From 1885753690c4ebf47022d8c65159d52e518ed37a Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Mon, 2 Mar 2026 20:39:17 +0000
Subject: [PATCH 1/4] Initial plan


From cfea2db8bfe878d21d4dfbbb45331514624d4738 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Mon, 2 Mar 2026 21:14:34 +0000
Subject: [PATCH 2/4] Update sgraph and seq_plugin: egraph ref, hash matrix,
 subst cache, sgraph pointer

Co-authored-by: NikolajBjorner <3085284+NikolajBjorner@users.noreply.github.com>
---
 .gitignore                     |  1 +
 src/ast/euf/euf_seq_plugin.cpp | 20 +++++++-
 src/ast/euf/euf_seq_plugin.h   | 10 +++-
 src/ast/euf/euf_sgraph.cpp     | 88 ++++++++++++++++++++++++++++++----
 src/ast/euf/euf_sgraph.h       | 10 ++--
 src/ast/euf/euf_snode.h        | 12 +++++
 src/test/euf_seq_plugin.cpp    | 21 +++++---
 src/test/euf_sgraph.cpp        | 54 ++++++++++++++-------
 8 files changed, 174 insertions(+), 42 deletions(-)

diff --git a/.gitignore b/.gitignore
index ee5df58ff..bf7762077 100644
--- a/.gitignore
+++ b/.gitignore
@@ -117,3 +117,4 @@ genaisrc/genblogpost.genai.mts
 bazel-*
 # Local issue tracking
 .beads
+build/
diff --git a/src/ast/euf/euf_seq_plugin.cpp b/src/ast/euf/euf_seq_plugin.cpp
index da366eb1f..2a711a765 100644
--- a/src/ast/euf/euf_seq_plugin.cpp
+++ b/src/ast/euf/euf_seq_plugin.cpp
@@ -22,6 +22,7 @@ Author:
 
 #include "ast/euf/euf_seq_plugin.h"
 #include "ast/euf/euf_egraph.h"
+#include "ast/euf/euf_sgraph.h"
 #include "ast/ast_pp.h"
 
 namespace euf {
@@ -46,6 +47,12 @@ namespace euf {
     }
 
     unsigned enode_concat_hash::operator()(enode* n) const {
+        sgraph* sg = sg_ptr ? *sg_ptr : nullptr;
+        if (sg) {
+            snode* sn = sg->find(n->get_expr());
+            if (sn && sn->has_cached_hash())
+                return sn->assoc_hash();
+        }
         if (!is_any_concat(n, seq))
             return n->get_id();
         enode_vector leaves;
@@ -71,13 +78,22 @@ namespace euf {
         return true;
     }
 
-    seq_plugin::seq_plugin(egraph& g):
+    seq_plugin::seq_plugin(egraph& g, sgraph* sg):
         plugin(g),
         m_seq(g.get_manager()),
         m_rewriter(g.get_manager()),
-        m_concat_hash(m_seq),
+        m_sg(sg),
+        m_sg_owned(sg == nullptr),
+        m_concat_hash(m_seq, &m_sg),
         m_concat_eq(m_seq),
         m_concat_table(DEFAULT_HASHTABLE_INITIAL_CAPACITY, m_concat_hash, m_concat_eq) {
+        if (!m_sg)
+            m_sg = alloc(sgraph, g.get_manager(), g, false);
+    }
+
+    seq_plugin::~seq_plugin() {
+        if (m_sg_owned && m_sg)
+            dealloc(m_sg);
     }
 
     void seq_plugin::register_node(enode* n) {
diff --git a/src/ast/euf/euf_seq_plugin.h b/src/ast/euf/euf_seq_plugin.h
index 8fc435892..b6f2637ba 100644
--- a/src/ast/euf/euf_seq_plugin.h
+++ b/src/ast/euf/euf_seq_plugin.h
@@ -47,14 +47,17 @@ Author:
 namespace euf {
 
     class egraph;
+    class sgraph;
 
     // Associativity-respecting hash for enode concat trees.
     // Flattens concat(concat(a,b),c) and concat(a,concat(b,c))
     // to the same leaf sequence [a,b,c] before hashing.
     // Handles both str.++ (OP_SEQ_CONCAT) and re.++ (OP_RE_CONCAT).
+    // When an sgraph is available, uses cached snode hash matrices.
     struct enode_concat_hash {
         seq_util const& seq;
-        enode_concat_hash(seq_util const& s) : seq(s) {}
+        sgraph* const* sg_ptr;  // pointer to the sgraph* member in seq_plugin
+        enode_concat_hash(seq_util const& s, sgraph* const* sg = nullptr) : seq(s), sg_ptr(sg) {}
         unsigned operator()(enode* n) const;
     };
 
@@ -75,6 +78,8 @@ namespace euf {
 
         seq_util         m_seq;
         seq_rewriter     m_rewriter;
+        sgraph*          m_sg = nullptr;   // sgraph (may or may not be owned)
+        bool             m_sg_owned = false; // whether we own the sgraph
         svector<undo_kind> m_undo;
 
         // queue of merges and registrations to process
@@ -150,7 +155,8 @@ namespace euf {
         bool same_loop_body(enode* a, enode* b, unsigned& lo1, unsigned& hi1, unsigned& lo2, unsigned& hi2);
 
     public:
-        seq_plugin(egraph& g);
+        seq_plugin(egraph& g, sgraph* sg = nullptr);
+        ~seq_plugin() override;
 
         theory_id get_id() const override { return m_seq.get_family_id(); }
 
diff --git a/src/ast/euf/euf_sgraph.cpp b/src/ast/euf/euf_sgraph.cpp
index 3168c2263..a080ab3fa 100644
--- a/src/ast/euf/euf_sgraph.cpp
+++ b/src/ast/euf/euf_sgraph.cpp
@@ -23,15 +23,35 @@ Author:
 
 namespace euf {
 
-    sgraph::sgraph(ast_manager& m):
+    // substitution cache stored on snode for ZIPT-style optimization
+    struct snode_subst_cache {
+        struct entry {
+            unsigned var_id;
+            unsigned repl_id;
+            snode* result;
+        };
+        svector<entry> m_entries;
+        snode* find(unsigned var_id, unsigned repl_id) const {
+            for (auto const& e : m_entries)
+                if (e.var_id == var_id && e.repl_id == repl_id)
+                    return e.result;
+            return nullptr;
+        }
+        void insert(unsigned var_id, unsigned repl_id, snode* result) {
+            m_entries.push_back({var_id, repl_id, result});
+        }
+    };
+
+    sgraph::sgraph(ast_manager& m, egraph& eg, bool add_plugin):
         m(m),
         m_seq(m),
         m_rewriter(m),
-        m_egraph(m),
-        m_exprs(m),
-        m_str_sort(m_seq.str.mk_string_sort(), m) {
+        m_egraph(eg),
+        m_str_sort(m_seq.str.mk_string_sort(), m),
+        m_add_plugin(add_plugin) {
         // create seq_plugin and register it with the egraph
-        m_egraph.add_plugin(alloc(seq_plugin, m_egraph));
+        if (add_plugin)
+            m_egraph.add_plugin(alloc(seq_plugin, m_egraph, this));
         // register on_make callback so sgraph creates snodes for new enodes
         std::function<void(enode*)> on_make = [this](enode* n) {
             expr* e = n->get_expr();
@@ -42,6 +62,8 @@ namespace euf {
     }
 
     sgraph::~sgraph() {
+        for (auto* c : m_subst_caches)
+            dealloc(c);
     }
 
     snode_kind sgraph::classify(expr* e) const {
@@ -265,16 +287,49 @@ namespace euf {
         }
     }
 
+    static const unsigned HASH_BASE = 31;
+
+    void sgraph::compute_hash_matrix(snode* n) {
+        if (n->is_empty()) {
+            // identity matrix: concat with empty is identity
+            n->m_hash_matrix[0][0] = 1;
+            n->m_hash_matrix[0][1] = 0;
+            n->m_hash_matrix[1][0] = 0;
+            n->m_hash_matrix[1][1] = 1;
+        }
+        else if (n->is_concat()) {
+            snode* l = n->arg(0);
+            snode* r = n->arg(1);
+            if (l->has_cached_hash() && r->has_cached_hash()) {
+                // 2x2 matrix multiplication: M(L) * M(R)
+                n->m_hash_matrix[0][0] = l->m_hash_matrix[0][0] * r->m_hash_matrix[0][0] + l->m_hash_matrix[0][1] * r->m_hash_matrix[1][0];
+                n->m_hash_matrix[0][1] = l->m_hash_matrix[0][0] * r->m_hash_matrix[0][1] + l->m_hash_matrix[0][1] * r->m_hash_matrix[1][1];
+                n->m_hash_matrix[1][0] = l->m_hash_matrix[1][0] * r->m_hash_matrix[0][0] + l->m_hash_matrix[1][1] * r->m_hash_matrix[1][0];
+                n->m_hash_matrix[1][1] = l->m_hash_matrix[1][0] * r->m_hash_matrix[0][1] + l->m_hash_matrix[1][1] * r->m_hash_matrix[1][1];
+            }
+        }
+        else {
+            // leaf/token: [[HASH_BASE, value], [0, 1]]
+            unsigned v = n->get_expr() ? n->get_expr()->get_id() + 1 : n->id() + 1;
+            n->m_hash_matrix[0][0] = HASH_BASE;
+            n->m_hash_matrix[0][1] = v;
+            n->m_hash_matrix[1][0] = 0;
+            n->m_hash_matrix[1][1] = 1;
+        }
+    }
+
     snode* sgraph::mk_snode(expr* e, snode_kind k, unsigned num_args, snode* const* args) {
         unsigned id = m_nodes.size();
         snode* n = snode::mk(m_region, e, k, id, num_args, args);
         compute_metadata(n);
+        compute_hash_matrix(n);
         m_nodes.push_back(n);
-        m_exprs.push_back(e);
         if (e) {
             unsigned eid = e->get_id();
             m_expr2snode.reserve(eid + 1, nullptr);
             m_expr2snode[eid] = n;
+            // pin expression via egraph (the egraph has an expr trail)
+            mk_enode(e);
         }
         ++m_stats.m_num_nodes;
         return n;
@@ -350,7 +405,6 @@ namespace euf {
             }
         }
         m_nodes.shrink(old_sz);
-        m_exprs.shrink(old_sz);
         m_scopes.shrink(new_lvl);
         m_num_scopes = new_lvl;
         m_egraph.pop(num_scopes);
@@ -418,9 +472,23 @@ namespace euf {
             return replacement;
         if (n->is_empty() || n->is_char())
             return n;
-        if (n->is_concat())
-            return mk_concat(subst(n->arg(0), var, replacement),
-                             subst(n->arg(1), var, replacement));
+        if (n->is_concat()) {
+            // check substitution cache (ZIPT-style optimization)
+            if (n->m_subst_cache) {
+                snode* cached = n->m_subst_cache->find(var->id(), replacement->id());
+                if (cached)
+                    return cached;
+            }
+            snode* result = mk_concat(subst(n->arg(0), var, replacement),
+                                      subst(n->arg(1), var, replacement));
+            // cache the result
+            if (!n->m_subst_cache) {
+                n->m_subst_cache = alloc(snode_subst_cache);
+                m_subst_caches.push_back(n->m_subst_cache);
+            }
+            n->m_subst_cache->insert(var->id(), replacement->id(), result);
+            return result;
+        }
         // for non-concat compound nodes (power, star, etc.), no substitution into children
         return n;
     }
diff --git a/src/ast/euf/euf_sgraph.h b/src/ast/euf/euf_sgraph.h
index 9ef7b86d1..4d7231924 100644
--- a/src/ast/euf/euf_sgraph.h
+++ b/src/ast/euf/euf_sgraph.h
@@ -82,14 +82,17 @@ namespace euf {
         ast_manager&     m;
         seq_util         m_seq;
         seq_rewriter     m_rewriter;
-        egraph           m_egraph;
+        egraph&          m_egraph;
         region           m_region;
         snode_vector     m_nodes;
-        expr_ref_vector  m_exprs;       // pin expressions
         sort_ref         m_str_sort;    // cached string sort
         unsigned_vector  m_scopes;
         unsigned         m_num_scopes = 0;
         stats            m_stats;
+        bool             m_add_plugin; // whether sgraph created the seq_plugin
+
+        // tracks allocated subst caches for cleanup
+        ptr_vector<snode_subst_cache> m_subst_caches;
 
         // maps expression id to snode
         ptr_vector<snode> m_expr2snode;
@@ -97,10 +100,11 @@ namespace euf {
         snode* mk_snode(expr* e, snode_kind k, unsigned num_args, snode* const* args);
         snode_kind classify(expr* e) const;
         void compute_metadata(snode* n);
+        void compute_hash_matrix(snode* n);
         void collect_re_predicates(snode* re, expr_ref_vector& preds);
 
     public:
-        sgraph(ast_manager& m);
+        sgraph(ast_manager& m, egraph& eg, bool add_plugin = true);
         ~sgraph();
 
         ast_manager& get_manager() const { return m; }
diff --git a/src/ast/euf/euf_snode.h b/src/ast/euf/euf_snode.h
index 26de2d3ec..bf9ce293e 100644
--- a/src/ast/euf/euf_snode.h
+++ b/src/ast/euf/euf_snode.h
@@ -32,6 +32,7 @@ namespace euf {
 
     class sgraph;
     class snode;
+    struct snode_subst_cache;
 
     typedef ptr_vector<snode> snode_vector;
 
@@ -68,6 +69,13 @@ namespace euf {
         unsigned    m_level      = 0;      // tree depth/level (0 for empty, 1 for singletons)
         unsigned    m_length     = 0;      // token count, number of leaf tokens in the tree
 
+        // hash matrix for associativity-respecting hashing (2x2 polynomial hash matrix)
+        // all zeros means not cached, non-zero means cached
+        unsigned    m_hash_matrix[2][2] = {{0,0},{0,0}};
+
+        // substitution cache (lazy-initialized, owned by sgraph)
+        snode_subst_cache* m_subst_cache = nullptr;
+
         snode*      m_args[0]; // variable-length array, allocated via get_snode_size(num_args)
 
         friend class sgraph;
@@ -101,6 +109,10 @@ namespace euf {
         unsigned level()     const { return m_level; }
         unsigned length()    const { return m_length; }
 
+        // associativity-respecting hash: cached if the 2x2 matrix is non-zero
+        bool has_cached_hash() const { return m_hash_matrix[0][0] != 0; }
+        unsigned assoc_hash()  const { return m_hash_matrix[0][1]; }
+
         bool is_empty()   const { return m_kind == snode_kind::s_empty; }
         bool is_char()    const { return m_kind == snode_kind::s_char; }
         bool is_var()     const { return m_kind == snode_kind::s_var; }
diff --git a/src/test/euf_seq_plugin.cpp b/src/test/euf_seq_plugin.cpp
index e933d87c4..4beaa012e 100644
--- a/src/test/euf_seq_plugin.cpp
+++ b/src/test/euf_seq_plugin.cpp
@@ -31,7 +31,8 @@ static void test_sgraph_basic() {
     std::cout << "test_sgraph_basic\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -72,7 +73,8 @@ static void test_sgraph_backtrack() {
     std::cout << "test_sgraph_backtrack\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -99,7 +101,8 @@ static void test_seq_plugin_assoc() {
     std::cout << "test_seq_plugin_assoc\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     euf::egraph& g = sg.get_egraph();
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
@@ -125,7 +128,8 @@ static void test_seq_plugin_empty() {
     std::cout << "test_seq_plugin_empty\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     euf::egraph& g = sg.get_egraph();
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
@@ -150,7 +154,8 @@ static void test_seq_plugin_star_merge() {
     std::cout << "test_seq_plugin_star_merge\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     euf::egraph& g = sg.get_egraph();
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
@@ -178,7 +183,8 @@ static void test_seq_plugin_nullable_absorb() {
     std::cout << "test_seq_plugin_nullable_absorb\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     euf::egraph& g = sg.get_egraph();
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
@@ -202,7 +208,8 @@ static void test_sgraph_egraph_sync() {
     std::cout << "test_sgraph_egraph_sync\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     euf::egraph& g = sg.get_egraph();
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
diff --git a/src/test/euf_sgraph.cpp b/src/test/euf_sgraph.cpp
index 028573bec..569e8fd75 100644
--- a/src/test/euf_sgraph.cpp
+++ b/src/test/euf_sgraph.cpp
@@ -27,7 +27,8 @@ static void test_sgraph_classify() {
     std::cout << "test_sgraph_classify\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -85,7 +86,8 @@ static void test_sgraph_regex() {
     std::cout << "test_sgraph_regex\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -160,7 +162,8 @@ static void test_sgraph_power() {
     std::cout << "test_sgraph_power\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     arith_util arith(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
@@ -185,7 +188,8 @@ static void test_sgraph_push_pop() {
     std::cout << "test_sgraph_push_pop\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -221,7 +225,8 @@ static void test_sgraph_nested_scopes() {
     std::cout << "test_sgraph_nested_scopes\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -260,7 +265,8 @@ static void test_sgraph_find_idempotent() {
     std::cout << "test_sgraph_find_idempotent\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -276,7 +282,8 @@ static void test_sgraph_mk_concat() {
     std::cout << "test_sgraph_mk_concat\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -310,7 +317,8 @@ static void test_sgraph_mk_power() {
     std::cout << "test_sgraph_mk_power\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     arith_util arith(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
@@ -335,7 +343,8 @@ static void test_sgraph_first_last() {
     std::cout << "test_sgraph_first_last\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -372,7 +381,8 @@ static void test_sgraph_concat_metadata() {
     std::cout << "test_sgraph_concat_metadata\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -414,7 +424,8 @@ static void test_sgraph_display() {
     std::cout << "test_sgraph_display\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -436,7 +447,8 @@ static void test_sgraph_factory() {
     std::cout << "test_sgraph_factory\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
 
     // mk_var
     euf::snode* x = sg.mk_var(symbol("x"));
@@ -485,7 +497,8 @@ static void test_sgraph_indexing() {
     std::cout << "test_sgraph_indexing\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
 
     euf::snode* a = sg.mk_char('A');
     euf::snode* b = sg.mk_char('B');
@@ -532,7 +545,8 @@ static void test_sgraph_drop() {
     std::cout << "test_sgraph_drop\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
 
     euf::snode* a = sg.mk_char('A');
     euf::snode* b = sg.mk_char('B');
@@ -586,7 +600,8 @@ static void test_sgraph_subst() {
     std::cout << "test_sgraph_subst\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
 
     euf::snode* x = sg.mk_var(symbol("x"));
     euf::snode* y = sg.mk_var(symbol("y"));
@@ -620,7 +635,8 @@ static void test_sgraph_complex_concat() {
     std::cout << "test_sgraph_complex_concat\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
 
     // build a string "HELLO" = concat(H, concat(E, concat(L, concat(L, O))))
     euf::snode* h = sg.mk_char('H');
@@ -678,7 +694,8 @@ static void test_sgraph_brzozowski() {
     std::cout << "test_sgraph_brzozowski\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 
@@ -713,7 +730,8 @@ static void test_sgraph_minterms() {
     std::cout << "test_sgraph_minterms\n";
     ast_manager m;
     reg_decl_plugins(m);
-    euf::sgraph sg(m);
+    euf::egraph eg(m);
+    euf::sgraph sg(m, eg);
     seq_util seq(m);
     sort_ref str_sort(seq.str.mk_string_sort(), m);
 

From fbd3d445e28464cda5de89fd1eb125641ccc3ab8 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Mon, 2 Mar 2026 21:16:46 +0000
Subject: [PATCH 3/4] Add clarifying comments for hash matrix overflow safety
 and cache validity

Co-authored-by: NikolajBjorner <3085284+NikolajBjorner@users.noreply.github.com>
---
 src/ast/euf/euf_sgraph.cpp | 5 +++++
 src/ast/euf/euf_snode.h    | 4 +++-
 2 files changed, 8 insertions(+), 1 deletion(-)

diff --git a/src/ast/euf/euf_sgraph.cpp b/src/ast/euf/euf_sgraph.cpp
index a080ab3fa..eff926004 100644
--- a/src/ast/euf/euf_sgraph.cpp
+++ b/src/ast/euf/euf_sgraph.cpp
@@ -289,6 +289,10 @@ namespace euf {
 
     static const unsigned HASH_BASE = 31;
 
+    // Compute a 2x2 polynomial hash matrix for associativity-respecting hashing.
+    // Unsigned overflow is intentional and well-defined (mod 2^32).
+    // M[0][0] tracks HASH_BASE^(num_leaves), which is always nonzero since
+    // HASH_BASE is odd. M[0][1] is the actual hash value.
     void sgraph::compute_hash_matrix(snode* n) {
         if (n->is_empty()) {
             // identity matrix: concat with empty is identity
@@ -310,6 +314,7 @@ namespace euf {
         }
         else {
             // leaf/token: [[HASH_BASE, value], [0, 1]]
+            // +1 avoids zero hash values; wraps safely on unsigned overflow
             unsigned v = n->get_expr() ? n->get_expr()->get_id() + 1 : n->id() + 1;
             n->m_hash_matrix[0][0] = HASH_BASE;
             n->m_hash_matrix[0][1] = v;
diff --git a/src/ast/euf/euf_snode.h b/src/ast/euf/euf_snode.h
index bf9ce293e..955dd3788 100644
--- a/src/ast/euf/euf_snode.h
+++ b/src/ast/euf/euf_snode.h
@@ -109,7 +109,9 @@ namespace euf {
         unsigned level()     const { return m_level; }
         unsigned length()    const { return m_length; }
 
-        // associativity-respecting hash: cached if the 2x2 matrix is non-zero
+        // associativity-respecting hash: cached if the 2x2 matrix is non-zero.
+        // M[0][0] = HASH_BASE^(num_leaves) which is always nonzero since HASH_BASE
+        // is odd and gcd(odd, 2^32) = 1, so the check is safe.
         bool has_cached_hash() const { return m_hash_matrix[0][0] != 0; }
         unsigned assoc_hash()  const { return m_hash_matrix[0][1]; }
 

From 7c8f2d93bbc5eb870fef95a5b0fbb69d9e38e7e0 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Tue, 3 Mar 2026 00:07:53 +0000
Subject: [PATCH 4/4] Change seq_plugin m_sg to sgraph& reference,
 enode_concat_hash takes sgraph& directly

Co-authored-by: NikolajBjorner <3085284+NikolajBjorner@users.noreply.github.com>
---
 src/ast/euf/euf_seq_plugin.cpp | 19 +++++++------------
 src/ast/euf/euf_seq_plugin.h   | 10 ++++------
 2 files changed, 11 insertions(+), 18 deletions(-)

diff --git a/src/ast/euf/euf_seq_plugin.cpp b/src/ast/euf/euf_seq_plugin.cpp
index 2a711a765..618b1699a 100644
--- a/src/ast/euf/euf_seq_plugin.cpp
+++ b/src/ast/euf/euf_seq_plugin.cpp
@@ -47,12 +47,9 @@ namespace euf {
     }
 
     unsigned enode_concat_hash::operator()(enode* n) const {
-        sgraph* sg = sg_ptr ? *sg_ptr : nullptr;
-        if (sg) {
-            snode* sn = sg->find(n->get_expr());
-            if (sn && sn->has_cached_hash())
-                return sn->assoc_hash();
-        }
+        snode* sn = sg.find(n->get_expr());
+        if (sn && sn->has_cached_hash())
+            return sn->assoc_hash();
         if (!is_any_concat(n, seq))
             return n->get_id();
         enode_vector leaves;
@@ -82,18 +79,16 @@ namespace euf {
         plugin(g),
         m_seq(g.get_manager()),
         m_rewriter(g.get_manager()),
-        m_sg(sg),
+        m_sg(sg ? *sg : *alloc(sgraph, g.get_manager(), g, false)),
         m_sg_owned(sg == nullptr),
-        m_concat_hash(m_seq, &m_sg),
+        m_concat_hash(m_seq, m_sg),
         m_concat_eq(m_seq),
         m_concat_table(DEFAULT_HASHTABLE_INITIAL_CAPACITY, m_concat_hash, m_concat_eq) {
-        if (!m_sg)
-            m_sg = alloc(sgraph, g.get_manager(), g, false);
     }
 
     seq_plugin::~seq_plugin() {
-        if (m_sg_owned && m_sg)
-            dealloc(m_sg);
+        if (m_sg_owned)
+            dealloc(&m_sg);
     }
 
     void seq_plugin::register_node(enode* n) {
diff --git a/src/ast/euf/euf_seq_plugin.h b/src/ast/euf/euf_seq_plugin.h
index b6f2637ba..8f72a6837 100644
--- a/src/ast/euf/euf_seq_plugin.h
+++ b/src/ast/euf/euf_seq_plugin.h
@@ -50,14 +50,12 @@ namespace euf {
     class sgraph;
 
     // Associativity-respecting hash for enode concat trees.
-    // Flattens concat(concat(a,b),c) and concat(a,concat(b,c))
-    // to the same leaf sequence [a,b,c] before hashing.
+    // Uses cached snode hash matrices from the sgraph for O(1) hashing.
     // Handles both str.++ (OP_SEQ_CONCAT) and re.++ (OP_RE_CONCAT).
-    // When an sgraph is available, uses cached snode hash matrices.
     struct enode_concat_hash {
         seq_util const& seq;
-        sgraph* const* sg_ptr;  // pointer to the sgraph* member in seq_plugin
-        enode_concat_hash(seq_util const& s, sgraph* const* sg = nullptr) : seq(s), sg_ptr(sg) {}
+        sgraph& sg;
+        enode_concat_hash(seq_util const& s, sgraph& sg) : seq(s), sg(sg) {}
         unsigned operator()(enode* n) const;
     };
 
@@ -78,7 +76,7 @@ namespace euf {
 
         seq_util         m_seq;
         seq_rewriter     m_rewriter;
-        sgraph*          m_sg = nullptr;   // sgraph (may or may not be owned)
+        sgraph&          m_sg;
         bool             m_sg_owned = false; // whether we own the sgraph
         svector<undo_kind> m_undo;