set::iterator it  = m_r2clause_set.begin(r);
+        r2clause_set::iterator end = m_r2clause_set.end(r);
+        for (; it != end; ++it) {
+            clause_pos_pair & pair  = *it;
+            clause * aux_cls        = pair.first;
+            unsigned aux_idx        = pair.second >> 1;
+            literal const & aux_lit = aux_cls->get_literal(aux_idx);
+            if (aux_lit.sign() == m_lit->sign())
+                continue;
+            if (aux_lit.atom() != r)
+                continue;
+            if (!aux_cls->is_eligible_for_resolution(m_order, aux_lit, 1, &m_subst))
+                continue;
+            literal_buffer new_literals(m_manager);
+            m_subst.reset_cache();
+            copy_literals(m_clause, m_idx, 0, new_literals);
+            copy_literals(aux_cls, aux_idx, 1, new_literals);
+            mk_res_clause(new_literals.size(), new_literals.c_ptr(), m_clause->get_justification(), aux_cls->get_justification());
+        }
+    }
+        
+    void superposition::operator()(clause * cls, ptr_vector<clause> & new_clauses) {
+        m_subst.reserve_vars(cls->get_num_vars());
+        m_clause      = cls;
+        m_new_clauses = &new_clauses;
+        SASSERT(m_deltas[0] == 0);
+        m_deltas[1]   = m_clause->get_num_vars();
+        unsigned num_lits = cls->get_num_literals();
+        for (m_idx = 0; m_idx < num_lits; m_idx++) {
+            m_lit      = &(cls->get_literal(m_idx));
+            bool is_eq = m_manager.is_eq(m_lit->atom());
+            if (!m_lit->sign() && m_lit->is_p_indexed() && is_eq)
+                try_superposition_main();
+            if (m_lit->is_r_indexed()) {
+                try_superposition_aux();
+                if (!is_eq)
+                    try_resolution();
+            }
+        }
+    }
+
+};
+
+

src/spc/spc_superposition.h

@@ -0,0 +1,147 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_superposition.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-15.
+
+Revision History:
+
+--*/
+#ifndef _SPC_SUPERPOSITION_H_
+#define _SPC_SUPERPOSITION_H_
+
+#include"spc_clause.h"
+#include"spc_clause_pos_set.h"
+#include"substitution_tree.h"
+#include"obj_hashtable.h"
+#include"sparse_use_list.h"
+#include"normalize_vars.h"
+#include"spc_statistics.h"
+
+namespace spc {
+
+    /**
+       \brief Functor for applying the superposition right/left rules.
+
+       - Superposition Left
+       s = t or S,  u != v or R
+       ==>
+       sigma(u[p<-t] != v or S or R)
+       
+       sigma is the mgu(u|p, s)
+       sigma(s) not greater than sigma(t)
+       sigma(u) not greater than sigma(v)
+       sigma(s = t) is eligible for paramodulation
+       sigma(u != v) is eligible for resolution
+       u|p is not a variable
+       
+
+       - Superposition Right
+       s = t or S,  u = v or R
+       ==>
+       sigma(u[p<-t] != v or S or R)
+       
+       Same restrictions of Superposition Left
+    
+       This functor also applied binary resolution rule.
+
+       We say the left clause is the main clause in the superposition.
+    */
+    class superposition {
+        ast_manager &     m_manager;
+        order &           m_order;
+        statistics &      m_stats;
+        substitution      m_subst;
+
+        // indexes for left clause
+        substitution_tree m_p; // potential left hand sides for superposition
+        typedef sparse_use_list<expr, svector<clause_pos_pair> > p2clause_set;
+        p2clause_set      m_p2clause_set;
+
+        void insert_p(clause * cls, expr * lhs, unsigned i);
+        void insert_p(clause * cls, literal & l, unsigned i);
+
+        void erase_p(clause * cls, expr * lhs, unsigned i);
+        void erase_p(clause * cls, literal & l, unsigned i);
+
+        // indexes for right clause
+        substitution_tree m_r; // potential targets for superposition 
+        typedef sparse_use_list<expr, clause_pos_set> r2clause_set;
+        r2clause_set      m_r2clause_set;
+        ptr_vector<app>   m_todo;
+
+        void insert_r(clause * cls, expr * n, unsigned i, bool lhs);
+        void insert_r(clause * cls, literal & l, unsigned i);
+        void erase_r(clause * cls, literal & l, unsigned i);
+
+        normalize_vars       m_normalize_vars;
+        
+        // temporary fields...
+        ptr_vector<clause> * m_new_clauses;
+        clause *             m_clause;
+        literal *            m_lit;
+        expr *               m_lhs;
+        expr *               m_rhs;
+        app *                m_target;
+        unsigned             m_idx;
+        unsigned             m_deltas[2];
+        family_id            m_spc_fid;
+
+        void normalize_literals(unsigned num_lits, literal * lits, literal_buffer & result);
+        void copy_literals(clause * s, unsigned idx, unsigned offset, literal_buffer & result);
+        void mk_sp_clause(unsigned num_lits, literal * lits, justification * p1, justification * p2);
+        void mk_res_clause(unsigned num_lits, literal * lits, justification * p1, justification * p2);
+        void try_superposition_main(expr * lhs, expr * rhs);
+        void try_superposition_main();
+        void found_r(expr * r);
+        void try_superposition_aux(expr * lhs, expr * rhs);
+        void try_superposition_aux();
+        void found_p(expr * p);
+        void try_resolution();
+        void found_res(expr * r);
+
+        friend struct r_visitor;
+        struct r_visitor : public st_visitor {
+            superposition & m_owner;
+            r_visitor(superposition & o, substitution & s):st_visitor(s), m_owner(o) {}
+            virtual bool operator()(expr * e) { m_owner.found_r(e); return true; /* continue */ }
+        };
+
+        friend struct p_visitor;
+        struct p_visitor : public st_visitor {
+            superposition & m_owner;
+            p_visitor(superposition & o, substitution & s):st_visitor(s), m_owner(o) {}
+            virtual bool operator()(expr * e) { m_owner.found_p(e); return true; /* continue */ }
+        };
+
+        friend struct res_visitor;
+        struct res_visitor : public st_visitor {
+            superposition & m_owner;
+            res_visitor(superposition & o, substitution & s):st_visitor(s), m_owner(o) {}
+            virtual bool operator()(expr * e) { m_owner.found_res(e); return true; /* continue */ }
+        };
+
+    public:
+        superposition(ast_manager & m, order & o, statistics & stats);
+        ~superposition();
+
+        void insert(clause * cls);
+        void erase(clause * cls);
+        void reset();
+        
+        void operator()(clause * cls, ptr_vector<clause> & new_clauses);
+    };
+
+};
+
+#endif /* _SPC_SUPERPOSITION_H_ */
+

src/spc/spc_unary_inference.cpp

@@ -0,0 +1,52 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_unary_inference.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-14.
+
+Revision History:
+
+--*/
+#include"spc_unary_inference.h"
+
+namespace spc {
+
+    unary_inference::unary_inference(ast_manager & m, order & ord):
+        m_manager(m),
+        m_order(ord),
+        m_subst(m),
+        m_unifier(m) {
+        m_subst.reserve_offsets(1);
+    }
+
+    /**
+       \brief Create the result clause. The literal at position j of \c cls in removed,
+       and the substitution m_subst is applied to the resultant clause.
+    */
+    clause * unary_inference::mk_result(clause * cls, unsigned j) {
+        sbuffer<literal> new_literals;
+        unsigned num = cls->get_num_literals();
+        for (unsigned i = 0; i < num; i++) {
+            if (i != j) {
+                literal const & l = cls->get_literal(i);
+                expr_ref new_atom(m_manager);
+                m_subst.apply(l.atom(), new_atom);
+                new_literals.push_back(literal(new_atom, l.sign()));
+            }
+        }
+        
+        justification * js = mk_justification(cls->get_justification(), new_literals.size(), new_literals.c_ptr());
+        clause * new_cls = clause::mk(m_manager, new_literals.size(), new_literals.c_ptr(), js, cls->get_scope_lvl());
+        return new_cls;
+    }
+
+};

src/spc/spc_unary_inference.h

@@ -0,0 +1,48 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    spc_unary_inference.h
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-14.
+
+Revision History:
+
+--*/
+#ifndef _SPC_UNARY_INFERENCE_H_
+#define _SPC_UNARY_INFERENCE_H_
+
+#include"spc_clause.h"
+#include"unifier.h"
+
+namespace spc {
+
+    /**
+       \brief Superclass for eq_resolution and factoring.
+    */
+    class unary_inference {
+    protected:
+        ast_manager & m_manager;
+        order &       m_order;
+        substitution  m_subst;
+        unifier       m_unifier; 
+        
+        clause * mk_result(clause * cls, unsigned j);
+        virtual justification * mk_justification(justification * parent, unsigned num_lits, literal * new_lits) = 0;
+    public:
+        unary_inference(ast_manager & m, order & ord);
+        virtual ~unary_inference() {}
+    };
+
+};
+
+
+#endif /* _SPC_UNARY_INFERENCE_H_ */
+

src/spc/splay_tree.h

@@ -0,0 +1,180 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    splay_tree.h
+
+Abstract:
+
+    Splay trees
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-31.
+
+Revision History:
+
+--*/
+#ifndef _SPLAY_TREE_H_
+#define _SPLAY_TREE_H_
+
+#include"util.h"
+#include"buffer.h"
+
+template<typename Key, typename Compare>
+class splay_tree : private Compare {
+    struct cell {
+        Key    m_key;
+        cell * m_left;
+        cell * m_right;
+
+        cell():m_left(0), m_right(0) {}
+        cell(Key const & k, cell * l = 0, cell * r = 0):
+            m_key(k), m_left(l), m_right(r) {}
+    };
+
+    cell * m_root;
+    int compare(Key const & k1, Key const & k2) const { return Compare::operator()(k1, k2); }
+    cell * splay(cell * c, Key const & k);
+
+    void display_core(std::ostream & out, cell * c) const {
+        if (c) {
+            out << "(" << c->m_key << " ";
+            display_core(out, c->m_left);
+            out << " ";
+            display_core(out, c->m_right);
+            out << ")";
+        }
+        else 
+            out << "null";
+    }
+
+public:
+    splay_tree(Compare const & c = Compare()):
+        Compare(c),
+        m_root(0) {}
+    
+    ~splay_tree() {
+        m_root = 0;
+    }
+
+    void insert(Key const & k);
+
+    bool find(Key const & k, Key & r) const;
+
+    void erase(Key const & k);
+
+    void reset();
+
+    bool empty() const { return m_root == 0; }
+
+    bool singleton() const { return m_root != 0 && m_root->m_left == 0 && m_root->m_right == 0; }
+
+    /**
+       \brief Visit nodes in the splay tree in ascending order.
+       The Visitor functor should provide the following methods:
+       
+       - bool visit_left(Key const & k)
+         return true if the left child should be visited
+
+       - bool visit_right(Key const & k)
+         return true if the right child should be visited
+
+       - void operator()(Key const & k)
+         do something with the key.
+    */
+    template<typename Visitor>
+    void visit_core(Visitor & v) {
+        typedef std::pair<cell *, bool> entry;
+        if (m_root) {
+            buffer<entry> todo;
+            todo.push_back(entry(m_root, false));
+            while (!todo.empty()) {
+                entry & curr = todo.back();
+                cell * c = curr.first;
+                if (!curr.second) {
+                    curr.second = true;
+                    if (c->m_left && v.visit_left(c->m_key))
+                        todo.push_back(entry(c->m_left, false));
+                }
+                else {
+                    v(c->m_key);
+                    todo.pop_back();
+                    if (c->m_right && v.visit_right(c->m_key))
+                        todo.push_back(entry(c->m_right, false));
+                }
+            }
+        }
+    }
+
+    template<typename Visitor>
+    struct all_visitor_wrapper {
+        Visitor & m_visitor;
+        all_visitor_wrapper(Visitor & v):m_visitor(v) {}
+        bool visit_right(Key const & k) { return true; }
+        bool visit_left(Key const & k) { return true; }
+        void operator()(Key const & k) { m_visitor.operator()(k); }
+    };
+
+    /**
+       \brief Visit all nodes in the splay tree in ascending order.
+
+       - void operator()(Key const & k)
+         do something with the key pair.
+    */
+    template<typename Visitor>
+    void visit(Visitor & v) {
+        all_visitor_wrapper<Visitor> w(v);
+        visit_core(w);
+    }
+
+    template<typename Visitor, bool LE>
+    struct visitor_wrapper {
+        Visitor &    m_visitor;
+        splay_tree & m_tree;
+        Key          m_key;
+        visitor_wrapper(Visitor & v, splay_tree & t, Key const & k):m_visitor(v), m_tree(t), m_key(k) {}
+        bool visit_left(Key const & k) { 
+            return LE || m_tree.compare(k, m_key) > 0;
+        }
+        bool visit_right(Key const & k) { 
+            return !LE || m_tree.compare(k, m_key) < 0;
+        }
+        void operator()(Key const & k) {
+            if ((LE  && m_tree.compare(k, m_key) <= 0) ||
+                (!LE && m_tree.compare(k, m_key) >= 0)) 
+                m_visitor.operator()(k);
+        }
+    };
+
+    /**
+       \brief Visit all nodes with keys less than or equal to k.
+
+       - void operator()(Key const & k)
+         do something with the key.
+    */
+    template<typename Visitor>
+    void visit_le(Visitor & v, Key const & k) {
+        visitor_wrapper<Visitor, true> w(v, *this, k);
+        visit_core(w);
+    }
+
+    /**
+       \brief Visit all nodes with keys greater than or equal to k.
+
+       - void operator()(Key const & k)
+         do something with the key.
+    */
+    template<typename Visitor>
+    void visit_ge(Visitor & v, Key const & k) {
+        visitor_wrapper<Visitor, false> w(v, *this, k);
+        visit_core(w);
+    }
+
+    void display(std::ostream & out) const {
+        display_core(out, m_root);
+    }
+};
+
+#endif

src/spc/splay_tree_def.h

@@ -0,0 +1,152 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    splay_tree_def.h
+
+Abstract:
+
+    Splay trees
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-31.
+
+Revision History:
+
+--*/
+#ifndef _SPLAY_TREE_DEF_H_
+#define _SPLAY_TREE_DEF_H_
+
+#include"splay_tree.h"
+
+template<typename Key, typename Compare>
+typename splay_tree<Key, Compare>::cell * splay_tree<Key, Compare>::splay(cell * root, Key const & k) {
+    if (!root) 
+        return 0;
+    
+    cell aux;
+    cell * tmp;
+    cell * left  = &aux;
+    cell * right = &aux;
+    cell * t     = root;
+    
+    while (true) {
+        int r = compare(k, t->m_key);
+        if (r < 0) {
+            if (!t->m_left)
+                break;
+            if (compare(k, t->m_left->m_key) < 0) {
+                tmp          = t->m_left;
+                t->m_left    = tmp->m_right;
+                tmp->m_right = t;
+                t            = tmp;
+                if (!t->m_left)
+                    break;
+            }
+            right->m_left = t;
+            right         = t;
+            t             = t->m_left;
+        }
+        else if (r > 0) {
+            if (!t->m_right)
+                break;
+            if (compare(k, t->m_right->m_key) > 0) {
+                tmp          = t->m_right;
+                t->m_right   = tmp->m_left;
+                tmp->m_left  = t;
+                t            = tmp;
+                if (!t->m_right)
+                    break;
+                
+            }
+            left->m_right = t;
+            left          = t;
+            t             = t->m_right;
+        }
+        else 
+            break;
+    }
+    
+    left->m_right = t->m_left;
+    right->m_left = t->m_right;
+    t->m_left     = aux.m_right;
+    t->m_right    = aux.m_left;
+    
+    return t;
+}
+
+template<typename Key, typename Compare>
+void splay_tree<Key, Compare>::insert(Key const & k) {
+    if (!m_root) 
+        m_root = alloc(cell, k);
+    else {
+        m_root = splay(m_root, k);
+        int r = compare(k, m_root->m_key);
+        if (r < 0) {
+            cell * new_cell   = alloc(cell, k, m_root->m_left, m_root);
+            m_root->m_left    = 0;
+            m_root            = new_cell;
+        }
+        else if (r > 0) {
+            cell * new_cell   = alloc(cell, k, m_root, m_root->m_right);
+            m_root->m_right   = 0;
+            m_root            = new_cell;
+        }
+        else
+            m_root->m_key     = k;
+    }
+}
+
+template<typename Key, typename Compare>
+bool splay_tree<Key, Compare>::find(Key const & k, Key & r) const {
+    if (m_root) {
+        splay_tree<Key, Compare> * _this = const_cast<splay_tree<Key, Compare> *>(this);
+        _this->m_root = _this->splay(m_root, k);
+        if (compare(k, m_root->m_key) == 0) {
+            r = m_root->m_key;
+            return true;
+        }
+    }
+    return false;
+}
+
+template<typename Key, typename Compare>
+void splay_tree<Key, Compare>::erase(Key const & k) {
+    if (m_root) {
+        m_root = splay(m_root, k);
+        if (compare(k, m_root->m_key) == 0) {
+            cell * to_delete = m_root;
+            if (m_root->m_left) {
+                cell * aux = splay(m_root->m_left, k);
+                SASSERT(!aux->m_right);
+                aux->m_right = m_root->m_right;
+                m_root       = aux;
+            }
+            else
+                m_root = m_root->m_right;
+
+            dealloc(to_delete);
+        }
+    }
+}
+
+template<typename Key, typename Compare>
+void splay_tree<Key, Compare>::reset() {
+    ptr_buffer<cell> todo;
+    if (m_root) 
+        todo.push_back(m_root);
+    while (!todo.empty()) {
+        cell * c = todo.back();
+        todo.pop_back();
+        if (c->m_left)
+            todo.push_back(c->m_left);
+        if (c->m_right)
+            todo.push_back(c->m_right);
+        dealloc(c);
+    }
+    m_root = 0;
+}
+
+#endif /* _SPLAY_TREE_DEF_H_ */

src/spc/splay_tree_map.h

@@ -0,0 +1,114 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    splay_tree_map.h
+
+Abstract:
+
+    A mapping as a splay tree.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-02.
+
+Revision History:
+
+--*/
+#ifndef _SPLAY_TREE_MAP_H_
+#define _SPLAY_TREE_MAP_H_
+
+#include"splay_tree.h"
+
+template<typename Key, typename Data, typename Compare>
+class splay_tree_map {
+    typedef std::pair<Key, Data> entry;
+
+    struct entry_compare : private Compare {
+        entry_compare(Compare const & c):Compare(c) {}
+        int operator()(entry const & e1, entry const & e2) const {
+            return Compare::operator()(e1.first, e2.first);
+        }
+    };
+
+    typedef splay_tree<entry, entry_compare> tree;
+    
+    tree m_tree;
+    
+    template<typename Visitor>
+    struct core_visitor_wrapper {
+        Visitor & m_visitor;
+        core_visitor_wrapper(Visitor & v):m_visitor(v) {}
+        bool visit_right(entry const & k) { return m_visitor.visit_right(k.first); }
+        bool visit_left(entry const & k) {  return m_visitor.visit_left(k.first); }
+        void operator()(entry const & k) { m_visitor.operator()(k.first, k.second); }
+    };
+
+    template<typename Visitor>
+    struct visitor_wrapper {
+        Visitor & m_visitor;
+        visitor_wrapper(Visitor & v):m_visitor(v) {}
+        void operator()(entry const & k) { m_visitor.operator()(k.first, k.second); }
+    };
+
+public:
+    splay_tree_map(Compare const & c = Compare()):
+        m_tree(entry_compare(c)) {}
+
+    void insert(Key const & k, Data const & d) {
+        m_tree.insert(entry(k, d));
+    }
+
+    bool find(Key const & k, Data & r) const {
+        entry e(k, r);
+        if (m_tree.find(e, e)) {
+            r = e.second;
+            return true;
+        }
+        return false;
+    }
+
+    void erase(Key const & k) {
+        entry e;
+        e.first = k;
+        m_tree.erase(e);
+    }
+
+    void reset() { m_tree.reset(); }
+
+    bool empty() const { return m_tree.empty(); }
+
+    void display(std::ostream & out) const { m_tree.display(out); }
+
+    template<typename Visitor>
+    void visit_core(Visitor & v) {
+        core_visitor_wrapper<Visitor> w(v);
+        m_tree.visit_core(w);
+    }
+
+    template<typename Visitor>
+    void visit(Visitor & v) {
+        visitor_wrapper<Visitor> w(v);
+        m_tree.visit(w);
+    }
+
+    template<typename Visitor>
+    void visit_le(Visitor & v, Key const & k) {
+        visitor_wrapper<Visitor> w(v);
+        entry e;
+        e.first = k;
+        m_tree.visit_le(w, e);
+    }
+
+    template<typename Visitor>
+    void visit_ge(Visitor & v, Key const & k) {
+        visitor_wrapper<Visitor> w(v);
+        entry e;
+        e.first = k;
+        m_tree.visit_ge(w, e);
+    }
+};
+
+#endif /* _SPLAY_TREE_MAP_H_ */
+

src/spc/substitution.cpp

@@ -0,0 +1,286 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    substitution.cpp
+
+Abstract:
+
+    A substitution, that is, a mapping from (variable, offset) to (expr, offset).
+    We use offsets in order to avoid creating variants of terms.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-01.
+
+Revision History:
+
+--*/
+#include"substitution.h"
+#include"ast_pp.h"
+#include"ast_ll_pp.h"
+
+substitution::substitution(ast_manager & m):
+    m_manager(m),
+    m_new_exprs(m) {
+}
+
+void substitution::reset() {
+    reset_subst();
+    reset_cache();
+}
+
+void substitution::reset_subst() { 
+    m_subst.reset(); 
+    m_vars.reset();
+    m_scopes.reset();
+}
+
+void substitution::reset_cache() {
+    TRACE("subst_bug", tout << "substitution::reset_cache\n";
+          for (unsigned i = 0; i < m_new_exprs.size(); i++) { tout << mk_pp(m_new_exprs.get(i), m_manager) << "\nref_count: " << m_new_exprs.get(i)->get_ref_count() << "\n"; });
+
+    m_apply_cache.reset();
+    m_new_exprs.reset();
+}
+
+void substitution::pop_scope(unsigned num_scopes) {
+    unsigned lvl     = m_scopes.size();
+    SASSERT(num_scopes <= lvl);
+    unsigned new_lvl = lvl - num_scopes;
+    unsigned old_sz  = m_scopes[new_lvl];
+    unsigned curr_sz = m_vars.size();
+    SASSERT(old_sz <= curr_sz);
+    for (unsigned i = old_sz; i < curr_sz; i++) {
+        var_offset & curr = m_vars[i];
+        m_subst.erase(curr.first, curr.second);
+    }
+    m_vars.shrink(old_sz);
+    m_scopes.shrink(new_lvl);
+}
+
+inline void substitution::apply_visit(expr_offset const & n, bool & visited) {
+    if (!m_apply_cache.contains(n)) {
+        m_todo.push_back(n);
+        visited = false;
+    }
+}
+
+void substitution::apply(unsigned num_actual_offsets, unsigned const * deltas, expr_offset const & n, 
+                         expr_offset const & s, expr_offset const & t, expr_ref & result) {
+    
+    TRACE("subst_bug", tout << "BEGIN substitution::apply\n";);
+
+    // It is incorrect to cache results between different calls if we are applying a substitution
+    // modulo a substitution s -> t.
+    if (s != expr_offset(0,0))
+        reset_cache();
+
+    unsigned         j;
+    expr *           e;
+    unsigned         off;
+    expr_offset      n1;
+    bool             visited;
+    unsigned         num_args;
+    ptr_buffer<expr> new_args;
+
+    m_todo.push_back(n);
+    while (!m_todo.empty()) {
+        expr_offset n = m_todo.back();
+        TRACE("subst_bug", tout << "n: " << mk_pp(n.get_expr(), m_manager) << " : " << n.get_offset() << "\n";);
+        if (m_apply_cache.contains(n)) {
+            m_todo.pop_back();
+            continue;
+        }
+        expr_offset n_prime = n == s ? t : n;
+        TRACE("subst_bug", tout << "n_prime: " << mk_pp(n_prime.get_expr(), m_manager) << " : " << n_prime.get_offset() << "\n";);
+        visited = true;
+        e   = n_prime.get_expr();
+        off = n_prime.get_offset();
+        switch (e->get_kind()) {
+        case AST_VAR:
+            if (find(to_var(e)->get_idx(), off, n1)) {
+                apply_visit(n1, visited);
+                TRACE("subst_bug", tout << "visited: " << visited << ", n1: " << mk_pp(n1.get_expr(), m_manager) << " : " << n1.get_offset() << "\n";);
+                if (visited) {
+                    m_todo.pop_back();
+                    expr * new_expr;
+                    m_apply_cache.find(n1, new_expr);
+                    m_apply_cache.insert(n, new_expr);
+                    TRACE("subst_bug", tout << "1. insert n: " << mk_pp(n.get_expr(), m_manager) << " : " << n.get_offset() 
+                          << " --> " << mk_pp(new_expr, m_manager) << "\n";);
+                }
+            }
+            else {
+                m_todo.pop_back();
+                SASSERT(off < num_actual_offsets);
+                unsigned delta = deltas[off];
+                expr * new_expr = e;
+                if (delta > 0) { 
+                    new_expr = m_manager.mk_var(to_var(e)->get_idx() + delta, to_var(e)->get_sort());
+                    m_new_exprs.push_back(new_expr);
+                }
+                m_apply_cache.insert(n, new_expr);
+                TRACE("subst_bug", tout << "2. insert n: " << mk_pp(n.get_expr(), m_manager) << " : " << n.get_offset() 
+                      << " --> " << mk_pp(new_expr, m_manager) << "\n";);
+            }
+            break;
+        case AST_APP:
+            num_args = to_app(e)->get_num_args();
+            j        = num_args;
+            while (j > 0) {
+                --j;
+                apply_visit(expr_offset(to_app(e)->get_arg(j), off), visited);
+            }
+            if (visited) {
+                m_todo.pop_back();
+                new_args.reset();
+                bool has_new_args = false;
+                for (unsigned i = 0; i < num_args; i++) {
+                    expr * arg     = to_app(e)->get_arg(i);
+                    expr * new_arg;
+                    
+                    m_apply_cache.find(expr_offset(arg, off), new_arg);
+                    new_args.push_back(new_arg);
+                    if (arg != new_arg)
+                        has_new_args = true;
+                }
+                if (!has_new_args) {
+                    m_apply_cache.insert(n, e);
+                    TRACE("subst_bug", tout << "3. insert n: " << mk_pp(n.get_expr(), m_manager) << " : " << n.get_offset() 
+                          << " --> " << mk_pp(e, m_manager) << "\n";);
+                }
+                else {
+                    expr * new_expr = m_manager.mk_app(to_app(e)->get_decl(), new_args.size(), new_args.c_ptr());
+                    m_new_exprs.push_back(new_expr);
+                    m_apply_cache.insert(n, new_expr);
+                    TRACE("subst_bug", tout << "3. insert n: " << mk_pp(n.get_expr(), m_manager) << " : " << n.get_offset() 
+                          << " --> " << mk_pp(new_expr, m_manager) << "\n";);
+                }
+            }
+            break;
+        default:
+            UNREACHABLE();
+        }
+    }
+    SASSERT(m_apply_cache.contains(n));
+    m_apply_cache.find(n, e);
+    m_new_exprs.push_back(e);
+    result = e;
+    
+    if (s != expr_offset(0,0))
+        reset_cache();
+    
+    TRACE("subst_bug", tout << "END substitution::apply\nresult:\n" << mk_pp(e, m_manager) << "\nref_count: " << e->get_ref_count() << "\n";);
+}
+
+inline substitution::color substitution::get_color(expr_offset const & p) const {
+    color c;
+    if (m_color.find(p, c)) 
+        return c;
+    return White;
+}
+
+inline void substitution::set_color(expr_offset const & p, color c) {
+    m_color.insert(p, c);
+}
+
+inline void substitution::visit(expr_offset const & p, bool & visited) {
+    if (get_color(p) != Black) {
+        m_todo.push_back(p);
+        visited = false;
+    }
+}
+
+bool substitution::visit_children(expr_offset const & p) {
+    bool visited = true;
+    expr * n     = p.get_expr();
+    unsigned off;
+    expr_offset p1;
+    unsigned j;
+    switch (n->get_kind()) {
+    case AST_VAR:
+        if (find(p, p1) && p != p1)
+            visit(p1, visited);
+        break;
+    case AST_APP:
+        off  = p.get_offset();
+        j    = to_app(n)->get_num_args();
+        while (j > 0) {
+            --j;
+            visit(expr_offset(to_app(n)->get_arg(j), off), visited);
+        }
+        break;
+    default:
+        UNREACHABLE();
+    }
+    return visited;
+}
+
+bool substitution::acyclic(expr_offset p) {
+    if (get_color(p) == Black)
+        return true;
+    m_todo.reset();
+    m_todo.push_back(p);
+    while (!m_todo.empty()) {
+        expr_offset p = m_todo.back();
+        switch (get_color(p)) {
+        case Black:
+            m_todo.pop_back();
+            break;
+        case White:
+            set_color(p, Grey);
+            if (visit_children(p)) {
+                set_color(p, Black);
+                SASSERT(m_todo.back() == p);
+                m_todo.pop_back();
+            }
+            break;
+        case Grey:
+            if (!visit_children(p))
+                return false;
+            set_color(p, Black);
+            SASSERT(m_todo.back() == p);
+            m_todo.pop_back();
+            break;
+        }
+    }
+    return true;
+}
+
+bool substitution::acyclic() {
+    m_color.reset();
+    expr_offset r;
+    svector<var_offset>::iterator it  = m_vars.begin();
+    svector<var_offset>::iterator end = m_vars.end();
+    for (; it != end; ++it) {
+        m_subst.find(it->first, it->second, r);
+        if (!acyclic(r))
+            return false;
+    }
+    return true;
+}
+
+void substitution::display(std::ostream & out, unsigned num_actual_offsets, unsigned const * deltas) {
+    reset_cache();
+    for (unsigned i = 0; i < num_actual_offsets; i++)
+        for (unsigned j = 0; j < m_subst.vars_capacity(); j++) {
+            expr_offset r;
+            if (find(j, i, r)) {
+                expr_ref tmp(m_manager);
+                apply(num_actual_offsets, deltas, r, tmp);
+                out << "VAR " << j << ":" << i << " -->\n" << mk_pp(tmp, m_manager) << "\n";
+            }
+        }
+}
+
+void substitution::display(std::ostream & out) {
+    for (unsigned i = 0; i < m_subst.offsets_capacity(); i++)
+        for (unsigned j = 0; j < m_subst.vars_capacity(); j++) {
+            expr_offset r;
+            if (find(j, i, r))
+                out << "VAR " << j << ":" << i << " --> " << r.get_offset() << "\n" << mk_pp(r.get_expr(), m_manager) << "\n";
+        }
+}
+

src/spc/substitution.h

@@ -0,0 +1,202 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    substitution.h
+
+Abstract:
+
+    A substitution, that is, a mapping from (variable, offset) to (expr, offset).
+    We use offsets in order to avoid creating variants of terms.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-01.
+
+Revision History:
+
+--*/
+#ifndef _SUBSTITUTION_H_
+#define _SUBSTITUTION_H_
+
+#include"expr_offset_map.h"
+#include"var_offset_map.h"
+#include"ast_pp.h"
+
+/**
+   \brief A mapping from (variable,offset) to expr_offset.
+*/
+class substitution {
+    ast_manager &               m_manager;
+    var_offset_map<expr_offset> m_subst;
+
+    // field for backtracking
+    typedef std::pair<unsigned, unsigned> var_offset;
+    svector<var_offset>         m_vars;
+    unsigned_vector             m_scopes;
+
+    // fields for applying substitutions
+    svector<expr_offset>        m_todo;
+    expr_offset_map<expr *>     m_apply_cache;
+    expr_ref_vector             m_new_exprs;
+
+    // fields for checking for cycles
+    enum color { White, Grey, Black };
+    expr_offset_map<color>      m_color;
+
+#ifdef Z3DEBUG
+    unsigned                    m_max_offset_since_reset;
+#endif 
+    void apply_visit(expr_offset const & n, bool & visited);
+
+
+    color get_color(expr_offset const & p) const;
+    void set_color(expr_offset const & p, color c);
+    
+    void visit(expr_offset const & p, bool & visited);
+    bool visit_children(expr_offset const & p);
+    bool acyclic(expr_offset p);
+
+public:
+    substitution(ast_manager & m);
+    ast_manager & get_manager() const { return m_manager; }
+
+    // -----------------------------------
+    //
+    // Reserve memory for the given number of 
+    // offsets and variables.
+    //
+    // -----------------------------------
+
+    void reserve(unsigned num_offsets, unsigned num_vars) { m_subst.reserve(num_offsets, num_vars); }
+    void reserve_offsets(unsigned num_offsets) { m_subst.reserve_offsets(num_offsets); }
+    void reserve_vars(unsigned num_vars) { m_subst.reserve_vars(num_vars); }
+
+    // -----------------------------------
+    //
+    // Reset functions
+    //
+    // -----------------------------------
+
+    // reset everything
+    void reset();
+    // reset only the mapping from variables to expressions
+    void reset_subst();
+    // reset only the substitution application cache
+    void reset_cache();
+
+    // -----------------------------------
+    //
+    // Backtracking
+    //
+    // -----------------------------------
+
+    void push_scope() { m_scopes.push_back(m_vars.size()); }
+    void pop_scope(unsigned num_scopes = 1);
+    unsigned get_scope_lvl() { return m_scopes.size(); }
+    bool top_scope_has_bindings() const { return m_scopes.empty() ? !m_vars.empty() : m_scopes.back() < m_vars.size(); }
+    unsigned get_num_bindings() const { return m_vars.size(); }
+    
+
+    // -----------------------------------
+    //
+    // Cycle detection
+    //
+    // -----------------------------------
+
+    bool acyclic();
+
+    // -----------------------------------
+    //
+    // Insertion & Lookup
+    // 
+    // get_binding supplies a way to inspect the substitution.
+    //
+    // -----------------------------------
+
+    void insert(unsigned v_idx, unsigned offset, expr_offset const & t) {
+        TRACE("subst_insert", tout << "inserting: #" << v_idx << ":" << offset << " --> " << mk_pp(t.get_expr(), m_manager)
+              << ":" << t.get_offset() << "\n";);
+        m_vars.push_back(var_offset(v_idx, offset));
+        m_subst.insert(v_idx, offset, t);
+    }
+    void insert(var * v, unsigned offset, expr_offset const & t) { insert(v->get_idx(), offset, t); }
+    void insert(expr_offset v, expr_offset const & t) {
+        SASSERT(is_var(v.get_expr()));
+        insert(to_var(v.get_expr()), v.get_offset(), t);
+    }
+
+    bool find(unsigned v_idx, unsigned offset, expr_offset & r) const { return m_subst.find(v_idx, offset, r); }
+    bool find(var * v, unsigned offset, expr_offset & r) const { return find(v->get_idx(), offset, r); }
+    bool find(expr_offset v, expr_offset & r) const { 
+        SASSERT(is_var(v.get_expr()));
+        return find(to_var(v.get_expr()), v.get_offset(), r);
+    }
+
+    void get_binding(unsigned binding_num, var_offset& var, expr_offset& r) {
+        var = m_vars[binding_num];
+        VERIFY(m_subst.find(var.first, var.second, r));
+    }
+
+    bool contains(var * v, unsigned offset) { expr_offset r; return find(v, offset, r); }
+
+    // -----------------------------------
+    //
+    // Application
+    //
+    // -----------------------------------
+    
+    /**
+       \brief Apply the current substitution to the given
+       expression+offset. The result is an expression.
+       
+       The argument num_actual_offsets is the maximum offset used in a
+       insert method since the last reset.
+       
+       The argument deltas is an array of size num_actual_offsets. It contains
+       the variable delta for each offset. A free variable x:i in an expression offset t:j is mapped
+       to the variable x+delta[i]. 
+    */
+    void apply(unsigned num_actual_offsets, unsigned const * deltas, expr_offset const & n, expr_ref & result) {
+        apply(num_actual_offsets, deltas, n, expr_offset(0, 0), expr_offset(0, 0), result);
+    }
+    
+    /**
+       \brief Similar to the previous method, but occurrences of s in n are substituted by t.
+       If s != expr_offset(0,0), then the cache is reset before and after the execution of this procedure.
+    */
+    void apply(unsigned num_actual_offsets, unsigned const* deltas, expr_offset const & n, expr_offset const & s, expr_offset const & t, expr_ref & result);
+
+    void apply(expr * n, expr_ref & result) {
+        unsigned deltas[1] = { 0 };
+        apply(1, deltas, expr_offset(n, 0), result);
+    }
+
+    // -----------------------------------
+    //
+    // Debugging
+    //
+    // -----------------------------------
+ 
+    /**
+       \brief Dump the current substitution (for debugging purposes).
+    */
+    void display(std::ostream & out, unsigned num_actual_offsets, unsigned const * deltas);
+
+    /**
+       \brief Dump the current substitution without normalizing expressions (for debugging purposes).
+    */
+    void display(std::ostream & out);
+
+    
+    // -----------------------------------
+    //
+    // Compare terms modulo a substitution
+    //
+    // -----------------------------------
+    bool compare(expr_offset t1, expr_offset t2);
+
+};
+
+#endif

src/spc/substitution_tree.cpp

@@ -0,0 +1,899 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    substitution_tree.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-04.
+
+Revision History:
+
+--*/
+#include"substitution_tree.h"
+#include"ast_pp.h"
+#include"ast_smt2_pp.h"
+
+/**
+   \brief Return the next available register.
+*/
+unsigned substitution_tree::next_reg() {
+    while(true) {
+        unsigned curr = m_next_reg;
+        if (curr > m_max_reg)
+            m_max_reg = curr;
+        m_next_reg++;
+        if (curr >= m_used_regs.size() || !m_used_regs.get(curr))
+            return curr;
+    }
+}
+
+inline void substitution_tree::push(svector<subst> & sv, subst const & s) {
+    sv.push_back(s);
+    m_manager.inc_ref(s.first);
+    m_manager.inc_ref(s.second);
+}
+
+inline expr * substitution_tree::get_reg_value(unsigned ridx) {
+    return m_registers.get(ridx, 0);
+}
+
+inline void substitution_tree::set_reg_value(unsigned ridx, expr * e) {
+    m_registers.setx(ridx, e, 0);
+}
+
+inline void substitution_tree::erase_reg_from_todo(unsigned ridx) {
+    SASSERT(m_registers[ridx]);
+    m_registers[ridx] = 0;
+    SASSERT(m_todo.contains(ridx));
+    m_todo.erase(ridx);
+}
+
+/**
+   \brief Linearize the expressions in the registers stored in m_todo.
+   Store the result in \c result.
+
+   Example:
+     m_todo = { 3, 4 }
+     m_registers[3] = (f (g a))
+     m_registers[4] = b
+     next_regs are 5 6 7
+     
+     result:
+     #3 -> (f #5); #4 -> b; #5 -> (g #6); #6 -> a
+*/
+void substitution_tree::linearize(svector<subst> & result) {
+    ptr_buffer<expr> new_args;
+    for (unsigned i = 0; i < m_todo.size(); i++) {
+        unsigned ireg_idx = m_todo[i];
+        expr * n          = get_reg_value(ireg_idx);
+        var * ireg        = m_manager.mk_var(ireg_idx, m_manager.get_sort(n));
+        if (is_var(n))
+            push(result, subst(ireg, n));
+        else {
+            SASSERT(is_app(n));
+            app * new_app;
+            unsigned num = to_app(n)->get_num_args();
+            if (num == 0)
+                new_app = to_app(n);
+            else {
+                for (unsigned j = 0; j < num; j++) {
+                    unsigned oreg     = next_reg();
+                    set_reg_value(oreg, to_app(n)->get_arg(j));
+                    m_todo.push_back(oreg);
+                    sort * s          = m_manager.get_sort(get_reg_value(oreg));
+                    new_args.push_back(m_manager.mk_var(oreg, s));
+                }
+                new_app = m_manager.mk_app(to_app(n)->get_decl(), new_args.size(), new_args.c_ptr());
+                new_args.reset();
+            }
+            push(result, subst(ireg, new_app));
+        }
+    }
+}
+
+/**
+   \brief Process the pair in := (f t_1 ... t_n) and out := (f r_1 ... r_n),
+   where r_i's are variables (register ids), and t_i's are arbitrary expressions.
+   The r_i's are added to the m_todo list, and m_registers[r_i] is assigned to t_i.
+
+   If save_set_registers == true, then r_i's are stored in m_to_reset.
+*/
+void substitution_tree::process_args(app * in, app * out) {
+    CTRACE("subst_tree_bug", in->get_num_args() != out->get_num_args(), tout << mk_ismt2_pp(in, m_manager) << "\n" 
+           << mk_ismt2_pp(out, m_manager) << "\n";);
+    unsigned num = out->get_num_args();
+    for (unsigned i = 0; i < num; i++) {
+        expr * in_arg  = in->get_arg(i);
+        expr * out_arg = out->get_arg(i);
+        SASSERT(is_var(out_arg));
+        unsigned oreg  = to_var(out_arg)->get_idx();
+        set_reg_value(oreg, in_arg);
+        m_todo.push_back(oreg);
+    }
+}
+
+/**
+   \brief Reset registers in m_todo at [old_size, m_todo.size())
+*/
+void substitution_tree::reset_registers(unsigned old_size) {
+    SASSERT(m_todo.size() >= old_size);
+    unsigned_vector::iterator it2  = m_todo.begin() + old_size;
+    unsigned_vector::iterator end2 = m_todo.end();
+    for (; it2 != end2; ++it2)
+        m_registers[*it2] = 0;
+    m_todo.shrink(old_size);
+}
+
+/**
+   \brief Return a measure on how compatible sv and the expressions to be processed are.
+*/
+unsigned substitution_tree::get_compatibility_measure(svector<subst> const & sv) {
+    unsigned old_size = m_todo.size();
+    unsigned measure  = 0;
+    svector<subst>::const_iterator it  = sv.begin();
+    svector<subst>::const_iterator end = sv.end();
+    for (; it != end; ++it) {
+        subst const & s = *it;
+        unsigned ireg = s.first->get_idx();
+        expr * out    = s.second;
+        expr * in     = get_reg_value(ireg);
+
+        if (is_var(out)) {
+            if (out == in)
+                measure += 1;
+        }
+        else {
+            SASSERT(is_app(out));
+            if (in && is_app(in) && to_app(out)->get_decl() == to_app(in)->get_decl()) {
+                measure += 2;
+                process_args(to_app(in), to_app(out));
+            }
+        }
+    }
+    reset_registers(old_size);
+    return measure;
+}
+
+/**
+   \brief Find the child of r that is most compatible with the expressions stored
+   in the registers in m_todo.
+
+   Return 0 if none of the children has any compatible substitution entry.
+*/
+substitution_tree::node * substitution_tree::find_best_child(node * r) {
+    SASSERT(!r->m_leaf);
+#ifdef Z3DEBUG
+    unsigned todo_size   = m_todo.size();
+#endif
+    node * best_child    = 0;
+    unsigned max_measure = 0;
+    node * curr_child    = r->m_first_child;
+    while (curr_child) {
+        unsigned measure = get_compatibility_measure(curr_child->m_subst);
+        if (measure > max_measure) {
+            best_child  = curr_child;
+            max_measure = measure;
+        }
+        curr_child = curr_child->m_next_sibling;
+    }
+    SASSERT(todo_size == m_todo.size());
+    return best_child;
+}
+
+/**
+   \brief Reset datastructures used to insert/erase elements from the substitution tree.
+*/
+void substitution_tree::reset_compiler() {
+    m_todo.reset();
+    m_used_regs.reset();
+    m_next_reg = 1; // register 0 is reserved for input.
+    DEBUG_CODE({
+        ptr_vector<expr>::iterator it  = m_registers.begin();
+        ptr_vector<expr>::iterator end = m_registers.end();
+        for (; it != end; ++it) {
+            SASSERT(*it == 0);
+        }
+    });
+}
+
+/**
+   \brief Create a node with the linearization for all registers in todo.
+   Attach new_expr to it.
+*/
+substitution_tree::node * substitution_tree::mk_node_for(expr * new_expr) {
+    node * n = alloc(node, true);
+    linearize(n->m_subst);
+    n->m_expr = new_expr;
+    m_manager.inc_ref(new_expr);
+    return n;
+}
+
+/**
+   \brief Mark register ridx as used.
+*/
+void substitution_tree::mark_used_reg(unsigned ridx) {
+    if (ridx >= m_used_regs.size()) 
+        m_used_regs.resize(ridx+1);
+    m_used_regs.set(ridx);
+}
+
+/**
+   \brief Mark (m_used_regs) all registers used in \c sv.
+*/
+void substitution_tree::mark_used_regs(svector<subst> const & sv) {
+    svector<subst>::const_iterator it  = sv.begin();
+    svector<subst>::const_iterator end = sv.end();
+    for (; it != end; ++it) {
+        subst const & s = *it;
+        mark_used_reg(s.first->get_idx());
+        if (is_app(s.second)) {
+            unsigned num_args = to_app(s.second)->get_num_args();
+            for (unsigned i = 0; i < num_args; i++) {
+                expr * arg = to_app(s.second)->get_arg(i);
+                SASSERT(is_var(arg));
+                mark_used_reg(to_var(arg)->get_idx());
+            }
+        }
+    }
+}
+
+/**
+   \brief Insert a new expression in the substitution tree.
+*/
+void substitution_tree::insert(expr * new_expr) {
+    if (is_app(new_expr)) {
+        insert(to_app(new_expr));
+    }
+    else {
+        SASSERT(is_var(new_expr));
+        sort * s    = to_var(new_expr)->get_sort();
+        unsigned id = s->get_decl_id();
+        if (id >= m_vars.size())
+            m_vars.resize(id+1, 0);
+        if (m_vars[id] == 0)
+            m_vars[id] = alloc(var_ref_vector, m_manager);
+        var_ref_vector * v = m_vars[id];
+        if (!v->contains(to_var(new_expr)))
+            v->push_back(to_var(new_expr));
+    }
+}
+
+/**
+   \brief Insert a new application in the substitution tree.
+*/
+void substitution_tree::insert(app * new_expr) {
+    reset_compiler();
+    set_reg_value(0, new_expr);
+    m_todo.push_back(0);
+
+    func_decl * d = new_expr->get_decl();
+    unsigned id   = d->get_decl_id();
+    
+    if (id >= m_roots.size()) 
+        m_roots.resize(id+1, 0);
+
+    if (!m_roots[id]) {
+        // there is no tree for the function symbol heading new_expr
+        m_roots[id] = mk_node_for(new_expr);
+        reset_registers(0);
+        m_size++;
+        return;
+    }
+
+    node * r = m_roots[id];
+    
+    while (true) {
+        m_compatible.reset();
+        m_incompatible.reset();
+        svector<subst> & sv = r->m_subst;
+        // separate sv in the set of compatible & incompatible instructions
+        svector<subst>::iterator it  = sv.begin();
+        svector<subst>::iterator end = sv.end();
+        for (; it != end; ++it) {
+            subst & s = *it;
+            unsigned ireg = s.first->get_idx();
+            expr * out = s.second;
+            expr * in  = get_reg_value(ireg);
+            SASSERT(is_var(out) || is_app(out));
+            if (is_var(out)) {
+                if (out == in) {
+                    erase_reg_from_todo(ireg);
+                    m_compatible.push_back(s);
+                }
+                else {
+                    m_incompatible.push_back(s);
+                }
+            }
+            else {
+                if (in && is_app(in) && to_app(out)->get_decl() == to_app(in)->get_decl()) {
+                    erase_reg_from_todo(ireg);
+                    m_compatible.push_back(s);
+                    process_args(to_app(in), to_app(out));
+                }
+                else {
+                    m_incompatible.push_back(s);
+                }
+            }
+        }
+
+        // process m_compatible & m_incompatible
+        if (m_incompatible.empty()) {
+            if (m_todo.empty()) {
+                // nothing else to process
+                // new_expr is already in the substitution tree
+                SASSERT(r->m_leaf && r->m_expr == new_expr);
+                reset_registers(0);
+                return;
+            }
+            else {
+                mark_used_regs(r->m_subst);
+                node * best_child = find_best_child(r);
+                if (best_child == 0) {
+                    // there is no compatible child
+                    node * n          = mk_node_for(new_expr);
+                    n->m_next_sibling = r->m_first_child;
+                    r->m_first_child  = n;
+                    reset_registers(0);
+                    m_size++;
+                    return;
+                }
+                else {
+                    // continue with best_child
+                    r = best_child;
+                }
+            }
+        }
+        else {
+            SASSERT(!m_compatible.empty());
+            SASSERT(!m_incompatible.empty());
+
+            mark_used_regs(m_compatible);
+
+            r->m_subst.swap(m_compatible);
+            
+            node * n      = mk_node_for(new_expr);
+
+            node * incomp = alloc(node, r->m_leaf);
+            incomp->m_subst.swap(m_incompatible);
+            if (r->m_leaf) {
+                incomp->m_expr = r->m_expr;
+                r->m_leaf      = false;
+            }
+            else 
+                incomp->m_first_child  = r->m_first_child;
+            incomp->m_next_sibling = n; 
+
+            SASSERT(!r->m_leaf);
+            r->m_first_child = incomp;
+            reset_registers(0);
+            m_size++;
+            return;
+        }
+    }
+}
+
+/**
+   \brief Return true if sv is fully compatible with the expressions in the registers in m_todo.
+*/
+bool substitution_tree::is_fully_compatible(svector<subst> const & sv) {
+    unsigned old_size = m_todo.size();
+    svector<subst>::const_iterator it  = sv.begin();
+    svector<subst>::const_iterator end = sv.end();
+    for (; it != end; ++it) {
+        subst const & s = *it;
+        unsigned ireg = s.first->get_idx();
+        expr * out    = s.second;
+        expr * in     = get_reg_value(ireg);
+        if (is_var(out)) {
+            if (out != in) {
+                reset_registers(old_size);
+                return false;
+            }
+        }
+        else {
+            if (!in || !is_app(in) || to_app(in)->get_decl() != to_app(out)->get_decl()) {
+                reset_registers(old_size);
+                return false;
+            }
+            process_args(to_app(in), to_app(out));
+        }
+    }
+    reset_registers(old_size);
+    return true;
+}
+
+/**
+   \brief Return a child of r that is fully compatible with the expressions in the registers in m_todo.
+*/
+bool substitution_tree::find_fully_compatible_child(node * r, node * & prev, node * & child) {
+    SASSERT(!r->m_leaf);
+    prev  = 0;
+    child = r->m_first_child;
+    while (child) {
+        if (is_fully_compatible(child->m_subst))
+            return true;
+        prev  = child;
+        child = child->m_next_sibling;
+    }
+    return false;
+}
+
+inline bool substitution_tree::at_least_3_children(node * r) {
+    return !r->m_leaf && r->m_first_child->m_next_sibling && r->m_first_child->m_next_sibling->m_next_sibling;
+}
+
+/**
+   \brief Remove expression from the substitution tree.
+   Do nothing, if n is not in the tree.
+*/
+void substitution_tree::erase(expr * e) {
+    if (is_app(e))
+        erase(to_app(e));
+    else {
+        SASSERT(is_var(e));
+        sort * s = to_var(e)->get_sort();
+        unsigned id = s->get_decl_id();
+        if (id >= m_vars.size() || m_vars[id] == 0)
+            return;
+        var_ref_vector * v = m_vars[id];
+        v->erase(to_var(e));
+    }
+}
+
+/**
+   \brief Remove application from the substitution tree.
+   Do nothing, if n is not in the tree.
+*/
+void substitution_tree::erase(app * e) {
+    func_decl * d = e->get_decl();
+    unsigned id   = d->get_decl_id();
+    if (id >= m_roots.size() || !m_roots[id])
+        return;
+
+    reset_compiler();
+    set_reg_value(0, e);
+    m_todo.push_back(0);
+    
+    node * r = m_roots[id];
+    node * parent = 0;
+    node * prev   = 0;
+
+    while (true) {
+        svector<subst> & sv = r->m_subst;
+        svector<subst>::iterator it  = sv.begin();
+        svector<subst>::iterator end = sv.end();
+        for (; it != end; ++it) {
+            subst & s     = *it;
+            unsigned ireg = s.first->get_idx();
+            expr * out    = s.second;
+            expr * in     = get_reg_value(ireg);
+            SASSERT(is_var(out) || is_app(out));
+            if (is_var(out)) {
+                if (out != in) {
+                    reset_registers(0);
+                    return; // node is not in the substitution tree
+                }
+                erase_reg_from_todo(ireg);
+            }
+            else {
+                if (!in || !is_app(in) || to_app(out)->get_decl() != to_app(in)->get_decl()) {
+                    reset_registers(0);
+                    return; // node is not in the substitution tree
+                }
+                erase_reg_from_todo(ireg);
+                process_args(to_app(in), to_app(out));
+            }
+        }
+
+        if (m_todo.empty()) {
+            reset_registers(0);
+            SASSERT(r->m_expr == e);
+            if (parent == 0) {
+                delete_node(r);
+                m_roots[id] = 0;
+            }
+            else if (at_least_3_children(parent)) {
+                if (prev == 0)
+                    parent->m_first_child = r->m_next_sibling;
+                else 
+                    prev->m_next_sibling  = r->m_next_sibling;
+                delete_node(r);
+            }
+            else {
+                SASSERT(parent->m_first_child && parent->m_first_child->m_next_sibling && !parent->m_first_child->m_next_sibling->m_next_sibling);
+                node * other_child = prev ? prev : r->m_next_sibling;
+                SASSERT(other_child);
+                parent->m_subst.append(other_child->m_subst);
+                parent->m_leaf = other_child->m_leaf;
+                if (other_child->m_leaf)
+                    parent->m_expr = other_child->m_expr;
+                else 
+                    parent->m_first_child = other_child->m_first_child;
+                delete_node(r);
+                dealloc(other_child); // Remark: I didn't use delete_node since its resources were sent to parent.
+            }
+            m_size --;
+            return;
+        }
+        else {
+            parent = r;
+            if (!find_fully_compatible_child(r, prev, r)) {
+                // node is not in the substitution tree
+                reset_registers(0);
+                return;
+            }
+            // continue with fully compatible child
+        }
+    }
+}
+
+void substitution_tree::delete_node(node * n) {
+    ptr_buffer<node> todo;
+    SASSERT(todo.empty());
+    todo.push_back(n);
+    while (!todo.empty()) {
+        node * n = todo.back();
+        todo.pop_back();
+        svector<subst>::iterator it2  = n->m_subst.begin();
+        svector<subst>::iterator end2 = n->m_subst.end();
+        for (; it2 != end2; ++it2) {
+            m_manager.dec_ref(it2->first);
+            m_manager.dec_ref(it2->second);
+        }
+        if (n->m_leaf) 
+            m_manager.dec_ref(n->m_expr);
+        else {
+            node * c = n->m_first_child;
+            while (c) {
+                todo.push_back(c);
+                c = c->m_next_sibling;
+            }
+        }
+        dealloc(n);
+    }
+}
+
+void substitution_tree::reset() {
+    ptr_vector<node>::iterator it  = m_roots.begin();
+    ptr_vector<node>::iterator end = m_roots.end();
+    for (; it != end; ++it) {
+        if (*it)
+            delete_node(*it);
+    }
+    m_roots.reset();
+    std::for_each(m_vars.begin(), m_vars.end(), delete_proc<var_ref_vector>());
+    m_vars.reset();
+    m_size = 0;
+}
+
+void substitution_tree::display(std::ostream & out, subst const & s) const {
+    out << "r!" << s.first->get_idx() << " -> ";
+    if (is_app(s.second)) {
+        unsigned num = to_app(s.second)->get_num_args();
+        if (num == 0)
+            out << to_app(s.second)->get_decl()->get_name();
+        else {
+            out << "(" << to_app(s.second)->get_decl()->get_name();
+            for (unsigned i = 0; i < num; i++)
+                out << " r!" << to_var(to_app(s.second)->get_arg(i))->get_idx();
+            out << ")";
+        }
+    }
+    else {
+        out << mk_pp(s.second, m_manager);
+    }
+}
+
+void substitution_tree::display(std::ostream & out, svector<subst> const & sv) const {
+    svector<subst>::const_iterator it  = sv.begin();
+    svector<subst>::const_iterator end = sv.end();
+    for (bool first = true; it != end; ++it, first = false) {
+        subst const & s = *it;
+        if (!first) 
+            out << "; ";
+        display(out, s);
+    }
+}
+
+void substitution_tree::display(std::ostream & out, node * n, unsigned delta) const {
+    for (unsigned i = 0; i < delta; i++) 
+        out << "  ";
+    display(out, n->m_subst);
+    if (n->m_leaf) {
+        pp_params p;
+        p.m_pp_single_line = true;
+        out << "  ==> ";
+        ast_pp(out, n->m_expr, m_manager, p);
+        out << "\n";
+    }
+    else {
+        out << "\n";
+        node * c = n->m_first_child;
+        while (c) {
+            display(out, c, delta+1);
+            c = c->m_next_sibling;
+        }
+    }
+}
+
+bool substitution_tree::backtrack() {
+    while (!m_bstack.empty()) {
+        TRACE("st", tout << "backtracking...\n";);
+        m_subst->pop_scope();
+
+        node * n = m_bstack.back();
+        if (n->m_next_sibling) {
+            m_bstack.back() = n->m_next_sibling;
+            return true;
+        }
+        m_bstack.pop_back();
+    }
+    return false;
+}
+
+inline expr_offset substitution_tree::find(expr_offset p) {
+    TRACE("substitution_tree_bug", tout << "find...\n";);
+    while (is_var(p.get_expr())) {
+        TRACE("substitution_tree_bug", tout << mk_pp(p.get_expr(), m_manager) << " " << p.get_offset() << "\n";);
+        if (!m_subst->find(to_var(p.get_expr()), p.get_offset(), p))
+            return p;
+    }
+    return p;
+}
+
+template<substitution_tree::st_visit_mode Mode>
+bool substitution_tree::bind_var(var * v, unsigned offset, expr_offset const & p) {
+    TRACE("st", tout << "bind_var: " << mk_pp(v, m_manager) << " " << offset << "\n" << 
+          mk_pp(p.get_expr(), m_manager) << " " << p.get_offset() << "\n";);
+    if (Mode == STV_INST && offset == m_st_offset) {
+        SASSERT(!is_var(p.get_expr()) || p.get_offset() != m_reg_offset);
+        if (is_var(p.get_expr()) && p.get_offset() == m_in_offset) {
+            m_subst->insert(p, expr_offset(v, offset));
+            return true;
+        }
+        return false;
+    }
+    if (Mode == STV_GEN && offset == m_in_offset) {
+        SASSERT(!is_var(p.get_expr()) || p.get_offset() != m_reg_offset);
+        if (is_var(p.get_expr()) && p.get_offset() == m_st_offset) {
+            m_subst->insert(p, expr_offset(v, offset));
+            return true;
+        }
+        return false;
+    }
+    m_subst->insert(v, offset, p);
+    TRACE("st_bug", tout << "substitution updated\n"; m_subst->display(tout););
+    return true;
+}
+
+template<substitution_tree::st_visit_mode Mode>
+bool substitution_tree::unify_match(expr_offset p1, expr_offset p2) {
+    svector<entry> & todo = m_visit_todo;
+    todo.reset();
+    todo.push_back(entry(p1, p2));
+    while (!todo.empty()) {
+        entry const & e = todo.back();
+        p1 = find(e.first);
+        p2 = find(e.second);
+        todo.pop_back();
+        if (p1 != p2) {
+            expr * n1 = p1.get_expr();
+            expr * n2 = p2.get_expr();
+            SASSERT(!is_quantifier(n1));
+            SASSERT(!is_quantifier(n2));
+            bool v1 = is_var(n1);
+            bool v2 = is_var(n2);
+            TRACE("st", 
+                  tout << "n1: " << mk_pp(n1, m_manager) << " " << p1.get_offset() << "\n";
+                  tout << "n2: " << mk_pp(n2, m_manager) << " " << p2.get_offset() << "\n";);
+            if (v1 && v2) {
+                if (p2.get_offset() == m_reg_offset)
+                    std::swap(p1, p2);
+                if (!bind_var<Mode>(to_var(p1.get_expr()), p1.get_offset(), p2))
+                    return false;
+            }
+            else if (v1) { 
+                if (!bind_var<Mode>(to_var(n1), p1.get_offset(), p2))
+                    return false;
+            }
+            else if (v2) {
+                if (!bind_var<Mode>(to_var(n2), p2.get_offset(), p1))
+                    return false;
+            }
+            else {
+                app * a1 = to_app(n1);
+                app * a2 = to_app(n2);
+                unsigned off1 = p1.get_offset();
+                unsigned off2 = p2.get_offset();
+                if (a1->get_decl() != a2->get_decl() || a1->get_num_args() != a2->get_num_args())
+                    return false;
+                unsigned j = a1->get_num_args();
+                while (j > 0) {
+                    --j;
+                    entry new_e(expr_offset(a1->get_arg(j), off1),
+                                expr_offset(a2->get_arg(j), off2));
+                    todo.push_back(new_e);
+                }
+            }
+        }
+    }
+    return true;
+}
+
+template<substitution_tree::st_visit_mode Mode>
+bool substitution_tree::visit_vars(expr * e, st_visitor & st) {
+    if (m_vars.empty())
+        return true; // continue
+    sort * s      = m_manager.get_sort(e);
+    unsigned s_id = s->get_decl_id();
+    if (s_id < m_vars.size()) {
+        var_ref_vector * v = m_vars[s_id];
+        if (v && !v->empty()) {
+            unsigned sz = v->size();
+            for (unsigned i = 0; i < sz; i++) {
+                var * curr = v->get(i);
+                m_subst->push_scope();
+                if (unify_match<Mode>(expr_offset(curr, m_st_offset), expr_offset(e, m_in_offset))) {
+                    if (Mode != STV_UNIF || m_subst->acyclic()) {
+                        if (!st(curr)) {
+                            m_subst->pop_scope();
+                            return false; // stop
+                        }
+                    }
+                }
+                m_subst->pop_scope();
+            }
+        }
+    }
+    return true; // continue
+}
+
+template<substitution_tree::st_visit_mode Mode>
+bool substitution_tree::visit(svector<subst> const & sv) {
+    svector<subst>::const_iterator it  = sv.begin();
+    svector<subst>::const_iterator end = sv.end();
+    for (; it != end; ++it) {
+        subst const & s = *it;
+        TRACE("st", tout << "processing subst:\n"; display(tout, s); tout << "\n";);
+        var *  rin  = s.first;
+        expr * out  = s.second; 
+        expr_offset p1(rin, m_reg_offset);
+        expr_offset p2(out, is_var(out) ? m_st_offset : m_reg_offset);
+        if (!unify_match<Mode>(p1, p2)) 
+            return false;
+    }
+    return true;
+}
+
+template<substitution_tree::st_visit_mode Mode>
+bool substitution_tree::visit(expr * e, st_visitor & st, node * r) {
+    m_bstack.reset();
+    m_bstack.push_back(r);
+    m_subst->push_scope();
+    m_subst->insert(static_cast<unsigned>(0), m_reg_offset, expr_offset(e, m_in_offset));
+
+    while (true) {
+        node * n = m_bstack.back();
+        TRACE("st", tout << "push scope...\n";);
+        m_subst->push_scope();
+        TRACE("st", tout << "processing node:\n"; display(tout, n->m_subst); tout << "\n";);
+        if (visit<Mode>(n->m_subst)) {
+            if (n->m_leaf) {
+                // if searching for unifiers and the substitution is cyclic, then backtrack.
+                if (Mode == STV_UNIF && !m_subst->acyclic()) {
+                    if (!backtrack())
+                        break;
+                }
+                else {
+                    TRACE("st_bug", tout << "found match:\n"; m_subst->display(tout); tout << "m_subst: " << m_subst << "\n";);
+                    if (!st(n->m_expr))
+                        return false;
+                    if (!backtrack())
+                        break;
+                }
+            }
+            else {
+                m_bstack.push_back(n->m_first_child);
+            }
+        }
+        else if (!backtrack())
+            break;
+    }
+    while (!m_bstack.empty()) {
+        m_subst->pop_scope();
+        m_bstack.pop_back();
+    }
+    m_subst->pop_scope();
+    return true;
+}
+
+template<substitution_tree::st_visit_mode Mode>
+void substitution_tree::visit(expr * e, st_visitor & st, unsigned in_offset, unsigned st_offset, unsigned reg_offset) {
+    m_in_offset  = in_offset;
+    m_st_offset  = st_offset;
+    m_reg_offset = reg_offset;
+
+    m_subst = &(st.get_substitution());
+    m_subst->reserve_vars(get_approx_num_regs());
+
+    if (visit_vars<Mode>(e, st)) {
+        if (is_app(e)) {
+            func_decl * d  = to_app(e)->get_decl();
+            unsigned id    = d->get_decl_id();
+            node * r       = m_roots.get(id, 0);
+            if (r)  
+                visit<Mode>(e, st, r);
+        }
+        else {
+            SASSERT(is_var(e));
+            ptr_vector<node>::iterator it  = m_roots.begin();
+            ptr_vector<node>::iterator end = m_roots.end();
+            for (; it != end; ++it) {
+                node * r = *it;
+                if (r != 0) {
+                    var * v = r->m_subst[0].first;
+                    if (v->get_sort() == to_var(e)->get_sort())
+                        if (!visit<Mode>(e, st, r))
+                            break;
+                }
+            }
+        }
+    }
+}
+
+void substitution_tree::unify(expr * e, st_visitor & v, unsigned in_offset, unsigned st_offset, unsigned reg_offset) {
+    visit<STV_UNIF>(e, v, in_offset, st_offset, reg_offset);
+}
+
+void substitution_tree::inst(expr * e, st_visitor & v, unsigned in_offset, unsigned st_offset, unsigned reg_offset) {
+    visit<STV_INST>(e, v, in_offset, st_offset, reg_offset);
+}
+
+void substitution_tree::gen(expr * e, st_visitor & v, unsigned in_offset, unsigned st_offset, unsigned reg_offset) {
+    visit<STV_GEN>(e, v, in_offset, st_offset, reg_offset);
+}
+
+void substitution_tree::display(std::ostream & out) const {
+    out << "substitution tree:\n";
+    ptr_vector<node>::const_iterator it  = m_roots.begin();
+    ptr_vector<node>::const_iterator end = m_roots.end();
+    for (; it != end; ++it)
+        if (*it)
+            display(out, *it, 0);
+    bool found_var = false;
+    ptr_vector<var_ref_vector>::const_iterator it2  = m_vars.begin();
+    ptr_vector<var_ref_vector>::const_iterator end2 = m_vars.end();
+    for (; it2 != end2; ++it2) {
+        var_ref_vector * v = *it2;
+        if (v == 0)
+            continue; // m_vars may contain null pointers. See substitution_tree::insert.
+        unsigned num = v->size();
+        for (unsigned i = 0; i < num; i++) {
+            if (!found_var) {
+                found_var = true;
+                out << "vars: ";
+            }
+            out << mk_pp(v->get(i), m_manager) << " ";
+        }
+    }
+    if (found_var)
+        out << "\n";
+}
+
+
+substitution_tree::substitution_tree(ast_manager & m):
+    m_manager(m),
+    m_max_reg(0),
+    m_size(0) {
+}
+
+substitution_tree::~substitution_tree() {
+    reset();
+}

src/spc/substitution_tree.h

@@ -0,0 +1,148 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    substitution_tree.h
+
+Abstract:
+
+    Substitution Trees 
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-03.
+
+Revision History:
+
+--*/
+#ifndef _SUBSTITUTION_TREE_H_
+#define _SUBSTITUTION_TREE_H_
+
+#include"ast.h"
+#include"substitution.h"
+
+/**
+   \brief Substitution tree visitor.
+*/
+class st_visitor {
+protected:
+    substitution & m_subst;
+public:
+    st_visitor(substitution & s):m_subst(s) {}
+    virtual ~st_visitor() {}
+    substitution & get_substitution() { return m_subst; }
+    virtual bool operator()(expr * e) { return true; }
+};
+
+/**
+   \brief Substitution tree term index.
+*/
+class substitution_tree {
+    
+    typedef std::pair<var *, expr *> subst;
+
+    struct node {
+        bool   m_leaf;
+        svector<subst>   m_subst;
+        node *           m_next_sibling;
+        union {
+            node *           m_first_child;
+            expr *           m_expr;
+        };
+        node(bool leaf):m_leaf(leaf), m_next_sibling(0), m_first_child(0) {}
+    };
+
+    ast_manager &     m_manager;
+    ptr_vector<node>  m_roots;
+    unsigned          m_max_reg;
+    ptr_vector<expr>  m_registers;
+    unsigned          m_size;
+    ptr_vector<var_ref_vector> m_vars; // mapping from decl_id to var_ref_vector
+
+    // Compilation time fields
+    unsigned          m_next_reg;
+    bit_vector        m_used_regs;
+    unsigned_vector   m_todo;
+    svector<subst>    m_compatible;
+    svector<subst>    m_incompatible;
+    
+    // Execution time fields
+    substitution *    m_subst;
+    ptr_vector<node>  m_bstack;
+    unsigned          m_in_offset;
+    unsigned          m_st_offset;
+    unsigned          m_reg_offset;
+    typedef std::pair<expr_offset, expr_offset> entry;
+    svector<entry>    m_visit_todo;
+
+    unsigned next_reg();
+    void push(svector<subst> & sv, subst const & s);
+    expr * get_reg_value(unsigned ridx);
+    void set_reg_value(unsigned ridx, expr * e);
+    void erase_reg_from_todo(unsigned ridx);
+
+    void linearize(svector<subst> & result);
+    void process_args(app * in, app * out);
+    void reset_registers(unsigned old_size);
+    unsigned get_compatibility_measure(svector<subst> const & sv);
+    node * find_best_child(node * r);
+    void reset_compiler();
+    node * mk_node_for(expr * new_expr);
+    void mark_used_reg(unsigned ridx);
+    void mark_used_regs(svector<subst> const & sv);
+
+    bool is_fully_compatible(svector<subst> const & sv);
+    bool find_fully_compatible_child(node * r, node * & prev, node * & child);
+    static bool at_least_3_children(node * r);
+    void delete_node(node * n);
+
+    void display(std::ostream & out, subst const & s) const;
+    void display(std::ostream & out, svector<subst> const & sv) const;
+    void display(std::ostream & out, node * n, unsigned delta) const;
+
+    enum st_visit_mode {
+        STV_UNIF,
+        STV_INST,
+        STV_GEN
+    };
+
+    expr_offset find(expr_offset p);
+    bool backtrack();
+    
+    template<st_visit_mode Mode>
+    bool bind_var(var * v, unsigned offset, expr_offset const & p);
+    template<st_visit_mode Mode>
+    bool unify_match(expr_offset p1, expr_offset p2);
+    template<substitution_tree::st_visit_mode Mode>
+    bool visit_vars(expr * e, st_visitor & st);
+    template<st_visit_mode Mode>
+    bool visit(svector<subst> const & s);
+    template<st_visit_mode Mode>
+    bool visit(expr * e, st_visitor & st, node * r);
+    template<st_visit_mode Mode>
+    void visit(expr * e, st_visitor & st, unsigned in_offset, unsigned st_offset, unsigned reg_offset);
+
+public:
+    substitution_tree(ast_manager & m);
+    ~substitution_tree();
+
+    void insert(app * n);
+    void insert(expr * n);
+    void erase(app * n);
+    void erase(expr * n);
+    void reset();
+    
+    bool empty() const { return m_size == 0; }
+
+    void unify(expr * e, st_visitor & v, unsigned in_offset = 0, unsigned st_offset = 1, unsigned reg_offset = 2);
+    void inst(expr * e, st_visitor & v, unsigned in_offset = 0, unsigned st_offset = 1, unsigned reg_offset = 2);
+    void gen(expr * e, st_visitor & v, unsigned in_offset = 0, unsigned st_offset = 1, unsigned reg_offset = 2);
+
+    unsigned get_approx_num_regs() const { return m_max_reg + 1; }
+
+    void display(std::ostream & out) const;
+};
+
+#endif /* _SUBSTITUTION_TREE_H_ */
+

src/spc/unifier.cpp

@@ -0,0 +1,183 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    unifier.cpp
+
+Abstract:
+
+    <abstract>
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-28.
+
+Revision History:
+
+--*/
+#include"unifier.h"
+#include"ast_pp.h"
+
+void unifier::reset(unsigned num_offsets) {
+    m_todo.reset();
+    m_find.reset();
+    m_size.reset();
+}
+
+/**
+   \brief Find with path compression.
+*/
+expr_offset unifier::find(expr_offset p) {
+    buffer<expr_offset> path;
+    expr_offset next;
+    while (m_find.find(p, next)) {
+        path.push_back(p);
+        p = next;
+    }
+    buffer<expr_offset>::iterator it  = path.begin();
+    buffer<expr_offset>::iterator end = path.end();
+    for (; it != end; ++it) {
+        expr_offset & prev = *it;
+        m_find.insert(prev, p);
+    }
+    return p;
+}
+
+void unifier::save_var(expr_offset const & p, expr_offset const & t) {
+    expr * n = p.get_expr();
+    if (is_var(n)) {
+        unsigned off = p.get_offset();
+        m_subst->insert(to_var(n)->get_idx(), off, t);
+    }
+}
+
+
+/**
+   \brief Merge the equivalence classes of n1 and n2. n2 will be the
+   root of the resultant equivalence class.
+*/
+void unifier::union1(expr_offset const & n1, expr_offset const & n2) {
+    DEBUG_CODE({
+        expr_offset f;
+        SASSERT(!m_find.find(n1, f));
+        SASSERT(!m_find.find(n2, f));
+    });
+    unsigned sz1 = 1;
+    unsigned sz2 = 1;
+    m_size.find(n1, sz1);
+    m_size.find(n2, sz2);
+    m_find.insert(n1, n2);
+    m_size.insert(n2, sz1 + sz2);
+    save_var(n1, n2);
+}
+
+/**
+   \brief Merge the equivalence classes of n1 and n2. The root of the
+   resultant equivalence class is the one with more elements.
+*/
+void unifier::union2(expr_offset n1, expr_offset n2) {
+    DEBUG_CODE({
+        expr_offset f;
+        SASSERT(!m_find.find(n1, f));
+        SASSERT(!m_find.find(n2, f));
+    });
+    unsigned sz1 = 1;
+    unsigned sz2 = 1;
+    m_size.find(n1, sz1);
+    m_size.find(n2, sz2);
+    if (sz1 > sz2) 
+        std::swap(n1, n2);
+    m_find.insert(n1, n2);
+    m_size.insert(n2, sz1 + sz2);
+    save_var(n1, n2);
+}
+
+bool unifier::unify_core(expr_offset p1, expr_offset p2) {
+    entry e(p1, p2);
+    m_todo.push_back(e);
+    while (!m_todo.empty()) {
+        entry const & e = m_todo.back();
+        p1 = find(e.first);
+        p2 = find(e.second);
+        m_todo.pop_back();
+        if (p1 != p2) {
+            expr * n1 = p1.get_expr();
+            expr * n2 = p2.get_expr();
+            SASSERT(!is_quantifier(n1));
+            SASSERT(!is_quantifier(n2));
+            bool v1 = is_var(n1);
+            bool v2 = is_var(n2);
+            if (v1 && v2) {
+                union2(p1, p2);
+            }
+            else if (v1) {
+                union1(p1, p2);
+            }
+            else if (v2) {
+                union1(p2, p1);
+            }
+            else {
+                app * a1 = to_app(n1);
+                app * a2 = to_app(n2);
+
+                unsigned off1 = p1.get_offset();
+                unsigned off2 = p2.get_offset();
+                if (a1->get_decl() != a2->get_decl() || a1->get_num_args() != a2->get_num_args())
+                    return false;
+                union2(p1, p2);
+                unsigned j = a1->get_num_args();
+                while (j > 0) {
+                    --j;
+                    entry new_e(expr_offset(a1->get_arg(j), off1),
+                                expr_offset(a2->get_arg(j), off2));
+                    m_todo.push_back(new_e);
+                }
+            }
+        }
+    }
+    return true;
+}
+
+bool unifier::operator()(unsigned num_exprs, expr ** es, substitution & s, bool use_offsets) {
+    SASSERT(num_exprs > 0);
+    unsigned num_offsets = use_offsets ? num_exprs : 1;
+    reset(num_offsets);
+    m_subst = &s;    
+#if 1
+    TRACE("unifier", for (unsigned i = 0; i < num_exprs; ++i) tout << mk_pp(es[i], m_manager) << "\n";);
+    for (unsigned i = s.get_num_bindings(); i > 0; ) {
+        --i;
+        std::pair<unsigned,unsigned> bound;
+        expr_offset root, child;
+        s.get_binding(i, bound, root);
+        TRACE("unifier", tout << bound.first << " |-> " << mk_pp(root.get_expr(), m_manager) << "\n";);
+        if (is_var(root.get_expr())) {
+            var* v = m_manager.mk_var(bound.first,to_var(root.get_expr())->get_sort());
+            child = expr_offset(v, bound.second);
+            unsigned sz1 = 1;
+            unsigned sz2 = 1;
+            m_size.find(child, sz1);
+            m_size.find(root, sz2);
+            m_find.insert(child, root);
+            m_size.insert(root, sz1 + sz2);
+        }
+    }
+#endif
+    for (unsigned i = 0; i < num_exprs - 1; i++) {
+        if (!unify_core(expr_offset(es[i], use_offsets ? i : 0), 
+                        expr_offset(es[i+1], use_offsets ? i + 1 : 0))) {
+            m_last_call_succeeded = false;
+            return m_last_call_succeeded;
+        }
+    }
+    
+    m_last_call_succeeded = m_subst->acyclic();
+    return m_last_call_succeeded;
+}
+
+bool unifier::operator()(expr * e1, expr * e2, substitution & s, bool use_offsets) {
+    expr * es[2] = { e1, e2 };
+    return operator()(2, es, s, use_offsets);
+}
+    

src/spc/unifier.h

@@ -0,0 +1,70 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    unifier.h
+
+Abstract:
+
+    Quasi-linear unification.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-01-28.
+
+Revision History:
+
+--*/
+#ifndef _UNIFIER_H_
+#define _UNIFIER_H_
+
+#include"ast.h"
+#include"substitution.h"
+
+/**
+   \brief Functor for unifying expressions.
+   It implements a quasi-linear unification algorithm.
+
+   It has support for two different variable banks: left and right.
+   That is, variable i in left bank is considered different from
+   variable i in the right bank.  This feature allows us to avoid
+   unnecessary variable renaming.
+*/
+class unifier {
+    typedef std::pair<expr_offset, expr_offset> entry;
+    
+    ast_manager &                m_manager;
+    substitution *               m_subst;
+    
+    svector<entry>               m_todo;
+    expr_offset_map<expr_offset> m_find;
+    expr_offset_map<unsigned>    m_size;
+
+    bool                         m_last_call_succeeded;
+
+    expr_offset find(expr_offset n);
+    void save_var(expr_offset const & p, expr_offset const & t);
+    void union1(expr_offset const & n1, expr_offset const & n2);
+    void union2(expr_offset n1, expr_offset n2);
+    void reset(unsigned num_offsets);
+
+    bool unify_core(expr_offset p1, expr_offset p2);
+    
+public:
+    unifier(ast_manager & m):m_manager(m), m_last_call_succeeded(false) {}
+
+    /**
+       \brief Unify the given expressions. Return true if succeeded,
+       and store the result in the given substitution.
+       
+       If use_offsets is true, then the variables in the given expressions are assumed to be 
+       in different banks.
+    */
+    bool operator()(unsigned num_exprs, expr ** es, substitution & s, bool use_offsets = true);
+
+    bool operator()(expr * e1, expr * e2, substitution & s, bool use_offsets = true);
+};
+
+#endif /* _UNIFIER_H_ */
+

src/spc/use_list.cpp

@@ -0,0 +1,90 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    use_list.cpp
+
+Abstract:
+
+    Use list term index.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-04.
+
+Revision History:
+
+--*/
+#include"use_list.h"
+
+void app_use_list::inc_ref(app * n) {
+    if (n->get_num_args() == 0)
+        return; // ignore constants
+    unsigned id = n->get_id();
+    unsigned c  = m_ref_counter.get(id, 0);
+    m_ref_counter.setx(id, c+1, 0);
+    if (c == 0)
+        m_todo.push_back(n);
+}
+
+void app_use_list::dec_ref(app * n) {
+    if (n->get_num_args() == 0)
+        return; // ignore constants
+    unsigned id = n->get_id();
+    SASSERT(m_ref_counter[id] > 0);
+    m_ref_counter[id]--;
+    if (m_ref_counter[id] == 0) 
+        m_todo.push_back(n);
+}
+
+void app_use_list::insert(expr * n) {
+    if (is_var(n)) 
+        return; // nothing to index
+    SASSERT(m_todo.empty());
+    inc_ref(to_app(n));
+    while (!m_todo.empty()) {
+        app * n = m_todo.back();
+        unsigned num_args = n->get_num_args();
+        for (unsigned i = 0; i < num_args; i++) {
+            expr * c = n->get_arg(i);
+            if (is_var(c)) {
+                if (!m_ignore_vars) 
+                    use_list<app*>::insert(n, c);
+            }
+            else {
+                SASSERT(is_app(c));
+                use_list<app*>::insert(n, c);
+                inc_ref(to_app(c));
+            }
+        }
+    }
+}
+
+void app_use_list::erase(expr * n) {
+    if (is_var(n)) 
+        return; // nothing to index
+    SASSERT(m_todo.empty());
+    dec_ref(to_app(n));
+    while (!m_todo.empty()) {
+        app * n = m_todo.back();
+        unsigned num_args = n->get_num_args();
+        for (unsigned i = 0; i < num_args; i++) {
+            expr * c = n->get_arg(i);
+            if (is_var(c)) {
+                if (!m_ignore_vars) 
+                    use_list<app*>::erase(n, c);
+            }
+            else {
+                SASSERT(is_app(c));
+                use_list<app*>::erase(n, c);
+                dec_ref(to_app(c));
+            }
+        }
+    }
+}
+    
+void app_use_list::reset() {
+    use_list<app*>::reset();
+    m_ref_counter.reset();
+}

src/spc/use_list.h

@@ -0,0 +1,120 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    use_list.h
+
+Abstract:
+
+    Use list expression index.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-04.
+
+Revision History:
+
+--*/
+#ifndef _USE_LIST_H_
+#define _USE_LIST_H_
+
+#include"ast.h"
+#include"vector.h"
+
+/**
+   \brief Generic use-list data-structure.
+*/
+template<typename T>
+class use_list {
+    typedef vector<T> set;
+    vector<set> m_use_list;
+public:
+    typedef typename set::const_iterator iterator;
+    use_list() {}
+    
+    void insert(T const & parent, expr * child) {
+        unsigned id  = child->get_id();
+        if (id >= m_use_list.size())
+            m_use_list.resize(id+1, set());
+        set & s = m_use_list[id];
+        s.push_back(parent);
+    }
+    
+    void erase(T const & parent, expr * child) {
+        unsigned id  = child->get_id();
+        if (id >= m_use_list.size())
+            return;
+        set & s = m_use_list[id];
+        s.erase(parent);
+    }
+
+    void reset() {
+        m_use_list.reset();
+    }
+    
+    iterator begin(expr * e) const {
+        unsigned id = e->get_id();
+        if (id >= m_use_list.size())
+            return 0;
+        return m_use_list[id].begin();
+    }
+    
+    iterator end(expr * e) const {
+        unsigned id = e->get_id();
+        if (id >= m_use_list.size())
+            return 0;
+        return m_use_list[id].end();
+    }
+
+    bool empty(expr * e) const {
+        unsigned id = e->get_id();
+        if (id >= m_use_list.size())
+            return true;
+        return m_use_list[id].empty();
+    }
+};
+
+/**
+   \brief Index for tracking the uses of an expression. It is a
+   mapping from expressions to expressions. For example, consider the
+   term (f a (g a)), the constant a is used by f and g applications.
+   
+   \remark The expressions inserted in this index should not contain
+   quantifiers.
+
+   \warning This index will not increase the reference counter of the
+   indexed expressions.
+*/
+class app_use_list : use_list<app*> {
+    
+    bool m_ignore_vars; //!< when true, variables are not indexed
+    unsigned_vector  m_ref_counter;
+    ptr_vector<app> m_todo;
+
+    void inc_ref(app * n);
+    void dec_ref(app * n);
+
+public:
+    /**
+       \brief If ignore_vars = true, then the index will not track
+       the use of variables.
+    */
+    app_use_list(bool ignore_vars = true):
+        m_ignore_vars(ignore_vars) {
+    }
+
+    /**
+       \brief Update the use list of all direct/indirect children of n.
+    */
+    void insert(expr * n);
+
+    /**
+       \brief Remove n (and its unreachable direct/indirect children) from the index.
+    */
+    void erase(expr * n);
+    
+    void reset();
+};
+
+#endif /* _USE_LIST_H_ */

src/spc/var_offset_map.h

@@ -0,0 +1,114 @@
+/*++
+Copyright (c) 2006 Microsoft Corporation
+
+Module Name:
+
+    var_offset_map.h
+
+Abstract:
+
+    A generic mapping from (var, offset) to a value T.
+
+Author:
+
+    Leonardo de Moura (leonardo) 2008-02-01.
+
+Revision History:
+
+--*/
+#ifndef _VAR_OFFSET_MAP_H_
+#define _VAR_OFFSET_MAP_H_
+
+#include"ast.h"
+#include"vector.h"
+
+/**
+   \brief A mapping from variable-id + offset to some value of type T.
+*/
+template<typename T>
+class var_offset_map {
+protected:
+    struct data {
+        T        m_data;
+        unsigned m_timestamp;
+        data():m_timestamp(0) {}
+    };
+    
+    svector<data>           m_map;
+    unsigned                m_num_offsets;
+    unsigned                m_num_vars;
+    unsigned                m_timestamp;
+
+public:
+    var_offset_map():
+        m_num_offsets(0),
+        m_num_vars(0),
+        m_timestamp(1) {
+    }
+
+    void reset() {
+        m_timestamp++;
+        if (m_timestamp == UINT_MAX) {
+            typename svector<data>::iterator it  = m_map.begin();
+            typename svector<data>::iterator end = m_map.end();
+            for (; it != end; ++it)
+                it->m_timestamp = 0;
+            m_timestamp = 1;
+        }
+    }
+
+    unsigned offsets_capacity() const { return m_num_offsets; }
+
+    unsigned vars_capacity() const { return m_num_vars; }
+
+    void reserve(unsigned num_offsets, unsigned num_vars) {
+        if (num_offsets > m_num_offsets || num_vars > m_num_vars) {
+            unsigned sz = num_offsets * num_vars;
+            m_map.resize(sz);
+            m_num_vars    = num_vars;
+            m_num_offsets = num_offsets;
+        }
+        reset();
+    }
+
+    void reserve_offsets(unsigned num_offsets) { reserve(num_offsets, m_num_vars); }
+
+    void reserve_vars(unsigned num_vars) { reserve(m_num_offsets, num_vars); }
+
+    void insert(unsigned v_idx, unsigned offset, T const & t) {
+        SASSERT(v_idx < m_num_vars);
+        SASSERT(offset < m_num_offsets);
+        unsigned idx  = v_idx + offset * m_num_vars;
+        SASSERT(idx < m_map.size());
+        data & d = m_map[idx];
+        d.m_data      = t;
+        d.m_timestamp = m_timestamp;
+    }
+
+    void insert(var * v, unsigned offset, T const & t) { insert(v->get_idx(), offset, t); }
+    
+    bool find(unsigned v_idx, unsigned offset, T & r) const {
+        SASSERT(v_idx < m_num_vars);
+        SASSERT(offset < m_num_offsets);
+        unsigned idx  = v_idx + offset * m_num_vars;
+        data const & d = m_map[idx];
+        SASSERT(d.m_timestamp <= m_timestamp);
+        if (d.m_timestamp == m_timestamp) {
+            r = d.m_data;
+            return true;
+        }
+        return false;
+    }
+    
+    bool find(var * v, unsigned offset, T & r) const { return find(v->get_idx(), offset, r); }
+
+    void erase(unsigned v_idx, unsigned offset) {
+        SASSERT(v_idx < m_num_vars);
+        SASSERT(offset < m_num_offsets);
+        unsigned idx  = v_idx + offset * m_num_vars;
+        m_map[idx].m_timestamp = 0;
+    }
+
+};
+
+#endif /* _VAR_OFFSET_MAP_H_ */