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diff --git a/proofsearch/src/proof/proof_trace.cc b/proofsearch/src/proof/proof_trace.cc
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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/>.
+ */
+
+#include "atp/proof/proof_trace.h"
+
+#include "atp/core/clause.h"
+#include "atp/core/clause_store.h"
+#include "atp/core/symbol_table.h"
+#include "atp/core/term.h"
+#include "atp/core/term_bank.h"
+#include "atp/core/types.h"
+
+#include <algorithm>
+#include <cstddef>
+#include <stack>
+#include <string>
+#include <unordered_map>
+#include <unordered_set>
+#include <vector>
+
+namespace atp {
+
+// extractProof
+// DFS from empty clause back to axioms, collect all participating clauses,
+// then reverse for topological order (axioms first).
+
+std::vector<ProofStep> extractProof(const ClauseStore& store, ClauseId empty_clause_id) {
+    std::vector<ProofStep> steps;
+    std::unordered_set<ClauseId> visited;
+    std::stack<ClauseId> worklist;
+    worklist.push(empty_clause_id);
+
+    while (!worklist.empty()) {
+        const ClauseId cid = worklist.top();
+        worklist.pop();
+
+        if (cid == kInvalidId) {
+            continue;
+        }
+        if (visited.contains(cid)) {
+            continue;
+        }
+        visited.insert(cid);
+
+        const Clause& c = store.getClause(cid);
+        steps.push_back({
+            .clause_id_ = cid,
+            .rule_ = c.rule_,
+            .parent1_ = c.parent1_,
+            .parent2_ = c.parent2_,
+        });
+
+        // Recurse into parents (axioms have kInvalidId parents, which are
+        // filtered by the check at the top of the loop)
+        worklist.push(c.parent1_);
+        worklist.push(c.parent2_);
+    }
+
+    // Reverse: axioms (leaves) first → empty clause (root) last
+    std::ranges::reverse(steps);
+    return steps;
+}
+
+// termToString
+
+std::string termToString(TermId id, const TermBank& bank) {
+    const Term& t = bank.getTerm(id);
+    std::string name(bank.symbols().getName(t.symbol_id_));
+
+    if (t.args_.empty()) {
+        return name;
+    }
+
+    std::string result = name + "(";
+    for (size_t i = 0; i < t.args_.size(); ++i) {
+        if (i > 0) {
+            result += ",";
+        }
+        result += termToString(t.args_[i], bank);
+    }
+    result += ")";
+    return result;
+}
+
+// Pretty variable names
+
+namespace {
+
+/// Generate a human-readable variable name from an index.
+/// 0→X, 1→Y, 2→Z, 3→W, 4→V, 5→U, 6→X1, 7→Y1, ...
+std::string prettyVarName(size_t index) {
+    constexpr size_t kBaseCount = 6;
+    const size_t base_index = index % kBaseCount;
+    const char base = [&]() {
+        switch (base_index) {
+            case 0:
+                return 'X';
+            case 1:
+                return 'Y';
+            case 2:
+                return 'Z';
+            case 3:
+                return 'W';
+            case 4:
+                return 'V';
+            case 5:
+                return 'U';
+            default:
+                return 'X';
+        }
+    }();
+
+    const size_t suffix = index / kBaseCount;
+    if (suffix == 0) {
+        return std::string{base};
+    }
+    return std::string{base} + std::to_string(suffix);
+}
+
+/// Pretty-print a term, renaming variables to readable names.
+/// `var_map` is built incrementally: each new variable gets the next name.
+std::string termToPretty(TermId id, const TermBank& bank,
+                         std::unordered_map<TermId, std::string>& var_map) {
+    if (bank.isVariable(id)) {
+        auto it = var_map.find(id);
+        if (it != var_map.end()) {
+            return it->second;
+        }
+        const std::string name = prettyVarName(var_map.size());
+        var_map[id] = name;
+        return name;
+    }
+
+    const Term& t = bank.getTerm(id);
+    std::string name(bank.symbols().getName(t.symbol_id_));
+
+    if (t.args_.empty()) {
+        return name;
+    }
+
+    std::string result = name + "(";
+    for (size_t i = 0; i < t.args_.size(); ++i) {
+        if (i > 0) {
+            result += ",";
+        }
+        result += termToPretty(t.args_[i], bank, var_map);
+    }
+    result += ")";
+    return result;
+}
+
+// ruleToString
+
+std::string ruleToString(InferenceRule rule) {
+    switch (rule) {
+        case InferenceRule::kInput:
+            return "input";
+        case InferenceRule::kResolution:
+            return "resolution";
+        case InferenceRule::kFactoring:
+            return "factoring";
+        default:
+            return "unknown";
+    }
+}
+
+}  // namespace
+
+// clauseToString
+
+std::string clauseToString(const Clause& clause, const TermBank& bank) {
+    if (clause.isEmpty()) {
+        return "□";
+    }
+
+    // Per-clause variable renaming: _R109 → X, _R286 → Y, etc.
+    std::unordered_map<TermId, std::string> var_map;
+
+    std::string result;
+    for (size_t i = 0; i < clause.literals_.size(); ++i) {
+        if (i > 0) {
+            result += " ∨ ";
+        }
+        if (!clause.literals_[i].is_positive_) {
+            result += "¬";
+        }
+        result += termToPretty(clause.literals_[i].atom_, bank, var_map);
+    }
+    return result;
+}
+
+// formatProof
+
+std::string formatProof(const std::vector<ProofStep>& proof, const ClauseStore& store,
+                        const TermBank& bank) {
+    // Build a local renumbering: internal ClauseId → sequential proof step number
+    std::unordered_map<ClauseId, size_t> id_to_step;
+    for (size_t i = 0; i < proof.size(); ++i) {
+        id_to_step[proof[i].clause_id_] = i + 1;  // 1-based
+    }
+
+    auto local_name = [&](ClauseId cid) -> std::string {
+        auto it = id_to_step.find(cid);
+        if (it != id_to_step.end()) {
+            return std::to_string(it->second);
+        }
+        // Fallback: shouldn't happen in a valid proof
+        return "?" + std::to_string(cid);
+    };
+
+    std::string result;
+    for (size_t i = 0; i < proof.size(); ++i) {
+        const auto& step = proof[i];
+        const Clause& c = store.getClause(step.clause_id_);
+
+        result += std::to_string(i + 1) + ". ";
+        result += "[" + ruleToString(step.rule_) + "] ";
+        result += clauseToString(c, bank);
+
+        // Show parents for derived clauses
+        if (step.parent1_ != kInvalidId) {
+            result += "  [" + local_name(step.parent1_);
+            if (step.parent2_ != kInvalidId) {
+                result += ", " + local_name(step.parent2_);
+            }
+            result += "]";
+        }
+
+        result += "\n";
+    }
+    return result;
+}
+
+}  // namespace atp