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diff --git a/proofsearch/include/atp/infer/substitution.h b/proofsearch/include/atp/infer/substitution.h
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+/*
+ * richard-atp-engine: A saturation-based automated theorem prover for first-order logic.
+ * Copyright (C) 2026 Richard Qin
+ *
+ * This program is free software: you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation, either version 3 of the License, or
+ * (at your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program.  If not, see <https://www.gnu.org/licenses/>.
+ */
+
+#pragma once
+
+/// @file substitution.h
+/// @brief Variable binding map: Var -> Term.
+///
+/// Short-lived object created during unification, applied to build resolvents,
+/// then discarded. Never stored long-term.
+
+#include "atp/core/term_bank.h"
+#include "atp/core/types.h"
+
+#include <unordered_map>
+
+namespace atp {
+
+/// A substitution σ maps variable TermIds to their bound TermIds.
+///
+/// Example: σ = { X → f(a), Y → Z }
+///   lookup(X) = f(a)     (direct binding)
+///   resolve(X) = f(a)    (same — f(a) is not a variable)
+///   resolve(Y) = Z       (if Z is unbound, stops at Z)
+///   resolve(Y) = f(b)    (if Z → f(b) also in σ, walks the chain)
+class Substitution {
+  public:
+    Substitution() = default;
+
+    /// Bind a variable to a term. Returns false if already bound to a different term.
+    bool bind(TermId var, TermId term);
+
+    /// Look up what a variable is directly bound to. Returns kInvalidId if unbound.
+    [[nodiscard]] TermId lookup(TermId var) const;
+
+    /// Walk the binding chain to the final value.
+    /// If var → Y → f(a), returns f(a).
+    /// Requires TermBank to check whether intermediate values are variables.
+    [[nodiscard]] TermId resolve(TermId id, const TermBank& bank) const;
+
+    /// Apply this substitution to a term recursively, creating new terms in the bank.
+    /// Free (unbound) variables are left as-is.
+    [[nodiscard]] TermId apply(TermId term, TermBank& bank) const;
+
+    /// Check if a variable occurs anywhere inside a term (for occurs check).
+    [[nodiscard]] bool occursIn(TermId var, TermId term, const TermBank& bank) const;
+
+    /// Clear all bindings.
+    void clear();
+
+    /// Number of bindings.
+    [[nodiscard]] size_t size() const { return bindings_.size(); }
+
+    /// Check if empty.
+    [[nodiscard]] bool empty() const { return bindings_.empty(); }
+
+  private:
+    std::unordered_map<TermId, TermId> bindings_;
+};
+
+}  // namespace atp