initial pattern abstraction and move matching to src

src/ast/euf/.#euf_justification.h

@@ -0,0 +1 @@
+nbjorner@LAPTOP-04AEAFKH.38072:1751392111

src/ast/euf/CMakeLists.txt

@@ -10,6 +10,7 @@ z3_add_component(euf
     euf_mam.cpp
     euf_plugin.cpp
     euf_specrel_plugin.cpp
+    ho_matcher.cpp
   COMPONENT_DEPENDENCIES
     ast
     util

src/ast/euf/ho_matcher.cpp

@@ -0,0 +1,807 @@
+/*++
+Copyright (c) 2025 Microsoft Corporation
+
+Module Name:
+
+    ho_matcher
+
+Abstract:
+
+    second and higher-order matcher
+    
+Author:
+
+    Nikolaj Bjorner (nbjorner) 2025-07-07
+
+   Ho-matcher performs a tree search over match_goals.
+   Let Q be the current set of match_goals
+   Let B be the current state of a backtracking stack.
+   Each element in Q is a work item. 
+   The workflow is as follows:
+   elements w of Q are removed and added to B, 
+   if processing of w results in a unitary match, it is removed from B and the match_goals are added to Q.
+
+   match_goals in Q can be processed independently on when they are generated.
+   If a subgoal of Q fails to match, the algorithm backtracks.
+   Backtracking can be tuned by keeping track of dependencies.
+
+   Elements in Q are simplified relative to a current substitution. 
+   The level where the current substitution affects simplification determines 
+   the persistence level of simplification.
+  
+   A v1 ignores dependencies and assumes always that side-effects are relative to the current backtracking scope.
+   A v2 should address and measure the overhead.
+
+   Elements in Q need to persist relative to changes made within a backtracking scope.
+   Every operation on Q should be replayed. 
+   Thus, elements removed from Q are re-inserted.
+
+   (pat, offset) (t, offset)
+   - a variable is considered bound if it's index is below offset.
+   - meta variables are adjusted relative to offset
+
+--*/
+
+#include "ast/euf/ho_matcher.h"
+
+
+
+namespace euf {
+
+
+    void ho_matcher::operator()(expr* pat, expr* t, unsigned num_vars) {
+        (*this)(pat, t, 0, num_vars);
+    }
+
+    void ho_matcher::operator()(expr* pat, expr* t, unsigned num_bound, unsigned num_vars) {               
+        m_trail.push_scope();
+        m_subst.resize(num_vars);
+        m_goals.reset();
+        m_goals.push(0, num_bound, pat, t); 
+        search();
+        m_trail.pop_scope(1);
+    }
+
+    void ho_matcher::search() {
+        IF_VERBOSE(1, display(verbose_stream()));
+
+        while (m.inc()) {
+            // Q, B -> Q', B'. Push work on the backtrack stack and new work items
+            // e, Bw -> Q', B'. Consume backtrack stack
+            if (!m_goals.empty())
+                push_backtrack();
+            else if (!m_backtrack.empty())
+                resume();
+            else
+                break;
+        }
+
+        IF_VERBOSE(1, display(verbose_stream() << "ho_matcher: done\n"));
+    }
+
+    void ho_matcher::backtrack() {
+        SASSERT(!m_backtrack.empty());
+        auto wi = m_backtrack.back();
+        if (wi->in_scope()) 
+            m_trail.pop_scope(1);        
+        m_backtrack.pop_back();
+    }
+
+    void ho_matcher::resume() {
+        while (!m_backtrack.empty()) {
+            auto& wi = *m_backtrack.back();
+            bool st = consume_work(wi);
+            IF_VERBOSE(3, display(verbose_stream() << "ho_matcher::consume_work: " << wi.pat << " =?= " << wi.t << " -> " << (st?"true":"false") << "\n"););
+            if (st) {
+                if (m_goals.empty()) 
+                    m_on_match(m_subst);                
+                break;
+            }
+            else 
+                backtrack();            
+        }
+    }
+
+    void ho_matcher::push_backtrack() {
+        SASSERT(!m_goals.empty());      
+        m_backtrack.push_back(m_goals.pop());
+        resume();
+    }
+
+    lbool ho_matcher::are_equal(unsigned o1, expr* p, unsigned o2, expr* t) const {
+        SASSERT(p->get_sort() == t->get_sort());
+        if (o1 == o2 && p == t)
+            return l_true;
+
+        if (is_ground(p) && is_ground(t)) 
+            return to_lbool(p == t);
+        
+        if (is_lambda(p) && is_lambda(t)) {
+            auto q1 = to_quantifier(p);
+            auto q2 = to_quantifier(t);
+            SASSERT(q1->get_num_decls() == q2->get_num_decls());
+            return are_equal(o1 + q1->get_num_decls(), q1->get_expr(), o2 + q2->get_num_decls(), q2->get_expr());
+        }
+
+        if (is_meta_var(p, o1)) {
+            auto p1 = m_subst.get(to_var(p)->get_idx() - o1);
+            if (p1) 
+                return are_equal(0, p1, o2, t);
+            return l_undef;
+        }
+
+        if (is_meta_var(t, o2)) {
+            auto t1 = m_subst.get(to_var(t)->get_idx() - o2);
+            if (t1) 
+                return are_equal(o1, p, 0, t1);
+            return l_undef;
+        }       
+
+
+        if (m_unitary.is_flex(o1, p)) {
+            expr_ref h = whnf(p, o1);
+            if (h != p)
+                return are_equal(o1, h, o2, t);
+            return l_undef;
+        }
+        if (m_unitary.is_flex(o2, t)) {
+            expr_ref h = whnf(t, o2);
+            if (h != t)
+                return are_equal(o1, p, o2, h);
+            return l_undef;
+        }
+        if (m_array.is_select(p))
+            return l_undef; // TODO: interleave check with whnf expansion
+        
+        if (is_app(p) && is_app(t)) {          
+            auto a1 = to_app(p);
+            auto a2 = to_app(t);
+            if (a1->get_decl() != a2->get_decl())
+                return l_false;
+            if (a1->get_num_args() != a2->get_num_args())
+                return l_false;
+            for (unsigned i = 0; i < a1->get_num_args(); ++i) {
+                auto r = are_equal(o1, a1->get_arg(i), o2, a2->get_arg(i));
+                if (r != l_true)
+                    return r;
+            }
+            return l_true;
+        }
+        if (is_bound_var(p, o1) || is_bound_var(t, o2))
+            return to_lbool(p == t);
+
+        return l_undef;
+    }
+
+    //
+    // reduces (M N)
+    // and recurisvely ((M N) K)
+    // such that M is not a lambda
+    //
+    expr_ref ho_matcher::whnf(expr* e, unsigned offset) const {
+        expr_ref r(e, m), t(m);
+//        verbose_stream() << "ho_matcher::whnf: " << r << "\n";
+        if (is_meta_var(r, offset)) {
+            auto v = to_var(e);
+            auto idx = v->get_idx();
+            if (idx >= offset && m_subst.get(idx - offset)) {
+                auto e = m_subst.get(idx - offset);
+                r = e;
+                if (offset > 0) {
+                    var_shifter sh(m);
+                    sh(e, offset, r);
+                }
+                return r;
+            }
+        }
+        while (m_array.is_select(r)) {
+            app* a = to_app(r);
+            auto arg0 = a->get_arg(0);
+            //  apply substitution:
+            if (is_meta_var(arg0, offset)) {
+                auto v = to_var(arg0);
+                auto idx = v->get_idx();
+                if (idx >= offset && m_subst.get(idx - offset)) {
+                    auto e = m_subst.get(idx - offset);
+                    if (offset > 0) {
+                        var_shifter sh(m);
+                        sh(e, offset, r);
+                        e = r;
+                    }
+                    expr_ref_vector args(m);
+                    args.push_back(e);
+                    for (unsigned i = 1; i < a->get_num_args(); ++i) 
+                        args.push_back(a->get_arg(i));
+                    
+                    r = m.mk_app(a->get_decl(), args.size(), args.data());
+  //                  verbose_stream() << "ho_matcher::whnf: " << r << "\n";
+                    continue;
+                }
+            }
+            if (m_array.is_select(arg0)) {
+                t = whnf(arg0, offset);
+                if (t != arg0) {
+                    ptr_vector<expr> args(a->get_num_args(), a->get_args());
+                    args[0] = t;
+                    r = m_array.mk_select(args.size(), args.data());
+   //                 verbose_stream() << "ho_matcher::whnf-rec: " << r << "\n";
+                    continue;
+                }
+            }
+            // outer beta reduction
+            auto st = m_rewriter.mk_app_core(a->get_decl(), a->get_num_args(), a->get_args(), t);
+            if (st != BR_FAILED)
+                r = t;
+            else
+                break;
+        }
+        return r;    
+    }
+
+    // We assume that m_rewriter should produce
+    // something amounting to weak-head normal form WHNF
+
+    void ho_matcher::reduce(match_goal& wi) {
+        while (true) {
+            expr_ref r = whnf(wi.pat, wi.pat_offset());
+            if (r == wi.pat)
+                break;
+            IF_VERBOSE(3, verbose_stream() << "ho_matcher::reduce: " << wi.pat << " -> " << r << "\n";);
+            wi.pat = r;
+        } 
+
+        while (true) {
+            expr_ref r = whnf(wi.t, wi.term_offset());
+            if (r == wi.t)
+                break;
+            IF_VERBOSE(3, verbose_stream() << "ho_matcher::reduce: " << wi.t << " -> " << r << "\n";);
+            wi.t = r;
+        } 
+    }
+
+    bool ho_matcher::consume_work(match_goal &wi) {
+//        IF_VERBOSE(1, display(verbose_stream() << "ho_matcher::consume_work: " << wi.pat << " =?= " << wi.t << "\n"););
+
+        if (wi.in_scope()) 
+            m_trail.pop_scope(1);
+        
+        wi.set_in_scope();
+        m_trail.push_scope();
+        
+        if (wi.is_done())
+            return false;
+
+        reduce(wi);
+
+        auto t = wi.t;
+        auto p = wi.pat;
+
+        switch (are_equal(wi.pat_offset(), p, wi.term_offset(), t)) {
+        case l_false:
+            wi.set_done();
+            return false;
+        case l_true:
+            wi.set_done();
+            return true;
+        case l_undef:
+            break;
+        }
+        
+
+        // v >= offset
+        // v - offset |-> t
+        if (is_meta_var(p, wi.pat_offset()) && is_closed(t, 0, wi.term_offset())) {
+            auto v = to_var(p);
+            auto idx = v->get_idx() - wi.pat_offset();
+            SASSERT(!m_subst.get(idx)); // reduce ensures meta variables are not in substitutions
+            add_binding(v, wi.pat_offset(), t);
+            wi.set_done();
+            return true;
+        }
+
+
+        // N = \ x. T => ((shift1 N) x) = T
+        if (is_lambda(t) && !is_lambda(p)) {
+            auto q = to_quantifier(t);
+            auto t_body = q->get_expr();
+            auto nd = q->get_num_decls();
+            var_shifter vs(m);
+            expr_ref r(m);
+            vs(p, nd, r);
+            expr_ref_vector args(m);
+            args.push_back(r);
+            for (unsigned i = 0; i < nd; ++i)
+                args.push_back(m.mk_var(nd - 1 - i, q->get_decl_sort(i)));
+            r = m_array.mk_select(args);
+            m_goals.push(wi.level, wi.term_offset() + nd, r, t_body);
+            wi.set_done();
+            return true;
+        }
+
+        // Flex head unitary
+        // H(pat) = t
+
+        if (m_array.is_select(p) && m_unitary.is_flex(wi.pat_offset(), p) && is_pattern(p, wi.pat_offset(), t)) {
+            auto lam = abstract_pattern(p, wi.pat_offset(), t);
+            while (m_array.is_select(p))
+                p = to_app(p)->get_arg(0);
+            SASSERT(is_meta_var(p, wi.pat_offset()));
+            add_binding(to_var(p), wi.pat_offset(), lam);
+            wi.set_done();
+            return true;
+        }       
+
+        // Flex head general case
+
+        if (m_array.is_select(p) && m_unitary.is_flex(wi.pat_offset(), p)) {
+            ptr_vector<app> pats;
+            auto p1 = p;
+            while (m_array.is_select(p1)) {
+                pats.push_back(to_app(p1));
+                p1 = to_app(p1)->get_arg(0);
+            }
+            auto v = to_var(p1);
+            if (wi.is_init())
+                wi.set_project();
+
+            if (wi.is_project()) {
+                // v -> \x\y . x_i
+                unsigned start = wi.index();
+                unsigned i = 0;
+                for (auto pa : pats) {
+                    for (auto pi : array_select_indices(pa)) {
+                        if (start <= i && pi->get_sort() == t->get_sort()) {                            
+                            auto eq = are_equal(wi.pat_offset(), pi, wi.term_offset(), t);
+                            if (eq == l_false) {
+                                ++i;
+                                continue;
+                            }
+                            auto e = mk_project(pats.size(), i, v->get_sort());
+                            add_binding(v, wi.pat_offset(), e);
+                            if (eq == l_undef)
+                                m_goals.push(wi.level + 1, wi.pat_offset(), pi, t);
+                            wi.set_index(i + 1);
+                            return true;
+                        }
+                        ++i;
+                    }
+                }
+            }
+
+            SASSERT(!is_lambda(t));            
+
+            if (!is_app(t))
+                return false;
+
+            auto ta = to_app(t);
+
+            if (wi.is_project())
+                wi.set_app();           
+
+            if (wi.is_app()) {
+                unsigned sz = ta->get_num_args();
+                if (sz > 0) {
+                    wi.inc_index();
+                    m_trail.push(undo_resize(m_subst));
+                }
+
+                // H (p1) (p2) = f(t1, .., tn)
+                // H -> \x1 \x2 f(H1(x1, x2), .., Hn(x1, x2))
+                // H1(p1, p2) = t1, .., Hn(p1, p2) = tn
+                ptr_vector<sort> domain, pat_domain;
+                ptr_vector<expr> pat_args;
+                expr_ref_vector args(m), pat_vars(m), bound_args(m);
+                vector<symbol> names;
+                pat_args.push_back(nullptr);
+                pat_vars.push_back(nullptr);
+                unsigned num_bound = 0;
+                expr_mark seen;
+                for (auto pat : pats) {
+                    for (auto pi : array_select_indices(pat)) {
+                        ++num_bound;
+                        domain.push_back(pi->get_sort());
+                        names.push_back(symbol(num_bound));
+                        if (seen.is_marked(pi))
+                            continue;
+                        pat_domain.push_back(pi->get_sort());
+                        pat_args.push_back(pi); 
+                        seen.mark(pi);
+                    }
+                }
+
+                for (unsigned i = pat_args.size(); i-- > 1; ) {
+                    auto pi = pat_args.get(i);
+                    pat_vars.push_back(m.mk_var(pat_args.size() - i - 1, pi->get_sort()));
+                }                        
+
+                for (auto ti : *ta) {
+                    sort* v_sort = m_array.mk_array_sort(pat_domain.size(), pat_domain.data(), ti->get_sort());
+                    auto v = m.mk_var(m_subst.size() + wi.pat_offset(), v_sort);
+                    auto w = m.mk_var(m_subst.size() + wi.pat_offset() + num_bound, v_sort); // shifted by number of bound
+                    m_subst.resize(m_subst.size() + 1);
+                    pat_args[0] = v;
+                    auto sel = m_array.mk_select(pat_args.size(), pat_args.data());
+                    m_goals.push(wi.level + 1, wi.term_offset(), sel, ti);
+                    pat_vars[0] = w;
+                    sel = m_array.mk_select(pat_vars.size(), pat_vars.data());
+                    bound_args.push_back(sel);
+                }
+
+                expr_ref lam(m);
+                lam = m.mk_app(ta->get_decl(), bound_args.size(), bound_args.data());
+
+                for (unsigned i = pats.size(); i-- > 0; ) {
+                    auto pa = pats[i];
+                    auto sz = pa->get_num_args() - 1;
+                    num_bound -= sz;
+                    lam = m.mk_lambda(sz, domain.data() + num_bound, names.data() + num_bound, lam);
+                }
+                add_binding(v, wi.pat_offset(), lam);
+                wi.set_done();
+                return true;
+            }
+            return false;
+        }
+
+        wi.set_done();
+
+        // first order match
+        if (is_app(t) && is_app(p)) {
+            auto ta = to_app(t);
+            auto tp = to_app(p);
+            if (ta->get_decl() != tp->get_decl())
+                return false;
+            if (ta->get_num_args() != tp->get_num_args())
+                return false;
+            for (unsigned i = 0; i < ta->get_num_args(); ++i)
+                m_goals.push(wi.level, wi.term_offset(), tp->get_arg(i), ta->get_arg(i));
+            return true;
+        }               
+                        
+        //
+        // lambda x . p == lambda x . t
+        // 
+        if (is_quantifier(p) && is_quantifier(t)) {
+            auto qp = to_quantifier(p);
+            auto qt = to_quantifier(t);
+            unsigned pd = qp->get_num_decls();
+            unsigned td = qt->get_num_decls();
+            if (qp->get_kind() != qt->get_kind())
+                return false;
+            if (pd != td)
+                return false;
+            for (unsigned i = 0; i < pd; ++i)
+                if (qp->get_decl_sort(i) != qt->get_decl_sort(i))
+                    return false;
+            m_goals.push(wi.level, wi.term_offset() + td, qp->get_expr(), qt->get_expr());
+            return true;
+        }
+
+        return false;		       
+    }
+
+    // M p1 p2 ... pk
+    // where M is a meta-variable and p1, .., pk are distinct bound variables
+    // and t does not contain a bound variable not mentioned in p1,..,pk
+    // or terms that don't occur in t.
+    bool ho_matcher::is_pattern(expr* p, unsigned offset, expr* t) {
+        uint_set vars;
+        while (m_array.is_select(p)) {
+            auto a = to_app(p);
+            for (unsigned i = 1; i < a->get_num_args(); ++i) {
+                auto arg = a->get_arg(i);
+                if (!is_bound_var(arg, offset))
+                    return false;
+                auto idx = to_var(arg)->get_idx();
+                if (vars.contains(idx))
+                    return false;
+                vars.insert(idx);
+            }
+            p = a->get_arg(0);
+        }
+        if (!is_meta_var(p, offset))
+            return false;
+        // check that every free variable in t is in vars.
+        vector<ptr_vector<expr>> btodo;
+        btodo.push_back(ptr_vector<expr>());
+        btodo[0].push_back(t);
+        for (unsigned i = 0; i < btodo.size(); ++i) {
+            expr_fast_mark1 visited;
+            for (unsigned j = 0; j < btodo[i].size(); ++j) {
+                auto t = btodo[i][j];
+                if (visited.is_marked(t))
+                    continue;
+                visited.mark(t);
+                
+                if (is_app(t)) {
+                    auto a = to_app(t);
+                    btodo[i].append(a->get_num_args(), a->get_args());
+                }
+                else if (is_var(t)) {
+                    auto idx = to_var(t)->get_idx();
+                    if (idx >= i && !vars.contains(idx - i))
+                        return false;
+                }
+                else {
+                    quantifier* q = to_quantifier(t);
+                    btodo.reserve(i + q->get_num_decls() + 1);
+                    btodo[i + q->get_num_decls()].push_back(q->get_expr());
+                }
+            }
+        }
+
+        return true;
+    }
+
+    // create a lambda abstraction for the meta variable such that
+    // when applied to patterns, the result is t.
+    // pre-condition: is_pattern(p, offset, t);
+    // H (v1, v2) (v3, v4, v5)
+    // lambda x1 x2 . lambda x3 x4 x5. t[v5 -> 0, v4 -> 1, v3 -> 2, v2 -> 3, v1 -> 4] 
+    expr_ref ho_matcher::abstract_pattern(expr* p, unsigned offset, expr* t) {
+        unsigned num_bound = 0;
+        expr* p1 = p;
+        ptr_vector<app> pats;
+        while (m_array.is_select(p1)) {
+            pats.push_back(to_app(p1));
+            num_bound += to_app(p1)->get_num_args() - 1;
+            p1 = to_app(p1)->get_arg(0);            
+        }
+        expr_ref_vector pat2bound(m);        
+        for (auto a : pats) {
+            unsigned sz = a->get_num_args();
+            for (unsigned i = 1; i < sz; ++i) {
+                auto arg = a->get_arg(i);
+                SASSERT(is_bound_var(arg, offset));
+                auto idx = to_var(arg)->get_idx();
+                pat2bound.reserve(idx + 1);
+                pat2bound[idx] = m.mk_var(--num_bound, arg->get_sort());
+            }   
+            p1 = a->get_arg(0);
+        }
+        var_subst sub(m, false);
+        expr_ref lam = sub(t, pat2bound);
+        for (unsigned i = pats.size(), j = 0; i-- > 0; ) {
+            vector<symbol> names;
+            ptr_vector<sort> sorts;
+            for (unsigned k = 1; k < pats[i]->get_num_args(); ++k) {
+                names.push_back(symbol(j++));
+                sorts.push_back(pats[i]->get_arg(k)->get_sort());
+            }
+            lam = m.mk_lambda(names.size(), sorts.data(), names.data(), lam);
+        }
+        return lam;
+    }
+
+    //
+    // keep track of number of internal scopes and offset to non-capture variables.
+    // a variable is captured if it's index is in the interval [scopes, offset[.
+    //
+    bool ho_matcher::is_closed(expr* v, unsigned scopes, unsigned offset) const {
+        if (is_ground(v))
+            return true;
+        if (is_app(v)) 
+            return all_of(*to_app(v), [&](expr* arg) { return is_closed(arg, scopes, offset); });
+        if (is_var(v)) 
+            return to_var(v)->get_idx() < scopes || to_var(v)->get_idx() >= offset;
+        if (is_quantifier(v)) {
+            auto q = to_quantifier(v);
+            return is_closed(q->get_expr(), scopes + q->get_num_decls(), offset + q->get_num_decls());
+        }
+        return false;
+    }
+
+    // (T1, T2,.., Tn) -> (Tn+1,.., Ti,.., Tm) -> Ti => lambda x1...xn . lambda x_n+1,..x_m . x_i
+    expr_ref ho_matcher::mk_project(unsigned num_lambdas, unsigned i, sort* s) {
+        SASSERT(num_lambdas > 0);
+        SASSERT(m_array.is_array(s));
+        unsigned num_binders = 0;
+        expr_ref body(m);
+        sort* a = s, * var_sort = nullptr;
+        unsigned j = i;
+        for (unsigned k = 0; k < num_lambdas; ++k) {
+            auto arity = get_array_arity(a);
+            if (j >= arity)
+                j -= arity;
+            else if (j < arity && !var_sort)
+                var_sort = get_array_domain(a, j);
+            num_binders += arity;
+            a = get_array_range(a);
+        }
+        SASSERT(var_sort);
+        body = m.mk_var(num_binders - i - 1, var_sort);
+        bind_lambdas(num_lambdas, s, body);
+        return body;
+    }
+
+    void ho_matcher::bind_lambdas(unsigned j, sort* s, expr_ref& body) {
+        if (j > 1) 
+            bind_lambdas(j - 1, get_array_range(s), body);        
+        unsigned sz = get_array_arity(s);
+        ptr_vector<sort> decl_sorts;
+        vector<symbol> decl_names;
+        for (unsigned i = 0; i < sz; ++i) {
+            decl_sorts.push_back(get_array_domain(s, i));
+            decl_names.push_back(symbol(i));
+        }
+        body = m.mk_lambda(sz, decl_sorts.data(), decl_names.data(), body);
+    }
+
+    void ho_matcher::add_binding(var* v, unsigned offset, expr* t) {
+        SASSERT(v->get_idx() >= offset);
+        m_subst.set(v->get_idx() - offset, t);
+        IF_VERBOSE(1, verbose_stream() << "ho_matcher::add_binding: v" << v->get_idx() - offset << " -> " << mk_pp(t, m) << "\n";);
+        m_trail.push(undo_set(m_subst, v->get_idx() - offset));
+    }
+
+
+    app* ho_matcher::compile_ho_pattern(quantifier* q, app* p) {
+        app* p1 = nullptr;
+        if (m_pat2hopat.find(p, p1))
+            return p1;
+        auto is_ho = any_of(subterms::all(expr_ref(p, m)), [&](expr* t) { return m_unitary.is_flex(0, t); });
+        if (!is_ho)
+            return p;
+        ptr_vector<expr> todo;
+        expr_ref_vector cache(m);
+        unsigned nb = q->get_num_decls();
+        todo.push_back(p);
+        while (!todo.empty()) {
+            auto t = todo.back();
+            if (is_var(t)) {
+                cache.setx(t->get_id(), t);
+                todo.pop_back();
+                continue;
+            }
+            if (m_unitary.is_flex(0, t)) {
+                m_pat2abs.insert_if_not_there(p, svector<std::pair<unsigned, expr*>>()).push_back({ nb, t });
+                cache.setx(t->get_id(), m.mk_var(nb++, t->get_sort()));
+                todo.pop_back();
+                continue;
+            }
+            if (is_app(t)) {
+                auto a = to_app(t);
+                unsigned sz = a->get_num_args();
+                ptr_buffer<expr> args;
+                for (auto arg : *a) {
+                    cache.reserve(arg->get_id() + 1);
+                    expr* arg1 = cache.get(arg->get_id());
+                    if (!arg1)
+                        todo.push_back(arg);
+                    else
+                        args.push_back(arg1);
+                }
+                if (sz != args.size())
+                    continue;
+                todo.pop_back();
+                cache.setx(t->get_id(), m.mk_app(a->get_decl(), args.size(), args.data()));
+            }
+            if (is_quantifier(t)) {
+                m_pat2abs.remove(p);
+                return p;
+            }
+        }
+
+        p1 = to_app(cache.get(p->get_id()));
+        m_pat2hopat.insert(p, p1);
+        m_hopat2pat.insert(p1, p);
+        m_ho_patterns.push_back(p1);
+        trail().push(push_back_vector(m_ho_patterns));
+        trail().push(insert_map(m_pat2hopat, p));
+        trail().push(insert_map(m_hopat2pat, p1));
+        trail().push(insert_map(m_pat2abs, p));
+        return p1;
+    }
+
+    bool ho_matcher::is_ho_pattern(app* p) {
+        return m_hopat2pat.contains(p);
+    }
+
+    void ho_matcher::refine_ho_match(app* p, expr_ref_vector const& s) {
+        auto fo_pat = m_hopat2pat[p];
+        m_trail.push_scope();
+        m_subst.resize(s.size());
+        m_goals.reset();
+        for (unsigned i = 0; i < s.size(); ++i) {
+            if (s[i])
+                m_subst.set(i, s[i]);
+        }
+
+        unsigned num_bound = 0, level = 0;
+        for (auto [v, pat] : m_pat2abs[fo_pat]) {
+            var_subst sub(m, false);
+            auto pat_refined = sub(pat, s);
+            IF_VERBOSE(1, verbose_stream() << mk_pp(pat, m) << " -> " << pat_refined << "\n");
+            m_goals.push(level, num_bound, pat_refined, s[v]);
+        }
+
+        search();
+        m_trail.pop_scope(1);
+    }
+
+    std::ostream& ho_matcher::display(std::ostream& out) const {
+        m_subst.display(out << "subst\n");
+        m_goals.display(out << "goals\n");
+        out << "stack\n";
+        for (auto const* mi : m_backtrack)
+            mi->display(out);       
+        return out;
+    }
+
+    struct retire_match_goal : public trail {
+        match_goal& mg;        
+        retire_match_goal(match_goal& mg) : mg(mg) {}
+        void undo() override {
+            mg.reset();
+        }
+    };
+
+    void match_goals::push(unsigned level, unsigned offset, expr_ref const& pat, expr_ref const& t) {
+        match_goal* wi = new (ho.trail().get_region()) match_goal(level, offset, pat, t);
+        ho.trail().push(retire_match_goal(*wi)); // reset on undo
+        wi->init(wi);
+        if (ho.is_cheap(*wi)) {
+            match_goal::push_to_front(m_cheap, wi);
+            ho.trail().push(remove_dll(m_cheap, wi));    // remove from cheap
+        }
+        else {
+            ho.reduce(*wi);
+            if (ho.is_cheap(*wi)) {
+                match_goal::push_to_front(m_cheap, wi);
+                ho.trail().push(remove_dll(m_cheap, wi));
+            }
+            else {
+                match_goal::push_to_front(m_expensive, wi);
+                ho.trail().push(remove_dll(m_expensive, wi)); // remove from expensive
+            }
+        }
+    }
+
+    match_goal* match_goals::pop() {
+        SASSERT(!empty());
+        if (m_cheap)
+            return pop(m_cheap);
+        auto* wi = m_expensive;
+        do {
+            expr_ref pat(wi->pat), t(wi->t);
+            ho.reduce(*wi);
+            if (pat == wi->pat && t == wi->t)
+                continue;
+            if (pat != wi->pat)
+                ho.trail().push(update_value(wi->pat, pat));
+            if (t != wi->t)
+                ho.trail().push(update_value(wi->t, t));
+            if (ho.is_cheap(*wi)) {
+                dll_base<match_goal>::remove_from(m_expensive, wi);
+                match_goal::push_to_front(m_cheap, wi);
+
+                ho.trail().push(insert_dll(m_expensive, wi)); // remove from expensive
+                ho.trail().push(remove_dll(m_cheap, wi));
+                return pop(m_cheap);
+            }
+            wi = wi->next();
+        } 
+        while (wi != m_expensive);
+                
+        return pop(m_expensive);
+    }    
+
+    match_goal* match_goals::pop(match_goal* &ws) {
+        SASSERT(ws);
+        auto* wi = ws->pop(ws);
+        ho.trail().push(insert_dll(ws, wi)); // insert wi into ws   
+        wi->set_init();
+        return wi;
+    }
+
+    std::ostream& match_goals::display(std::ostream& out) const {
+        for (auto const& wi : dll_elements(m_cheap))
+            wi.display(out << "c ");
+        
+        for (auto const& wi : dll_elements(m_expensive))
+            wi.display(out << "e ");
+        
+        return out;
+    }
+
+}

src/ast/euf/ho_matcher.h

@@ -0,0 +1,393 @@
+/*++
+Copyright (c) 2025 Microsoft Corporation
+
+Module Name:
+
+    ho_matcher
+
+Abstract:
+
+    second and higher-order matcher
+    
+Author:
+
+    Nikolaj Bjorner (nbjorner) 2025-07-07
+
+--*/
+
+#pragma once
+
+#include "util/trail.h"
+#include "util/dlist.h"
+#include "util/uint_set.h"
+#include "ast/array_decl_plugin.h"
+#include "ast/arith_decl_plugin.h"
+#include "ast/for_each_expr.h"
+#include "ast/reg_decl_plugins.h"
+#include "ast/ast_pp.h"
+#include "ast/rewriter/array_rewriter.h"
+#include "ast/rewriter/var_subst.h"
+
+
+namespace euf {
+
+    class ho_matcher;
+
+    class work_state {
+        enum state {
+            init_s,
+            project_s,
+            app_s, 
+            done_s
+        };
+        state m_state = state::init_s;
+        unsigned m_index = 0;
+        bool m_in_scope = false;
+
+    public:
+        void set_init() {
+            m_state = state::init_s;
+            m_index = 0;
+            m_in_scope = false;
+        }
+        bool is_init() const { return m_state == state::init_s; }
+        bool is_project() const { return m_state == state::project_s; }
+        bool is_app() const { return m_state == state::app_s && m_index == 0; }
+        bool is_done() const { return m_state == state::done_s; }
+        void set_project() { m_state = state::project_s; m_index = 0; }
+        void set_app() { m_state = state::app_s; m_index = 0; }
+        void set_done() { m_state = state::done_s; }
+        void inc_index() { ++m_index; }
+        void set_index(unsigned i) { m_index = i; }
+        unsigned index() const { return m_index; }
+        bool in_scope() const { return m_in_scope; }
+        void set_in_scope(bool f = true) { m_in_scope = f; }
+    };
+
+    class match_goal : public dll_base<match_goal>, public work_state {
+        unsigned base_offset = 0;
+        unsigned delta_offset = 0; // offset of term
+    public:
+        expr_ref pat, t;
+        unsigned level = 0; // level backtrack level
+       
+        void reset() {
+            pat.reset();
+            t.reset();
+            level = 0;
+            base_offset = 0;
+            delta_offset = 0;
+        }
+
+        bool operator==(match_goal const& other) const {
+            return pat == other.pat && t == other.t && base_offset == other.base_offset && delta_offset == other.delta_offset;
+        }
+
+        bool operator!=(match_goal const& other) const {
+            return !(*this == other);
+        }
+
+        match_goal(unsigned level, unsigned offset, expr_ref const& pat, expr_ref const& t) noexcept : 
+            base_offset(offset), pat(pat), t(t),  level(level)  {}
+
+        unsigned term_offset() const { return base_offset + delta_offset; }
+        unsigned pat_offset() const { return base_offset + delta_offset; }
+
+        std::ostream& display(std::ostream& out) const {
+            return out << "[" << level << ":" << base_offset + delta_offset << "] " << pat << " ~ " << t << "\n";
+        }
+    };
+
+    class match_goals {
+    protected:
+        ast_manager &m;
+        ho_matcher& ho;
+        
+        match_goal* m_expensive = nullptr, *m_cheap = nullptr;
+        match_goal* pop(match_goal*& q);
+        
+    public:
+        match_goals(ho_matcher& em, ast_manager &m) : m(m), ho(em) {}
+        bool empty() const { return m_cheap == nullptr && m_expensive == nullptr; }
+        void reset() { m_cheap = m_expensive = nullptr; }
+        void push(unsigned level, unsigned offset, expr_ref const& pat, expr_ref const& t);
+        void push(unsigned level, unsigned offset, expr* p, expr* t) { push(level, offset, expr_ref(p, m), expr_ref(t, m)); }
+        match_goal* pop();
+
+        std::ostream& display(std::ostream& out) const;
+
+    };
+
+    class ho_subst {
+        expr_ref_vector m_subst;
+    public:
+        ho_subst(ast_manager& m) :
+            m_subst(m) {            
+        }
+        void resize(unsigned n) {
+            m_subst.resize(n, nullptr);
+        }
+
+        unsigned size() const {
+            return m_subst.size();
+        }
+
+        void set(unsigned v, expr* t) {
+            SASSERT(!m_subst.get(v));
+            m_subst[v] = t;
+        }
+
+        expr* get(unsigned v) const {
+            return m_subst.get(v);
+        }
+
+        void unset(unsigned v) {
+            SASSERT(m_subst.get(v));
+            m_subst[v] = nullptr;
+        }
+
+        std::ostream& display(std::ostream& out) const {
+            auto& m = m_subst.get_manager();
+            for (unsigned i = 0; i < m_subst.size(); ++i) {
+                if (m_subst.get(i)) {
+                    out << "v" << i << " -> " << mk_pp(m_subst.get(i), m) << "\n";
+                }
+            }
+            return out;
+        }
+    };
+
+    class unitary_patterns {
+        ast_manager& m;
+        array_util a;
+        vector<expr_ref_vector> m_patterns;
+        vector<svector<lbool>>  m_is_unitary;
+        svector<std::pair<unsigned, expr*>> m_todo;
+
+        lbool find(unsigned offset, expr* p) const {
+            if (offset >= m_is_unitary.size())
+                return l_undef;
+            if (p->get_id() >= m_is_unitary[offset].size())
+                return l_undef;
+            return m_is_unitary[offset][p->get_id()];
+        }
+
+        void set_unitary(unsigned offset, expr* p, lbool is_unitary) {
+            if (offset >= m_is_unitary.size())
+                m_is_unitary.resize(offset + 1);
+            if (p->get_id() >= m_is_unitary[offset].size())
+                m_is_unitary[offset].resize(p->get_id() + 1, l_undef);
+            m_is_unitary[offset][p->get_id()] = is_unitary;
+        }
+
+    public:
+        unitary_patterns(ast_manager& m) : m(m), a(m) {}
+
+        bool is_unitary(unsigned offset, expr* p) const {
+            return find(offset, p) == l_true;
+        }
+
+        bool is_flex(unsigned offset, expr* e) const {
+            bool is_select = false;
+            expr* t = e;
+            while (a.is_select(t))
+                t = to_app(t)->get_arg(0), is_select = true;
+            if (is_select && is_var(t) && to_var(t)->get_idx() >= offset) {
+                // check if e is a pattern.
+                return true;
+            }
+            return false;
+        }
+
+        void add_pattern(expr* p) {
+            m_todo.push_back({ 0, p });
+            while (!m_todo.empty()) {
+                auto [o, p] = m_todo.back();
+                if (l_undef != find(o, p)) {
+                    m_todo.pop_back();
+                    continue;
+                }
+
+                // ((M N) K)
+                // if M is a meta variable and N, K are not patterns it is non_unitary
+                // otherwise we declare it as (locally) unitary. It will be inserted into the "cheap" queue.
+                // subterms can be non-unitary.
+
+                if (is_app(p)) {
+                    auto a = to_app(p);
+                    auto sz = m_todo.size();
+                    lbool is_unitary = l_true;
+                    for (auto arg : *a) {
+                        switch (find(o, arg)) {
+                        case l_undef:
+                            m_todo.push_back({ o, arg });
+                            break;                       
+                        default:
+                            break;
+                        }
+                    }
+                    if (sz != m_todo.size())
+                        continue;
+                    if (is_flex(o, p))
+                        is_unitary = l_false;
+                    set_unitary(o, p, is_unitary);
+                    m_todo.pop_back();
+                }
+                else if (is_var(p)) 
+                    set_unitary(o, p, l_true);              
+                else {
+                    auto q = to_quantifier(p);
+                    unsigned nd = q->get_num_decls();
+                    auto body = q->get_expr();
+                    switch (find(o + nd, body)) {
+                    case l_undef:
+                        m_todo.push_back({ o + nd, body });
+                        break;
+                    default:
+                        m_todo.pop_back();
+                        set_unitary(o, p, l_true);
+                        break;
+                    }
+                }
+            }
+        }
+
+    };
+
+    struct undo_set : public trail {
+        ho_subst& s;
+        unsigned v;
+        undo_set(ho_subst& s, unsigned v) : s(s), v(v) {}
+        void undo() override {
+            s.unset(v);
+        }
+    };
+
+    struct undo_resize : public trail {
+        ho_subst& s;
+        unsigned old_size;
+        undo_resize(ho_subst& s) : s(s), old_size(s.size()) {}
+        void undo() override {
+            s.resize(old_size);
+        }
+    };
+
+
+    template<typename V>
+    class update_value : public trail {
+        V& lval;
+        V  rval;
+    public:
+        update_value(V& lval, V rval) : lval(lval), rval(rval) {}
+        void undo() override {
+            lval = rval;
+            rval.reset();
+        }
+    };
+
+    struct remove_dll : public trail {
+        match_goal*& m_list;
+        match_goal* m_value;
+        remove_dll(match_goal*& list, match_goal* value) : m_list(list), m_value(value) {}
+        void undo() override {
+            dll_base<match_goal>::remove_from(m_list, m_value);
+        }
+    };
+
+    struct insert_dll : public trail {
+        match_goal*& m_list;
+        match_goal* m_value;
+        insert_dll(match_goal*& list, match_goal* value) : m_list(list), m_value(value) {}
+        void undo() override {
+            dll_base<match_goal>::push_to_front(m_list, m_value);
+        }
+    };
+
+    class ho_matcher {
+    protected:
+        friend class match_goals;
+        friend class test_ho_matcher;
+        ast_manager &m;
+        trail_stack& m_trail;
+        ho_subst         m_subst;
+        match_goals      m_goals;
+        unitary_patterns m_unitary;
+        ptr_vector<match_goal> m_backtrack;
+        mutable array_rewriter   m_rewriter;
+        array_util       m_array;
+        obj_map<app, app*>     m_pat2hopat, m_hopat2pat;
+        obj_map<app, svector<std::pair<unsigned, expr*>>> m_pat2abs;
+        expr_ref_vector        m_ho_patterns;
+
+    	void resume();
+
+        void push_backtrack();
+
+        void backtrack();
+
+        bool consume_work(match_goal& wi);
+
+        expr_ref whnf(expr* e, unsigned offset) const;
+        
+        bool is_bound_var(expr* v, unsigned offset) const { return is_var(v) && to_var(v)->get_idx() < offset; }
+
+        bool is_meta_var(expr* v, unsigned offset) const { return is_var(v) && to_var(v)->get_idx() >= offset; }
+
+        bool is_closed(expr* v, unsigned scopes, unsigned offset) const;
+
+        void add_binding(var* v, unsigned offset, expr* t);
+
+        expr_ref mk_project(unsigned num_lambdas, unsigned xi, sort* array_sort);
+
+        void bind_lambdas(unsigned j, sort* s, expr_ref& body);
+
+        lbool are_equal(unsigned o1, expr* p, unsigned o2, expr* t) const;
+
+        bool is_pattern(expr* p, unsigned offset, expr* t);
+
+        expr_ref abstract_pattern(expr* p, unsigned offset, expr* t);
+
+        void reduce(match_goal& wi);
+
+        trail_stack& trail() { return m_trail; }
+
+        std::ostream& display(std::ostream& out) const;
+
+        bool is_cheap(match_goal const& g) const { return m_unitary.is_unitary(g.pat_offset(), g.pat); }
+
+        void add_pattern(expr* p) {
+            m_unitary.add_pattern(p);
+        }
+
+        void search();
+
+        std::function<void(ho_subst&)> m_on_match;
+
+    public:
+
+        ho_matcher(ast_manager& m, trail_stack &trail) : 
+            m(m),
+            m_subst(m),
+            m_trail(trail),
+            m_goals(*this, m),
+            m_unitary(m),
+            m_rewriter(m),
+            m_array(m),
+            m_ho_patterns(m)
+        {
+        }
+
+        void set_on_match(std::function<void(ho_subst&)>& on_match) { m_on_match = on_match; }
+
+        void operator()(expr *pat, expr *t, unsigned num_vars);
+
+        void operator()(expr* pat, expr* t, unsigned num_bound, unsigned num_vars);
+
+        app* compile_ho_pattern(quantifier* q, app* p);
+
+        bool is_ho_pattern(app* p);
+
+        void refine_ho_match(app* p, expr_ref_vector const& s);
+
+    };
+}

src/ast/simplifiers/euf_completion.cpp

@@ -487,25 +487,35 @@ namespace euf {
     void completion::apply_binding(binding& b) {
         if (should_stop())
             return;
+#if 0
+        if (is_ho_binding(b)) 
+            apply_ho_binding(b);        
+        else 
+#endif
+        {
+            expr_ref_vector _binding(m);
+            quantifier* q = b.m_q;
+            for (unsigned i = 0; i < q->get_num_decls(); ++i)
+                _binding.push_back(b.m_nodes[i]->get_expr());
+            apply_binding(b, _binding);
+        }
+    }
+
+    void completion::apply_binding(binding& b, expr_ref_vector const& s) {
         var_subst subst(m);
-        expr_ref_vector _binding(m);
         quantifier* q = b.m_q;
-        for (unsigned i = 0; i < q->get_num_decls(); ++i)
+        expr_ref r = subst(q->get_expr(), s);
-            _binding.push_back(b.m_nodes[i]->get_expr());
-
-        expr_ref r = subst(q->get_expr(), _binding);
-
         scoped_generation sg(*this, b.m_max_top_generation + 1);
         auto [pr, d] = get_dependency(q);
         if (pr)
-            pr = m.mk_quant_inst(m.mk_or(m.mk_not(q), r), _binding.size(), _binding.data());
+            pr = m.mk_quant_inst(m.mk_or(m.mk_not(q), r), s.size(), s.data());
         add_constraint(r, pr, d);
         propagate_rules();
         m_egraph.propagate();
         m_should_propagate = true;
-        ++m_stats.m_num_instances;
     }
 
+
     void completion::read_egraph() {
         if (m_egraph.inconsistent()) {
             auto* d = explain_conflict();

src/ast/simplifiers/euf_completion.h

@@ -176,6 +176,7 @@ namespace euf {
         binding* alloc_binding(quantifier* q, app* pat, euf::enode* const* _binding, unsigned max_generation, unsigned min_top, unsigned max_top);
         void insert_binding(binding* b);
         void apply_binding(binding& b);
+        void apply_binding(binding& b, expr_ref_vector const& s);
         void flush_binding_queue();
         vector<ptr_vector<binding>> m_queue;