2 files changed, 102 insertions, 0 deletions
diff --git a/include/atp/rewrite/ordering.h b/include/atp/rewrite/ordering.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 ordering.h
+/// @brief Term orderings for equality reasoning (KBO, LPO).
+///
+/// Term orderings are the foundation of equality-based theorem proving.
+/// They ensure that rewriting always terminates by enforcing a well-founded
+/// ordering on terms. Required by paramodulation, demodulation, and
+/// ordered resolution.
+
+#include "atp/core/term_bank.h"
+#include "atp/core/types.h"
+
+namespace atp {
+
+/// Ordering comparison result.
+enum class OrderResult : uint8_t {
+    kLessThan,       ///< t1 < t2
+    kEqual,          ///< t1 = t2
+    kGreaterThan,    ///< t1 > t2
+    kIncomparable,   ///< t1 and t2 are incomparable
+};
+
+/// Abstract base for term orderings.
+/// Concrete implementations: KBO (Knuth-Bendix), LPO (Lexicographic Path).
+class TermOrdering {
+  public:
+    virtual ~TermOrdering() = default;
+
+    /// Compare two terms under the ordering.
+    [[nodiscard]] virtual OrderResult compare(const TermBank& bank, TermId t1,
+                                              TermId t2) const = 0;
+};
+
+}  // namespace atp
diff --git a/include/atp/rewrite/paramodulation.h b/include/atp/rewrite/paramodulation.h
new file mode 100644
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+++ b/include/atp/rewrite/paramodulation.h
@@ -0,0 +1,48 @@
+/*
+ * 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 paramodulation.h
+/// @brief Paramodulation inference rule for equality reasoning.
+///
+/// Paramodulation replaces a subterm in one clause with an equal term
+/// from an equation in another clause. Combined with term orderings,
+/// this gives a complete calculus for first-order logic with equality
+/// (the Superposition calculus).
+
+#include "atp/core/clause.h"
+#include "atp/core/term_bank.h"
+#include "atp/rewrite/ordering.h"
+
+#include <vector>
+
+namespace atp {
+
+/// Generate all paramodulants between two clauses.
+/// Uses the term ordering to restrict to ordered paramodulation (superposition).
+std::vector<Clause> paramodulate(const TermBank& bank, const Clause& c1, const Clause& c2,
+                                 const TermOrdering& ordering);
+
+/// Simplify a clause by demodulation (rewriting with unit equalities).
+/// Returns true if the clause was modified.
+bool demodulate(TermBank& bank, Clause& clause, const Clause& unit_eq,
+                const TermOrdering& ordering);
+
+}  // namespace atp