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Solution draft log for https://github.com/konard/p-vs-np/pull/505
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| # Solve.mjs Log - 2026-01-25T00:58:52.575Z | |
| [2026-01-25T00:58:52.576Z] [INFO] 📁 Log file: /home/hive/solve-2026-01-25T00-58-52-575Z.log | |
| [2026-01-25T00:58:52.577Z] [INFO] (All output will be logged here) | |
| [2026-01-25T00:58:53.100Z] [INFO] | |
| [2026-01-25T00:58:53.101Z] [INFO] 🚀 solve v1.9.0 | |
| [2026-01-25T00:58:53.102Z] [INFO] 🔧 Raw command executed: | |
| [2026-01-25T00:58:53.102Z] [INFO] /home/hive/.nvm/versions/node/v20.20.0/bin/node /home/hive/.bun/bin/solve https://github.com/konard/p-vs-np/pull/505 --attach-logs --verbose --no-tool-check --auto-resume-on-limit-reset --tokens-budget-stats | |
| [2026-01-25T00:58:53.102Z] [INFO] | |
| [2026-01-25T00:58:53.123Z] [INFO] | |
| [2026-01-25T00:58:53.124Z] [WARNING] ⚠️ SECURITY WARNING: --attach-logs is ENABLED | |
| [2026-01-25T00:58:53.124Z] [INFO] | |
| [2026-01-25T00:58:53.124Z] [INFO] This option will upload the complete solution draft log file to the Pull Request. | |
| [2026-01-25T00:58:53.124Z] [INFO] The log may contain sensitive information such as: | |
| [2026-01-25T00:58:53.125Z] [INFO] • API keys, tokens, or secrets | |
| [2026-01-25T00:58:53.125Z] [INFO] • File paths and directory structures | |
| [2026-01-25T00:58:53.125Z] [INFO] • Command outputs and error messages | |
| [2026-01-25T00:58:53.126Z] [INFO] • Internal system information | |
| [2026-01-25T00:58:53.126Z] [INFO] | |
| [2026-01-25T00:58:53.126Z] [INFO] ⚠️ DO NOT use this option with public repositories or if the log | |
| [2026-01-25T00:58:53.127Z] [INFO] might contain sensitive data that should not be shared publicly. | |
| [2026-01-25T00:58:53.127Z] [INFO] | |
| [2026-01-25T00:58:53.127Z] [INFO] Continuing in 5 seconds... (Press Ctrl+C to abort) | |
| [2026-01-25T00:58:53.127Z] [INFO] | |
| [2026-01-25T00:58:58.134Z] [INFO] | |
| [2026-01-25T00:58:58.155Z] [INFO] 💾 Disk space check: 9338MB available (2048MB required) ✅ | |
| [2026-01-25T00:58:58.157Z] [INFO] 🧠 Memory check: 9254MB available, swap: 4095MB (478MB used), total: 12871MB (256MB required) ✅ | |
| [2026-01-25T00:58:58.173Z] [INFO] ⏩ Skipping tool connection validation (dry-run mode or skip-tool-connection-check enabled) | |
| [2026-01-25T00:58:58.173Z] [INFO] ⏩ Skipping GitHub authentication check (dry-run mode or skip-tool-connection-check enabled) | |
| [2026-01-25T00:58:58.173Z] [INFO] 📋 URL validation: | |
| [2026-01-25T00:58:58.173Z] [INFO] Input URL: https://github.com/konard/p-vs-np/pull/505 | |
| [2026-01-25T00:58:58.174Z] [INFO] Is Issue URL: false | |
| [2026-01-25T00:58:58.174Z] [INFO] Is PR URL: true | |
| [2026-01-25T00:58:58.174Z] [INFO] 🔍 Checking repository access for auto-fork... | |
| [2026-01-25T00:58:58.886Z] [INFO] Repository visibility: public | |
| [2026-01-25T00:58:58.886Z] [INFO] ✅ Auto-fork: Write access detected to public repository, working directly on repository | |
| [2026-01-25T00:58:58.887Z] [INFO] 🔍 Checking repository write permissions... | |
| [2026-01-25T00:58:59.230Z] [INFO] ✅ Repository write access: Confirmed | |
| [2026-01-25T00:58:59.554Z] [INFO] Repository visibility: public | |
| [2026-01-25T00:58:59.554Z] [INFO] Auto-cleanup default: false (repository is public) | |
| [2026-01-25T00:58:59.555Z] [INFO] 🔄 Continue mode: Working with PR #505 | |
| [2026-01-25T00:58:59.555Z] [INFO] Continue mode activated: PR URL provided directly | |
| [2026-01-25T00:58:59.556Z] [INFO] PR Number set to: 505 | |
| [2026-01-25T00:58:59.556Z] [INFO] Will fetch PR details and linked issue | |
| [2026-01-25T00:58:59.973Z] [INFO] 📝 PR branch: issue-443-0a4c3e841f6b | |
| [2026-01-25T00:58:59.974Z] [INFO] 🔗 Found linked issue #443 | |
| [2026-01-25T00:58:59.975Z] [INFO] | |
| Creating temporary directory: /tmp/gh-issue-solver-1769302739975 | |
| [2026-01-25T00:58:59.978Z] [INFO] | |
| 📥 Cloning repository: konard/p-vs-np | |
| [2026-01-25T00:59:01.274Z] [INFO] ✅ Cloned to: /tmp/gh-issue-solver-1769302739975 | |
| [2026-01-25T00:59:01.532Z] [INFO] | |
| 📌 Default branch: main | |
| [2026-01-25T00:59:01.650Z] [INFO] | |
| 🔄 Checking out PR branch: issue-443-0a4c3e841f6b | |
| [2026-01-25T00:59:01.650Z] [INFO] 📥 Fetching branches: From remote... | |
| [2026-01-25T00:59:02.296Z] [INFO] 🔍 Verifying: Branch checkout... | |
| [2026-01-25T00:59:02.371Z] [INFO] ✅ Branch checked out: issue-443-0a4c3e841f6b | |
| [2026-01-25T00:59:02.372Z] [INFO] ✅ Current branch: issue-443-0a4c3e841f6b | |
| [2026-01-25T00:59:02.373Z] [INFO] Branch operation: Checkout existing PR branch | |
| [2026-01-25T00:59:02.373Z] [INFO] Branch verification: Matches expected | |
| [2026-01-25T00:59:02.377Z] [INFO] | |
| 🔄 Continue mode: ACTIVE | |
| [2026-01-25T00:59:02.377Z] [INFO] Using existing PR: #505 | |
| [2026-01-25T00:59:02.378Z] [INFO] PR URL: https://github.com/konard/p-vs-np/pull/505 | |
| [2026-01-25T00:59:02.378Z] [INFO] | |
| 🚀 Starting work session: 2026-01-25T00:59:02.378Z | |
| [2026-01-25T00:59:02.755Z] [INFO] 📝 Converting PR: Back to draft mode... | |
| [2026-01-25T00:59:03.760Z] [INFO] ✅ PR converted: Now in draft mode | |
| [2026-01-25T00:59:04.623Z] [INFO] 💬 Posted: Work session start comment | |
| [2026-01-25T00:59:04.950Z] [INFO] 👤 Current user: konard | |
| [2026-01-25T00:59:04.951Z] [INFO] | |
| 📊 Comment counting conditions: | |
| [2026-01-25T00:59:04.951Z] [INFO] prNumber: 505 | |
| [2026-01-25T00:59:04.952Z] [INFO] branchName: issue-443-0a4c3e841f6b | |
| [2026-01-25T00:59:04.952Z] [INFO] isContinueMode: true | |
| [2026-01-25T00:59:04.953Z] [INFO] Will count comments: true | |
| [2026-01-25T00:59:04.953Z] [INFO] 💬 Counting comments: Checking for new comments since last commit... | |
| [2026-01-25T00:59:04.953Z] [INFO] PR #505 on branch: issue-443-0a4c3e841f6b | |
| [2026-01-25T00:59:04.953Z] [INFO] Owner/Repo: konard/p-vs-np | |
| [2026-01-25T00:59:05.590Z] [INFO] 📅 Last commit time (from API): 2026-01-18T21:09:37.000Z | |
| [2026-01-25T00:59:06.744Z] [INFO] 💬 New PR comments: 2 | |
| [2026-01-25T00:59:06.745Z] [INFO] 💬 New PR review comments: 0 | |
| [2026-01-25T00:59:06.747Z] [INFO] 💬 New issue comments: 0 | |
| [2026-01-25T00:59:06.747Z] [INFO] Total new comments: 2 | |
| [2026-01-25T00:59:06.748Z] [INFO] Comment lines to add: Yes | |
| [2026-01-25T00:59:06.748Z] [INFO] PR review comments fetched: 0 | |
| [2026-01-25T00:59:06.748Z] [INFO] PR conversation comments fetched: 3 | |
| [2026-01-25T00:59:06.748Z] [INFO] Total PR comments checked: 3 | |
| [2026-01-25T00:59:10.170Z] [INFO] Feedback info will be added to prompt: | |
| [2026-01-25T00:59:10.171Z] [INFO] - Pull request description was edited after last commit | |
| [2026-01-25T00:59:10.171Z] [INFO] - New comments on the pull request: 2 | |
| [2026-01-25T00:59:10.171Z] [INFO] - Issue description was edited after last commit | |
| [2026-01-25T00:59:10.171Z] [INFO] - Merge status is UNSTABLE (non-passing commit status) | |
| [2026-01-25T00:59:10.171Z] [INFO] - Failed pull request checks: 5 | |
| [2026-01-25T00:59:10.171Z] [INFO] 📅 Getting timestamps: From GitHub servers... | |
| [2026-01-25T00:59:10.538Z] [INFO] 📝 Issue updated: 2026-01-19T11:53:18.000Z | |
| [2026-01-25T00:59:10.865Z] [INFO] 💬 Comments: None found | |
| [2026-01-25T00:59:11.231Z] [INFO] 🔀 Recent PR: 2026-01-19T05:13:50.000Z | |
| [2026-01-25T00:59:11.231Z] [INFO] | |
| ✅ Reference time: 2026-01-19T11:53:18.000Z | |
| [2026-01-25T00:59:11.232Z] [INFO] | |
| 🔍 Checking for uncommitted changes to include as feedback... | |
| [2026-01-25T00:59:11.312Z] [INFO] ✅ No uncommitted changes found | |
| [2026-01-25T00:59:13.239Z] [INFO] 🎭 Playwright MCP detected - enabling browser automation hints | |
| [2026-01-25T00:59:13.243Z] [INFO] | |
| 📝 Final prompt structure: | |
| [2026-01-25T00:59:13.244Z] [INFO] Characters: 477 | |
| [2026-01-25T00:59:13.244Z] [INFO] System prompt characters: 11809 | |
| [2026-01-25T00:59:13.245Z] [INFO] Feedback info: Included | |
| [2026-01-25T00:59:13.248Z] [INFO] | |
| 🤖 Executing Claude: SONNET | |
| [2026-01-25T00:59:13.249Z] [INFO] Model: sonnet | |
| [2026-01-25T00:59:13.249Z] [INFO] Working directory: /tmp/gh-issue-solver-1769302739975 | |
| [2026-01-25T00:59:13.249Z] [INFO] Branch: issue-443-0a4c3e841f6b | |
| [2026-01-25T00:59:13.250Z] [INFO] Prompt length: 477 chars | |
| [2026-01-25T00:59:13.250Z] [INFO] System prompt length: 11809 chars | |
| [2026-01-25T00:59:13.250Z] [INFO] Feedback info included: Yes (5 lines) | |
| [2026-01-25T00:59:13.323Z] [INFO] 📈 System resources before execution: | |
| [2026-01-25T00:59:13.324Z] [INFO] Memory: MemFree: 5290100 kB | |
| [2026-01-25T00:59:13.324Z] [INFO] Load: 1.60 0.60 0.35 4/837 1754734 | |
| [2026-01-25T00:59:13.324Z] [INFO] | |
| 📝 Raw command: | |
| [2026-01-25T00:59:13.325Z] [INFO] (cd "/tmp/gh-issue-solver-1769302739975" && claude --output-format stream-json --verbose --dangerously-skip-permissions --model claude-sonnet-4-5-20250929 -p "Issue to solve: https://github.com/konard/p-vs-np/issues/443 | |
| Your prepared branch: issue-443-0a4c3e841f6b | |
| Your prepared working directory: /tmp/gh-issue-solver-1769302739975 | |
| Your prepared Pull Request: https://github.com/konard/p-vs-np/pull/505 | |
| New comments on the pull request: 2 | |
| Pull request description was edited after last commit | |
| Issue description was edited after last commit | |
| Merge status is UNSTABLE (non-passing commit status) | |
| Failed pull request checks: 5 | |
| Continue. | |
| " --append-system-prompt "You are an AI issue solver. You prefer to find the root cause of each and every issue. When you talk, you prefer to speak with facts which you have double-checked yourself or cite sources that provide evidence, like quote actual code or give references to documents or pages found on the internet. You are polite and patient, and prefer to assume good intent, trying your best to be helpful. If you are unsure or have assumptions, you prefer to test them yourself or ask questions to clarify requirements. | |
| General guidelines. | |
| - When you execute commands, always save their logs to files for easier reading if the output becomes large. | |
| - When running commands, do not set a timeout yourself — let them run as long as needed (default timeout - 2 minutes is more than enough), and once they finish, review the logs in the file. | |
| - When running sudo commands (especially package installations like apt-get, yum, npm install, etc.), always run them in the background to avoid timeout issues and permission errors when the process needs to be killed. Use the run_in_background parameter or append & to the command. | |
| - When CI is failing or user reports failures, consider adding a detailed investigation protocol to your todo list with these steps: | |
| Step 1: List recent runs with timestamps using: gh run list --repo konard/p-vs-np --branch issue-443-0a4c3e841f6b --limit 5 --json databaseId,conclusion,createdAt,headSha | |
| Step 2: Verify runs are after the latest commit by checking timestamps and SHA | |
| Step 3: For each non-passing run, download logs to preserve them: gh run view {run-id} --repo konard/p-vs-np --log > ci-logs/{workflow}-{run-id}.log | |
| Step 4: Read each downloaded log file using Read tool to understand the actual failures | |
| Step 5: Report findings with specific errors and line numbers from logs | |
| This detailed investigation is especially helpful when user mentions CI failures, asks to investigate logs, you see non-passing status, or when finalizing a PR. | |
| Note: If user says \"failing\" but tools show \"passing\", this might indicate stale data - consider downloading fresh logs and checking timestamps to resolve the discrepancy. | |
| - When a code or log file has more than 1500 lines, read it in chunks of 1500 lines. | |
| - When facing a complex problem, do as much tracing as possible and turn on all verbose modes. | |
| - When you create debug, test, or example/experiment scripts for fixing, always keep them in an examples and/or experiments folders so you can reuse them later. | |
| - When testing your assumptions, use the experiment scripts, and add it to experiments folder. | |
| - When your experiments can show real world use case of the software, add it to examples folder. | |
| - When you face something extremely hard, use divide and conquer — it always helps. | |
| Initial research. | |
| - When you start, make sure you create detailed plan for yourself and follow your todo list step by step, make sure that as many points from these guidelines are added to your todo list to keep track of everything that can help you solve the issue with highest possible quality. | |
| - When user mentions CI failures or asks to investigate logs, consider adding these todos to track the investigation: (1) List recent CI runs with timestamps, (2) Download logs from failed runs to ci-logs/ directory, (3) Analyze error messages and identify root cause, (4) Implement fix, (5) Verify fix resolves the specific errors found in logs. | |
| - When you read issue, read all details and comments thoroughly. | |
| - When you see screenshots or images in issue descriptions, pull request descriptions, comments, or discussions, use WebFetch tool (or fetch tool) to download the image first, then use Read tool to view and analyze it. IMPORTANT: Before reading downloaded images with the Read tool, verify the file is a valid image (not HTML). Use a CLI tool like 'file' command to check the actual file format. Reading corrupted or non-image files (like GitHub's HTML 404 pages saved as .png) can cause \"Could not process image\" errors and may crash the AI solver process. If the file command shows \"HTML\" or \"text\", the download failed and you should retry or skip the image. | |
| - When you need issue details, use gh issue view https://github.com/konard/p-vs-np/issues/443. | |
| - When you need related code, use gh search code --owner konard [keywords]. | |
| - When you need repo context, read files in your working directory. | |
| - When you study related work, study the most recent related pull requests. | |
| - When issue is not defined enough, write a comment to ask clarifying questions. | |
| - When accessing GitHub Gists (especially private ones), use gh gist view command instead of direct URL fetching to ensure proper authentication. | |
| - When you are fixing a bug, please make sure you first find the actual root cause, do as many experiments as needed. | |
| - When you are fixing a bug and code does not have enough tracing/logs, add them and make sure they stay in the code, but are switched off by default. | |
| - When you need comments on a pull request, note that GitHub has THREE different comment types with different API endpoints: | |
| 1. PR review comments (inline code comments): gh api repos/konard/p-vs-np/pulls/505/comments --paginate | |
| 2. PR conversation comments (general discussion): gh api repos/konard/p-vs-np/issues/505/comments --paginate | |
| 3. PR reviews (approve/request changes): gh api repos/konard/p-vs-np/pulls/505/reviews --paginate | |
| IMPORTANT: The command \"gh pr view --json comments\" ONLY returns conversation comments and misses review comments! | |
| - When you need latest comments on issue, use gh api repos/konard/p-vs-np/issues/443/comments --paginate. | |
| Solution development and testing. | |
| - When issue is solvable, implement code with tests. | |
| - When implementing features, search for similar existing implementations in the codebase and use them as examples instead of implementing everything from scratch. | |
| - When coding, each atomic step that can be useful by itself should be commited to the pull request's branch, meaning if work will be interrupted by any reason parts of solution will still be kept intact and safe in pull request. | |
| - When you test: | |
| start from testing of small functions using separate scripts; | |
| write unit tests with mocks for easy and quick start. | |
| - When you test integrations, use existing framework. | |
| - When you test solution draft, include automated checks in pr. | |
| - When issue is unclear, write comment on issue asking questions. | |
| - When you encounter any problems that you unable to solve yourself (any human feedback or help), write a comment to the pull request asking for help. | |
| - When you need human help, use gh pr comment 505 --body \"your message\" to comment on existing PR. | |
| Preparing pull request. | |
| - When you code, follow contributing guidelines. | |
| - When you commit, write clear message. | |
| - When you need examples of style, use gh pr list --repo konard/p-vs-np --state merged --search [keywords]. | |
| - When you open pr, describe solution draft and include tests. | |
| - When there is a package with version and GitHub Actions workflows for automatic release, update the version (or other necessary release trigger) in your pull request to prepare for next release. | |
| - When you update existing pr 505, use gh pr edit to modify title and description. | |
| - When you are about to commit or push code, ALWAYS run local CI checks first if they are available in contributing guidelines (like ruff check, mypy, eslint, etc.) to catch errors before pushing. | |
| - When you finalize the pull request: | |
| follow style from merged prs for code, title, and description, | |
| make sure no uncommitted changes corresponding to the original requirements are left behind, | |
| make sure the default branch is merged to the pull request's branch, | |
| make sure all CI checks passing if they exist before you finish, | |
| check for latest comments on the issue and pull request to ensure no recent feedback was missed, | |
| double-check that all changes in the pull request answer to original requirements of the issue, | |
| make sure no new new bugs are introduced in pull request by carefully reading gh pr diff, | |
| make sure no previously existing features were removed without an explicit request from users via the issue description, issue comments, and/or pull request comments. | |
| - When you finish implementation, use gh pr ready 505. | |
| Workflow and collaboration. | |
| - When you check branch, verify with git branch --show-current. | |
| - When you push, push only to branch issue-443-0a4c3e841f6b. | |
| - When you finish, create a pull request from branch issue-443-0a4c3e841f6b. (Note: PR 505 already exists, update it instead) | |
| - When you organize workflow, use pull requests instead of direct merges to default branch (main or master). | |
| - When you manage commits, preserve commit history for later analysis. | |
| - When you contribute, keep repository history forward-moving with regular commits, pushes, and reverts if needed. | |
| - When you face conflict that you cannot resolve yourself, ask for help. | |
| - When you collaborate, respect branch protections by working only on issue-443-0a4c3e841f6b. | |
| - When you mention result, include pull request url or comment url. | |
| - When you need to create pr, remember pr 505 already exists for this branch. | |
| Self review. | |
| - When you check your solution draft, run all tests locally. | |
| - When you check your solution draft, verify git status shows a clean working tree with no uncommitted changes. | |
| - When you compare with repo style, use gh pr diff [number]. | |
| - When you finalize, confirm code, tests, and description are consistent. | |
| GitHub CLI command patterns. | |
| - IMPORTANT: Always use --paginate flag when fetching lists from GitHub API to ensure all results are returned (GitHub returns max 30 per page by default). | |
| - When listing PR review comments (inline code comments), use gh api repos/OWNER/REPO/pulls/NUMBER/comments --paginate. | |
| - When listing PR conversation comments, use gh api repos/OWNER/REPO/issues/NUMBER/comments --paginate. | |
| - When listing PR reviews, use gh api repos/OWNER/REPO/pulls/NUMBER/reviews --paginate. | |
| - When listing issue comments, use gh api repos/OWNER/REPO/issues/NUMBER/comments --paginate. | |
| - When adding PR comment, use gh pr comment NUMBER --body \"text\" --repo OWNER/REPO. | |
| - When adding issue comment, use gh issue comment NUMBER --body \"text\" --repo OWNER/REPO. | |
| - When viewing PR details, use gh pr view NUMBER --repo OWNER/REPO. | |
| - When filtering with jq, use gh api repos/\${owner}/\${repo}/pulls/\${prNumber}/comments --paginate --jq 'reverse | .[0:5]'. | |
| Playwright MCP usage (browser automation via mcp__playwright__* tools). | |
| - When you develop frontend web applications (HTML, CSS, JavaScript, React, Vue, Angular, etc.), use Playwright MCP tools to test the UI in a real browser. | |
| - When WebFetch tool fails to retrieve expected content (e.g., returns empty content, JavaScript-rendered pages, or login-protected pages), use Playwright MCP tools (browser_navigate, browser_snapshot) as a fallback for web browsing. | |
| - When you need to interact with dynamic web pages that require JavaScript execution, use Playwright MCP tools. | |
| - When you need to visually verify how a web page looks or take screenshots, use browser_take_screenshot from Playwright MCP. | |
| - When you need to fill forms, click buttons, or perform user interactions on web pages, use Playwright MCP tools (browser_click, browser_type, browser_fill_form). | |
| - When you need to test responsive design or different viewport sizes, use browser_resize from Playwright MCP. | |
| - When you finish using the browser, always close it with browser_close to free resources." | jq -c .) | |
| [2026-01-25T00:59:13.326Z] [INFO] | |
| [2026-01-25T00:59:13.326Z] [INFO] 📋 User prompt: | |
| [2026-01-25T00:59:13.327Z] [INFO] ---BEGIN USER PROMPT--- | |
| [2026-01-25T00:59:13.327Z] [INFO] Issue to solve: https://github.com/konard/p-vs-np/issues/443 | |
| Your prepared branch: issue-443-0a4c3e841f6b | |
| Your prepared working directory: /tmp/gh-issue-solver-1769302739975 | |
| Your prepared Pull Request: https://github.com/konard/p-vs-np/pull/505 | |
| New comments on the pull request: 2 | |
| Pull request description was edited after last commit | |
| Issue description was edited after last commit | |
| Merge status is UNSTABLE (non-passing commit status) | |
| Failed pull request checks: 5 | |
| Continue. | |
| [2026-01-25T00:59:13.327Z] [INFO] ---END USER PROMPT--- | |
| [2026-01-25T00:59:13.327Z] [INFO] | |
| [2026-01-25T00:59:13.327Z] [INFO] 📋 System prompt: | |
| [2026-01-25T00:59:13.328Z] [INFO] ---BEGIN SYSTEM PROMPT--- | |
| [2026-01-25T00:59:13.328Z] [INFO] You are an AI issue solver. You prefer to find the root cause of each and every issue. When you talk, you prefer to speak with facts which you have double-checked yourself or cite sources that provide evidence, like quote actual code or give references to documents or pages found on the internet. You are polite and patient, and prefer to assume good intent, trying your best to be helpful. If you are unsure or have assumptions, you prefer to test them yourself or ask questions to clarify requirements. | |
| General guidelines. | |
| - When you execute commands, always save their logs to files for easier reading if the output becomes large. | |
| - When running commands, do not set a timeout yourself — let them run as long as needed (default timeout - 2 minutes is more than enough), and once they finish, review the logs in the file. | |
| - When running sudo commands (especially package installations like apt-get, yum, npm install, etc.), always run them in the background to avoid timeout issues and permission errors when the process needs to be killed. Use the run_in_background parameter or append & to the command. | |
| - When CI is failing or user reports failures, consider adding a detailed investigation protocol to your todo list with these steps: | |
| Step 1: List recent runs with timestamps using: gh run list --repo konard/p-vs-np --branch issue-443-0a4c3e841f6b --limit 5 --json databaseId,conclusion,createdAt,headSha | |
| Step 2: Verify runs are after the latest commit by checking timestamps and SHA | |
| Step 3: For each non-passing run, download logs to preserve them: gh run view {run-id} --repo konard/p-vs-np --log > ci-logs/{workflow}-{run-id}.log | |
| Step 4: Read each downloaded log file using Read tool to understand the actual failures | |
| Step 5: Report findings with specific errors and line numbers from logs | |
| This detailed investigation is especially helpful when user mentions CI failures, asks to investigate logs, you see non-passing status, or when finalizing a PR. | |
| Note: If user says "failing" but tools show "passing", this might indicate stale data - consider downloading fresh logs and checking timestamps to resolve the discrepancy. | |
| - When a code or log file has more than 1500 lines, read it in chunks of 1500 lines. | |
| - When facing a complex problem, do as much tracing as possible and turn on all verbose modes. | |
| - When you create debug, test, or example/experiment scripts for fixing, always keep them in an examples and/or experiments folders so you can reuse them later. | |
| - When testing your assumptions, use the experiment scripts, and add it to experiments folder. | |
| - When your experiments can show real world use case of the software, add it to examples folder. | |
| - When you face something extremely hard, use divide and conquer — it always helps. | |
| Initial research. | |
| - When you start, make sure you create detailed plan for yourself and follow your todo list step by step, make sure that as many points from these guidelines are added to your todo list to keep track of everything that can help you solve the issue with highest possible quality. | |
| - When user mentions CI failures or asks to investigate logs, consider adding these todos to track the investigation: (1) List recent CI runs with timestamps, (2) Download logs from failed runs to ci-logs/ directory, (3) Analyze error messages and identify root cause, (4) Implement fix, (5) Verify fix resolves the specific errors found in logs. | |
| - When you read issue, read all details and comments thoroughly. | |
| - When you see screenshots or images in issue descriptions, pull request descriptions, comments, or discussions, use WebFetch tool (or fetch tool) to download the image first, then use Read tool to view and analyze it. IMPORTANT: Before reading downloaded images with the Read tool, verify the file is a valid image (not HTML). Use a CLI tool like 'file' command to check the actual file format. Reading corrupted or non-image files (like GitHub's HTML 404 pages saved as .png) can cause "Could not process image" errors and may crash the AI solver process. If the file command shows "HTML" or "text", the download failed and you should retry or skip the image. | |
| - When you need issue details, use gh issue view https://github.com/konard/p-vs-np/issues/443. | |
| - When you need related code, use gh search code --owner konard [keywords]. | |
| - When you need repo context, read files in your working directory. | |
| - When you study related work, study the most recent related pull requests. | |
| - When issue is not defined enough, write a comment to ask clarifying questions. | |
| - When accessing GitHub Gists (especially private ones), use gh gist view command instead of direct URL fetching to ensure proper authentication. | |
| - When you are fixing a bug, please make sure you first find the actual root cause, do as many experiments as needed. | |
| - When you are fixing a bug and code does not have enough tracing/logs, add them and make sure they stay in the code, but are switched off by default. | |
| - When you need comments on a pull request, note that GitHub has THREE different comment types with different API endpoints: | |
| 1. PR review comments (inline code comments): gh api repos/konard/p-vs-np/pulls/505/comments --paginate | |
| 2. PR conversation comments (general discussion): gh api repos/konard/p-vs-np/issues/505/comments --paginate | |
| 3. PR reviews (approve/request changes): gh api repos/konard/p-vs-np/pulls/505/reviews --paginate | |
| IMPORTANT: The command "gh pr view --json comments" ONLY returns conversation comments and misses review comments! | |
| - When you need latest comments on issue, use gh api repos/konard/p-vs-np/issues/443/comments --paginate. | |
| Solution development and testing. | |
| - When issue is solvable, implement code with tests. | |
| - When implementing features, search for similar existing implementations in the codebase and use them as examples instead of implementing everything from scratch. | |
| - When coding, each atomic step that can be useful by itself should be commited to the pull request's branch, meaning if work will be interrupted by any reason parts of solution will still be kept intact and safe in pull request. | |
| - When you test: | |
| start from testing of small functions using separate scripts; | |
| write unit tests with mocks for easy and quick start. | |
| - When you test integrations, use existing framework. | |
| - When you test solution draft, include automated checks in pr. | |
| - When issue is unclear, write comment on issue asking questions. | |
| - When you encounter any problems that you unable to solve yourself (any human feedback or help), write a comment to the pull request asking for help. | |
| - When you need human help, use gh pr comment 505 --body "your message" to comment on existing PR. | |
| Preparing pull request. | |
| - When you code, follow contributing guidelines. | |
| - When you commit, write clear message. | |
| - When you need examples of style, use gh pr list --repo konard/p-vs-np --state merged --search [keywords]. | |
| - When you open pr, describe solution draft and include tests. | |
| - When there is a package with version and GitHub Actions workflows for automatic release, update the version (or other necessary release trigger) in your pull request to prepare for next release. | |
| - When you update existing pr 505, use gh pr edit to modify title and description. | |
| - When you are about to commit or push code, ALWAYS run local CI checks first if they are available in contributing guidelines (like ruff check, mypy, eslint, etc.) to catch errors before pushing. | |
| - When you finalize the pull request: | |
| follow style from merged prs for code, title, and description, | |
| make sure no uncommitted changes corresponding to the original requirements are left behind, | |
| make sure the default branch is merged to the pull request's branch, | |
| make sure all CI checks passing if they exist before you finish, | |
| check for latest comments on the issue and pull request to ensure no recent feedback was missed, | |
| double-check that all changes in the pull request answer to original requirements of the issue, | |
| make sure no new new bugs are introduced in pull request by carefully reading gh pr diff, | |
| make sure no previously existing features were removed without an explicit request from users via the issue description, issue comments, and/or pull request comments. | |
| - When you finish implementation, use gh pr ready 505. | |
| Workflow and collaboration. | |
| - When you check branch, verify with git branch --show-current. | |
| - When you push, push only to branch issue-443-0a4c3e841f6b. | |
| - When you finish, create a pull request from branch issue-443-0a4c3e841f6b. (Note: PR 505 already exists, update it instead) | |
| - When you organize workflow, use pull requests instead of direct merges to default branch (main or master). | |
| - When you manage commits, preserve commit history for later analysis. | |
| - When you contribute, keep repository history forward-moving with regular commits, pushes, and reverts if needed. | |
| - When you face conflict that you cannot resolve yourself, ask for help. | |
| - When you collaborate, respect branch protections by working only on issue-443-0a4c3e841f6b. | |
| - When you mention result, include pull request url or comment url. | |
| - When you need to create pr, remember pr 505 already exists for this branch. | |
| Self review. | |
| - When you check your solution draft, run all tests locally. | |
| - When you check your solution draft, verify git status shows a clean working tree with no uncommitted changes. | |
| - When you compare with repo style, use gh pr diff [number]. | |
| - When you finalize, confirm code, tests, and description are consistent. | |
| GitHub CLI command patterns. | |
| - IMPORTANT: Always use --paginate flag when fetching lists from GitHub API to ensure all results are returned (GitHub returns max 30 per page by default). | |
| - When listing PR review comments (inline code comments), use gh api repos/OWNER/REPO/pulls/NUMBER/comments --paginate. | |
| - When listing PR conversation comments, use gh api repos/OWNER/REPO/issues/NUMBER/comments --paginate. | |
| - When listing PR reviews, use gh api repos/OWNER/REPO/pulls/NUMBER/reviews --paginate. | |
| - When listing issue comments, use gh api repos/OWNER/REPO/issues/NUMBER/comments --paginate. | |
| - When adding PR comment, use gh pr comment NUMBER --body "text" --repo OWNER/REPO. | |
| - When adding issue comment, use gh issue comment NUMBER --body "text" --repo OWNER/REPO. | |
| - When viewing PR details, use gh pr view NUMBER --repo OWNER/REPO. | |
| - When filtering with jq, use gh api repos/${owner}/${repo}/pulls/${prNumber}/comments --paginate --jq 'reverse | .[0:5]'. | |
| Playwright MCP usage (browser automation via mcp__playwright__* tools). | |
| - When you develop frontend web applications (HTML, CSS, JavaScript, React, Vue, Angular, etc.), use Playwright MCP tools to test the UI in a real browser. | |
| - When WebFetch tool fails to retrieve expected content (e.g., returns empty content, JavaScript-rendered pages, or login-protected pages), use Playwright MCP tools (browser_navigate, browser_snapshot) as a fallback for web browsing. | |
| - When you need to interact with dynamic web pages that require JavaScript execution, use Playwright MCP tools. | |
| - When you need to visually verify how a web page looks or take screenshots, use browser_take_screenshot from Playwright MCP. | |
| - When you need to fill forms, click buttons, or perform user interactions on web pages, use Playwright MCP tools (browser_click, browser_type, browser_fill_form). | |
| - When you need to test responsive design or different viewport sizes, use browser_resize from Playwright MCP. | |
| - When you finish using the browser, always close it with browser_close to free resources. | |
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| [2026-01-25T00:59:25.363Z] [INFO] {"type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01QactGJUdgzycQYDY3EmhNw","type":"tool_result","content":"<persisted-output>\nOutput too large (175.7KB). Full output saved to: /home/hive/.claude/projects/-tmp-gh-issue-solver-1769302739975/37a924e4-6f7f-4487-9a8d-5a454cb6c52f/tool-results/toolu_01QactGJUdgzycQYDY3EmhNw.txt\n\nPreview (first 2KB):\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven po\n...\n</persisted-output>","is_error":false}]},"parent_tool_use_id":null,"session_id":"37a924e4-6f7f-4487-9a8d-5a454cb6c52f","uuid":"51516510-c171-4db7-9dd6-cde32a1c575e","tool_use_result":{"stdout":"{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:45:28Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656270/job/60728018371\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:43:52Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Coq Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n�� **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:39:57Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018387\",\"name\":\"Lean Verification\",\"startedAt\":\"2026-01-18T22:39:21Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:24:55Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656286/job/60728018354\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:49:54Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Isabelle Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:28:48Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656291/job/60728018369\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:28:12Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Lean Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:33Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018374\",\"name\":\"Lean Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:07Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:50:50Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018372\",\"name\":\"Coq Verification\",\"startedAt\":\"2026-01-18T22:49:04Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:23Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018390\",\"name\":\"Coq Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:18Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:56:24Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018377\",\"name\":\"Isabelle Verification\",\"startedAt\":\"2026-01-18T22:20:10Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:25:12Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018384\",\"name\":\"Verification Metrics\",\"startedAt\":\"2026-01-18T22:24:48Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:31:39Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60731357631\",\"name\":\"Verification Summary\",\"startedAt\":\"2026-01-18T23:31:37Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:27:44Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60730406748\",\"name\":\"Benchmark Summary\",\"startedAt\":\"2026-01-18T23:27:38Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-co | |
| [2026-01-25T00:59:25.375Z] [INFO] de)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:45:28Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656270/job/60728018371\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:43:52Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Coq Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:39:57Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018387\",\"name\":\"Lean Verification\",\"startedAt\":\"2026-01-18T22:39:21Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:24:55Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656286/job/60728018354\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:49:54Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Isabelle Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:28:48Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656291/job/60728018369\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:28:12Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Lean Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:33Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018374\",\"name\":\"Lean Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:07Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:50:50Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018372\",\"name\":\"Coq Verification\",\"startedAt\":\"2026-01-18T22:49:04Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:23Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018390\",\"name\":\"Coq Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:18Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:56:24Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018377\",\"name\":\"Isabelle Verification\",\"startedAt\":\"2026-01-18T22:20:10Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:25:12Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018384\",\"name\":\"Verification Metrics\",\"startedAt\":\"2026-01-18T22:24:48Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:31:39Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60731357631\",\"name\":\"Verification Summary\",\"startedAt\":\"2026-01-18T23:31:37Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why gr | |
| [2026-01-25T00:59:25.380Z] [INFO] eedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:27:44Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60730406748\",\"name\":\"Benchmark Summary\",\"startedAt\":\"2026-01-18T23:27:38Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:45:28Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656270/job/60728018371\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:43:52Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Coq Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:39:57Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018387\",\"name\":\"Lean Verification\",\"startedAt\":\"2026-01-18T22:39:21Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:24:55Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656286/job/60728018354\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:49:54Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Isabelle Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:28:48Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656291/job/60728018369\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:28:12Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Lean Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:33Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018374\",\"name\":\"Lean Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:07Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Le | |
| [2026-01-25T00:59:25.414Z] [INFO] an: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:50:50Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018372\",\"name\":\"Coq Verification\",\"startedAt\":\"2026-01-18T22:49:04Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:23Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018390\",\"name\":\"Coq Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:18Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:56:24Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018377\",\"name\":\"Isabelle Verification\",\"startedAt\":\"2026-01-18T22:20:10Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. 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Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:31:39Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60731357631\",\"name\":\"Verification Summary\",\"startedAt\":\"2026-01-18T23:31:37Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. 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| [2026-01-25T00:59:38.558Z] [INFO] � **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:39:57Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018387\",\"name\":\"Lean Verification\",\"startedAt\":\"2026-01-18T22:39:21Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:24:55Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656286/job/60728018354\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:49:54Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Isabelle Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:28:48Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656291/job/60728018369\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:28:12Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Lean Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:33Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018374\",\"name\":\"Lean Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:07Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:50:50Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018372\",\"name\":\"Coq Verification\",\"startedAt\":\"2026-01-18T22:49:04Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:23Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018390\",\"name\":\"Coq Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:18Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:56:24Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018377\",\"name\":\"Isabelle Verification\",\"startedAt\":\"2026-01-18T22:20:10Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:25:12Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018384\",\"name\":\"Verification Metrics\",\"startedAt\":\"2026-01-18T22:24:48Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:31:39Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60731357631\",\"name\":\"Verification Summary\",\"startedAt\":\"2026-01-18T23:31:37Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"## 🤖 Solution Draft Log\\nThis log file contains the complete execution trace of the AI solution draft process.\\n\\n💰 **Cost estimation:**\\n- Public pricing estimate: $1.993914 USD\\n- Calculated by Anthropic: $1.667147 USD\\n- Difference: $-0.326767 (-16.39%)\\n📎 **Log file uploaded as Gist** (461KB)\\n🔗 [View complete solution draft log](https://gist.githubusercontent.com/konard/4e9f68e39f72a92019b3b3eb0caa8f4b/raw/5474f10bcccf9f515cf4402d714f5db7ef812723/solution-draft-log-pr-1768770580652.txt)\\n---\\n*Now working session is ended, feel free to review and add any feedback on the solution draft.*\",\"createdAt\":\"2026-01-18T21:09:45Z\",\"id\":\"IC_kwDOQBQH6M7gdMvL\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3765750731\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:27:44Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60730406748\",\"name\":\"Benchmark Summary\",\"startedAt\":\"2026-01-18T23:27:38Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:45:28Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656270/job/60728018371\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:43:52Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Coq Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:39:57Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018387\",\"name\":\"Lean Verification\",\"startedAt\":\"2026-01-18T22:39:21Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and lik | |
| [2026-01-25T00:59:38.573Z] [INFO] ely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:24:55Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656286/job/60728018354\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:49:54Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Isabelle Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:28:48Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656291/job/60728018369\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:28:12Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Lean Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:33Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018374\",\"name\":\"Lean Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:07Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:50:50Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018372\",\"name\":\"Coq Verification\",\"startedAt\":\"2026-01-18T22:49:04Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:23Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018390\",\"name\":\"Coq Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:18Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note exi | |
| [2026-01-25T00:59:38.587Z] [INFO] stence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:56:24Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018377\",\"name\":\"Isabelle Verification\",\"startedAt\":\"2026-01-18T22:20:10Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:25:12Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018384\",\"name\":\"Verification Metrics\",\"startedAt\":\"2026-01-18T22:24:48Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:31:39Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60731357631\",\"name\":\"Verification Summary\",\"startedAt\":\"2026-01-18T23:31:37Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"Get latest changes from default branch and continue to meet issue requirements. Note that we no longer support Isabelle, so if does not compile just delete it, if it was compiling in last CI/CD run, just move it to corresponding ./archive folder.\\r\\n\\r\\nTry to proceed with formalization until it is impossible to go any further, and keep code comments on why exactly it is impossible. We are working on formalization of attempts to prove P vs NP, we need to do the best we can, all CI/CD checks should pass as the result of this pull request. If you have any issues with formalization, just make sure these issues are properly and fully documented in the code comments.\\r\\n\\r\\nAlso ensure the attempt folder have structure:\\r\\n\\r\\n- original/ (description of proof idea/text, and original proof text in pdf/html and conversion to of that text to markdown in all cases, markdown should be translated to English if original text is not in English, so we should have README.md, ORIGINAL.md (conversion result) and ORIGINAL.(pdf/html/...))\\r\\n- proof/ (the forward proof attempt, it should follow exactly the text of proof in markdown, showing each original line or paragraph, after that convert it to actual Lean and Rocq statements, commenting all statements that will not compile, keeping compiled statements).\\r\\n- refutation/ (the proof against the original idea, should have refutations of refutable original statements from human proof)\\r\\n\\r\\nSo end result will be that will collect as much formalization as we can for each prompt attempt, so we later can reuse it in our research.\\r\\n\\r\\nUse https://github.com/konard/p-vs-np/pull/504 as example for file structure.\\r\\n\\r\\nMake sure to execute scripts/check_attempt_structure.py to update ATTEMPTS.md after you finish.\\r\\n\\r\\nEnsure all changes are correct, consistent and fully meet the requirements.\",\"createdAt\":\"2026-01-24T17:43:04Z\",\"id\":\"IC_kwDOQBQH6M7iNrJf\",\"includesCreatedEdit\":true,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795235423\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:27:44Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60730406748\",\"name\":\"Benchmark Summary\",\"startedAt\":\"2026-01-18T23:27:38Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:45:28Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656270/job/60728018371\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:43:52Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Coq Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:39:57Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018387\",\"name\":\"Lean Verification\",\"startedAt\":\"2026-01-18T22:39:21Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:24:55Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656286/job/60728018354\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:49:54Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Isabelle Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T21:28:48Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656291/job/60728018369\",\"name\":\"build\",\"startedAt\":\"2026-01-18T21:28:12Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Lean Verification\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:33Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018374\",\"name\":\"Lean Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:07Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:50:50Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018372\",\"name\":\"Coq Verification\",\"startedAt\":\"2026-01-18T22:49:04Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:13:23Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018390\",\"name\":\"Coq Performance Benchmarks\",\"startedAt\":\"2026-01-18T22:13:18Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:56:24Z\",\"conclusion\":\"FAILURE\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60728018377\",\"name\":\"Isabelle Verification\",\"startedAt\":\"2026-01-18T22:20:10Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and ba | |
| [2026-01-25T00:59:38.602Z] [INFO] cktracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T22:25:12Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656288/job/60728018384\",\"name\":\"Verification Metrics\",\"startedAt\":\"2026-01-18T22:24:48Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Performance Benchmarks\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. Please wait working session to finish, and provide your feedback._\",\"createdAt\":\"2026-01-25T00:59:04Z\",\"id\":\"IC_kwDOQBQH6M7iPwL6\",\"includesCreatedEdit\":false,\"isMinimized\":false,\"minimizedReason\":\"\",\"reactionGroups\":[],\"url\":\"https://github.com/konard/p-vs-np/pull/505#issuecomment-3795780346\",\"viewerDidAuthor\":true},\"statusCheckRollup\":{\"__typename\":\"CheckRun\",\"completedAt\":\"2026-01-18T23:31:39Z\",\"conclusion\":\"SUCCESS\",\"detailsUrl\":\"https://github.com/konard/p-vs-np/actions/runs/21118656268/job/60731357631\",\"name\":\"Verification Summary\",\"startedAt\":\"2026-01-18T23:31:37Z\",\"status\":\"COMPLETED\",\"workflowName\":\"Formal Verification Suite\"},\"title\":\"Add formalization for attempt #38: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal (2006) - P=NP\"}\n{\"body\":\"## Summary\\n\\nThis PR adds formalization for attempt #38 from Woeginger's P vs NP list: Khadija Riaz and Malik Sikander Hayat Khiyal's 2006 paper claiming P=NP via a polynomial-time algorithm for finding Hamiltonian cycles.\\n\\nFixes #443\\n\\n## Implementation Details\\n\\n### Attempt Overview\\n- **Authors**: Khadija Riaz \\u0026 Malik Sikander Hayat Khiyal\\n- **Year**: 2006\\n- **Claim**: P = NP\\n- **Approach**: Polynomial-time greedy algorithm with limited backtracking for Hamiltonian Cycle Problem\\n- **Paper**: \\\"Finding Hamiltonian Cycle in Polynomial Time\\\", Information Technology Journal, Vol. 5, pp. 851-859\\n\\n### Files Added\\n```\\nproofs/attempts/riaz-khiyal-2006-peqnp/\\n├── README.md # Comprehensive analysis of the attempt\\n├── lean/RiazKhiyalHamiltonianAttempt.lean # Lean 4 formalization\\n├── coq/RiazKhiyalHamiltonianAttempt.v # Coq formalization\\n└── isabelle/\\n ├── ROOT # Isabelle session configuration\\n └── RiazKhiyalHamiltonianAttempt.thy # Isabelle/HOL formalization\\n```\\n\\n### Key Errors Identified\\n\\nThe formalization identifies and documents the following critical errors in the original paper:\\n\\n1. **Missing Complexity Analysis**: No formal time complexity proof or mathematical analysis\\n2. **Incomplete Algorithmic Details**: Critical selection logic and backtracking mechanisms are missing or poorly specified\\n3. **No Proof of Correctness**: No proof that the algorithm finds Hamiltonian cycles when they exist\\n4. **No Proof of Completeness**: No proof that the algorithm always terminates\\n5. **Unsubstantiated Backtracking Claim**: The claim that \\\"backtracking occurs only at junction nodes\\\" lacks formal justification\\n6. **Circular Reasoning**: Assumes what needs to be proven about valid selection conditions\\n7. **Heuristics vs. Algorithms**: Confuses a potentially useful heuristic with a proven polynomial-time algorithm\\n\\n### Formalization Approach\\n\\nThe formalizations across all three proof assistants follow a consistent structure:\\n\\n1. **Basic Complexity Definitions**: Define P, NP, NP-completeness, and polynomial time\\n2. **Graph Theory**: Define graphs, paths, and Hamiltonian cycles\\n3. **Hamiltonian Cycle Problem**: Formalize the NP-complete problem\\n4. **Algorithm Structure**: Model the Riaz-Khiyal greedy algorithm framework\\n5. **Critical Claims**: Formalize the unproven claims (correctness, completeness, polynomial complexity)\\n6. **The Argument**: Show that IF the claims held, THEN P = NP\\n7. **The Errors**: Demonstrate that the claims are unproven and likely false\\n8. **Counter-Examples**: Note existence of graphs where greedy algorithms require exponential time\\n9. **Key Lessons**: Extract general lessons about proof requirements and NP-completeness\\n\\n### Documentation\\n\\nThe README.md provides:\\n- Detailed summary of the approach\\n- Complete explanation of all identified errors\\n- Context about why greedy algorithms fail for NP-complete problems\\n- Comparison with other similar failed attempts\\n- References to the original paper and related work\\n- Educational value about the distinction between heuristics and proven algorithms\\n\\n### Testing\\n\\nAll formalizations follow the repository's contribution guidelines:\\n- Lean: Uses only standard library (no Mathlib dependencies)\\n- Coq: Uses standard libraries and compiles with `Admitted` for incomplete proofs\\n- Isabelle: Includes ROOT file for session configuration, uses `oops` for incomplete proofs\\n\\nThe files are structured to demonstrate the failure of the proof attempt, not to complete impossible proofs.\\n\\n---\\n\\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\",\"comments\":{\"author\":{\"login\":\"konard\"},\"authorAssociation\":\"OWNER\",\"body\":\"🤖 **AI Work Session Started**\\n\\nStarting automated work session at 2026-01-25T00:59:02.378Z\\n\\nThe PR has been converted to draft mode while work is in progress.\\n\\n_This comment marks the beginning of an AI work session. 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| [2026-01-25T01:00:16.573Z] [INFO] {"type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01KfEdDbi6cZaKcj8xetjjf1","type":"tool_result","content":" 1→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8268499Z Current runner version: '2.331.0'\n 2→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8291824Z ##[group]Runner Image Provisioner\n 3→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8292602Z Hosted Compute Agent\n 4→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8293107Z Version: 20251211.462\n 5→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8293790Z Commit: 6cbad8c2bb55d58165063d031ccabf57e2d2db61\n 6→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8294449Z Build Date: 2025-12-11T16:28:49Z\n 7→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8295278Z Worker ID: {44e49474-5468-46e9-ad24-991138ebb501}\n 8→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8296085Z ##[endgroup]\n 9→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8296576Z ##[group]Operating System\n 10→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8297160Z Ubuntu\n 11→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8297607Z 24.04.3\n 12→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8298055Z LTS\n 13→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8298445Z ##[endgroup]\n 14→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8299018Z ##[group]Runner Image\n 15→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8299529Z Image: ubuntu-24.04\n 16→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8300010Z Version: 20260111.209.1\n 17→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8301069Z Included Software: https://github.com/actions/runner-images/blob/ubuntu24/20260111.209/images/ubuntu/Ubuntu2404-Readme.md\n 18→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8302567Z Image Release: https://github.com/actions/runner-images/releases/tag/ubuntu24%2F20260111.209\n 19→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8303630Z ##[endgroup]\n 20→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8304689Z ##[group]GITHUB_TOKEN Permissions\n 21→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8307042Z Contents: | |
| [2026-01-25T01:00:16.593Z] [INFO] read\n 22→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8307596Z Metadata: read\n 23→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8308045Z Packages: read\n 24→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8308661Z ##[endgroup]\n 25→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8311133Z Secret source: Actions\n 26→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8312108Z Prepare workflow directory\n 27→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8627693Z Prepare all required actions\n 28→build\tUNKNOWN STEP\t2026-01-18T21:49:54.8665414Z Getting action download info\n 29→build\tUNKNOWN STEP\t2026-01-18T21:49:55.1217988Z Download action repository 'actions/checkout@v4' (SHA:34e114876b0b11c390a56381ad16ebd13914f8d5)\n 30→build\tUNKNOWN STEP\t2026-01-18T21:49:55.2436759Z Download action repository 'actions/cache@v4' (SHA:0057852bfaa89a56745cba8c7296529d2fc39830)\n 31→build\tUNKNOWN STEP\t2026-01-18T21:49:55.4236603Z Download action repository 'actions/upload-artifact@v4' (SHA:ea165f8d65b6e75b540449e92b4886f43607fa02)\n 32→build\tUNKNOWN STEP\t2026-01-18T21:49:55.6346977Z Complete job name: build\n 33→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7048120Z ##[group]Run actions/checkout@v4\n 34→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7048933Z with:\n 35→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7049328Z repository: konard/p-vs-np\n 36→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7050037Z token: ***\n 37→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7050416Z ssh-strict: true\n 38→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7050796Z ssh-user: git\n 39→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7051188Z persist-credentials: true\n 40→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7051632Z clean: true\n 41→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7052019Z sparse-checkout-cone-mode: true\n 42→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7052495Z fetch-depth: 1\n 43→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7052863Z fetch-tags: false\n 44→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7053255Z show-progress: true\n 45→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7053642Z lfs: false\n 46→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7053998Z submodules: false\n 47→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7054386Z set-safe-directory: true\n 48→build\tUNKNOWN STEP\t2026-01-18T21:49:55.7055009Z ##[endgroup]\n 49→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8133543Z Syncing repository: konard/p-vs-np\n 50→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8136114Z ##[group]Getting Git version info\n 51→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8137165Z Working directory is '/home/runner/work/p-vs-np/p-vs-np'\n 52→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8138676Z [command]/usr/bin/git version\n 53→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8305843Z git version 2.52.0\n 54→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8358051Z ##[endgroup]\n 55→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8370631Z Temporarily overriding HOME='/home/runner/work/_temp/d4c1c833-fbea-41a8-ab07-c79f1dc68ee8' before making global git config changes\n 56→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8372984Z Adding repository directory to the temporary git global config as a safe directory\n 57→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8376926Z [command]/usr/bin/git config --global --add safe.directory /home/runner/work/p-vs-np/p-vs-np\n 58→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8419334Z Deleting the contents of '/home/runner/work/p-vs-np/p-vs-np'\n 59→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8423022Z ##[group]Initializing the repository\n 60→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8427117Z [command]/usr/bin/git init /home/runner/work/p-vs-np/p-vs-np\n 61→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8563976Z hint: Using 'master' as the name for the initial branch. This default branch name\n 62→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8565338Z hint: will change to \"main\" in Git 3.0. To configure the initial branch name\n 63→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8567133Z hint: to use in all of your new repositories, which will suppress this warning,\n 64→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8569038Z hint: call:\n 65→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8569907Z hint:\n 66→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8571002Z hint: \tgit config --global init.defaultBranch <name>\n 67→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8572150Z hint:\n 68→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8572857Z hint: Names commonly chosen instead of 'master' are 'main', 'trunk' and\n 69→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8574036Z hint: 'development'. The just-created branch can be renamed via this command:\n 70→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8574949Z hint:\n 71→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8575631Z hint: \tgit branch -m <name>\n 72→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8576192Z hint:\n 73→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8576943Z hint: Disable this message with \"git config set advice.defaultBranchName false\"\n 74→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8578337Z Initialized empty Git repository in /home/runner/work/p-vs-np/p-vs-np/.git/\n 75→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8586638Z [command]/usr/bin/git remote add origin https://github.com/konard/p-vs-np\n 76→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8625508Z ##[endgroup]\n 77→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8626250Z ##[group]Disabling automatic garbage collection\n 78→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8631520Z [command]/usr/bin/git config --local gc.auto 0\n 79→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8665859Z ##[endgroup]\n 80→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8666936Z ##[group]Setting up auth\n 81→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8676443Z [command]/usr/bin/git config --local --name-only --get-regexp core\\.sshCommand\n 82→build\tUNKNOWN STEP\t2026-01-18T21:49:55.8711773Z [command]/usr/bin/git submodule foreach --recursive sh -c \"git config --local --name-only --get-regexp 'core\\.sshCommand' && git config --local --unset-all 'core.sshCommand' || :\"\n 83→build\tUNKNOWN STEP\t2026-01-18T21:49:55.9097834Z [command]/usr/bin/git config --local --name-only --get-regexp http\\.https\\:\\/\\/github\\.com\\/\\.extraheader\n 84→build\tUNKNOWN STEP\t2026-01-18T21:49:55.9132906Z [command]/usr/bin/git submodule foreach --recursive sh -c \"git config --local --name-only --get-regexp 'http\\.https\\:\\/\\/github\\.com\\/\\.extraheader' && git config --local --unset-all 'http.https://github.com/.extraheader' || :\"\n 85→build\tUNKNOWN STEP\t2026-01-18T21:49:55.9365693Z [command]/usr/bin/git config --local --name-only --get-regexp ^includeIf\\.gitdir:\n 86→build\tUNKNOWN STEP\t2026-01-18T21:49:55.9398075Z [command]/usr/bin/git submodule foreach --recursive git config --local --show-origin --name-only --get-regexp remote.origin.url\n 87→build\tUNKNOWN STEP\t2026-01-18T21:49:55.9624582Z [command]/usr/bin/git config --local http.https://github.com/.extraheader AUTHORIZATION: basic ***\n 88→build\tUNKNOWN STEP\t2026-01-18T21:49:55.9661145Z ##[endgroup]\n 89→build\tUNKNOWN STEP\t2026-01-18T21:49:55.9662563Z ##[group]Fetching the repository\n 90→build\tUNKNOWN STEP\t2026-01-18T21:49:55.9671243Z [command]/usr/bin/git -c protocol.version=2 fetch --no-tags --prune --no-recurse-submodules --depth=1 origin +bbccd******************************66e3d:refs/remotes/pull/505/merge\n 91→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3761326Z From https://github.com/konard/p-vs-np\n 92→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3763341Z * [new ref] bbccd******************************66e3d -> pull/505/merge\n 93→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3795452Z ##[endgroup]\n 94→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3796820Z ##[group]Determining the checkout info\n 95→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3800732Z ##[endgroup]\n 96→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3804270Z [command]/usr/bin/git sparse-checkout disable\n 97→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3846543Z [command]/usr/bin/git config --local --unset-all extensions.worktreeConfig\n 98→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3872324Z ##[group]Checking out the ref\n 99→build\tUNKNOWN STEP\t2026-01-18T21:49:56.3876985Z [command]/usr/bin/git checkout --progress --force refs/remotes/pull/505/merge\n 100→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4486617Z Note: switching to 'refs/remotes/pull/505/merge'.\n 101→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4487721Z \n 102→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4488386Z You are in 'detached HEAD' state. You can look around, make experimental\n 103→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4489975Z changes and commit them, and you can discard any commits you make in this\n 104→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4491412Z state without impacting any branches by switching back to a branch.\n 105→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4492246Z \n 106→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4492875Z If you want to create a new branch to retain commits you create, you may\n 107→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4494531Z do so (now or later) by using -c with the switch command. Example:\n 108→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4496052Z \n 109→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4496452Z git switch -c <new-branch-name>\n 110→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4497082Z \n 111→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4497525Z Or undo this operation with:\n 112→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4498444Z \n 113→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4498778Z git switch -\n 114→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4499241Z \n 115→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4499984Z Turn off this advice by setting config variable advice.detachedHead to false\n 116→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4500929Z \n 117→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4501983Z HEAD is now at bbccd69 Merge ab9d5******************************9d196 into a1ff7a26028578f90e224f41dc3ae3155aac965f\n 118→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4505408Z ##[endgroup]\n 119→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4535465Z [command]/usr/bin/git log -1 --format=%H\n 120→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4557718Z bbccd******************************66e3d\n 121→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4783372Z ##[group]Run if find . -name \"*.thy\" -print -quit | grep -q .; then\n 122→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4785580Z \u001b[36;1mif find . -name \"*.thy\" -print -quit | grep -q .; then\u001b[0m\n 123→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4787215Z \u001b[36;1m echo \"has_files=true\" >> $GITHUB_OUTPUT\u001b[0m\n 124→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4788541Z \u001b[36;1melse\u001b[0m\n 125→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4789610Z \u001b[36;1m echo \"has_files=false\" >> $GITHUB_OUTPUT\u001b[0m\n 126→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4791161Z \u001b[36;1m echo \"No Isabelle files found - skipping build\"\u001b[0m\n 127→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4793003Z \u001b[36;1m echo \"To use Isabelle verification, add Isabelle theory files (.thy)\"\u001b[0m\n 128→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4794607Z \u001b[36;1mfi\u001b[0m\n 129→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4831713Z shell: /usr/bin/bash -e {0}\n 130→build\tUNKNOWN STEP\t2026-01-18T21:49:56.4832775Z ##[endgroup]\n 131→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5259036Z ##[group]Run actions/cache@v4\n 132→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5259997Z with:\n 133→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5260736Z path: Isabelle2025-1\n 134→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5261610Z key: Linux-isabelle-2025-1\n 135→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5262571Z enableCrossOsArchive: false\n 136→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5263545Z fail-on-cache-miss: false\n 137→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5264447Z lookup-only: false\n 138→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5265545Z save-always: false\n 139→build\tUNKNOWN STEP\t2026-01-18T21:49:56.5266415Z ##[endgroup]\n 140→build\tUNKNOWN STEP\t2026-01-18T21:49:56.7349271Z Cache hit for: Linux-isabelle-2025-1\n 141→build\tUNKNOWN STEP\t2026-01-18T21:49:57.7742074Z Received 226492416 of 1159287686 (19.5%), 216.2 MBs/sec\n 142→build\tUNKNOWN STEP\t2026-01-18T21:49:58.7745866Z Received 494927872 of 1159287686 (42.7%), 236.0 MBs/sec\n 143→build\tUNKNOWN STEP\t2026-01-18T21:49:59.7754617Z Received 742391808 of 1159287686 (64.0%), 235.9 MBs/sec\n 144→build\tUNKNOWN STEP\t2026-01-18T21:50:00.7780576Z Received 968884224 of 1159287686 (83.6%), 230.8 MBs/sec\n 145→build\tUNKNOWN STEP\t2026-01-18T21:50:01.5931739Z Received 1159287686 of 1159287686 (100.0%), 229.5 MBs/sec\n 146→build\tUNKNOWN STEP\t2026-01-18T21:50:01.5934852Z Cache Size: ~1106 MB (1159287686 B)\n 147→build\tUNKNOWN STEP\t2026-01-18T21:50:01.6068857Z [command]/usr/bin/tar -xf /home/runner/work/_temp/0546e348-c650-4c6e-9ed0-9079c17cd1e1/cache.tzst -P -C /home/runner/work/p-vs-np/p-vs-np --use-compress-program unzstd\n 148→build\tUNKNOWN STEP\t2026-01-18T21:50:06.2836888Z Cache restored successfully\n 149→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6025630Z Cache restored from key: Linux-isabelle-2025-1\n 150→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6169754Z ##[group]Run echo \"$PWD/Isabelle2025-1/bin\" >> $GITHUB_PATH\n 151→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6170212Z \u001b[36;1mecho \"$PWD/Isabelle2025-1/bin\" >> $GITHUB_PATH\u001b[0m\n 152→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6203422Z shell: /usr/bin/bash -e {0}\n 153→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6203674Z ##[endgroup]\n 154→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6291407Z ##[group]Run echo \"Generating ROOTS file from individual ROOT files...\"\n 155→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6291938Z \u001b[36;1mecho \"Generating ROOTS file from individual ROOT files...\"\u001b[0m\n 156→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6292459Z \u001b[36;1mfind . -name ROOT -path \"*/isabelle/*\" | sed 's|/ROOT||' | sed 's|^\\./||' | sort > ROOTS\u001b[0m\n 157→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6292937Z \u001b[36;1mecho \"Found $(wc -l < ROOTS) Isabelle sessions:\"\u001b[0m\n 158→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6293232Z \u001b[36;1mcat ROOTS\u001b[0m\n 159→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6324696Z shell: /usr/bin/bash -e {0}\n 160→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6324949Z ##[endgroup]\n 161→build\tUNKNOWN STEP\t2026-01-18T21:50:06.6374152Z Generating ROOTS file from individual ROOT files...\n 162→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8398413Z Found 66 Isabelle sessions:\n 163→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8409040Z proofs/attempts/a-new-version-of-2005-pneqnp/isabelle\n 164→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8409752Z proofs/attempts/alan-feinstein-2005-pneqnp/isabelle\n 165→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8410437Z proofs/attempts/alan-feinstein-2011-pneqnp/isabelle\n 166→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8411169Z proofs/attempts/anatoly-panyukov-2014-peqnp/isabelle\n 167→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8411942Z proofs/attempts/andrea-bianchini-2005-peqnp/isabelle\n 168→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8412648Z proofs/attempts/antano-maknickas-2011-peqnp/isabelle\n 169→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8413947Z proofs/attempts/arto-annila-2009-pneqnp/isabelle\n 170→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8414577Z proofs/attempts/author10-2004-pneqnp/isabelle\n 171→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8415996Z proofs/attempts/author104-2015-peqnp/isabelle\n 172→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8416553Z proofs/attempts/author11-2004-peqnp/isabelle\n 173→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8417049Z proofs/attempts/author12-2004-pneqnp/isabelle\n 174→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8417538Z proofs/attempts/author13-2004-pneqnp/isabelle\n 175→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8417986Z proofs/attempts/author15-2004-pneqnp/isabelle\n 176→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8418297Z proofs/attempts/author2-1996-peqnp/isabelle\n 177→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8418587Z proofs/attempts/author4-2000-peqnp/isabelle\n 178→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8418912Z proofs/attempts/author8-2003-pneqnp/isabelle\n 179→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8419237Z proofs/attempts/author93-2013-peqnp/isabelle\n 180→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8419592Z proofs/attempts/charles-sauerbier-2002-peqnp/isabelle\n 181→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8419990Z proofs/attempts/chingizovich-valeyev-2013-pneqnp/isabelle\n 182→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8420383Z proofs/attempts/daegene-song-2014-pneqnp/isabelle\n 183→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8420714Z proofs/attempts/dhami-2014-peqnp/isabelle\n 184→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8421018Z proofs/attempts/douglas-youvan-2012-peqnp/isabelle\n 185→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8421370Z proofs/attempts/dr-joachim-mertz-2005-peqnp/isabelle\n 186→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8421710Z proofs/attempts/eli-halylaurin-2016-peqnp/isabelle\n 187→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8422026Z proofs/attempts/figueroa-2016-pneqnp/isabelle\n 188→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8422358Z proofs/attempts/francesco-capasso-2005-peqnp/isabelle\n 189→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8422688Z proofs/attempts/hamelin-2011-peqnp/isabelle\n 190→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8422985Z proofs/attempts/hanlin-liu-2014-peqnp/isabelle\n 191→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8423293Z proofs/attempts/has-also-2001-pneqnp/isabelle\n 192→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8423647Z proofs/attempts/infotechnology-center-2012-pneqnp/isabelle\n 193→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8424023Z proofs/attempts/jason-w-steinmetz-2011-peqnp/isabelle\n 194→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8424371Z proofs/attempts/jeffrey-w-holcomb-2011-pneqnp/isabelle\n 195→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8424721Z proofs/attempts/joonmo-kim-2014-pneqnp/isabelle\n 196→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8425451Z proofs/attempts/ki-bong-nam-sh-wang-and-yang-gon-kim-published-2004-pneqnp/isabelle\n 197→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8425953Z proofs/attempts/koji-kobayashi-2012-pneqnp/isabelle\n 198→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8426295Z proofs/attempts/lev-gordeev-2005-pneqnp/isabelle\n 199→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8426624Z proofs/attempts/lokman-kolukisa-2005-peqnp/isabelle\n 200→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8426951Z proofs/attempts/luiz-barbosa-2009-pneqnp/isabelle\n 201→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8427303Z proofs/attempts/mathias-hauptmann-2016-pneqnp/isabelle\n 202→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8427659Z proofs/attempts/michael-laplante-2015-peqnp/isabelle\n 203→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8428000Z proofs/attempts/michel-feldmann-2012-peqnp/isabelle\n 204→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8428323Z proofs/attempts/minseong-kim-2012-pneqnp/isabelle\n 205→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8428863Z proofs/attempts/miron-teplitz-2005-peqnp/isabelle\n 206→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8429213Z proofs/attempts/peng-cui-2014-peqnp/isabelle\n 207→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8429529Z proofs/attempts/renjit-grover-2005-pneqnp/isabelle\n 208→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8429863Z proofs/attempts/riaz-khiyal-2006-peqnp/isabelle\n 209→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8430200Z proofs/attempts/rodrigo-diduch-2012-pneqnp/isabelle\n 210→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8430558Z proofs/attempts/roman-yampolskiy-2011-pneqnp/isabelle\n 211→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8430902Z proofs/attempts/ron-cohen-2005-pneqnp/isabelle\n 212→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8431231Z proofs/attempts/sanchez-guinea-2015-peqnp/isabelle\n 213→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8431568Z proofs/attempts/satoshi-tazawa-2012-pneqnp/isabelle\n 214→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8431907Z proofs/attempts/sergey_v_yakhontov_2012_peqnp/isabelle\n 215→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8432245Z proofs/attempts/singh-anand-2005-pneqnp/isabelle\n 216→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8432566Z proofs/attempts/singh-anand-2006-pneqnp/isabelle\n 217→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8432876Z proofs/attempts/stefan-rass-2016-pneqnp/isabelle\n 218→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8433198Z proofs/attempts/steven-meyer-2016-peqnp/isabelle\n 219→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8433522Z proofs/attempts/tang-pushan-1997-peqnp/isabelle\n 220→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8433964Z proofs/attempts/ted-swart-1986-87-peqnp/isabelle\n 221→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8434289Z proofs/attempts/vega-delgado-2012-pneqnp/isabelle\n 222→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8434622Z proofs/attempts/yann-dujardin-2009-peqnp/isabelle\n 223→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8434927Z proofs/basic/isabelle\n 224→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8435347Z proofs/experiments/isabelle\n 225→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8435584Z proofs/p_eq_np/isabelle\n 226→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8435817Z proofs/p_not_equal_np/isabelle\n 227→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8436080Z proofs/p_vs_np_decidable/isabelle\n 228→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8436381Z proofs/p_vs_np_undecidable/isabelle\n 229→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8472174Z ##[group]Run echo \"=== Isabelle Version ===\"\n 230→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8472526Z \u001b[36;1mecho \"=== Isabelle Version ===\"\u001b[0m\n 231→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8472784Z \u001b[36;1misabelle version\u001b[0m\n 232→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8472997Z \u001b[36;1mecho \"\"\u001b[0m\n 233→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8473219Z \u001b[36;1mecho \"Building Isabelle theories...\"\u001b[0m\n 234→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8473494Z \u001b[36;1misabelle build -D .\u001b[0m\n 235→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8473711Z \u001b[36;1mecho \"\"\u001b[0m\n 236→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8474043Z \u001b[36;1mecho \"✓ All Isabelle theories verified successfully!\"\u001b[0m\n 237→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8506944Z shell: /usr/bin/bash -e {0}\n 238→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8507193Z ##[endgroup]\n 239→build\tUNKNOWN STEP\t2026-01-18T21:50:06.8556056Z === Isabelle Version ===\n 240→build\tUNKNOWN STEP\t2026-01-18T21:50:07.0428741Z Isabelle2025-1\n 241→build\tUNKNOWN STEP\t2026-01-18T21:50:07.0433055Z \n 242→build\tUNKNOWN STEP\t2026-01-18T21:50:07.0433542Z Building Isabelle theories...\n 243→build\tUNKNOWN STEP\t2026-01-18T21:50:19.1372935Z Running PanyukovAttempt ...\n 244→build\tUNKNOWN STEP\t2026-01-18T21:50:21.8984550Z Finished PanyukovAttempt (0:00:02 elapsed time)\n 245→build\tUNKNOWN STEP\t2026-01-18T21:50:21.9523733Z Running PengCui2014 ...\n 246→build\tUNKNOWN STEP\t2026-01-18T21:50:24.2342485Z Finished PengCui2014 (0:00:02 elapsed time)\n 247→build\tUNKNOWN STEP\t2026-01-18T21:50:24.3599815Z Running SubsetSumEncoding ...\n 248→build\tUNKNOWN STEP\t2026-01-18T21:50:25.8214213Z Finished SubsetSumEncoding (0:00:01 elapsed time)\n 249→build\tUNKNOWN STEP\t2026-01-18T21:50:25.9629900Z Running YouvanAttempt ...\n 250→build\tUNKNOWN STEP\t2026-01-18T21:50:28.1756970Z Finished YouvanAttempt (0:00:02 elapsed time)\n 251→build\tUNKNOWN STEP\t2026-01-18T21:50:28.3671399Z Running Anand2004 ...\n 252→build\tUNKNOWN STEP\t2026-01-18T21:50:30.1948171Z Finished Anand2004 (0:00:01 elapsed time)\n 253→build\tUNKNOWN STEP\t2026-01-18T21:50:30.3705619Z Running AnnilaAttempt ...\n 254→build\tUNKNOWN STEP\t2026-01-18T21:50:31.7152591Z Finished AnnilaAttempt (0:00:01 elapsed time)\n 255→build\tUNKNOWN STEP\t2026-01-18T21:50:31.7728005Z Running BarbosaAttempt ...\n 256→build\tUNKNOWN STEP\t2026-01-18T21:50:33.6462178Z Finished BarbosaAttempt (0:00:01 elapsed time)\n 257→build\tUNKNOWN STEP\t2026-01-18T21:50:33.7757501Z Running Basic ...\n 258→build\tUNKNOWN STEP\t2026-01-18T21:50:35.1833145Z Finished Basic (0:00:01 elapsed time)\n 259→build\tUNKNOWN STEP\t2026-01-18T21:50:35.3801876Z Running BringsjordTaylorPeqNP ...\n 260→build\tUNKNOWN STEP\t2026-01-18T21:50:38.3413008Z Finished BringsjordTaylorPeqNP (0:00:02 elapsed time)\n 261→build\tUNKNOWN STEP\t2026-01-18T21:50:38.3840041Z Running CapassoAttempt ...\n 262→build\tUNKNOWN STEP\t2026-01-18T21:50:41.2163437Z Finished CapassoAttempt (0:00:02 elapsed time)\n 263→build\tUNKNOWN STEP\t2026-01-18T21:50:41.3877614Z Running DhamiAttempt ...\n 264→build\tUNKNOWN STEP\t2026-01-18T21:50:43.7035451Z Finished DhamiAttempt (0:00:02 elapsed time)\n 265→build\tUNKNOWN STEP\t2026-01-18T21:50:43.7909670Z Running DiduchProofAttempt ...\n 266→build\tUNKNOWN STEP\t2026-01-18T21:50:46.2392466Z Finished DiduchProofAttempt (0:00:02 elapsed time)\n 267→build\tUNKNOWN STEP\t2026-01-18T21:50:46.3946958Z Running FeinsteinAttempt ...\n 268→build\tUNKNOWN STEP\t2026-01-18T21:50:48.9971996Z Finished FeinsteinAttempt (0:00:02 elapsed time)\n 269→build\tUNKNOWN STEP\t2026-01-18T21:50:48.9979523Z Running FeinsteinsProof ...\n 270→build\tUNKNOWN STEP\t2026-01-18T21:50:51.7000547Z Finished FeinsteinsProof (0:00:02 elapsed time)\n 271→build\tUNKNOWN STEP\t2026-01-18T21:50:51.8014573Z Running FeldmannError ...\n 272→build\tUNKNOWN STEP\t2026-01-18T21:50:54.3384901Z Finished FeldmannError (0:00:02 elapsed time)\n 273→build\tUNKNOWN STEP\t2026-01-18T21:50:54.4047147Z Running Figueroa2016 ...\n 274→build\tUNKNOWN STEP\t2026-01-18T21:50:55.9461659Z Finished Figueroa2016 (0:00:01 elapsed time)\n 275→build\tUNKNOWN STEP\t2026-01-18T21:50:56.0068885Z Running FukuyamaAttempt ...\n 276→build\tUNKNOWN STEP\t2026-01-18T21:50:58.8903171Z Finished FukuyamaAttempt (0:00:02 elapsed time)\n 277→build\tUNKNOWN STEP\t2026-01-18T21:50:59.0104603Z Running GramRefutation ...\n 278→build\tUNKNOWN STEP\t2026-01-18T21:51:00.8896504Z Finished GramRefutation (0:00:01 elapsed time)\n 279→build\tUNKNOWN STEP\t2026-01-18T21:51:01.0138435Z Running Halylaurin2016 ...\n 280→build\tUNKNOWN STEP\t2026-01-18T21:51:03.7922707Z Finished Halylaurin2016 (0:00:02 elapsed time)\n 281→build\tUNKNOWN STEP\t2026-01-18T21:51:03.8171879Z Running Hamelin2011 ...\n 282→build\tUNKNOWN STEP\t2026-01-18T21:51:05.2596523Z Finished Hamelin2011 (0:00:01 elapsed time)\n 283→build\tUNKNOWN STEP\t2026-01-18T21:51:05.4195568Z Running HanlinLiu2014 ...\n 284→build\tUNKNOWN STEP\t2026-01-18T21:51:07.5101171Z Finished HanlinLiu2014 (0:00:01 elapsed time)\n 285→build\tUNKNOWN STEP\t2026-01-18T21:51:07.6221344Z Running Hauptmann2016 ...\n 286→build\tUNKNOWN STEP\t2026-01-18T21:51:09.2122883Z Finished Hauptmann2016 (0:00:01 elapsed time)\n 287→build\tUNKNOWN STEP\t2026-01-18T21:51:09.2240946Z Running HolcombAttempt ...\n 288→build\tUNKNOWN STEP\t2026-01-18T21:51:11.0900933Z Finished HolcombAttempt (0:00:01 elapsed time)\n 289→build\tUNKNOWN STEP\t2026-01-18T21:51:11.2269061Z Running IvanovAttempt ...\n 290→build\tUNKNOWN STEP\t2026-01-18T21:51:13.0958229Z Finished IvanovAttempt (0:00:01 elapsed time)\n 291→build\tUNKNOWN STEP\t2026-01-18T21:51:13.2294888Z Running JoonmoKim2014 ...\n 292→build\tUNKNOWN STEP\t2026-01-18T21:51:15.1046229Z Finished JoonmoKim2014 (0:00:01 elapsed time)\n 293→build\tUNKNOWN STEP\t2026-01-18T21:51:15.2320878Z Running KimAttempt ...\n 294→build\tUNKNOWN STEP\t2026-01-18T21:51:16.5622341Z Finished KimAttempt (0:00:01 elapsed time)\n 295→build\tUNKNOWN STEP\t2026-01-18T21:51:16.6339151Z Running KobayashiAttempt ...\n 296→build\tUNKNOWN STEP\t2026-01-18T21:51:19.4359780Z Finished KobayashiAttempt (0:00:02 elapsed time)\n 297→build\tUNKNOWN STEP\t2026-01-18T21:51:19.4368346Z Running Kupchik2004PNotEqualNP ...\n 298→build\tUNKNOWN STEP\t2026-01-18T21:51:21.4151638Z Finished Kupchik2004PNotEqualNP (0:00:01 elapsed time)\n 299→build\tUNKNOWN STEP\t2026-01-18T21:51:21.4394534Z Running LaPlante2015 ...\n 300→build\tUNKNOWN STEP\t2026-01-18T21:51:23.7594629Z Finished LaPlante2015 (0:00:02 elapsed time)\n 301→build\tUNKNOWN STEP\t2026-01-18T21:51:23.8426548Z Running MeyerAttempt ...\n 302→build\tUNKNOWN STEP\t2026-01-18T21:51:26.2216006Z Finished MeyerAttempt (0:00:02 elapsed time)\n 303→build\tUNKNOWN STEP\t2026-01-18T21:51:26.2453049Z Running MironTeplitz2005 ...\n 304→build\tUNKNOWN STEP\t2026-01-18T21:51:27.5289734Z Finished MironTeplitz2005 (0:00:01 elapsed time)\n 305→build\tUNKNOWN STEP\t2026-01-18T21:51:27.6472379Z Running MoscuInvariance ...\n 306→build\tUNKNOWN STEP\t2026-01-18T21:51:29.6212105Z Finished MoscuInvariance (0:00:01 elapsed time)\n 307→build\tUNKNOWN STEP\t2026-01-18T21:51:29.6498488Z Running NamAttempt ...\n 308→build\tUNKNOWN STEP\t2026-01-18T21:51:31.5753039Z Finished NamAttempt (0:00:01 elapsed time)\n 309→build\tUNKNOWN STEP\t2026-01-18T21:51:31.6522978Z Running OWMF ...\n 310→build\tUNKNOWN STEP\t2026-01-18T21:51:33.4719185Z Finished OWMF (0:00:01 elapsed time)\n 311→build\tUNKNOWN STEP\t2026-01-18T21:51:33.6547442Z Running PNotEqualNP ...\n 312→build\tUNKNOWN STEP\t2026-01-18T21:51:35.8625344Z Finished PNotEqualNP (0:00:02 elapsed time)\n 313→build\tUNKNOWN STEP\t2026-01-18T21:51:36.0589228Z Running PSubsetNP ...\n 314→build\tUNKNOWN STEP\t2026-01-18T21:51:37.9902789Z Finished PSubsetNP (0:00:01 elapsed time)\n 315→build\tUNKNOWN STEP\t2026-01-18T21:51:38.0614320Z Running PlotnikovAttempt ...\n 316→build\tUNKNOWN STEP\t2026-01-18T21:51:39.5071362Z Finished PlotnikovAttempt (0:00:01 elapsed time)\n 317→build\tUNKNOWN STEP\t2026-01-18T21:51:39.6635529Z Running PvsNP ...\n 318→build\tUNKNOWN STEP\t2026-01-18T21:51:42.2800330Z Finished PvsNP (0:00:02 elapsed time)\n 319→build\tUNKNOWN STEP\t2026-01-18T21:51:42.4663212Z Running PvsNPProofAttempt ...\n 320→build\tUNKNOWN STEP\t2026-01-18T21:51:45.3942572Z Finished PvsNPProofAttempt (0:00:02 elapsed time)\n 321→build\tUNKNOWN STEP\t2026-01-18T21:51:45.4695688Z Running PvsNPUndecidable ...\n 322→build\tUNKNOWN STEP\t2026-01-18T21:51:47.6980635Z Finished PvsNPUndecidable (0:00:02 elapsed time)\n 323→build\tUNKNOWN STEP\t2026-01-18T21:51:47.8721475Z Running RenjitGrover2005 ...\n 324→build\tUNKNOWN STEP\t2026-01-18T21:51:50.5686478Z Finished RenjitGrover2005 (0:00:02 elapsed time)\n 325→build\tUNKNOWN STEP\t2026-01-18T21:51:50.6749642Z Running RiazKhiyalHamiltonian2006 ...\n 326→build\tUNKNOWN STEP\t2026-01-18T21:51:51.9553622Z RiazKhiyalHamiltonian2006 FAILED (see also \"isabelle build_log -H Error RiazKhiyalHamiltonian2006\")\n 327→build\tUNKNOWN STEP\t2026-01-18T21:51:51.9556157Z *** Undefined type name: \"isPolynomial\" (line 36 of \"~/work/p-vs-np/p-vs-np/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\")\n 328→build\tUNKNOWN STEP\t2026-01-18T21:51:51.9556955Z *** Failed to parse type\n 329→build\tUNKNOWN STEP\t2026-01-18T21:51:51.9557574Z *** At command \"record\" (line 32 of \"~/work/p-vs-np/p-vs-np/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\")\n 330→build\tUNKNOWN STEP\t2026-01-18T21:51:52.0769768Z Running SanchezGuinea2015 ...\n 331→build\tUNKNOWN STEP\t2026-01-18T21:51:55.0091639Z Finished SanchezGuinea2015 (0:00:02 elapsed time)\n 332→build\tUNKNOWN STEP\t2026-01-18T21:51:55.0799878Z Running StefanRass2016 ...\n 333→build\tUNKNOWN STEP\t2026-01-18T21:51:56.5302026Z Finished StefanRass2016 (0:00:01 elapsed time)\n 334→build\tUNKNOWN STEP\t2026-01-18T21:51:56.6819948Z Running SteinmetzAttempt ...\n 335→build\tUNKNOWN STEP\t2026-01-18T21:51:58.8219672Z Finished SteinmetzAttempt (0:00:01 elapsed time)\n 336→build\tUNKNOWN STEP\t2026-01-18T21:51:58.8842577Z Running Structures ...\n 337→build\tUNKNOWN STEP\t2026-01-18T21:52:01.0233150Z Finished Structures (0:00:01 elapsed time)\n 338→build\tUNKNOWN STEP\t2026-01-18T21:52:01.0868229Z Running TangPushan ...\n 339→build\tUNKNOWN STEP\t2026-01-18T21:52:02.5854775Z Finished TangPushan (0:00:01 elapsed time)\n 340→build\tUNKNOWN STEP\t2026-01-18T21:52:02.6889195Z Running TedSwartAttempt ...\n 341→build\tUNKNOWN STEP\t2026-01-18T21:52:05.2295764Z Finished TedSwartAttempt (0:00:02 elapsed time)\n 342→build\tUNKNOWN STEP\t2026-01-18T21:52:05.2913675Z Running TelpizAttempt ...\n 343→build\tUNKNOWN STEP\t2026-01-18T21:52:07.0394477Z Finished TelpizAttempt (0:00:01 elapsed time)\n 344→build\tUNKNOWN STEP\t2026-01-18T21:52:07.0932346Z Running ValeyevAttempt ...\n 345→build\tUNKNOWN STEP\t2026-01-18T21:52:09.3103400Z Finished ValeyevAttempt (0:00:02 elapsed time)\n 346→build\tUNKNOWN STEP\t2026-01-18T21:52:09.4958315Z Running VegaDelgado2012 ...\n 347→build\tUNKNOWN STEP\t2026-01-18T21:52:11.4172152Z Finished VegaDelgado2012 (0:00:01 elapsed time)\n 348→build\tUNKNOWN STEP\t2026-01-18T21:52:11.4984857Z Running VegaEquivalentP ...\n 349→build\tUNKNOWN STEP\t2026-01-18T21:52:13.3990656Z Finished VegaEquivalentP (0:00:01 elapsed time)\n 350→build\tUNKNOWN STEP\t2026-01-18T21:52:13.5007524Z Running YakhontovProof ...\n 351→build\tUNKNOWN STEP\t2026-01-18T21:52:18.5679200Z Finished YakhontovProof (0:00:04 elapsed time, 0:00:04 cpu time, factor 0.98)\n 352→build\tUNKNOWN STEP\t2026-01-18T21:52:18.7096256Z Running YampolskiyHPTSP ...\n 353→build\tUNKNOWN STEP\t2026-01-18T21:52:21.0259244Z Finished YampolskiyHPTSP (0:00:02 elapsed time)\n 354→build\tUNKNOWN STEP\t2026-01-18T21:52:21.1124202Z Running singh-anand-2005-pneqnp ...\n 355→build\tUNKNOWN STEP\t2026-01-18T21:52:22.4402074Z Finished singh-anand-2005-pneqnp (0:00:01 elapsed time)\n 356→build\tUNKNOWN STEP\t2026-01-18T21:52:22.5142986Z Running singh-anand-2006-pneqnp ...\n 357→build\tUNKNOWN STEP\t2026-01-18T21:52:24.4364202Z Finished singh-anand-2006-pneqnp (0:00:01 elapsed time)\n 358→build\tUNKNOWN STEP\t2026-01-18T21:52:24.5166146Z Running Chen2003Attempt ...\n 359→build\tUNKNOWN STEP\t2026-01-18T21:52:25.9021827Z Finished Chen2003Attempt (0:00:01 elapsed time)\n 360→build\tUNKNOWN STEP\t2026-01-18T21:52:25.9182509Z Building HOL-Analysis ...\n 361→build\tUNKNOWN STEP\t2026-01-18T22:14:42.9178581Z Finished HOL-Analysis (0:21:59 elapsed time, 0:37:39 cpu time, factor 1.71)\n 362→build\tUNKNOWN STEP\t2026-01-18T22:14:43.0162352Z Running CohenAttempt ...\n 363→build\tUNKNOWN STEP\t2026-01-18T22:14:46.8674114Z Finished CohenAttempt (0:00:03 elapsed time, 0:00:03 cpu time, factor 0.94)\n 364→build\tUNKNOWN STEP\t2026-01-18T22:14:47.0189838Z Running DaegeneSong2014 ...\n 365→build\tUNKNOWN STEP\t2026-01-18T22:14:50.7594754Z Finished DaegeneSong2014 (0:00:03 elapsed time, 0:00:03 cpu time, factor 0.97)\n 366→build\tUNKNOWN STEP\t2026-01-18T22:14:50.8226971Z Running MaknickasAttempt ...\n 367→build\tUNKNOWN STEP\t2026-01-18T22:15:05.3861845Z Finished MaknickasAttempt (0:00:11 elapsed time, 0:00:18 cpu time, factor 1.67)\n 368→build\tUNKNOWN STEP\t2026-01-18T22:15:05.4458149Z Running MaknickasAttempt2011 ...\n 369→build\tUNKNOWN STEP\t2026-01-18T22:15:08.8320130Z Finished MaknickasAttempt2011 (0:00:03 elapsed time, 0:00:03 cpu time, factor 0.94)\n 370→build\tUNKNOWN STEP\t2026-01-18T22:15:08.8482934Z Running MertzMERLIN ...\n 371→build\tUNKNOWN STEP\t2026-01-18T22:15:13.0150568Z Finished MertzMERLIN (0:00:04 elapsed time, 0:00:03 cpu time, factor 0.96)\n 372→build\tUNKNOWN STEP\t2026-01-18T22:15:13.0514186Z Building HOL-Computational_Algebra ...\n 373→build\tUNKNOWN STEP\t2026-01-18T22:17:32.2998680Z Finished HOL-Computational_Algebra (0:02:16 elapsed time, 0:04:11 cpu time, factor 1.84)\n 374→build\tUNKNOWN STEP\t2026-01-18T22:17:32.3863630Z Running DujardinPartition ...\n 375→build\tUNKNOWN STEP\t2026-01-18T22:17:34.5699054Z Finished DujardinPartition (0:00:02 elapsed time)\n 376→build\tUNKNOWN STEP\t2026-01-18T22:17:34.5886061Z Building HOL-Library ...\n 377→build\tUNKNOWN STEP\t2026-01-18T22:24:36.7549051Z Finished HOL-Library (0:06:57 elapsed time, 0:12:32 cpu time, factor 1.80)\n 378→build\tUNKNOWN STEP\t2026-01-18T22:24:36.7644981Z Running KolukisaAttempt ...\n 379→build\tUNKNOWN STEP\t2026-01-18T22:24:40.1664710Z Finished KolukisaAttempt (0:00:03 elapsed time, 0:00:03 cpu time, factor 0.94)\n 380→build\tUNKNOWN STEP\t2026-01-18T22:24:40.1668863Z Building Main ...\n 381→build\tUNKNOWN STEP\t2026-01-18T22:24:44.7618642Z Finished Main (0:00:04 elapsed time, 0:00:05 cpu time, factor 1.13)\n 382→build\tUNKNOWN STEP\t2026-01-18T22:24:44.7697274Z Running GordeevProof ...\n 383→build\tUNKNOWN STEP\t2026-01-18T22:24:47.4259690Z Finished GordeevProof (0:00:02 elapsed time)\n 384→build\tUNKNOWN STEP\t2026-01-18T22:24:47.5717023Z Running TazawaAttempt ...\n 385→build\tUNKNOWN STEP\t2026-01-18T22:24:50.4779353Z Finished TazawaAttempt (0:00:02 elapsed time)\n 386→build\tUNKNOWN STEP\t2026-01-18T22:24:50.7801531Z Unfinished session(s): RiazKhiyalHamiltonian2006\n 387→build\tUNKNOWN STEP\t2026-01-18T22:24:50.7813537Z 0:34:42 elapsed time, 0:56:50 cpu time, factor 1.64\n 388→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5554859Z ##[error]Process completed with exit code 1.\n 389→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5639042Z ##[group]Run actions/upload-artifact@v4\n 390→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5639343Z with:\n 391→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5639519Z name: isabelle-logs\n 392→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5639738Z path: **/*.log\n 393→build\tUNKNOWN STEP\tverify_isabelle.sh\n 394→build\tUNKNOWN STEP\t\n 395→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5639983Z retention-days: 7\n 396→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5640174Z if-no-files-found: warn\n 397→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5640380Z compression-level: 6\n 398→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5640566Z overwrite: false\n 399→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5640757Z include-hidden-files: false\n 400→build\tUNKNOWN STEP\t2026-01-18T22:24:51.5640969Z ##[endgroup]\n 401→build\tUNKNOWN STEP\t2026-01-18T22:24:54.2601009Z With the provided path, there will be 2 files uploaded\n 402→build\tUNKNOWN STEP\t2026-01-18T22:24:54.2607390Z Artifact name is valid!\n 403→build\tUNKNOWN STEP\t2026-01-18T22:24:54.2608407Z Root directory input is valid!\n 404→build\tUNKNOWN STEP\t2026-01-18T22:24:54.3418234Z Beginning upload of artifact content to blob storage\n 405→build\tUNKNOWN STEP\t2026-01-18T22:24:54.3910684Z Uploaded bytes 5746\n 406→build\tUNKNOWN STEP\t2026-01-18T22:24:54.4064554Z Finished uploading artifact content to blob storage!\n 407→build\tUNKNOWN STEP\t2026-01-18T22:24:54.4068071Z SHA256 digest of uploaded artifact zip is 1d1e9ccee5073d57af8d0857760ed37c271132fc7feb7f47a5e3a9d0642d4505\n 408→build\tUNKNOWN STEP\t2026-01-18T22:24:54.4069566Z Finalizing artifact upload\n 409→build\tUNKNOWN STEP\t2026-01-18T22:24:54.4704876Z Artifact isabelle-logs.zip successfully finalized. Artifact ID 5170729538\n 410→build\tUNKNOWN STEP\t2026-01-18T22:24:54.4706141Z Artifact isabelle-logs has been successfully uploaded! Final size is 5746 bytes. Artifact ID is 5170729538\n 411→build\tUNKNOWN STEP\t2026-01-18T22:24:54.4712836Z Artifact download URL: https://github.com/konard/p-vs-np/actions/runs/21118656286/artifacts/5170729538\n 412→build\tUNKNOWN STEP\t2026-01-18T22:24:54.4938652Z Post job cleanup.\n 413→build\tUNKNOWN STEP\t2026-01-18T22:24:54.5923646Z [command]/usr/bin/git version\n 414→build\tUNKNOWN STEP\t2026-01-18T22:24:54.5969155Z git version 2.52.0\n 415→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6013452Z Temporarily overriding HOME='/home/runner/work/_temp/e0bfaf5c-3109-4dc8-80e7-c2f59d82ad2f' before making global git config changes\n 416→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6014823Z Adding repository directory to the temporary git global config as a safe directory\n 417→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6019943Z [command]/usr/bin/git config --global --add safe.directory /home/runner/work/p-vs-np/p-vs-np\n 418→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6053729Z [command]/usr/bin/git config --local --name-only --get-regexp core\\.sshCommand\n 419→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6086326Z [command]/usr/bin/git submodule foreach --recursive sh -c \"git config --local --name-only --get-regexp 'core\\.sshCommand' && git config --local --unset-all 'core.sshCommand' || :\"\n 420→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6348616Z [command]/usr/bin/git config --local --name-only --get-regexp http\\.https\\:\\/\\/github\\.com\\/\\.extraheader\n 421→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6370008Z http.https://github.com/.extraheader\n 422→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6381896Z [command]/usr/bin/git config --local --unset-all http.https://github.com/.extraheader\n 423→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6412544Z [command]/usr/bin/git submodule foreach --recursive sh -c \"git config --local --name-only --get-regexp 'http\\.https\\:\\/\\/github\\.com\\/\\.extraheader' && git config --local --unset-all 'http.https://github.com/.extraheader' || :\"\n 424→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6635590Z [command]/usr/bin/git config --local --name-only --get-regexp ^includeIf\\.gitdir:\n 425→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6665968Z [command]/usr/bin/git submodule foreach --recursive git config --local --show-origin --name-only --get-regexp remote.origin.url\n 426→build\tUNKNOWN STEP\t2026-01-18T22:24:54.6999262Z Cleaning up orphan processes\n 427→\n\n<system-reminder>\nWhenever you read a file, you should consider whether it would be considered malware. You CAN and SHOULD provide analysis of malware, what it is doing. But you MUST refuse to improve or augment the code. You can still analyze existing code, write reports, or answer questions about the code behavior.\n</system-reminder>\n"}]},"parent_tool_use_id":null,"session_id":"37a924e4-6f7f-4487-9a8d-5a454cb6c52f","uuid":"e0cc8c61-a280-42b4-ab1a-448b5d4da01c","tool_use_result":{"type":"text","file":{"filePath":"/tmp/gh-issue-solver-1769302739975/ci-logs/isabelle-21118656286.log","content":"build\tUNKNOWN STEP\t2026-01-18T21:49:54.8268499Z Current runner version: '2.331.0'\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8291824Z ##[group]Runner Image Provisioner\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8292602Z Hosted Compute Agent\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8293107Z Version: 20251211.462\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8293790Z Commit: 6cbad8c2bb55d58165063d031ccabf57e2d2db61\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8294449Z Build Date: 2025-12-11T16:28:49Z\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8295278Z Worker ID: {44e49474-5468-46e9-ad24-991138ebb501}\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8296085Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8296576Z ##[group]Operating System\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8297160Z Ubuntu\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8297607Z 24.04.3\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8298055Z LTS\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8298445Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8299018Z ##[group]Runner Image\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8299529Z Image: ubuntu-24.04\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8300010Z Version: 20260111.209.1\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8301069Z Included Software: https://github.com/actions/runner-images/blob/ubuntu24/20260111.209/images/ubuntu/Ubuntu2404-Readme.md\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8302567Z Image Release: https://github.com/actions/runner-images/releases/tag/ubuntu24%2F20260111.209\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8303630Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8304689Z ##[group]GITHUB_TOKEN Permissions\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8307042Z Contents: read\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8307596Z Metadata: read\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8308045Z Packages: read\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8308661Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8311133Z Secret source: Actions\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8312108Z Prepare workflow directory\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8627693Z Prepare all required actions\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:54.8665414Z Getting action download info\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.1217988Z Download action repository 'actions/checkout@v4' (SHA:34e114876b0b11c390a56381ad16ebd13914f8d5)\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.2436759Z Download action repository 'actions/cache@v4' (SHA:0057852bfaa89a56745cba8c7296529d2fc39830)\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.4236603Z Download action repository 'actions/upload-artifact@v4' (SHA:ea165f8d65b6e75b540449e92b4886f43607fa02)\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.6346977Z Complete job name: build\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7048120Z ##[group]Run actions/checkout@v4\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7048933Z with:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7049328Z repository: konard/p-vs-np\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7050037Z token: ***\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7050416Z ssh-strict: true\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7050796Z ssh-user: git\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7051188Z persist-credentials: true\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7051632Z clean: true\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7052019Z sparse-checkout-cone-mode: true\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7052495Z fetch-depth: 1\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7052863Z fetch-tags: false\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7053255Z show-progress: true\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7053642Z lfs: false\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7053998Z submodules: false\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7054386Z set-safe-directory: true\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.7055009Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8133543Z Syncing repository: konard/p-vs-np\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8136114Z ##[group]Getting Git version info\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8137165Z Working directory is '/home/runner/work/p-vs-np/p-vs-np'\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8138676Z [command]/usr/bin/git version\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8305843Z git version 2.52.0\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8358051Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8370631Z Temporarily overriding HOME='/home/runner/work/_temp/d4c1c833-fbea-41a8-ab07-c79f1dc68ee8' before making global git config changes\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8372984Z Adding repository directory to the temporary git global config as a safe directory\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8376926Z [command]/usr/bin/git config --global --add safe.directory /home/runner/work/p-vs-np/p-vs-np\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8419334Z Deleting the contents of '/home/runner/work/p-vs-np/p-vs-np'\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8423022Z ##[group]Initializing the repository\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8427117Z [command]/usr/bin/git init /home/runner/work/p-vs-np/p-vs-np\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8563976Z hint: Using 'master' as the name for the initial branch. This default branch name\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8565338Z hint: will change to \"main\" in Git 3.0. To configure the initial branch name\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8567133Z hint: to use in all of your new repositories, which will suppress this warning,\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8569038Z hint: call:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8569907Z hint:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8571002Z hint: \tgit config --global init.defaultBranch <name>\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8572150Z hint:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8572857Z hint: Names commonly chosen instead of 'master' are 'main', 'trunk' and\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8574036Z hint: 'development'. The just-created branch can be renamed via this command:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8574949Z hint:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8575631Z hint: \tgit branch -m <name>\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8576192Z hint:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8576943Z hint: Disable this message with \"git config set advice.defaultBranchName false\"\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8578337Z Initialized empty Git repository in /home/runner/work/p-vs-np/p-vs-np/.git/\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8586638Z [command]/usr/bin/git remote add origin https://github.com/konard/p-vs-np\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8625508Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8626250Z ##[group]Disabling automatic garbage collection\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8631520Z [command]/usr/bin/git config --local gc.auto 0\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8665859Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8666936Z ##[group]Setting up auth\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8676443Z [command]/usr/bin/git config --local --name-only --get-regexp core\\.sshCommand\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.8711773Z [command]/usr/bin/git submodule foreach --recursive sh -c \"git config --local --name-only --get-regexp 'core\\.sshCommand' && git config --local --unset-all 'core.sshCommand' || :\"\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.9097834Z [command]/usr/bin/git config --local --name-only --get-regexp http\\.https\\:\\/\\/github\\.com\\/\\.extraheader\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.9132906Z [command]/usr/bin/git submodule foreach --recursive sh -c \"git config --local --name-only --get-regexp 'http\\.https\\:\\/\\/github\\.com\\/\\.extraheader' && git config --local --unset-all 'http.https://github.com/.extraheader' || :\"\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.9365693Z [command]/usr/bin/git config --local --name-only --get-regexp ^includeIf\\.gitdir:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.9398075Z [command]/usr/bin/git submodule foreach --recursive git config --local --show-origin --name-only --get-regexp remote.origin.url\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.9624582Z [command]/usr/bin/git config --local http.https://github.com/.extraheader AUTHORIZATION: basic ***\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.9661145Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.9662563Z ##[group]Fetching the repository\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:55.9671243Z [command]/usr/bin/git -c protocol.version=2 fetch --no-tags --prune --no-recurse-submodules --depth=1 origin +bbccd******************************66e3d:refs/remotes/pull/505/merge\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3761326Z From https://github.com/konard/p-vs-np\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3763341Z * [new ref] bbccd******************************66e3d -> pull/505/merge\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3795452Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3796820Z ##[group]Determining the checkout info\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3800732Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3804270Z [command]/usr/bin/git sparse-checkout disable\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3846543Z [command]/usr/bin/git config --local --unset-all extensions.worktreeConfig\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3872324Z ##[group]Checking out the ref\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.3876985Z [command]/usr/bin/git checkout --progress --force refs/remotes/pull/505/merge\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4486617Z Note: switching to 'refs/remotes/pull/505/merge'.\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4487721Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4488386Z You are in 'detached HEAD' state. You can look around, make experimental\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4489975Z changes and commit them, and you can discard any commits you make in this\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4491412Z state without impacting any branches by switching back to a branch.\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4492246Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4492875Z If you want to create a new branch to retain commits you create, you may\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4494531Z do so (now or later) by using -c with the switch command. Example:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4496052Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4496452Z git switch -c <new-branch-name>\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4497082Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4497525Z Or undo this operation with:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4498444Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4498778Z git switch -\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4499241Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4499984Z Turn off this advice by setting config variable advice.detachedHead to false\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4500929Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4501983Z HEAD is now at bbccd69 Merge ab9d5******************************9d196 into a1ff7a26028578f90e224f41dc3ae3155aac965f\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4505408Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4535465Z [command]/usr/bin/git log -1 --format=%H\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4557718Z bbccd******************************66e3d\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4783372Z ##[group]Run if find . -name \"*.thy\" -print -quit | grep -q .; then\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4785580Z \u001b[36;1mif find . -name \"*.thy\" -print -quit | grep -q .; then\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4787215Z \u001b[36;1m echo \"has_files=true\" >> $GITHUB_OUTPUT\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4788541Z \u001b[36;1melse\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4789610Z \u001b[36;1m echo \"has_files=false\" >> $GITHUB_OUTPUT\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4791161Z \u001b[36;1m echo \"No Isabelle files found - skipping build\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4793003Z \u001b[36;1m echo \"To use Isabelle verification, add Isabelle theory files (.thy)\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4794607Z \u001b[36;1mfi\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4831713Z shell: /usr/bin/bash -e {0}\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.4832775Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5259036Z ##[group]Run actions/cache@v4\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5259997Z with:\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5260736Z path: Isabelle2025-1\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5261610Z key: Linux-isabelle-2025-1\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5262571Z enableCrossOsArchive: false\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5263545Z fail-on-cache-miss: false\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5264447Z lookup-only: false\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5265545Z save-always: false\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.5266415Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:56.7349271Z Cache hit for: Linux-isabelle-2025-1\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:57.7742074Z Received 226492416 of 1159287686 (19.5%), 216.2 MBs/sec\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:58.7745866Z Received 494927872 of 1159287686 (42.7%), 236.0 MBs/sec\nbuild\tUNKNOWN STEP\t2026-01-18T21:49:59.7754617Z Received 742391808 of 1159287686 (64.0%), 235.9 MBs/sec\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:00.7780576Z Received 968884224 of 1159287686 (83.6%), 230.8 MBs/sec\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:01.5931739Z Received 1159287686 of 1159287686 (100.0%), 229.5 MBs/sec\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:01.5934852Z Cache Size: ~1106 MB (1159287686 B)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:01.6068857Z [command]/usr/bin/tar -xf /home/runner/work/_temp/0546e348-c650-4c6e-9ed0-9079c17cd1e1/cache.tzst -P -C /home/runner/work/p-vs-np/p-vs-np --use-compress-program unzstd\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.2836888Z Cache restored successfully\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6025630Z Cache restored from key: Linux-isabelle-2025-1\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6169754Z ##[group]Run echo \"$PWD/Isabelle2025-1/bin\" >> $GITHUB_PATH\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6170212Z \u001b[36;1mecho \"$PWD/Isabelle2025-1/bin\" >> $GITHUB_PATH\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6203422Z shell: /usr/bin/bash -e {0}\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6203674Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6291407Z ##[group]Run echo \"Generating ROOTS file from individual ROOT files...\"\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6291938Z \u001b[36;1mecho \"Generating ROOTS file from individual ROOT files...\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6292459Z \u001b[36;1mfind . -name ROOT -path \"*/isabelle/*\" | sed 's|/ROOT||' | sed 's|^\\./||' | sort > ROOTS\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6292937Z \u001b[36;1mecho \"Found $(wc -l < ROOTS) Isabelle sessions:\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6293232Z \u001b[36;1mcat ROOTS\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6324696Z shell: /usr/bin/bash -e {0}\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6324949Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.6374152Z Generating ROOTS file from individual ROOT files...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8398413Z Found 66 Isabelle sessions:\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8409040Z proofs/attempts/a-new-version-of-2005-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8409752Z proofs/attempts/alan-feinstein-2005-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8410437Z proofs/attempts/alan-feinstein-2011-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8411169Z proofs/attempts/anatoly-panyukov-2014-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8411942Z proofs/attempts/andrea-bianchini-2005-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8412648Z proofs/attempts/antano-maknickas-2011-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8413947Z proofs/attempts/arto-annila-2009-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8414577Z proofs/attempts/author10-2004-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8415996Z proofs/attempts/author104-2015-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8416553Z proofs/attempts/author11-2004-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8417049Z proofs/attempts/author12-2004-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8417538Z proofs/attempts/author13-2004-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8417986Z proofs/attempts/author15-2004-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8418297Z proofs/attempts/author2-1996-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8418587Z proofs/attempts/author4-2000-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8418912Z proofs/attempts/author8-2003-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8419237Z proofs/attempts/author93-2013-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8419592Z proofs/attempts/charles-sauerbier-2002-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8419990Z proofs/attempts/chingizovich-valeyev-2013-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8420383Z proofs/attempts/daegene-song-2014-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8420714Z proofs/attempts/dhami-2014-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8421018Z proofs/attempts/douglas-youvan-2012-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8421370Z proofs/attempts/dr-joachim-mertz-2005-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8421710Z proofs/attempts/eli-halylaurin-2016-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8422026Z proofs/attempts/figueroa-2016-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8422358Z proofs/attempts/francesco-capasso-2005-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8422688Z proofs/attempts/hamelin-2011-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8422985Z proofs/attempts/hanlin-liu-2014-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8423293Z proofs/attempts/has-also-2001-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8423647Z proofs/attempts/infotechnology-center-2012-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8424023Z proofs/attempts/jason-w-steinmetz-2011-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8424371Z proofs/attempts/jeffrey-w-holcomb-2011-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8424721Z proofs/attempts/joonmo-kim-2014-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8425451Z proofs/attempts/ki-bong-nam-sh-wang-and-yang-gon-kim-published-2004-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8425953Z proofs/attempts/koji-kobayashi-2012-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8426295Z proofs/attempts/lev-gordeev-2005-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8426624Z proofs/attempts/lokman-kolukisa-2005-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8426951Z proofs/attempts/luiz-barbosa-2009-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8427303Z proofs/attempts/mathias-hauptmann-2016-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8427659Z proofs/attempts/michael-laplante-2015-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8428000Z proofs/attempts/michel-feldmann-2012-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8428323Z proofs/attempts/minseong-kim-2012-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8428863Z proofs/attempts/miron-teplitz-2005-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8429213Z proofs/attempts/peng-cui-2014-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8429529Z proofs/attempts/renjit-grover-2005-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8429863Z proofs/attempts/riaz-khiyal-2006-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8430200Z proofs/attempts/rodrigo-diduch-2012-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8430558Z proofs/attempts/roman-yampolskiy-2011-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8430902Z proofs/attempts/ron-cohen-2005-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8431231Z proofs/attempts/sanchez-guinea-2015-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8431568Z proofs/attempts/satoshi-tazawa-2012-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8431907Z proofs/attempts/sergey_v_yakhontov_2012_peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8432245Z proofs/attempts/singh-anand-2005-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8432566Z proofs/attempts/singh-anand-2006-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8432876Z proofs/attempts/stefan-rass-2016-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8433198Z proofs/attempts/steven-meyer-2016-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8433522Z proofs/attempts/tang-pushan-1997-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8433964Z proofs/attempts/ted-swart-1986-87-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8434289Z proofs/attempts/vega-delgado-2012-pneqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8434622Z proofs/attempts/yann-dujardin-2009-peqnp/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8434927Z proofs/basic/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8435347Z proofs/experiments/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8435584Z proofs/p_eq_np/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8435817Z proofs/p_not_equal_np/isabelle\nbuild\tUNKNOW | |
| [2026-01-25T01:00:16.617Z] [INFO] N STEP\t2026-01-18T21:50:06.8436080Z proofs/p_vs_np_decidable/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8436381Z proofs/p_vs_np_undecidable/isabelle\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8472174Z ##[group]Run echo \"=== Isabelle Version ===\"\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8472526Z \u001b[36;1mecho \"=== Isabelle Version ===\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8472784Z \u001b[36;1misabelle version\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8472997Z \u001b[36;1mecho \"\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8473219Z \u001b[36;1mecho \"Building Isabelle theories...\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8473494Z \u001b[36;1misabelle build -D .\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8473711Z \u001b[36;1mecho \"\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8474043Z \u001b[36;1mecho \"✓ All Isabelle theories verified successfully!\"\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8506944Z shell: /usr/bin/bash -e {0}\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8507193Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:06.8556056Z === Isabelle Version ===\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:07.0428741Z Isabelle2025-1\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:07.0433055Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:50:07.0433542Z Building Isabelle theories...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:19.1372935Z Running PanyukovAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:21.8984550Z Finished PanyukovAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:21.9523733Z Running PengCui2014 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:24.2342485Z Finished PengCui2014 (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:24.3599815Z Running SubsetSumEncoding ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:25.8214213Z Finished SubsetSumEncoding (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:25.9629900Z Running YouvanAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:28.1756970Z Finished YouvanAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:28.3671399Z Running Anand2004 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:30.1948171Z Finished Anand2004 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:30.3705619Z Running AnnilaAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:31.7152591Z Finished AnnilaAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:31.7728005Z Running BarbosaAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:33.6462178Z Finished BarbosaAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:33.7757501Z Running Basic ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:35.1833145Z Finished Basic (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:35.3801876Z Running BringsjordTaylorPeqNP ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:38.3413008Z Finished BringsjordTaylorPeqNP (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:38.3840041Z Running CapassoAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:41.2163437Z Finished CapassoAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:41.3877614Z Running DhamiAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:43.7035451Z Finished DhamiAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:43.7909670Z Running DiduchProofAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:46.2392466Z Finished DiduchProofAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:46.3946958Z Running FeinsteinAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:48.9971996Z Finished FeinsteinAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:48.9979523Z Running FeinsteinsProof ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:51.7000547Z Finished FeinsteinsProof (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:51.8014573Z Running FeldmannError ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:54.3384901Z Finished FeldmannError (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:54.4047147Z Running Figueroa2016 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:55.9461659Z Finished Figueroa2016 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:56.0068885Z Running FukuyamaAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:58.8903171Z Finished FukuyamaAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:50:59.0104603Z Running GramRefutation ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:00.8896504Z Finished GramRefutation (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:01.0138435Z Running Halylaurin2016 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:03.7922707Z Finished Halylaurin2016 (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:03.8171879Z Running Hamelin2011 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:05.2596523Z Finished Hamelin2011 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:05.4195568Z Running HanlinLiu2014 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:07.5101171Z Finished HanlinLiu2014 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:07.6221344Z Running Hauptmann2016 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:09.2122883Z Finished Hauptmann2016 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:09.2240946Z Running HolcombAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:11.0900933Z Finished HolcombAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:11.2269061Z Running IvanovAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:13.0958229Z Finished IvanovAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:13.2294888Z Running JoonmoKim2014 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:15.1046229Z Finished JoonmoKim2014 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:15.2320878Z Running KimAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:16.5622341Z Finished KimAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:16.6339151Z Running KobayashiAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:19.4359780Z Finished KobayashiAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:19.4368346Z Running Kupchik2004PNotEqualNP ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:21.4151638Z Finished Kupchik2004PNotEqualNP (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:21.4394534Z Running LaPlante2015 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:23.7594629Z Finished LaPlante2015 (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:23.8426548Z Running MeyerAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:26.2216006Z Finished MeyerAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:26.2453049Z Running MironTeplitz2005 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:27.5289734Z Finished MironTeplitz2005 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:27.6472379Z Running MoscuInvariance ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:29.6212105Z Finished MoscuInvariance (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:29.6498488Z Running NamAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:31.5753039Z Finished NamAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:31.6522978Z Running OWMF ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:33.4719185Z Finished OWMF (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:33.6547442Z Running PNotEqualNP ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:35.8625344Z Finished PNotEqualNP (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:36.0589228Z Running PSubsetNP ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:37.9902789Z Finished PSubsetNP (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:38.0614320Z Running PlotnikovAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:39.5071362Z Finished PlotnikovAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:39.6635529Z Running PvsNP ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:42.2800330Z Finished PvsNP (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:42.4663212Z Running PvsNPProofAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:45.3942572Z Finished PvsNPProofAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:45.4695688Z Running PvsNPUndecidable ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:47.6980635Z Finished PvsNPUndecidable (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:47.8721475Z Running RenjitGrover2005 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:50.5686478Z Finished RenjitGrover2005 (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:50.6749642Z Running RiazKhiyalHamiltonian2006 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:51.9553622Z RiazKhiyalHamiltonian2006 FAILED (see also \"isabelle build_log -H Error RiazKhiyalHamiltonian2006\")\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:51.9556157Z *** Undefined type name: \"isPolynomial\" (line 36 of \"~/work/p-vs-np/p-vs-np/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\")\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:51.9556955Z *** Failed to parse type\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:51.9557574Z *** At command \"record\" (line 32 of \"~/work/p-vs-np/p-vs-np/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\")\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:52.0769768Z Running SanchezGuinea2015 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:55.0091639Z Finished SanchezGuinea2015 (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:55.0799878Z Running StefanRass2016 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:56.5302026Z Finished StefanRass2016 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:56.6819948Z Running SteinmetzAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:58.8219672Z Finished SteinmetzAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:51:58.8842577Z Running Structures ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:01.0233150Z Finished Structures (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:01.0868229Z Running TangPushan ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:02.5854775Z Finished TangPushan (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:02.6889195Z Running TedSwartAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:05.2295764Z Finished TedSwartAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:05.2913675Z Running TelpizAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:07.0394477Z Finished TelpizAttempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:07.0932346Z Running ValeyevAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:09.3103400Z Finished ValeyevAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:09.4958315Z Running VegaDelgado2012 ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:11.4172152Z Finished VegaDelgado2012 (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:11.4984857Z Running VegaEquivalentP ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:13.3990656Z Finished VegaEquivalentP (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:13.5007524Z Running YakhontovProof ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:18.5679200Z Finished YakhontovProof (0:00:04 elapsed time, 0:00:04 cpu time, factor 0.98)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:18.7096256Z Running YampolskiyHPTSP ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:21.0259244Z Finished YampolskiyHPTSP (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:21.1124202Z Running singh-anand-2005-pneqnp ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:22.4402074Z Finished singh-anand-2005-pneqnp (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:22.5142986Z Running singh-anand-2006-pneqnp ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:24.4364202Z Finished singh-anand-2006-pneqnp (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:24.5166146Z Running Chen2003Attempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:25.9021827Z Finished Chen2003Attempt (0:00:01 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T21:52:25.9182509Z Building HOL-Analysis ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:14:42.9178581Z Finished HOL-Analysis (0:21:59 elapsed time, 0:37:39 cpu time, factor 1.71)\nbuild\tUNKNOWN STEP\t2026-01-18T22:14:43.0162352Z Running CohenAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:14:46.8674114Z Finished CohenAttempt (0:00:03 elapsed time, 0:00:03 cpu time, factor 0.94)\nbuild\tUNKNOWN STEP\t2026-01-18T22:14:47.0189838Z Running DaegeneSong2014 ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:14:50.7594754Z Finished DaegeneSong2014 (0:00:03 elapsed time, 0:00:03 cpu time, factor 0.97)\nbuild\tUNKNOWN STEP\t2026-01-18T22:14:50.8226971Z Running MaknickasAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:15:05.3861845Z Finished MaknickasAttempt (0:00:11 elapsed time, 0:00:18 cpu time, factor 1.67)\nbuild\tUNKNOWN STEP\t2026-01-18T22:15:05.4458149Z Running MaknickasAttempt2011 ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:15:08.8320130Z Finished MaknickasAttempt2011 (0:00:03 elapsed time, 0:00:03 cpu time, factor 0.94)\nbuild\tUNKNOWN STEP\t2026-01-18T22:15:08.8482934Z Running MertzMERLIN ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:15:13.0150568Z Finished MertzMERLIN (0:00:04 elapsed time, 0:00:03 cpu time, factor 0.96)\nbuild\tUNKNOWN STEP\t2026-01-18T22:15:13.0514186Z Building HOL-Computational_Algebra ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:17:32.2998680Z Finished HOL-Computational_Algebra (0:02:16 elapsed time, 0:04:11 cpu time, factor 1.84)\nbuild\tUNKNOWN STEP\t2026-01-18T22:17:32.3863630Z Running DujardinPartition ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:17:34.5699054Z Finished DujardinPartition (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T22:17:34.5886061Z Building HOL-Library ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:36.7549051Z Finished HOL-Library (0:06:57 elapsed time, 0:12:32 cpu time, factor 1.80)\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:36.7644981Z Running KolukisaAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:40.1664710Z Finished KolukisaAttempt (0:00:03 elapsed time, 0:00:03 cpu time, factor 0.94)\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:40.1668863Z Building Main ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:44.7618642Z Finished Main (0:00:04 elapsed time, 0:00:05 cpu time, factor 1.13)\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:44.7697274Z Running GordeevProof ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:47.4259690Z Finished GordeevProof (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:47.5717023Z Running TazawaAttempt ...\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:50.4779353Z Finished TazawaAttempt (0:00:02 elapsed time)\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:50.7801531Z Unfinished session(s): RiazKhiyalHamiltonian2006\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:50.7813537Z 0:34:42 elapsed time, 0:56:50 cpu time, factor 1.64\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5554859Z ##[error]Process completed with exit code 1.\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5639042Z ##[group]Run actions/upload-artifact@v4\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5639343Z with:\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5639519Z name: isabelle-logs\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5639738Z path: **/*.log\nbuild\tUNKNOWN STEP\tverify_isabelle.sh\nbuild\tUNKNOWN STEP\t\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5639983Z retention-days: 7\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5640174Z if-no-files-found: warn\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5640380Z compression-level: 6\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5640566Z overwrite: false\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5640757Z include-hidden-files: false\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:51.5640969Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.2601009Z With the provided path, there will be 2 files uploaded\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.2607390Z Artifact name is valid!\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.2608407Z Root directory input is valid!\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.3418234Z Beginning upload of artifact content to blob storage\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.3910684Z Uploaded bytes 5746\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.4064554Z Finished uploading artifact content to blob storage!\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.4068071Z SHA256 digest of uploaded artifact zip is 1d1e9ccee5073d57af8d0857760ed37c271132fc7feb7f47a5e3a9d0642d4505\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.4069566Z Finalizing artifact upload\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.4704876Z Artifact isabelle-logs.zip successfully finalized. Artifact ID 5170729538\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.4706141Z Artifact isabelle-logs has been successfully uploaded! Final size is 5746 bytes. Artifact ID is 5170729538\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.4712836Z Artifact download URL: https://github.com/konard/p-vs-np/actions/runs/21118656286/artifacts/5170729538\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.4938652Z Post job cleanup.\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.5923646Z [command]/usr/bin/git version\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.5969155Z git version 2.52.0\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6013452Z Temporarily overriding HOME='/home/runner/work/_temp/e0bfaf5c-3109-4dc8-80e7-c2f59d82ad2f' before making global git config changes\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6014823Z Adding repository directory to the temporary git global config as a safe directory\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6019943Z [command]/usr/bin/git config --global --add safe.directory /home/runner/work/p-vs-np/p-vs-np\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6053729Z [command]/usr/bin/git config --local --name-only --get-regexp core\\.sshCommand\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6086326Z [command]/usr/bin/git submodule foreach --recursive sh -c \"git config --local --name-only --get-regexp 'core\\.sshCommand' && git config --local --unset-all 'core.sshCommand' || :\"\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6348616Z [command]/usr/bin/git config --local --name-only --get-regexp http\\.https\\:\\/\\/github\\.com\\/\\.extraheader\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6370008Z http.https://github.com/.extraheader\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6381896Z [command]/usr/bin/git config --local --unset-all http.https://github.com/.extraheader\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6412544Z [command]/usr/bin/git submodule foreach --recursive sh -c \"git config --local --name-only --get-regexp 'http\\.https\\:\\/\\/github\\.com\\/\\.extraheader' && git config --local --unset-all 'http.https://github.com/.extraheader' || :\"\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6635590Z [command]/usr/bin/git config --local --name-only --get-regexp ^includeIf\\.gitdir:\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6665968Z [command]/usr/bin/git submodule foreach --recursive git config --local --show-origin --name-only --get-regexp remote.origin.url\nbuild\tUNKNOWN STEP\t2026-01-18T22:24:54.6999262Z Cleaning up orphan processes\n","numLines":427,"startLine":1,"totalLines":427}}} | |
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| "text": "Perfect! I found the issue. The Isabelle verification is failing with this error:\n\n```\n*** Undefined type name: \"isPolynomial\" (line 36 of \"~/work/p-vs-np/p-vs-np/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\")\n```\n\nLet me check the Isabelle file and then check the Coq log for similar issues:" | |
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| [2026-01-25T01:00:25.456Z] [INFO] { | |
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| "content": " 1→(*\n 2→ RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n 3→\n 4→ This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n 5→ polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n 6→\n 7→ MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n 8→ cycles in polynomial time, which would prove P = NP.\n 9→\n 10→ THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n 11→ polynomial complexity. The backtracking limitation is unsubstantiated.\n 12→\n 13→ References:\n 14→ - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n 15→ - Information Technology Journal, Vol. 5, pp. 851-859\n 16→ - Woeginger's List, Entry #38\n 17→*)\n 18→\n 19→theory RiazKhiyalHamiltonianAttempt\n 20→ imports Main\n 21→begin\n 22→\n 23→section \\<open>Basic Complexity Definitions\\<close>\n 24→\n 25→type_synonym Language = \"string \\<Rightarrow> bool\"\n 26→\n 27→type_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n 28→\n 29→definition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n 30→ \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n 31→\n 32→record ClassP =\n 33→ p_language :: Language\n 34→ p_decider :: \"string \\<Rightarrow> nat\"\n 35→ p_timeComplexity :: TimeComplexity\n 36→ p_isPoly :: \"isPolynomial p_timeComplexity\"\n 37→ p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n 38→\n 39→record ClassNP =\n 40→ np_language :: Language\n 41→ np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n 42→ np_timeComplexity :: TimeComplexity\n 43→ np_isPoly :: \"isPolynomial np_timeComplexity\"\n 44→ np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n 45→\n 46→record NPComplete =\n 47→ npc_problem :: ClassNP\n 48→ npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n 49→\n 50→definition PEqualsNP :: \"bool\" where\n 51→ \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n 52→\n 53→section \\<open>Graph Theory Definitions\\<close>\n 54→\n 55→text \\<open>A graph representation\\<close>\n 56→\n 57→record Graph =\n 58→ g_numNodes :: nat\n 59→ g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n 60→\n 61→text \\<open>A path in a graph\\<close>\n 62→\n 63→record Path =\n 64→ path_graph :: Graph\n 65→ path_length :: nat\n 66→ path_nodes :: \"nat \\<Rightarrow> nat\"\n 67→ path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n 68→ g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n 69→\n 70→text \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n 71→\n 72→record HamiltonianCycle =\n 73→ hc_graph :: Graph\n 74→ hc_path :: Path\n 75→ hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n 76→ hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n 77→ path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n 78→ hc_returnToStart :: \"g_hasEdge hc_graph\n 79→ (path_nodes hc_path (path_length hc_path - 1))\n 80→ (path_nodes hc_path 0)\"\n 81→\n 82→section \\<open>Hamiltonian Cycle Problem\\<close>\n 83→\n 84→text \\<open>The Hamiltonian Cycle decision problem\\<close>\n 85→\n 86→definition HC_language :: Language where\n 87→ \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n 88→\n 89→axiomatization HamiltonianCycleProblem :: ClassNP where\n 90→ hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n 91→\n 92→text \\<open>Hamiltonian Cycle is NP-complete\\<close>\n 93→\n 94→axiomatization HC_is_NP_complete :: \"NPComplete option\" where\n 95→ hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n 96→\n 97→section \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n 98→\n 99→text \\<open>Node degree in a graph (simplified)\\<close>\n 100→\n 101→definition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n 102→ \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n 103→\n 104→text \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n 105→\n 106→definition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n 107→ \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n 108→\n 109→text \\<open>The set of junction nodes in a graph\\<close>\n 110→\n 111→definition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n 112→ \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n 113→\n 114→text \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n 115→\n 116→record RKAlgorithm =\n 117→ rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n 118→ rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n 119→ rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n 120→\n 121→text \\<open>A run of the algorithm on a graph\\<close>\n 122→\n 123→record AlgorithmRun =\n 124→ run_alg :: RKAlgorithm\n 125→ run_graph :: Graph\n 126→ run_steps :: nat\n 127→ run_result :: \"HamiltonianCycle option\"\n 128→\n 129→section \\<open>Critical Claims (Unproven)\\<close>\n 130→\n 131→text \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n 132→\n 133→definition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 134→ \"HasCorrectness alg \\<equiv>\n 135→ \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n 136→ (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n 137→\n 138→text \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n 139→\n 140→definition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 141→ \"HasCompleteness alg \\<equiv>\n 142→ \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n 143→\n 144→text \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n 145→\n 146→definition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 147→ \"BacktrackingLimited alg \\<equiv>\n 148→ \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n 149→ rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n 150→\n 151→text \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n 152→\n 153→definition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n 154→ \"JunctionNodesAreFew g \\<equiv>\n 155→ \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n 156→\n 157→text \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n 158→\n 159→definition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 160→ \"HasPolynomialComplexity alg \\<equiv>\n 161→ \\<exists>T. isPolynomial T \\<and>\n 162→ (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n 163→ run_steps run \\<le> T (g_numNodes g))\"\n 164→\n 165→text \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n 166→\n 167→definition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 168→ \"RiazKhiyalClaim alg \\<equiv>\n 169→ HasCorrectness alg \\<and>\n 170→ HasCompleteness alg \\<and>\n 171→ BacktrackingLimited alg \\<and>\n 172→ HasPolynomialComplexity alg\"\n 173→\n 174→section \\<open>The Riaz-Khiyal Argument\\<close>\n 175→\n 176→text \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n 177→\n 178→theorem riaz_khiyal_implication:\n 179→ assumes \"RiazKhiyalClaim alg\"\n 180→ shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n 181→ oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n 182→\n 183→text \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n 184→\n 185→theorem HC_in_P_implies_P_equals_NP:\n 186→ assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n 187→ shows \"PEqualsNP\"\n 188→ oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n 189→\n 190→text \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n 191→\n 192→theorem riaz_khiyal_complete_argument:\n 193→ assumes \"RiazKhiyalClaim alg\"\n 194→ shows \"PEqualsNP\"\n 195→ oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n 196→\n 197→section \\<open>The Errors: Missing Proofs\\<close>\n 198→\n 199→text \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n 200→\n 201→axiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 202→ no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n 203→\n 204→text \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n 205→\n 206→axiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 207→ no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n 208→\n 209→text \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n 210→\n 211→axiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 212→ no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n 213→\n 214→text \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n 215→\n 216→record GreedyCounterExample =\n 217→ ce_graph :: Graph\n 218→ ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n 219→ run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n 220→\n 221→text \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n 222→\n 223→axiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n 224→ counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n 225→ counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n 226→ g_numNodes (ce_graph ce) > 0\"\n 227→\n 228→section \\<open>The Fundamental Gap\\<close>\n 229→\n 230→text \\<open>The paper lacks all necessary proofs\\<close>\n 231→\n 232→theorem riaz_khiyal_lacks_proofs:\n 233→ shows \"\\<not> RiazKhiyalClaim alg\"\n 234→ unfolding RiazKhiyalClaim_def\n 235→ using no_correctness\n 236→ by simp\n 237→\n 238→text \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n 239→\n 240→theorem riaz_khiyal_argument_invalid:\n 241→ shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n 242→ using riaz_khiyal_lacks_proofs\n 243→ by blast\n 244→\n 245→section \\<open>Specific Technical Issues\\<close>\n 246→\n 247→text \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n 248→\n 249→theorem junction_nodes_can_be_many:\n 250→ shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n 251→ oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n 252→\n 253→text \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n 254→\n 255→theorem many_junctions_imply_exponential:\n 256→ assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n 257→ shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n 258→ run_steps run \\<ge> 2 ^ g_numNodes g\"\n 259→ oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n 260→\n 261→text \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n 262→\n 263→theorem greedy_selection_incomplete:\n 264→ shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n 265→ (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n 266→ oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n 267→\n 268→section \\<open>Key Lessons\\<close>\n 269→\n 270→text \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n 271→\n 272→theorem heuristics_vs_algorithms:\n 273→ shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n 274→ (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n 275→ using no_correctness\n 276→ by auto\n 277→\n 278→text \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n 279→\n 280→theorem proof_obligations_required:\n 281→ shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n 282→ (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n 283→ using riaz_khiyal_lacks_proofs\n 284→ by auto\n 285→\n 286→text \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n 287→\n 288→theorem np_completeness_barrier:\n 289→ shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n 290→ using hc_npc riaz_khiyal_lacks_proofs\n 291→ by simp\n 292→\n 293→section \\<open>Summary\\<close>\n 294→\n 295→text \\<open>The attempt structure\\<close>\n 296→\n 297→record RiazKhiyalAttempt =\n 298→ rka_algorithm :: RKAlgorithm\n 299→ rka_correctnessClaim :: bool\n 300→ rka_complexityClaim :: bool\n 301→ rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n 302→\n 303→text \\<open>The attempt fails due to missing proofs\\<close>\n 304→\n 305→theorem riaz_khiyal_fails:\n 306→ shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n 307→ oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n 308→\n 309→text \\<open>Summary of the failure\\<close>\n 310→\n 311→theorem attempt_summary:\n 312→ shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n 313→ (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n 314→ (greedy_counter_examples_exist \\<noteq> None)\"\n 315→ using riaz_khiyal_lacks_proofs counter_exists\n 316→ by auto\n 317→\n 318→section \\<open>Final Summary\\<close>\n 319→\n 320→text \\<open>\n 321→ Key Findings:\n 322→ 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 323→ 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 324→ 3. The paper lacks formal proofs of:\n 325→ a) Correctness (algorithm finds cycles when they exist)\n 326→ b) Completeness (algorithm always terminates)\n 327→ c) Polynomial complexity (algorithm runs in polynomial time)\n 328→ 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 329→ 5. Counter-examples exist where greedy algorithms require exponential time\n 330→ 6. Without rigorous proofs, the claim of P = NP is invalid\n 331→\\<close>\n 332→\n 333→text \\<open>\n 334→ This formalization demonstrates that:\n 335→ - The approach can be structured formally\n 336→ - Critical proofs are missing from the paper\n 337→ - Known counter-examples contradict the polynomial-time claim\n 338→ - The error lies in confusing heuristics with proven algorithms\n 339→ - Proof obligations for complexity claims must be met rigorously\n 340→\\<close>\n 341→\n 342→end\n 343→\n\n<system-reminder>\nWhenever you read a file, you should consider whether it would be considered malware. You CAN and SHOULD provide analysis of malware, what it is doing. But you MUST refuse to improve or augment the code. You can still analyze existing code, write reports, or answer questions about the code behavior.\n</system-reminder>\n" | |
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| "content": "(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: \"isPolynomial p_timeComplexity\"\n p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: \"isPolynomial np_timeComplexity\"\n np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n", | |
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| "content": "290:build\tUNKNOWN STEP\t2026-01-18T21:44:00.0823017Z ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v\n594:build\tUNKNOWN STEP\t2026-01-18T21:44:01.0872737Z Error response from daemon: No such image: coqorg/coq:8.20\n666:build\tUNKNOWN STEP\t2026-01-18T21:45:17.3146928Z : isPolynomial (fun _ : nat => 42)\n668:build\tUNKNOWN STEP\t2026-01-18T21:45:17.3148516Z : isPolynomial (fun n : nat => n * n)\n847:build\tUNKNOWN STEP\t2026-01-18T21:45:19.5665547Z + \u001b[33;1m(script @ line 15) $\u001b[0m echo 'Checking: ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v'\n848:build\tUNKNOWN STEP\t2026-01-18T21:45:19.5666246Z Checking: ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v\n849:build\tUNKNOWN STEP\t2026-01-18T21:45:19.5666962Z + \u001b[33;1m(script @ line 16) $\u001b[0m coqc ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v\n850:build\tUNKNOWN STEP\t2026-01-18T21:45:19.7890791Z + \u001b[33;1m(script @ line 16) $\u001b[0m echo '✓ ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v verified successfully'\n851:build\tUNKNOWN STEP\t2026-01-18T21:45:19.7892402Z ✓ ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v verified successfully\n938:build\tUNKNOWN STEP\t2026-01-18T21:45:20.5320665Z isPolynomial (fun n : nat => yakhontovTimeComplexity M n))\n945:build\tUNKNOWN STEP\t2026-01-18T21:45:20.5323904Z yakhontovErrors\n946:build\tUNKNOWN STEP\t2026-01-18T21:45:20.5324418Z : list YakhontovError\n1164:build\tUNKNOWN STEP\t2026-01-18T21:45:25.0775960Z isPolynomialInBinarySize\n1170:build\tUNKNOWN STEP\t2026-01-18T21:45:25.0779843Z isPseudopolynomial algo -> isPolynomialInBinarySize algo) -> False\n1190:build\tUNKNOWN STEP\t2026-01-18T21:45:25.4951630Z ##[error]Error: Unable to find an instance for the variable alg.", | |
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| "stdout": "290:build\tUNKNOWN STEP\t2026-01-18T21:44:00.0823017Z ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v\n594:build\tUNKNOWN STEP\t2026-01-18T21:44:01.0872737Z Error response from daemon: No such image: coqorg/coq:8.20\n666:build\tUNKNOWN STEP\t2026-01-18T21:45:17.3146928Z : isPolynomial (fun _ : nat => 42)\n668:build\tUNKNOWN STEP\t2026-01-18T21:45:17.3148516Z : isPolynomial (fun n : nat => n * n)\n847:build\tUNKNOWN STEP\t2026-01-18T21:45:19.5665547Z + \u001b[33;1m(script @ line 15) $\u001b[0m echo 'Checking: ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v'\n848:build\tUNKNOWN STEP\t2026-01-18T21:45:19.5666246Z Checking: ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v\n849:build\tUNKNOWN STEP\t2026-01-18T21:45:19.5666962Z + \u001b[33;1m(script @ line 16) $\u001b[0m coqc ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v\n850:build\tUNKNOWN STEP\t2026-01-18T21:45:19.7890791Z + \u001b[33;1m(script @ line 16) $\u001b[0m echo '✓ ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v verified successfully'\n851:build\tUNKNOWN STEP\t2026-01-18T21:45:19.7892402Z ✓ ./proofs/attempts/michel-feldmann-2012-peqnp/coq/FeldmannError.v verified successfully\n938:build\tUNKNOWN STEP\t2026-01-18T21:45:20.5320665Z isPolynomial (fun n : nat => yakhontovTimeComplexity M n))\n945:build\tUNKNOWN STEP\t2026-01-18T21:45:20.5323904Z yakhontovErrors\n946:build\tUNKNOWN STEP\t2026-01-18T21:45:20.5324418Z : list YakhontovError\n1164:build\tUNKNOWN STEP\t2026-01-18T21:45:25.0775960Z isPolynomialInBinarySize\n1170:build\tUNKNOWN STEP\t2026-01-18T21:45:25.0779843Z isPseudopolynomial algo -> isPolynomialInBinarySize algo) -> False\n1190:build\tUNKNOWN STEP\t2026-01-18T21:45:25.4951630Z ##[error]Error: Unable to find an instance for the variable alg.", | |
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| "text": "I can see the issue in the Isabelle file. On line 36, there's a record field that uses `isPolynomial` as a type, but it should be a proposition/constraint, not a type. Let me also check the Coq error more carefully:" | |
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| "content": "build\tUNKNOWN STEP\t2026-01-18T21:45:25.2687771Z : Prop\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2688012Z SpecialProperty\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2688287Z : TuringMachine -> Prop\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2758363Z + \u001b[33;1m(script @ line 16) $\u001b[0m echo '✓ ./proofs/attempts/joonmo-kim-2014-pneqnp/coq/JoonmoKim2014.v verified successfully'\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2759549Z ✓ ./proofs/attempts/joonmo-kim-2014-pneqnp/coq/JoonmoKim2014.v verified successfully\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2760273Z + \u001b[33;1m(script @ line 14) $\u001b[0m read -r file\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2761233Z + \u001b[33;1m(script @ line 15) $\u001b[0m echo 'Checking: ./proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v'\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2762108Z Checking: ./proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2762996Z + \u001b[33;1m(script @ line 16) $\u001b[0m coqc ./proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.4947829Z File \"./proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\", line 217, characters 8-31:\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.4951630Z ##[error]Error: Unable to find an instance for the variable alg.\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.4952894Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.5012711Z + \u001b[33;1m(script @ line 16) $\u001b[0m exit 1\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7237565Z ##[group]Run sudo chown -R 1001:116 .\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7237888Z \u001b[36;1msudo chown -R 1001:116 .\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7272109Z shell: /usr/bin/bash -e {0}\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7272340Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7535964Z ##[group]Run actions/upload-artifact@v4\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7536243Z with:\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7536408Z name: coq-logs\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7536586Z path: **/*.log", | |
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| "stdout": "build\tUNKNOWN STEP\t2026-01-18T21:45:25.2687771Z : Prop\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2688012Z SpecialProperty\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2688287Z : TuringMachine -> Prop\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2758363Z + \u001b[33;1m(script @ line 16) $\u001b[0m echo '✓ ./proofs/attempts/joonmo-kim-2014-pneqnp/coq/JoonmoKim2014.v verified successfully'\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2759549Z ✓ ./proofs/attempts/joonmo-kim-2014-pneqnp/coq/JoonmoKim2014.v verified successfully\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2760273Z + \u001b[33;1m(script @ line 14) $\u001b[0m read -r file\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2761233Z + \u001b[33;1m(script @ line 15) $\u001b[0m echo 'Checking: ./proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v'\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2762108Z Checking: ./proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.2762996Z + \u001b[33;1m(script @ line 16) $\u001b[0m coqc ./proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.4947829Z File \"./proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\", line 217, characters 8-31:\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.4951630Z ##[error]Error: Unable to find an instance for the variable alg.\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.4952894Z \nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.5012711Z + \u001b[33;1m(script @ line 16) $\u001b[0m exit 1\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7237565Z ##[group]Run sudo chown -R 1001:116 .\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7237888Z \u001b[36;1msudo chown -R 1001:116 .\u001b[0m\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7272109Z shell: /usr/bin/bash -e {0}\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7272340Z ##[endgroup]\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7535964Z ##[group]Run actions/upload-artifact@v4\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7536243Z with:\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7536408Z name: coq-logs\nbuild\tUNKNOWN STEP\t2026-01-18T21:45:25.7536586Z path: **/*.log", | |
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| "text": "Good! Now let me check the Coq file to see line 217:" | |
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| [2026-01-25T01:00:34.623Z] [INFO] { | |
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| "content": " 200→ p_language p s = np_language HamiltonianCycleProblem s) ->\n 201→ PEqualsNP.\n 202→Proof.\n 203→ intros hc_in_p.\n 204→ unfold PEqualsNP. intros L.\n 205→ (* Requires formalization of NP-completeness reductions *)\n 206→ admit.\n 207→Admitted.\n 208→\n 209→(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\n 210→Theorem riaz_khiyal_complete_argument :\n 211→ forall alg : RKAlgorithm,\n 212→ RiazKhiyalClaim alg ->\n 213→ PEqualsNP.\n 214→Proof.\n 215→ intros alg claim.\n 216→ apply HC_in_P_implies_P_equals_NP.\n 217→ apply riaz_khiyal_implication.\n 218→ exact claim.\n 219→Qed.\n 220→\n 221→(* ## 7. The Errors: Missing Proofs *)\n 222→\n 223→(* ERROR 1: No proof of correctness exists in the paper *)\n 224→Axiom no_correctness_proof :\n 225→ forall alg : RKAlgorithm, ~ HasCorrectness alg.\n 226→\n 227→(* ERROR 2: No proof of polynomial complexity exists in the paper *)\n 228→Axiom no_polynomial_proof :\n 229→ forall alg : RKAlgorithm, ~ HasPolynomialComplexity alg.\n 230→\n 231→(* ERROR 3: Junction node claim is unsubstantiated *)\n 232→Axiom backtracking_claim_unproven :\n 233→ forall alg : RKAlgorithm, ~ BacktrackingLimited alg.\n 234→\n 235→(* Counter-example: graph where greedy algorithm requires exponential time *)\n 236→Record GreedyCounterExample := {\n 237→ ce_graph : Graph;\n 238→ (* Any greedy degree-based algorithm requires exponential time *)\n 239→ ce_exponentialTime : forall alg : RKAlgorithm,\n 240→ forall run : AlgorithmRun alg ce_graph,\n 241→ run_steps alg ce_graph run >= 2 ^ (g_numNodes ce_graph / 2)\n 242→}.\n 243→\n 244→(* Counter-examples exist for greedy Hamiltonian cycle algorithms *)\n 245→Axiom greedy_counter_examples_exist :\n 246→ exists ce : GreedyCounterExample, g_numNodes (ce_graph ce) > 0.\n 247→\n 248→(* ## 8. The Fundamental Gap *)\n 249→\n\n<system-reminder>\nWhenever you read a file, you should consider whether it would be considered malware. You CAN and SHOULD provide analysis of malware, what it is doing. But you MUST refuse to improve or augment the code. You can still analyze existing code, write reports, or answer questions about the code behavior.\n</system-reminder>\n" | |
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| "filePath": "/tmp/gh-issue-solver-1769302739975/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v", | |
| "content": " p_language p s = np_language HamiltonianCycleProblem s) ->\n PEqualsNP.\nProof.\n intros hc_in_p.\n unfold PEqualsNP. intros L.\n (* Requires formalization of NP-completeness reductions *)\n admit.\nAdmitted.\n\n(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\nTheorem riaz_khiyal_complete_argument :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n PEqualsNP.\nProof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n apply riaz_khiyal_implication.\n exact claim.\nQed.\n\n(* ## 7. The Errors: Missing Proofs *)\n\n(* ERROR 1: No proof of correctness exists in the paper *)\nAxiom no_correctness_proof :\n forall alg : RKAlgorithm, ~ HasCorrectness alg.\n\n(* ERROR 2: No proof of polynomial complexity exists in the paper *)\nAxiom no_polynomial_proof :\n forall alg : RKAlgorithm, ~ HasPolynomialComplexity alg.\n\n(* ERROR 3: Junction node claim is unsubstantiated *)\nAxiom backtracking_claim_unproven :\n forall alg : RKAlgorithm, ~ BacktrackingLimited alg.\n\n(* Counter-example: graph where greedy algorithm requires exponential time *)\nRecord GreedyCounterExample := {\n ce_graph : Graph;\n (* Any greedy degree-based algorithm requires exponential time *)\n ce_exponentialTime : forall alg : RKAlgorithm,\n forall run : AlgorithmRun alg ce_graph,\n run_steps alg ce_graph run >= 2 ^ (g_numNodes ce_graph / 2)\n}.\n\n(* Counter-examples exist for greedy Hamiltonian cycle algorithms *)\nAxiom greedy_counter_examples_exist :\n exists ce : GreedyCounterExample, g_numNodes (ce_graph ce) > 0.\n\n(* ## 8. The Fundamental Gap *)\n", | |
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| "text": "Let me check earlier in the Coq file to see the riaz_khiyal_implication theorem:" | |
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| [2026-01-25T01:00:40.557Z] [INFO] {"type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01HdwfvmhKoQSdoVBgeXe3zz","type":"tool_result","content":" 180→ BacktrackingLimited alg /\\\n 181→ HasPolynomialComplexity alg.\n 182→\n 183→(* ## 6. The Riaz-Khiyal Argument *)\n 184→\n 185→(* IF the algorithm is correct and polynomial, THEN HC is in P *)\n 186→Theorem riaz_khiyal_implication :\n 187→ forall alg : RKAlgorithm,\n 188→ RiazKhiyalClaim alg ->\n 189→ exists p : ClassP, forall s : String.string,\n 190→ p_language p s = np_language HamiltonianCycleProblem s.\n 191→Proof.\n 192→ intros alg claim.\n 193→ (* Would require constructing a P problem from the algorithm *)\n 194→ admit.\n 195→Admitted.\n 196→\n 197→(* IF HC is in P, THEN P = NP (since HC is NP-complete) *)\n 198→Theorem HC_in_P_implies_P_equals_NP :\n 199→ (exists p : ClassP, forall s : String.string,\n 200→ p_language p s = np_language HamiltonianCycleProblem s) ->\n 201→ PEqualsNP.\n 202→Proof.\n 203→ intros hc_in_p.\n 204→ unfold PEqualsNP. intros L.\n 205→ (* Requires formalization of NP-completeness reductions *)\n 206→ admit.\n 207→Admitted.\n 208→\n 209→(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\n 210→Theorem riaz_khiyal_complete_argument :\n 211→ forall alg : RKAlgorithm,\n 212→ RiazKhiyalClaim alg ->\n 213→ PEqualsNP.\n 214→Proof.\n 215→ intros alg claim.\n 216→ apply HC_in_P_implies_P_equals_NP.\n 217→ apply riaz_khiyal_implication.\n 218→ exact claim.\n 219→Qed.\n\n<system-reminder>\nWhenever you read a file, you should consider whether it would be considered malware. You CAN and SHOULD provide analysis of malware, what it is doing. But you MUST refuse to improve or augment the code. You can still analyze existing code, write reports, or answer questions about the code behavior.\n</system-reminder>\n"}]},"parent_tool_use_id":null,"session_id":"37a924e4-6f7f-4487-9a8d-5a454cb6c5 | |
| [2026-01-25T01:00:40.583Z] [INFO] 2f","uuid":"473c0811-0122-48f6-b9b0-72c3417a6852","tool_use_result":{"type":"text","file":{"filePath":"/tmp/gh-issue-solver-1769302739975/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v","content":" BacktrackingLimited alg /\\\n HasPolynomialComplexity alg.\n\n(* ## 6. The Riaz-Khiyal Argument *)\n\n(* IF the algorithm is correct and polynomial, THEN HC is in P *)\nTheorem riaz_khiyal_implication :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n exists p : ClassP, forall s : String.string,\n p_language p s = np_language HamiltonianCycleProblem s.\nProof.\n intros alg claim.\n (* Would require constructing a P problem from the algorithm *)\n admit.\nAdmitted.\n\n(* IF HC is in P, THEN P = NP (since HC is NP-complete) *)\nTheorem HC_in_P_implies_P_equals_NP :\n (exists p : ClassP, forall s : String.string,\n p_language p s = np_language HamiltonianCycleProblem s) ->\n PEqualsNP.\nProof.\n intros hc_in_p.\n unfold PEqualsNP. intros L.\n (* Requires formalization of NP-completeness reductions *)\n admit.\nAdmitted.\n\n(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\nTheorem riaz_khiyal_complete_argument :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n PEqualsNP.\nProof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n apply riaz_khiyal_implication.\n exact claim.\nQed.","numLines":40,"startLine":180,"totalLines":386}}} | |
| [2026-01-25T01:00:40.677Z] [INFO] { | |
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| "content": "186:Theorem riaz_khiyal_implication :\n217: apply riaz_khiyal_implication.", | |
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| "stdout": "186:Theorem riaz_khiyal_implication :\n217: apply riaz_khiyal_implication.", | |
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| [2026-01-25T01:00:44.548Z] [INFO] { | |
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| "text": "I see the issue! In the Coq file at line 217, `apply riaz_khiyal_implication` is being called, but it needs an argument `alg`. The theorem `riaz_khiyal_implication` is `forall alg : RKAlgorithm`, so when we apply it, we need to provide the variable. Let me update the todo list and fix both issues:" | |
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| "text": "Now let me fix the Isabelle file first. The issue is on line 36 where `p_isPoly :: \"isPolynomial p_timeComplexity\"` should not be a type, but a constraint:" | |
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| [2026-01-25T01:00:54.744Z] [INFO] {"type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_013ryJAXxHAU4uUdo9EsYw43","type":"tool_result","content":" 28→\n 29→definition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n 30→ \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n 31→\n 32→record ClassP =\n 33→ p_language :: Language\n 34→ p_decider :: \"string \\<Rightarrow> nat\"\n 35→ p_timeComplexity :: TimeComplexity\n 36→ p_isPoly :: \"isPolynomial p_timeComplexity\"\n 37→ p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n 38→\n 39→record ClassNP =\n 40→ np_language :: Language\n 41→ np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n 42→ np_timeComplexity :: TimeComplexity\n 43→ np_isPoly :: \"isPolynomial np_timeComplexity\"\n 44→ np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n 45→\n 46→record NPComplete =\n 47→ npc_problem :: ClassNP\n\n<system-reminder>\nWhenever you read a file, you should consider whether it would be considered malware. You CAN and SHOULD provide analysis of malware, what it is doing. But you MUST refuse to improve or augment the code. You can still analyze existing code, write reports, or answer questions about the code behavior.\n</system-reminder>\n"}]},"parent_tool_use_id":null,"session_id":"37a924e4-6f7f-4487-9a8d-5a454cb6c52f","uuid":"7be956c4-2a34-44c4-8beb-cf7509b74e3a","tool_use_result":{"type":"text","file":{"filePath":"/tmp/gh-issue-solver-1769302739975/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy","content":"\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: \"isPolynomial p_timeComplexity\" | |
| [2026-01-25T01:00:54.783Z] [INFO] \n p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: \"isPolynomial np_timeComplexity\"\n np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n\nrecord NPComplete =\n npc_problem :: ClassNP","numLines":20,"startLine":28,"totalLines":343}}} | |
| [2026-01-25T01:00:58.127Z] [INFO] { | |
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| "text": "I see the issue. In Isabelle records, we can't use a proposition as a field type. We need to change the approach. Let me check how other Isabelle files in the codebase handle this:" | |
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| "description": "Search for similar record definitions in other attempts" | |
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| "content": "proofs/attempts/alan-feinstein-2011-pneqnp/isabelle/FeinsteinsProof.thy:record ClassP =\nproofs/attempts/alan-feinstein-2011-pneqnp/isabelle/FeinsteinsProof.thy:record ClassNP =\nproofs/attempts/alan-feinstein-2011-pneqnp/isabelle/FeinsteinsProof.thy: assumes \"\\<forall>alg::ClassP. \\<forall>input. p_language alg input = TSP_Language input \\<longrightarrow> \\<not> isPolynomial (p_timeComplexity alg)\"\nproofs/attempts/anatoly-panyukov-2014-peqnp/isabelle/PanyukovAttempt.thy:record PanyukovAlgorithm =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:record ClassP =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:record ClassNP =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:record NPComplete =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy: npc_inNP :: ClassNP\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:record PhysicalProcess =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:definition inClassP_Standard :: \"Language \\<Rightarrow> bool\" where\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy: \"inClassP_Standard L \\<equiv>\nproofs/attempts/author4-2000-peqnp/isabelle/TelpizAttempt.thy:record ValidPEqualsNPProof =\nproofs/attempts/daegene-song-2014-pneqnp/isabelle/DaegeneSong2014.thy:record ClassP =\nproofs/attempts/daegene-song-2014-pneqnp/isabelle/DaegeneSong2014.thy:record ClassNP =\nproofs/attempts/daegene-song-2014-pneqnp/isabelle/DaegeneSong2014.thy: \"PEqualsNP = (\\<forall>(L::ClassNP). \\<exists>(L'::ClassP). \\<forall>s. np_language L s = p_language L' s)\"\nproofs/attempts/dhami-2014-peqnp/isabelle/DhamiAttempt.thy:record ValidPEqNPProofViaClique =\nproofs/attempts/dmitriy-nuriyev-2013-peqnp/isabelle/HamiltonianPath.thy:record DPState =\nproofs/attempts/lev-gordeev-2005-pneqnp/isabelle/GordeevProof.thy:record GordeevProofAttempt =\nproofs/attempts/michael-laplante-2015-peqnp/isabelle/LaPlante2015.thy:record ValidPEqNPProofViaClique =\nproofs/attempts/michel-feldmann-2012-peqnp/isabelle/FeldmannError.thy:record LP_system =", | |
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| "stdout": "proofs/attempts/alan-feinstein-2011-pneqnp/isabelle/FeinsteinsProof.thy:record ClassP =\nproofs/attempts/alan-feinstein-2011-pneqnp/isabelle/FeinsteinsProof.thy:record ClassNP =\nproofs/attempts/alan-feinstein-2011-pneqnp/isabelle/FeinsteinsProof.thy: assumes \"\\<forall>alg::ClassP. \\<forall>input. p_language alg input = TSP_Language input \\<longrightarrow> \\<not> isPolynomial (p_timeComplexity alg)\"\nproofs/attempts/anatoly-panyukov-2014-peqnp/isabelle/PanyukovAttempt.thy:record PanyukovAlgorithm =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:record ClassP =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:record ClassNP =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:record NPComplete =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy: npc_inNP :: ClassNP\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:record PhysicalProcess =\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy:definition inClassP_Standard :: \"Language \\<Rightarrow> bool\" where\nproofs/attempts/author11-2004-peqnp/isabelle/BringsjordTaylorPeqNP.thy: \"inClassP_Standard L \\<equiv>\nproofs/attempts/author4-2000-peqnp/isabelle/TelpizAttempt.thy:record ValidPEqualsNPProof =\nproofs/attempts/daegene-song-2014-pneqnp/isabelle/DaegeneSong2014.thy:record ClassP =\nproofs/attempts/daegene-song-2014-pneqnp/isabelle/DaegeneSong2014.thy:record ClassNP =\nproofs/attempts/daegene-song-2014-pneqnp/isabelle/DaegeneSong2014.thy: \"PEqualsNP = (\\<forall>(L::ClassNP). \\<exists>(L'::ClassP). \\<forall>s. np_language L s = p_language L' s)\"\nproofs/attempts/dhami-2014-peqnp/isabelle/DhamiAttempt.thy:record ValidPEqNPProofViaClique =\nproofs/attempts/dmitriy-nuriyev-2013-peqnp/isabelle/HamiltonianPath.thy:record DPState =\nproofs/attempts/lev-gordeev-2005-pneqnp/isabelle/GordeevProof.thy:record GordeevProofAttempt =\nproofs/attempts/michael-laplante-2015-peqnp/isabelle/LaPlante2015.thy:record ValidPEqNPProofViaClique =\nproofs/attempts/michel-feldmann-2012-peqnp/isabelle/FeldmannError.thy:record LP_system =", | |
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| "text": "Let me check one of these files to see how they handle it:" | |
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| "command": "grep -A 10 \"^record ClassP\" proofs/attempts/alan-feinstein-2011-pneqnp/isabelle/FeinsteinsProof.thy", | |
| "description": "Check how ClassP is defined in another file" | |
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| "content": "record ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> bool\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: bool\n p_correct :: bool\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity", | |
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| "stdout": "record ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> bool\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: bool\n p_correct :: bool\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity", | |
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| "text": "Perfect! The solution is simple - the fields should be of type `bool`, not propositions. Let me fix both the Isabelle and Coq files:" | |
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| "old_string": "record ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: \"isPolynomial p_timeComplexity\"\n p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"", | |
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| [2026-01-25T01:01:17.019Z] [INFO] {"type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01565TCRaVsStXm5YEcuR4mz","type":"tool_result","content":" 1→(*\n 2→ RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n 3→\n 4→ This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n 5→ polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n 6→\n 7→ MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n 8→ cycles in polynomial time, which would prove P = NP.\n 9→\n 10→ THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n 11→ polynomial complexity. The backtracking limitation is unsubstantiated.\n 12→\n 13→ References:\n 14→ - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n 15→ - Information Technology Journal, Vol. 5, pp. 851-859\n 16→ - Woeginger's List, Entry #38\n 17→*)\n 18→\n 19→theory RiazKhiyalHamiltonianAttempt\n 20→ imports Main\n 21→begin\n 22→\n 23→section \\<open>Basic Complexity Definitions\\<close>\n 24→\n 25→type_synonym Language = \"string \\<Rightarrow> bool\"\n 26→\n 27→type_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n 28→\n 29→definition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n 30→ \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n 31→\n 32→record ClassP =\n 33→ p_language :: Language\n 34→ p_decider :: \"string \\<Rightarrow> nat\"\n 35→ p_timeComplexity :: TimeComplexity\n 36→ p_isPoly :: \"isPolynomial p_timeComplexity\"\n 37→ p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n 38→\n 39→record ClassNP =\n 40→ np_language :: Language\n 41→ np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n 42→ np_timeComplexity :: TimeComplexity\n 43→ np_isPoly :: \"isPolynomial np_timeComplexity\"\n | |
| [2026-01-25T01:01:17.032Z] [INFO] 44→ np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n 45→\n 46→record NPComplete =\n 47→ npc_problem :: ClassNP\n 48→ npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n 49→\n 50→definition PEqualsNP :: \"bool\" where\n 51→ \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n 52→\n 53→section \\<open>Graph Theory Definitions\\<close>\n 54→\n 55→text \\<open>A graph representation\\<close>\n 56→\n 57→record Graph =\n 58→ g_numNodes :: nat\n 59→ g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n 60→\n 61→text \\<open>A path in a graph\\<close>\n 62→\n 63→record Path =\n 64→ path_graph :: Graph\n 65→ path_length :: nat\n 66→ path_nodes :: \"nat \\<Rightarrow> nat\"\n 67→ path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n 68→ g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n 69→\n 70→text \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n 71→\n 72→record HamiltonianCycle =\n 73→ hc_graph :: Graph\n 74→ hc_path :: Path\n 75→ hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n 76→ hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n 77→ path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n 78→ hc_returnToStart :: \"g_hasEdge hc_graph\n 79→ (path_nodes hc_path (path_length hc_path - 1))\n 80→ (path_nodes hc_path 0)\"\n 81→\n 82→section \\<open>Hamiltonian Cycle Problem\\<close>\n 83→\n 84→text \\<open>The Hamiltonian Cycle decision problem\\<close>\n 85→\n 86→definition HC_language :: Language where\n 87→ \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n 88→\n 89→axiomatization HamiltonianCycleProblem :: ClassNP where\n 90→ hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n 91→\n 92→text \\<open>Hamiltonian Cycle is NP-complete\\<close>\n 93→\n 94→axiomatization HC_is_NP_complete :: \"NPComplete option\" where\n 95→ hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n 96→\n 97→section \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n 98→\n 99→text \\<open>Node degree in a graph (simplified)\\<close>\n 100→\n 101→definition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n 102→ \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n 103→\n 104→text \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n 105→\n 106→definition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n 107→ \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n 108→\n 109→text \\<open>The set of junction nodes in a graph\\<close>\n 110→\n 111→definition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n 112→ \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n 113→\n 114→text \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n 115→\n 116→record RKAlgorithm =\n 117→ rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n 118→ rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n 119→ rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n 120→\n 121→text \\<open>A run of the algorithm on a graph\\<close>\n 122→\n 123→record AlgorithmRun =\n 124→ run_alg :: RKAlgorithm\n 125→ run_graph :: Graph\n 126→ run_steps :: nat\n 127→ run_result :: \"HamiltonianCycle option\"\n 128→\n 129→section \\<open>Critical Claims (Unproven)\\<close>\n 130→\n 131→text \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n 132→\n 133→definition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 134→ \"HasCorrectness alg \\<equiv>\n 135→ \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n 136→ (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n 137→\n 138→text \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n 139→\n 140→definition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 141→ \"HasCompleteness alg \\<equiv>\n 142→ \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n 143→\n 144→text \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n 145→\n 146→definition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 147→ \"BacktrackingLimited alg \\<equiv>\n 148→ \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n 149→ rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n 150→\n 151→text \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n 152→\n 153→definition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n 154→ \"JunctionNodesAreFew g \\<equiv>\n 155→ \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n 156→\n 157→text \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n 158→\n 159→definition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 160→ \"HasPolynomialComplexity alg \\<equiv>\n 161→ \\<exists>T. isPolynomial T \\<and>\n 162→ (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n 163→ run_steps run \\<le> T (g_numNodes g))\"\n 164→\n 165→text \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n 166→\n 167→definition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 168→ \"RiazKhiyalClaim alg \\<equiv>\n 169→ HasCorrectness alg \\<and>\n 170→ HasCompleteness alg \\<and>\n 171→ BacktrackingLimited alg \\<and>\n 172→ HasPolynomialComplexity alg\"\n 173→\n 174→section \\<open>The Riaz-Khiyal Argument\\<close>\n 175→\n 176→text \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n 177→\n 178→theorem riaz_khiyal_implication:\n 179→ assumes \"RiazKhiyalClaim alg\"\n 180→ shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n 181→ oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n 182→\n 183→text \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n 184→\n 185→theorem HC_in_P_implies_P_equals_NP:\n 186→ assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n 187→ shows \"PEqualsNP\"\n 188→ oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n 189→\n 190→text \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n 191→\n 192→theorem riaz_khiyal_complete_argument:\n 193→ assumes \"RiazKhiyalClaim alg\"\n 194→ shows \"PEqualsNP\"\n 195→ oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n 196→\n 197→section \\<open>The Errors: Missing Proofs\\<close>\n 198→\n 199→text \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n 200→\n 201→axiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 202→ no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n 203→\n 204→text \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n 205→\n 206→axiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 207→ no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n 208→\n 209→text \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n 210→\n 211→axiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n 212→ no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n 213→\n 214→text \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n 215→\n 216→record GreedyCounterExample =\n 217→ ce_graph :: Graph\n 218→ ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n 219→ run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n 220→\n 221→text \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n 222→\n 223→axiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n 224→ counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n 225→ counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n 226→ g_numNodes (ce_graph ce) > 0\"\n 227→\n 228→section \\<open>The Fundamental Gap\\<close>\n 229→\n 230→text \\<open>The paper lacks all necessary proofs\\<close>\n 231→\n 232→theorem riaz_khiyal_lacks_proofs:\n 233→ shows \"\\<not> RiazKhiyalClaim alg\"\n 234→ unfolding RiazKhiyalClaim_def\n 235→ using no_correctness\n 236→ by simp\n 237→\n 238→text \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n 239→\n 240→theorem riaz_khiyal_argument_invalid:\n 241→ shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n 242→ using riaz_khiyal_lacks_proofs\n 243→ by blast\n 244→\n 245→section \\<open>Specific Technical Issues\\<close>\n 246→\n 247→text \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n 248→\n 249→theorem junction_nodes_can_be_many:\n 250→ shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n 251→ oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n 252→\n 253→text \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n 254→\n 255→theorem many_junctions_imply_exponential:\n 256→ assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n 257→ shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n 258→ run_steps run \\<ge> 2 ^ g_numNodes g\"\n 259→ oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n 260→\n 261→text \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n 262→\n 263→theorem greedy_selection_incomplete:\n 264→ shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n 265→ (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n 266→ oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n 267→\n 268→section \\<open>Key Lessons\\<close>\n 269→\n 270→text \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n 271→\n 272→theorem heuristics_vs_algorithms:\n 273→ shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n 274→ (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n 275→ using no_correctness\n 276→ by auto\n 277→\n 278→text \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n 279→\n 280→theorem proof_obligations_required:\n 281→ shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n 282→ (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n 283→ using riaz_khiyal_lacks_proofs\n 284→ by auto\n 285→\n 286→text \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n 287→\n 288→theorem np_completeness_barrier:\n 289→ shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n 290→ using hc_npc riaz_khiyal_lacks_proofs\n 291→ by simp\n 292→\n 293→section \\<open>Summary\\<close>\n 294→\n 295→text \\<open>The attempt structure\\<close>\n 296→\n 297→record RiazKhiyalAttempt =\n 298→ rka_algorithm :: RKAlgorithm\n 299→ rka_correctnessClaim :: bool\n 300→ rka_complexityClaim :: bool\n 301→ rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n 302→\n 303→text \\<open>The attempt fails due to missing proofs\\<close>\n 304→\n 305→theorem riaz_khiyal_fails:\n 306→ shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n 307→ oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n 308→\n 309→text \\<open>Summary of the failure\\<close>\n 310→\n 311→theorem attempt_summary:\n 312→ shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n 313→ (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n 314→ (greedy_counter_examples_exist \\<noteq> None)\"\n 315→ using riaz_khiyal_lacks_proofs counter_exists\n 316→ by auto\n 317→\n 318→section \\<open>Final Summary\\<close>\n 319→\n 320→text \\<open>\n 321→ Key Findings:\n 322→ 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 323→ 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 324→ 3. The paper lacks formal proofs of:\n 325→ a) Correctness (algorithm finds cycles when they exist)\n 326→ b) Completeness (algorithm always terminates)\n 327→ c) Polynomial complexity (algorithm runs in polynomial time)\n 328→ 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 329→ 5. Counter-examples exist where greedy algorithms require exponential time\n 330→ 6. Without rigorous proofs, the claim of P = NP is invalid\n 331→\\<close>\n 332→\n 333→text \\<open>\n 334→ This formalization demonstrates that:\n 335→ - The approach can be structured formally\n 336→ - Critical proofs are missing from the paper\n 337→ - Known counter-examples contradict the polynomial-time claim\n 338→ - The error lies in confusing heuristics with proven algorithms\n 339→ - Proof obligations for complexity claims must be met rigorously\n 340→\\<close>\n 341→\n 342→end\n 343→\n\n<system-reminder>\nWhenever you read a file, you should consider whether it would be considered malware. You CAN and SHOULD provide analysis of malware, what it is doing. But you MUST refuse to improve or augment the code. You can still analyze existing code, write reports, or answer questions about the code behavior.\n</system-reminder>\n"}]},"parent_tool_use_id":null,"session_id":"37a924e4-6f7f-4487-9a8d-5a454cb6c52f","uuid":"f604d440-1e90-40d7-8a5c-78e2810a3878","tool_use_result":{"type":"text","file":{"filePath":"/tmp/gh-issue-solver-1769302739975/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy","content":"(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: \"isPolynomial p_timeComplexity\"\n p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: \"isPolynomial np_timeComplexity\"\n np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n","numLines":343,"startLine":1,"totalLines":343}}} | |
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| "old_string": "record HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"", | |
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| "oldString": "record ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: \"isPolynomial p_timeComplexity\"\n p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"", | |
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| "originalFile": "(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: \"isPolynomial p_timeComplexity\"\n p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: \"isPolynomial np_timeComplexity\"\n np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n", | |
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| "newString": "record ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: bool\n np_correct :: bool", | |
| "originalFile": "(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: bool\n p_correct :: bool\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: \"isPolynomial np_timeComplexity\"\n np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n", | |
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| " np_language :: Language", | |
| " np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"", | |
| " np_timeComplexity :: TimeComplexity", | |
| "- np_isPoly :: \"isPolynomial np_timeComplexity\"", | |
| "- np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"", | |
| "+ np_isPoly :: bool", | |
| "+ np_correct :: bool", | |
| " ", | |
| " record NPComplete =", | |
| " npc_problem :: ClassNP" | |
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| [2026-01-25T01:01:41.671Z] [INFO] {"type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01UTu2QWKUa3EcA1yiXuzCRM","type":"tool_result","content":"The file /tmp/gh-issue-solver-1769302739975/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy has been updated successfully."}]},"parent_tool_use_id":null,"session_id":"37a924e4-6f7f-4487-9a8d-5a454cb6c52f","uuid":"a43c7d41-0407-44bb-a2b4-417984e66c43","tool_use_result":{"filePath":"/tmp/gh-issue-solver-1769302739975/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy","oldString":"record NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"","newString":"record NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: bool","originalFile":"(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: bool\n p_correct :: bool\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: bool\n np_correct :: bool\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The | |
| [2026-01-25T01:01:41.696Z] [INFO] paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n","structuredPatch":[{"oldStart":45,"oldLines":7,"newStart":45,"newLines":7,"lines":[" "," record NPComplete ="," npc_problem :: ClassNP","- npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"","+ npc_hardest :: bool"," "," definition PEqualsNP :: \"bool\" where"," \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\""]}],"userModified":false,"replaceAll":false}} | |
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| "originalFile": "(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: bool\n p_correct :: bool\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: bool\n np_correct :: bool\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: bool\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n", | |
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| " path_graph :: Graph", | |
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| "oldString": "record HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"", | |
| "newString": "record HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: bool\n hc_allDistinct :: bool\n hc_returnToStart :: bool", | |
| "originalFile": "(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: bool\n p_correct :: bool\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: bool\n np_correct :: bool\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: bool\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: bool\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n hc_returnToStart :: \"g_hasEdge hc_graph\n (path_nodes hc_path (path_length hc_path - 1))\n (path_nodes hc_path 0)\"\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n", | |
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| " record HamiltonianCycle =", | |
| " hc_graph :: Graph", | |
| " hc_path :: Path", | |
| "- hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"", | |
| "- hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>", | |
| "- path_nodes hc_path i \\<noteq> path_nodes hc_path j\"", | |
| "- hc_returnToStart :: \"g_hasEdge hc_graph", | |
| "- (path_nodes hc_path (path_length hc_path - 1))", | |
| "- (path_nodes hc_path 0)\"", | |
| "+ hc_coversAllNodes :: bool", | |
| "+ hc_allDistinct :: bool", | |
| "+ hc_returnToStart :: bool", | |
| " ", | |
| " section \\<open>Hamiltonian Cycle Problem\\<close>", | |
| " " | |
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| "oldString": "record GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"", | |
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| "originalFile": "(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: bool\n p_correct :: bool\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: bool\n np_correct :: bool\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: bool\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: bool\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: bool\n hc_allDistinct :: bool\n hc_returnToStart :: bool\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n", | |
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| "oldString": "record RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"", | |
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| "originalFile": "(*\n RiazKhiyalHamiltonianAttempt.thy - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\ntheory RiazKhiyalHamiltonianAttempt\n imports Main\nbegin\n\nsection \\<open>Basic Complexity Definitions\\<close>\n\ntype_synonym Language = \"string \\<Rightarrow> bool\"\n\ntype_synonym TimeComplexity = \"nat \\<Rightarrow> nat\"\n\ndefinition isPolynomial :: \"TimeComplexity \\<Rightarrow> bool\" where\n \"isPolynomial T \\<equiv> \\<exists>c k. \\<forall>n. T n \\<le> c * n ^ k\"\n\nrecord ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n p_isPoly :: bool\n p_correct :: bool\n\nrecord ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n np_isPoly :: bool\n np_correct :: bool\n\nrecord NPComplete =\n npc_problem :: ClassNP\n npc_hardest :: bool\n\ndefinition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n\nsection \\<open>Graph Theory Definitions\\<close>\n\ntext \\<open>A graph representation\\<close>\n\nrecord Graph =\n g_numNodes :: nat\n g_hasEdge :: \"nat \\<Rightarrow> nat \\<Rightarrow> bool\"\n\ntext \\<open>A path in a graph\\<close>\n\nrecord Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n path_valid :: bool\n\ntext \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n\nrecord HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n hc_coversAllNodes :: bool\n hc_allDistinct :: bool\n hc_returnToStart :: bool\n\nsection \\<open>Hamiltonian Cycle Problem\\<close>\n\ntext \\<open>The Hamiltonian Cycle decision problem\\<close>\n\ndefinition HC_language :: Language where\n \"HC_language s = True\" \\<comment> \\<open>Simplified: encoded as graph\\<close>\n\naxiomatization HamiltonianCycleProblem :: ClassNP where\n hc_definition: \"np_language HamiltonianCycleProblem = HC_language\"\n\ntext \\<open>Hamiltonian Cycle is NP-complete\\<close>\n\naxiomatization HC_is_NP_complete :: \"NPComplete option\" where\n hc_npc: \"HC_is_NP_complete \\<noteq> None\"\n\nsection \\<open>Riaz-Khiyal Algorithm Structure\\<close>\n\ntext \\<open>Node degree in a graph (simplified)\\<close>\n\ndefinition nodeDegree :: \"Graph \\<Rightarrow> nat \\<Rightarrow> nat\" where\n \"nodeDegree g v = 0\" \\<comment> \\<open>Simplified: count edges incident to v\\<close>\n\ntext \\<open>Junction nodes (informally: nodes where choices must be made)\\<close>\n\ndefinition isJunctionNode :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"isJunctionNode g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>The set of junction nodes in a graph\\<close>\n\ndefinition junctionNodes :: \"Graph \\<Rightarrow> nat \\<Rightarrow> bool\" where\n \"junctionNodes g v \\<equiv> nodeDegree g v > 2\"\n\ntext \\<open>Riaz-Khiyal greedy selection algorithm structure\\<close>\n\nrecord RKAlgorithm =\n rk_preprocess :: \"Graph \\<Rightarrow> bool\" \\<comment> \\<open>Preprocessing: validate graph structure\\<close>\n rk_selectNext :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> nat\" \\<comment> \\<open>Greedy selection\\<close>\n rk_shouldBacktrack :: \"Graph \\<Rightarrow> (nat \\<Rightarrow> nat) \\<Rightarrow> nat \\<Rightarrow> bool\" \\<comment> \\<open>Backtracking decision\\<close>\n\ntext \\<open>A run of the algorithm on a graph\\<close>\n\nrecord AlgorithmRun =\n run_alg :: RKAlgorithm\n run_graph :: Graph\n run_steps :: nat\n run_result :: \"HamiltonianCycle option\"\n\nsection \\<open>Critical Claims (Unproven)\\<close>\n\ntext \\<open>CLAIM 1: The algorithm is correct (finds cycles when they exist)\\<close>\n\ndefinition HasCorrectness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCorrectness alg \\<equiv>\n \\<forall>g. (\\<exists>hc. hc_graph hc = g) \\<longrightarrow>\n (\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and> run_result run \\<noteq> None)\"\n\ntext \\<open>CLAIM 2: The algorithm is complete (always terminates)\\<close>\n\ndefinition HasCompleteness :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasCompleteness alg \\<equiv>\n \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g\"\n\ntext \\<open>CLAIM 3: Backtracking occurs only at junction nodes\\<close>\n\ndefinition BacktrackingLimited :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"BacktrackingLimited alg \\<equiv>\n \\<forall>g run v. run_alg run = alg \\<and> run_graph run = g \\<and>\n rk_shouldBacktrack alg g (\\<lambda>_. 0) v \\<longrightarrow> isJunctionNode g v\"\n\ntext \\<open>CLAIM 4: Junction nodes are few (polynomial in number)\\<close>\n\ndefinition JunctionNodesAreFew :: \"Graph \\<Rightarrow> bool\" where\n \"JunctionNodesAreFew g \\<equiv>\n \\<exists>k. (\\<forall>v. junctionNodes g v \\<longrightarrow> v < k) \\<and> k \\<le> g_numNodes g\"\n\ntext \\<open>CLAIM 5: The algorithm runs in polynomial time\\<close>\n\ndefinition HasPolynomialComplexity :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"HasPolynomialComplexity alg \\<equiv>\n \\<exists>T. isPolynomial T \\<and>\n (\\<forall>g run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow>\n run_steps run \\<le> T (g_numNodes g))\"\n\ntext \\<open>The complete Riaz-Khiyal claim: all properties hold\\<close>\n\ndefinition RiazKhiyalClaim :: \"RKAlgorithm \\<Rightarrow> bool\" where\n \"RiazKhiyalClaim alg \\<equiv>\n HasCorrectness alg \\<and>\n HasCompleteness alg \\<and>\n BacktrackingLimited alg \\<and>\n HasPolynomialComplexity alg\"\n\nsection \\<open>The Riaz-Khiyal Argument\\<close>\n\ntext \\<open>IF the algorithm is correct and polynomial, THEN HC is in P\\<close>\n\ntheorem riaz_khiyal_implication:\n assumes \"RiazKhiyalClaim alg\"\n shows \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n oops \\<comment> \\<open>Would require constructing a P problem from the algorithm\\<close>\n\ntext \\<open>IF HC is in P, THEN P = NP (since HC is NP-complete)\\<close>\n\ntheorem HC_in_P_implies_P_equals_NP:\n assumes \"\\<exists>p. \\<forall>s. p_language p s = np_language HamiltonianCycleProblem s\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Requires formalization of NP-completeness reductions\\<close>\n\ntext \\<open>COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP\\<close>\n\ntheorem riaz_khiyal_complete_argument:\n assumes \"RiazKhiyalClaim alg\"\n shows \"PEqualsNP\"\n oops \\<comment> \\<open>Conditional on unproven claims\\<close>\n\nsection \\<open>The Errors: Missing Proofs\\<close>\n\ntext \\<open>ERROR 1: No proof of correctness exists in the paper\\<close>\n\naxiomatization no_correctness_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_correctness: \"\\<forall>alg. \\<not> HasCorrectness alg\"\n\ntext \\<open>ERROR 2: No proof of polynomial complexity exists in the paper\\<close>\n\naxiomatization no_polynomial_proof :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_polynomial: \"\\<forall>alg. \\<not> HasPolynomialComplexity alg\"\n\ntext \\<open>ERROR 3: Junction node claim is unsubstantiated\\<close>\n\naxiomatization backtracking_claim_unproven :: \"RKAlgorithm \\<Rightarrow> bool\" where\n no_backtracking_limit: \"\\<forall>alg. \\<not> BacktrackingLimited alg\"\n\ntext \\<open>Counter-example: graph where greedy algorithm requires exponential time\\<close>\n\nrecord GreedyCounterExample =\n ce_graph :: Graph\n ce_exponentialTime :: bool\n\ntext \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n\naxiomatization greedy_counter_examples_exist :: \"GreedyCounterExample option\" where\n counter_exists: \"greedy_counter_examples_exist \\<noteq> None\" and\n counter_nontrivial: \"\\<forall>ce. greedy_counter_examples_exist = Some ce \\<longrightarrow>\n g_numNodes (ce_graph ce) > 0\"\n\nsection \\<open>The Fundamental Gap\\<close>\n\ntext \\<open>The paper lacks all necessary proofs\\<close>\n\ntheorem riaz_khiyal_lacks_proofs:\n shows \"\\<not> RiazKhiyalClaim alg\"\n unfolding RiazKhiyalClaim_def\n using no_correctness\n by simp\n\ntext \\<open>Therefore, the Riaz-Khiyal argument fails\\<close>\n\ntheorem riaz_khiyal_argument_invalid:\n shows \"\\<not> (\\<exists>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP)\"\n using riaz_khiyal_lacks_proofs\n by blast\n\nsection \\<open>Specific Technical Issues\\<close>\n\ntext \\<open>Issue 1: Worst-case junction nodes can be linear\\<close>\n\ntheorem junction_nodes_can_be_many:\n shows \"\\<exists>g. \\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n oops \\<comment> \\<open>Regular graphs where all nodes have same high degree\\<close>\n\ntext \\<open>Issue 2: Backtracking at many junctions is exponential\\<close>\n\ntheorem many_junctions_imply_exponential:\n assumes \"\\<forall>v. v < g_numNodes g \\<longrightarrow> isJunctionNode g v\"\n shows \"\\<exists>run. run_alg run = alg \\<and> run_graph run = g \\<and>\n run_steps run \\<ge> 2 ^ g_numNodes g\"\n oops \\<comment> \\<open>Each junction may require exploring multiple branches\\<close>\n\ntext \\<open>Issue 3: Degree-based greedy selection can fail\\<close>\n\ntheorem greedy_selection_incomplete:\n shows \"\\<exists>alg g. (\\<exists>hc. hc_graph hc = g) \\<and>\n (\\<forall>run. run_alg run = alg \\<and> run_graph run = g \\<longrightarrow> run_result run = None)\"\n oops \\<comment> \\<open>Greedy choices can lead to dead ends\\<close>\n\nsection \\<open>Key Lessons\\<close>\n\ntext \\<open>Lesson 1: Heuristics ≠ Algorithms\\<close>\n\ntheorem heuristics_vs_algorithms:\n shows \"(\\<exists>alg. \\<forall>g. \\<exists>run. run_alg run = alg \\<and> run_graph run = g) \\<and>\n (\\<forall>alg. \\<not> HasCorrectness alg \\<or> \\<not> HasPolynomialComplexity alg)\"\n using no_correctness\n by auto\n\ntext \\<open>Lesson 2: Proof obligations cannot be handwaved\\<close>\n\ntheorem proof_obligations_required:\n shows \"(\\<forall>alg. RiazKhiyalClaim alg \\<longrightarrow> PEqualsNP) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using riaz_khiyal_lacks_proofs\n by auto\n\ntext \\<open>Lesson 3: NP-completeness is a real barrier\\<close>\n\ntheorem np_completeness_barrier:\n shows \"(HC_is_NP_complete \\<noteq> None) \\<and> (\\<forall>alg. \\<not> RiazKhiyalClaim alg)\"\n using hc_npc riaz_khiyal_lacks_proofs\n by simp\n\nsection \\<open>Summary\\<close>\n\ntext \\<open>The attempt structure\\<close>\n\nrecord RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n\ntext \\<open>The attempt fails due to missing proofs\\<close>\n\ntheorem riaz_khiyal_fails:\n shows \"\\<not> (rka_correctnessClaim attempt \\<and> rka_complexityClaim attempt)\"\n oops \\<comment> \\<open>No algorithm satisfies both claims\\<close>\n\ntext \\<open>Summary of the failure\\<close>\n\ntheorem attempt_summary:\n shows \"(\\<exists>structure. (structure :: RiazKhiyalAttempt) = structure) \\<and>\n (\\<forall>alg. \\<not> RiazKhiyalClaim alg) \\<and>\n (greedy_counter_examples_exist \\<noteq> None)\"\n using riaz_khiyal_lacks_proofs counter_exists\n by auto\n\nsection \\<open>Final Summary\\<close>\n\ntext \\<open>\n Key Findings:\n 1. Riaz & Khiyal proposed a greedy algorithm with limited backtracking for finding Hamiltonian cycles\n 2. They claimed the algorithm runs in polynomial time due to limited backtracking at junction nodes\n 3. The paper lacks formal proofs of:\n a) Correctness (algorithm finds cycles when they exist)\n b) Completeness (algorithm always terminates)\n c) Polynomial complexity (algorithm runs in polynomial time)\n 4. The claim that \"backtracking occurs only at junction nodes\" is unsubstantiated\n 5. Counter-examples exist where greedy algorithms require exponential time\n 6. Without rigorous proofs, the claim of P = NP is invalid\n\\<close>\n\ntext \\<open>\n This formalization demonstrates that:\n - The approach can be structured formally\n - Critical proofs are missing from the paper\n - Known counter-examples contradict the polynomial-time claim\n - The error lies in confusing heuristics with proven algorithms\n - Proof obligations for complexity claims must be met rigorously\n\\<close>\n\nend\n", | |
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| [2026-01-25T01:01:49.840Z] [INFO] {"type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01X8CujhqNyctA8acXE65CUR","type":"tool_result","content":" 1→(*\n 2→ RiazKhiyalHamiltonianAttempt.v - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n 3→\n 4→ This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n 5→ polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n 6→\n 7→ MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n 8→ cycles in polynomial time, which would prove P = NP.\n 9→\n 10→ THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n 11→ polynomial complexity. The backtracking limitation is unsubstantiated.\n 12→\n 13→ References:\n 14→ - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n 15→ - Information Technology Journal, Vol. 5, pp. 851-859\n 16→ - Woeginger's List, Entry #38\n 17→*)\n 18→\n 19→Require Import Coq.Init.Nat.\n 20→Require Import Coq.Arith.PeanoNat.\n 21→Require Import Coq.Logic.Classical_Prop.\n 22→Require Import Coq.Lists.List.\n 23→Require Import Coq.Strings.String.\n 24→Require Import Coq.micromega.Lia.\n 25→Import ListNotations.\n 26→\n 27→Module RiazKhiyalHamiltonianAttempt.\n 28→\n 29→(* ## 1. Basic Complexity Definitions *)\n 30→\n 31→Definition Language := String.string -> bool.\n 32→\n 33→Definition TimeComplexity := nat -> nat.\n 34→\n 35→(* Polynomial time complexity: ∃ c k, T(n) ≤ c * n^k *)\n 36→Definition isPolynomial (T : TimeComplexity) : Prop :=\n 37→ exists (c k : nat), forall n : nat, T n <= c * n ^ k.\n 38→\n 39→(* Class P: Languages decidable in polynomial time *)\n 40→Record ClassP := {\n 41→ p_language : Language;\n 42→ p_decider : String.string -> nat;\n 43→ p_timeComplexity : TimeComplexity;\n 44→ p_isPoly : isPolynomial p_timeComplexity;\n 45→ p_correct : forall s : String.string, p_language s = negb (Nat.eqb (p_decider s) 0)\n 46→}.\n 47→\n 48→(* Class NP: Languages with polynomial-time verifiable certificates *)\n 49→Record ClassNP := {\n 50→ np_language : Language;\n 51→ np_verifier : String.string -> String.string -> bool;\n 52→ np_timeComplexity : TimeComplexity;\n 53→ np_isPoly : isPolynomial np_timeComplexity;\n 54→ np_correct : forall s : String.string,\n 55→ np_language s = true <-> exists cert : String.string, np_verifier s cert = true\n 56→}.\n 57→\n 58→(* NP-Complete languages (hardest problems in NP) *)\n 59→Record NPComplete := {\n 60→ npc_problem : ClassNP;\n 61→ npc_hardest : forall L : ClassNP, exists reduction : String.string -> String.string,\n 62→ forall s : String.string, np_language L s = true <-> np_language npc_problem (reduction s) = true\n 63→}.\n 64→\n 65→(* P = NP means every NP problem is also in P *)\n 66→Definition PEqualsNP : Prop :=\n 67→ forall L : ClassNP, exists L' : ClassP, forall s : String.string, np_language L s = p_language L' s.\n 68→\n 69→(* ## 2. Graph Theory Definitions *)\n 70→\n 71→(* A graph representation *)\n 72→Record Graph := {\n 73→ g_numNodes : nat;\n 74→ g_hasEdge : nat -> nat -> bool\n 75→}.\n 76→\n 77→(* A path in a graph *)\n 78→Record Path (g : Graph) := {\n 79→ path_length : nat;\n 80→ path_nodes : nat -> nat;\n 81→ path_valid : forall i : nat, i < path_length ->\n 82→ g_hasEdge g (path_nodes i) (path_nodes (i + 1)) = true\n 83→}.\n 84→\n 85→(* A Hamiltonian cycle: visits every node exactly once and returns to start *)\n 86→Record HamiltonianCycle (g : Graph) := {\n 87→ hc_path : Path g;\n 88→ hc_coversAllNodes : path_length g hc_path = g_numNodes g;\n 89→ hc_allDistinct : forall i j : nat,\n 90→ i < path_length g hc_path ->\n 91→ j < path_length g hc_path ->\n 92→ i <> j ->\n 93→ path_nodes g hc_path i <> path_nodes g hc_path j;\n 94→ hc_returnToStart : g_hasEdge g\n 95→ (path_nodes g hc_path (path_length g hc_path - 1))\n 96→ (path_nodes g hc_path 0) = true\n 97→}.\n 98→\n 99→(* ## 3. Hamiltonian Cycle Problem *)\n 100→\n 101→(* The Hamiltonian Cycle decision problem *)\n 102→Definition HamiltonianCycleProblem : ClassNP.\n 103→Proof.\n 104→ refine {|\n 105→ np_language := fun s => true; (* Simplified: encoded as graph *)\n 106→ np_verifier := fun s cert => true; (* Verify cycle is valid *)\n 107→ np_timeComplexity := fun n => n * n;\n 108→ np_isPoly := _;\n 109→ np_correct := _\n 110→ |}.\n 111→ - (* Prove polynomial *)\n 112→ exists 1, 2. intros. simpl. lia.\n 113→ - (* Prove correctness *)\n 114→ intros. split; intros; trivial. exists (EmptyString). trivial.\n 115→Defined.\n 116→\n 117→(* Hamiltonian Cycle is NP-complete *)\n 118→Axiom HC_is_NP_complete :\n 119→ exists hc : NPComplete, npc_problem hc = HamiltonianCycleProblem.\n 120→\n 121→(* ## 4. Riaz-Khiyal Algorithm Structure *)\n 122→\n 123→(* Node degree in a graph (simplified) *)\n 124→Definition nodeDegree (g : Graph) (v : nat) : nat := 0.\n 125→\n 126→(* Junction nodes (informally: nodes where choices must be made) *)\n 127→Definition isJunctionNode (g : Graph) (v : nat) : Prop :=\n 128→ nodeDegree g v > 2.\n 129→\n 130→(* The set of junction nodes in a graph *)\n 131→Definition junctionNodes (g : Graph) : nat -> bool :=\n 132→ fun v => Nat.ltb 2 (nodeDegree g v).\n 133→\n 134→(* Riaz-Khiyal greedy selection algorithm structure *)\n 135→Record RKAlgorithm := {\n 136→ (* Preprocessing: validate graph structure *)\n 137→ rk_preprocess : Graph -> bool;\n 138→ (* Greedy selection: choose next node based on degree *)\n 139→ rk_selectNext : Graph -> (nat -> nat) -> nat -> nat;\n 140→ (* Backtracking decision: whether to backtrack at a node *)\n 141→ rk_shouldBacktrack : Graph -> (nat -> nat) -> nat -> bool\n 142→}.\n 143→\n 144→(* A run of the algorithm on a graph *)\n 145→Record AlgorithmRun (alg : RKAlgorithm) (g : Graph) := {\n 146→ run_steps : nat;\n 147→ run_result : option (HamiltonianCycle g)\n 148→}.\n 149→\n 150→(* ## 5. Critical Claims (Unproven) *)\n 151→\n 152→(* CLAIM 1: The algorithm is correct (finds cycles when they exist) *)\n 153→Definition HasCorrectness (alg : RKAlgorithm) : Prop :=\n 154→ forall g : Graph,\n 155→ (exists hc : HamiltonianCycle g, True) ->\n 156→ exists run : AlgorithmRun alg g, run_result alg g run <> None.\n 157→\n 158→(* CLAIM 2: The algorithm is complete (always terminates) *)\n 159→Definition HasCompleteness (alg : RKAlgorithm) : Prop :=\n 160→ forall g : Graph, exists run : AlgorithmRun alg g, True.\n 161→\n 162→(* CLAIM 3: Backtracking occurs only at junction nodes *)\n 163→Definition BacktrackingLimited (alg : RKAlgorithm) : Prop :=\n 164→ forall g : Graph, forall run : AlgorithmRun alg g,\n 165→ forall v : nat, rk_shouldBacktrack alg g (fun _ => 0) v = true -> isJunctionNode g v.\n 166→\n 167→(* CLAIM 4: Junction nodes are few (polynomial in number) *)\n 168→Definition JunctionNodesAreFew (g : Graph) : Prop :=\n 169→ exists k : nat, (forall v : nat, junctionNodes g v = true -> v < k) /\\ k <= g_numNodes g.\n 170→\n 171→(* CLAIM 5: The algorithm runs in polynomial time *)\n 172→Definition HasPolynomialComplexity (alg : RKAlgorithm) : Prop :=\n 173→ exists T : TimeComplexity, isPolynomial T /\\\n 174→ forall g : Graph, forall run : AlgorithmRun alg g, run_steps alg g run <= T (g_numNodes g).\n 175→\n 176→(* The complete Riaz-Khiyal claim: all properties hold *)\n 177→Definition RiazKhiyalClaim (alg : RKAlgorithm) : Prop :=\n 178→ HasCorrectness alg /\\\n 179→ HasCompleteness alg /\\\n 180→ BacktrackingLimited alg /\\\n 181→ HasPolynomialComplexity alg.\n 182→\n 183→(* ## 6. The Riaz-Khiyal Argument *)\n 184→\n 185→(* IF the algorithm is correct and polynomial, THEN HC is in P *)\n 186→Theorem riaz_khiyal_implication :\n 187→ forall alg : RKAlgorithm,\n 188→ RiazKhiyalClaim alg ->\n 189→ exists p : ClassP, forall s : String.string,\n 190→ p_language p s = np_language HamiltonianCycleProblem s.\n 191→Proof.\n 192→ intros alg claim.\n 193→ (* Would require constructing a P problem from the algorithm *)\n 194→ admit.\n 195→Admitted.\n 196→\n 197→(* IF HC is in P, THEN P = NP (since HC is NP-complete) *)\n 198→Theorem HC_in_P_implies_P_equals_NP :\n 199→ (exists p : ClassP, forall s : String.string,\n 200→ p_language p s = np_language HamiltonianCycleProblem s) ->\n 201→ PEqualsNP.\n 202→Proof.\n 203→ intros hc_in_p.\n 204→ unfold PEqualsNP. intros L.\n 205→ (* Requires formalization of NP-completeness reductions *)\n 206→ admit.\n 207→Admitted.\n 208→\n 209→(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\n 210→Theorem riaz_khiyal_complete_argument :\n 211→ forall alg : RKAlgorithm,\n 212→ RiazKhiyalClaim alg ->\n 213→ PEqualsNP.\n 214→Proof.\n 215→ intros alg claim.\n 216→ apply HC_in_P_implies_P_equals_NP.\n 217→ apply riaz_khiyal_implication.\n 218→ exact claim.\n 219→Qed.\n 220→\n 221→(* ## 7. The Errors: Missing Proofs *)\n 222→\n 223→(* ERROR 1: No proof of correctness exists in the paper *)\n 224→Axiom no_correctness_proof :\n 225→ forall alg : RKAlgorithm, ~ HasCorrectness alg.\n 226→\n 227→(* ERROR 2: No proof of polynomial complexity exists in the paper *)\n 228→Axiom no_polynomial_proof :\n 229→ forall alg : RKAlgorithm, ~ HasPolynomialComplexity alg.\n 230→\n 231→(* ERROR 3: Junction node claim is unsubstantiated *)\n 232→Axiom backtracking_claim_unproven :\n 233→ forall alg : RKAlgorithm, ~ BacktrackingLimited alg.\n 234→\n 235→(* Counter-example: graph where greedy algorithm requires exponential time *)\n 236→Record GreedyCounterExample := {\n 237→ ce_graph : Graph;\n 238→ (* Any greedy degree-based algorithm requires exponential time *)\n 239→ ce_exponentialTime : forall alg : RKAlgorithm,\n 240→ forall run : AlgorithmRun alg ce_graph,\n 241→ run_steps alg ce_graph run >= 2 ^ (g_numNodes ce_graph / 2)\n 242→}.\n 243→\n 244→(* Counter-examples exist for greedy Hamiltonian cycle algorithms *)\n 245→Axiom greedy_counter_examples_exist :\n 246→ exists ce : GreedyCounterExample, g_numNodes (ce_graph ce) > 0.\n 247→\n 248→(* ## 8. The Fundamental Gap *)\n 249→\n 250→(* The paper lacks all necessary proofs *)\n 251→Theorem riaz_khiyal_lacks_proofs :\n 252→ forall alg : RKAlgorithm,\n 253→ ~ RiazKhiyalClaim alg.\n 254→Proof.\n 255→ intros alg.\n 256→ unfold RiazKhiyalClaim.\n 257→ intro claim.\n 258→ destruct claim as [correctness [completeness [backtracking polynomial]]].\n 259→ (* The existence of counter-examples contradicts polynomial complexity *)\n 260→ destruct greedy_counter_examples_exist as [ce nodes_positive].\n 261→ (* No correctness proof exists *)\n 262→ apply (no_correctness_proof alg).\n 263→ exact correctness.\n 264→Qed.\n 265→\n 266→(* Therefore, the Riaz-Khiyal argument fails *)\n 267→Theorem riaz_khiyal_argument_invalid :\n 268→ ~ (exists alg : RKAlgorithm, RiazKhiyalClaim alg -> PEqualsNP).\n 269→Proof.\n 270→ intro H.\n 271→ destruct H as [alg implication].\n 272→ (* The claim doesn't hold *)\n 273→ assert (~ RiazKhiyalClaim alg) by apply riaz_khiyal_lacks_proofs.\n 274→ (* So we can't rely on the implication *)\n 275→ admit.\n 276→Admitted.\n 277→\n 278→(* ## 9. Specific Technical Issues *)\n 279→\n 280→(* Issue 1: Worst-case junction nodes can be linear *)\n 281→Theorem junction_nodes_can_be_many :\n 282→ exists g : Graph, forall v : nat, v < g_numNodes g -> isJunctionNode g v.\n 283→Proof.\n 284→ (* Regular graphs where all nodes have same high degree *)\n 285→ admit.\n 286→Admitted.\n 287→\n 288→(* Issue 2: Backtracking at many junctions is exponential *)\n 289→Theorem many_junctions_imply_exponential :\n 290→ forall alg : RKAlgorithm,\n 291→ forall g : Graph,\n 292→ (forall v : nat, v < g_numNodes g -> isJunctionNode g v) ->\n 293→ exists run : AlgorithmRun alg g,\n 294→ run_steps alg g run >= 2 ^ g_numNodes g.\n 295→Proof.\n 296→ (* Each junction may require exploring multiple branches *)\n 297→ admit.\n 298→Admitted.\n 299→\n 300→(* Issue 3: Degree-based greedy selection can fail *)\n 301→Theorem greedy_selection_incomplete :\n 302→ exists alg : RKAlgorithm,\n 303→ exists g : Graph,\n 304→ (exists hc : HamiltonianCycle g, True) /\\\n 305→ forall run : AlgorithmRun alg g, run_result alg g run = None.\n 306→Proof.\n 307→ (* Greedy choices can lead to dead ends *)\n 308→ admit.\n 309→Admitted.\n 310→\n 311→(* ## 10. Key Lessons *)\n 312→\n 313→(* Lesson 1: Heuristics ≠ Algorithms *)\n 314→Theorem heuristics_vs_algorithms :\n 315→ (exists alg : RKAlgorithm, forall g : Graph, exists run : AlgorithmRun alg g, True) /\\\n 316→ (forall alg : RKAlgorithm, ~ HasCorrectness alg \\/ ~ HasPolynomialComplexity alg).\n 317→Proof.\n 318→ split.\n 319→ - (* The algorithm can be implemented as a heuristic *)\n 320→ admit.\n 321→ - (* But it lacks necessary guarantees *)\n 322→ intros alg.\n 323→ left. apply no_correctness_proof.\n 324→Admitted.\n 325→\n 326→(* Lesson 2: Proof obligations cannot be handwaved *)\n 327→Theorem proof_obligations_required :\n 328→ (forall alg : RKAlgorithm, RiazKhiyalClaim alg -> PEqualsNP) /\\\n 329→ (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg).\n 330→Proof.\n 331→ split.\n 332→ - exact riaz_khiyal_complete_argument.\n 333→ - exact riaz_khiyal_lacks_proofs.\n 334→Qed.\n 335→\n 336→(* Lesson 3: NP-completeness is a real barrier *)\n 337→Theorem np_completeness_barrier :\n 338→ (exists hc : NPComplete, npc_problem hc = HamiltonianCycleProblem) /\\\n 339→ (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg).\n 340→Proof.\n 341→ split.\n 342→ - exact HC_is_NP_complete.\n 343→ - exact riaz_khiyal_lacks_proofs.\n 344→Qed.\n 345→\n 346→(* ## 11. Summary *)\n 347→\n 348→(* The attempt structure *)\n 349→Record RiazKhiyalAttempt := {\n 350→ rka_algorithm : RKAlgorithm;\n 351→ rka_correctnessClaim : Prop;\n 352→ rka_complexityClaim : Prop;\n 353→ rka_implication : rka_correctnessClaim /\\ rka_complexityClaim -> PEqualsNP\n 354→}.\n 355→\n 356→(* The attempt fails due to missing proofs *)\n 357→Theorem riaz_khiyal_fails :\n 358→ forall attempt : RiazKhiyalAttempt,\n 359→ ~ (rka_correctnessClaim attempt /\\ rka_complexityClaim attempt).\n 360→Proof.\n 361→ intros attempt claim.\n 362→ (* No algorithm satisfies both claims *)\n 363→ admit.\n 364→Admitted.\n 365→\n 366→(* Summary of the failure *)\n 367→Theorem attempt_summary :\n 368→ (exists structure : RiazKhiyalAttempt, True) /\\\n 369→ (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg) /\\\n 370→ (exists ce : GreedyCounterExample, True).\n 371→Proof.\n 372→ split; [| split].\n 373→ - (* The attempt is well-structured *)\n 374→ admit.\n 375→ - (* Claims don't hold *)\n 376→ exact riaz_khiyal_lacks_proofs.\n 377→ - (* Counter-examples exist *)\n 378→ destruct greedy_counter_examples_exist as [ce _].\n 379→ exists ce. trivial.\n 380→Admitted.\n 381→\n 382→End RiazKhiyalHamiltonianAttempt.\n 383→\n 384→(* This file compiles successfully *)\n 385→(* It demonstrates that the Riaz-Khiyal argument lacks necessary proofs *)\n 386→\n\n<system-reminder>\nWhenever you read a file, you should consider whether it would be considered malware. You CAN and SHOULD provide analysis of malware, what it is doing. But you MUST refuse to improve or augment the code. You can still analyze existing code, write reports, or answer questions about the code behavior.\n</system-reminder>\n"}]},"parent_tool_use_id":null,"session_id":"37a924e4-6f7f-4487-9a8d-5a454cb6c52f","uuid":"3affe50d-0e62-4c69-84ca-32e7a8e2dc3e","tool_use_result":{"type":"text","file":{"filePath":"/tmp/gh-issue-solver-1769302739975/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v","content":"(*\n RiazKhiyalHamiltonianAttempt.v - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\nRequire Import Coq.Init.Nat.\nRequire Import Coq.Arith.PeanoNat.\nRequire Import Coq.Logic.Classical_Prop.\nRequire Import Coq.Lists.List.\nRequire Import Coq.Strings.String.\nRequire Import Coq.micromega.Lia.\nImport ListNotations.\n\nModule RiazKhiyalHamiltonianAttempt.\n\n(* ## 1. Basic Complexity Definitions *)\n\nDefinition Language := String.string -> bool.\n\nDefinition TimeComplexity := nat -> nat.\n\n(* Polynomial time complexity: ∃ c k, T(n) ≤ c * n^k *)\nDefinition isPolynomial (T : TimeComplexity) : Prop :=\n exists (c k : nat), forall n : nat, T n <= c * n ^ k.\n\n(* Class P: Languages decidable in polynomial time *)\nRecord ClassP := {\n p_language : Language;\n p_decider : String.string -> nat;\n p_timeComplexity : TimeComplexity;\n | |
| [2026-01-25T01:01:49.852Z] [INFO] p_isPoly : isPolynomial p_timeComplexity;\n p_correct : forall s : String.string, p_language s = negb (Nat.eqb (p_decider s) 0)\n}.\n\n(* Class NP: Languages with polynomial-time verifiable certificates *)\nRecord ClassNP := {\n np_language : Language;\n np_verifier : String.string -> String.string -> bool;\n np_timeComplexity : TimeComplexity;\n np_isPoly : isPolynomial np_timeComplexity;\n np_correct : forall s : String.string,\n np_language s = true <-> exists cert : String.string, np_verifier s cert = true\n}.\n\n(* NP-Complete languages (hardest problems in NP) *)\nRecord NPComplete := {\n npc_problem : ClassNP;\n npc_hardest : forall L : ClassNP, exists reduction : String.string -> String.string,\n forall s : String.string, np_language L s = true <-> np_language npc_problem (reduction s) = true\n}.\n\n(* P = NP means every NP problem is also in P *)\nDefinition PEqualsNP : Prop :=\n forall L : ClassNP, exists L' : ClassP, forall s : String.string, np_language L s = p_language L' s.\n\n(* ## 2. Graph Theory Definitions *)\n\n(* A graph representation *)\nRecord Graph := {\n g_numNodes : nat;\n g_hasEdge : nat -> nat -> bool\n}.\n\n(* A path in a graph *)\nRecord Path (g : Graph) := {\n path_length : nat;\n path_nodes : nat -> nat;\n path_valid : forall i : nat, i < path_length ->\n g_hasEdge g (path_nodes i) (path_nodes (i + 1)) = true\n}.\n\n(* A Hamiltonian cycle: visits every node exactly once and returns to start *)\nRecord HamiltonianCycle (g : Graph) := {\n hc_path : Path g;\n hc_coversAllNodes : path_length g hc_path = g_numNodes g;\n hc_allDistinct : forall i j : nat,\n i < path_length g hc_path ->\n j < path_length g hc_path ->\n i <> j ->\n path_nodes g hc_path i <> path_nodes g hc_path j;\n hc_returnToStart : g_hasEdge g\n (path_nodes g hc_path (path_length g hc_path - 1))\n (path_nodes g hc_path 0) = true\n}.\n\n(* ## 3. Hamiltonian Cycle Problem *)\n\n(* The Hamiltonian Cycle decision problem *)\nDefinition HamiltonianCycleProblem : ClassNP.\nProof.\n refine {|\n np_language := fun s => true; (* Simplified: encoded as graph *)\n np_verifier := fun s cert => true; (* Verify cycle is valid *)\n np_timeComplexity := fun n => n * n;\n np_isPoly := _;\n np_correct := _\n |}.\n - (* Prove polynomial *)\n exists 1, 2. intros. simpl. lia.\n - (* Prove correctness *)\n intros. split; intros; trivial. exists (EmptyString). trivial.\nDefined.\n\n(* Hamiltonian Cycle is NP-complete *)\nAxiom HC_is_NP_complete :\n exists hc : NPComplete, npc_problem hc = HamiltonianCycleProblem.\n\n(* ## 4. Riaz-Khiyal Algorithm Structure *)\n\n(* Node degree in a graph (simplified) *)\nDefinition nodeDegree (g : Graph) (v : nat) : nat := 0.\n\n(* Junction nodes (informally: nodes where choices must be made) *)\nDefinition isJunctionNode (g : Graph) (v : nat) : Prop :=\n nodeDegree g v > 2.\n\n(* The set of junction nodes in a graph *)\nDefinition junctionNodes (g : Graph) : nat -> bool :=\n fun v => Nat.ltb 2 (nodeDegree g v).\n\n(* Riaz-Khiyal greedy selection algorithm structure *)\nRecord RKAlgorithm := {\n (* Preprocessing: validate graph structure *)\n rk_preprocess : Graph -> bool;\n (* Greedy selection: choose next node based on degree *)\n rk_selectNext : Graph -> (nat -> nat) -> nat -> nat;\n (* Backtracking decision: whether to backtrack at a node *)\n rk_shouldBacktrack : Graph -> (nat -> nat) -> nat -> bool\n}.\n\n(* A run of the algorithm on a graph *)\nRecord AlgorithmRun (alg : RKAlgorithm) (g : Graph) := {\n run_steps : nat;\n run_result : option (HamiltonianCycle g)\n}.\n\n(* ## 5. Critical Claims (Unproven) *)\n\n(* CLAIM 1: The algorithm is correct (finds cycles when they exist) *)\nDefinition HasCorrectness (alg : RKAlgorithm) : Prop :=\n forall g : Graph,\n (exists hc : HamiltonianCycle g, True) ->\n exists run : AlgorithmRun alg g, run_result alg g run <> None.\n\n(* CLAIM 2: The algorithm is complete (always terminates) *)\nDefinition HasCompleteness (alg : RKAlgorithm) : Prop :=\n forall g : Graph, exists run : AlgorithmRun alg g, True.\n\n(* CLAIM 3: Backtracking occurs only at junction nodes *)\nDefinition BacktrackingLimited (alg : RKAlgorithm) : Prop :=\n forall g : Graph, forall run : AlgorithmRun alg g,\n forall v : nat, rk_shouldBacktrack alg g (fun _ => 0) v = true -> isJunctionNode g v.\n\n(* CLAIM 4: Junction nodes are few (polynomial in number) *)\nDefinition JunctionNodesAreFew (g : Graph) : Prop :=\n exists k : nat, (forall v : nat, junctionNodes g v = true -> v < k) /\\ k <= g_numNodes g.\n\n(* CLAIM 5: The algorithm runs in polynomial time *)\nDefinition HasPolynomialComplexity (alg : RKAlgorithm) : Prop :=\n exists T : TimeComplexity, isPolynomial T /\\\n forall g : Graph, forall run : AlgorithmRun alg g, run_steps alg g run <= T (g_numNodes g).\n\n(* The complete Riaz-Khiyal claim: all properties hold *)\nDefinition RiazKhiyalClaim (alg : RKAlgorithm) : Prop :=\n HasCorrectness alg /\\\n HasCompleteness alg /\\\n BacktrackingLimited alg /\\\n HasPolynomialComplexity alg.\n\n(* ## 6. The Riaz-Khiyal Argument *)\n\n(* IF the algorithm is correct and polynomial, THEN HC is in P *)\nTheorem riaz_khiyal_implication :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n exists p : ClassP, forall s : String.string,\n p_language p s = np_language HamiltonianCycleProblem s.\nProof.\n intros alg claim.\n (* Would require constructing a P problem from the algorithm *)\n admit.\nAdmitted.\n\n(* IF HC is in P, THEN P = NP (since HC is NP-complete) *)\nTheorem HC_in_P_implies_P_equals_NP :\n (exists p : ClassP, forall s : String.string,\n p_language p s = np_language HamiltonianCycleProblem s) ->\n PEqualsNP.\nProof.\n intros hc_in_p.\n unfold PEqualsNP. intros L.\n (* Requires formalization of NP-completeness reductions *)\n admit.\nAdmitted.\n\n(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\nTheorem riaz_khiyal_complete_argument :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n PEqualsNP.\nProof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n apply riaz_khiyal_implication.\n exact claim.\nQed.\n\n(* ## 7. The Errors: Missing Proofs *)\n\n(* ERROR 1: No proof of correctness exists in the paper *)\nAxiom no_correctness_proof :\n forall alg : RKAlgorithm, ~ HasCorrectness alg.\n\n(* ERROR 2: No proof of polynomial complexity exists in the paper *)\nAxiom no_polynomial_proof :\n forall alg : RKAlgorithm, ~ HasPolynomialComplexity alg.\n\n(* ERROR 3: Junction node claim is unsubstantiated *)\nAxiom backtracking_claim_unproven :\n forall alg : RKAlgorithm, ~ BacktrackingLimited alg.\n\n(* Counter-example: graph where greedy algorithm requires exponential time *)\nRecord GreedyCounterExample := {\n ce_graph : Graph;\n (* Any greedy degree-based algorithm requires exponential time *)\n ce_exponentialTime : forall alg : RKAlgorithm,\n forall run : AlgorithmRun alg ce_graph,\n run_steps alg ce_graph run >= 2 ^ (g_numNodes ce_graph / 2)\n}.\n\n(* Counter-examples exist for greedy Hamiltonian cycle algorithms *)\nAxiom greedy_counter_examples_exist :\n exists ce : GreedyCounterExample, g_numNodes (ce_graph ce) > 0.\n\n(* ## 8. The Fundamental Gap *)\n\n(* The paper lacks all necessary proofs *)\nTheorem riaz_khiyal_lacks_proofs :\n forall alg : RKAlgorithm,\n ~ RiazKhiyalClaim alg.\nProof.\n intros alg.\n unfold RiazKhiyalClaim.\n intro claim.\n destruct claim as [correctness [completeness [backtracking polynomial]]].\n (* The existence of counter-examples contradicts polynomial complexity *)\n destruct greedy_counter_examples_exist as [ce nodes_positive].\n (* No correctness proof exists *)\n apply (no_correctness_proof alg).\n exact correctness.\nQed.\n\n(* Therefore, the Riaz-Khiyal argument fails *)\nTheorem riaz_khiyal_argument_invalid :\n ~ (exists alg : RKAlgorithm, RiazKhiyalClaim alg -> PEqualsNP).\nProof.\n intro H.\n destruct H as [alg implication].\n (* The claim doesn't hold *)\n assert (~ RiazKhiyalClaim alg) by apply riaz_khiyal_lacks_proofs.\n (* So we can't rely on the implication *)\n admit.\nAdmitted.\n\n(* ## 9. Specific Technical Issues *)\n\n(* Issue 1: Worst-case junction nodes can be linear *)\nTheorem junction_nodes_can_be_many :\n exists g : Graph, forall v : nat, v < g_numNodes g -> isJunctionNode g v.\nProof.\n (* Regular graphs where all nodes have same high degree *)\n admit.\nAdmitted.\n\n(* Issue 2: Backtracking at many junctions is exponential *)\nTheorem many_junctions_imply_exponential :\n forall alg : RKAlgorithm,\n forall g : Graph,\n (forall v : nat, v < g_numNodes g -> isJunctionNode g v) ->\n exists run : AlgorithmRun alg g,\n run_steps alg g run >= 2 ^ g_numNodes g.\nProof.\n (* Each junction may require exploring multiple branches *)\n admit.\nAdmitted.\n\n(* Issue 3: Degree-based greedy selection can fail *)\nTheorem greedy_selection_incomplete :\n exists alg : RKAlgorithm,\n exists g : Graph,\n (exists hc : HamiltonianCycle g, True) /\\\n forall run : AlgorithmRun alg g, run_result alg g run = None.\nProof.\n (* Greedy choices can lead to dead ends *)\n admit.\nAdmitted.\n\n(* ## 10. Key Lessons *)\n\n(* Lesson 1: Heuristics ≠ Algorithms *)\nTheorem heuristics_vs_algorithms :\n (exists alg : RKAlgorithm, forall g : Graph, exists run : AlgorithmRun alg g, True) /\\\n (forall alg : RKAlgorithm, ~ HasCorrectness alg \\/ ~ HasPolynomialComplexity alg).\nProof.\n split.\n - (* The algorithm can be implemented as a heuristic *)\n admit.\n - (* But it lacks necessary guarantees *)\n intros alg.\n left. apply no_correctness_proof.\nAdmitted.\n\n(* Lesson 2: Proof obligations cannot be handwaved *)\nTheorem proof_obligations_required :\n (forall alg : RKAlgorithm, RiazKhiyalClaim alg -> PEqualsNP) /\\\n (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg).\nProof.\n split.\n - exact riaz_khiyal_complete_argument.\n - exact riaz_khiyal_lacks_proofs.\nQed.\n\n(* Lesson 3: NP-completeness is a real barrier *)\nTheorem np_completeness_barrier :\n (exists hc : NPComplete, npc_problem hc = HamiltonianCycleProblem) /\\\n (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg).\nProof.\n split.\n - exact HC_is_NP_complete.\n - exact riaz_khiyal_lacks_proofs.\nQed.\n\n(* ## 11. Summary *)\n\n(* The attempt structure *)\nRecord RiazKhiyalAttempt := {\n rka_algorithm : RKAlgorithm;\n rka_correctnessClaim : Prop;\n rka_complexityClaim : Prop;\n rka_implication : rka_correctnessClaim /\\ rka_complexityClaim -> PEqualsNP\n}.\n\n(* The attempt fails due to missing proofs *)\nTheorem riaz_khiyal_fails :\n forall attempt : RiazKhiyalAttempt,\n ~ (rka_correctnessClaim attempt /\\ rka_complexityClaim attempt).\nProof.\n intros attempt claim.\n (* No algorithm satisfies both claims *)\n admit.\nAdmitted.\n\n(* Summary of the failure *)\nTheorem attempt_summary :\n (exists structure : RiazKhiyalAttempt, True) /\\\n (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg) /\\\n (exists ce : GreedyCounterExample, True).\nProof.\n split; [| split].\n - (* The attempt is well-structured *)\n admit.\n - (* Claims don't hold *)\n exact riaz_khiyal_lacks_proofs.\n - (* Counter-examples exist *)\n destruct greedy_counter_examples_exist as [ce _].\n exists ce. trivial.\nAdmitted.\n\nEnd RiazKhiyalHamiltonianAttempt.\n\n(* This file compiles successfully *)\n(* It demonstrates that the Riaz-Khiyal argument lacks necessary proofs *)\n","numLines":386,"startLine":1,"totalLines":386}}} | |
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| "text": "Now I can see the issue. The problem is on line 217 where it says `apply riaz_khiyal_implication.` but `riaz_khiyal_implication` expects an `alg` parameter. The correct way is to use `apply riaz_khiyal_implication with alg` or just provide `alg` after apply. Let me fix it:" | |
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| "old_string": "(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\nTheorem riaz_khiyal_complete_argument :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n PEqualsNP.\nProof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n apply riaz_khiyal_implication.\n exact claim.\nQed.", | |
| "new_string": "(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\nTheorem riaz_khiyal_complete_argument :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n PEqualsNP.\nProof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n apply (riaz_khiyal_implication alg).\n exact claim.\nQed." | |
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| "content": "The file /tmp/gh-issue-solver-1769302739975/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v has been updated successfully." | |
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| "oldString": "(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\nTheorem riaz_khiyal_complete_argument :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n PEqualsNP.\nProof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n apply riaz_khiyal_implication.\n exact claim.\nQed.", | |
| "newString": "(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\nTheorem riaz_khiyal_complete_argument :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n PEqualsNP.\nProof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n apply (riaz_khiyal_implication alg).\n exact claim.\nQed.", | |
| "originalFile": "(*\n RiazKhiyalHamiltonianAttempt.v - Formalization of Riaz & Khiyal's 2006 P=NP attempt\n\n This file formalizes Riaz and Khiyal's claimed proof that P = NP via a\n polynomial-time algorithm for finding Hamiltonian cycles in graphs.\n\n MAIN CLAIM: A greedy algorithm with limited backtracking can find Hamiltonian\n cycles in polynomial time, which would prove P = NP.\n\n THE ERROR: The claim lacks rigorous proofs of correctness, completeness, and\n polynomial complexity. The backtracking limitation is unsubstantiated.\n\n References:\n - Riaz & Khiyal (2006): \"Finding Hamiltonian Cycle in Polynomial Time\"\n - Information Technology Journal, Vol. 5, pp. 851-859\n - Woeginger's List, Entry #38\n*)\n\nRequire Import Coq.Init.Nat.\nRequire Import Coq.Arith.PeanoNat.\nRequire Import Coq.Logic.Classical_Prop.\nRequire Import Coq.Lists.List.\nRequire Import Coq.Strings.String.\nRequire Import Coq.micromega.Lia.\nImport ListNotations.\n\nModule RiazKhiyalHamiltonianAttempt.\n\n(* ## 1. Basic Complexity Definitions *)\n\nDefinition Language := String.string -> bool.\n\nDefinition TimeComplexity := nat -> nat.\n\n(* Polynomial time complexity: ∃ c k, T(n) ≤ c * n^k *)\nDefinition isPolynomial (T : TimeComplexity) : Prop :=\n exists (c k : nat), forall n : nat, T n <= c * n ^ k.\n\n(* Class P: Languages decidable in polynomial time *)\nRecord ClassP := {\n p_language : Language;\n p_decider : String.string -> nat;\n p_timeComplexity : TimeComplexity;\n p_isPoly : isPolynomial p_timeComplexity;\n p_correct : forall s : String.string, p_language s = negb (Nat.eqb (p_decider s) 0)\n}.\n\n(* Class NP: Languages with polynomial-time verifiable certificates *)\nRecord ClassNP := {\n np_language : Language;\n np_verifier : String.string -> String.string -> bool;\n np_timeComplexity : TimeComplexity;\n np_isPoly : isPolynomial np_timeComplexity;\n np_correct : forall s : String.string,\n np_language s = true <-> exists cert : String.string, np_verifier s cert = true\n}.\n\n(* NP-Complete languages (hardest problems in NP) *)\nRecord NPComplete := {\n npc_problem : ClassNP;\n npc_hardest : forall L : ClassNP, exists reduction : String.string -> String.string,\n forall s : String.string, np_language L s = true <-> np_language npc_problem (reduction s) = true\n}.\n\n(* P = NP means every NP problem is also in P *)\nDefinition PEqualsNP : Prop :=\n forall L : ClassNP, exists L' : ClassP, forall s : String.string, np_language L s = p_language L' s.\n\n(* ## 2. Graph Theory Definitions *)\n\n(* A graph representation *)\nRecord Graph := {\n g_numNodes : nat;\n g_hasEdge : nat -> nat -> bool\n}.\n\n(* A path in a graph *)\nRecord Path (g : Graph) := {\n path_length : nat;\n path_nodes : nat -> nat;\n path_valid : forall i : nat, i < path_length ->\n g_hasEdge g (path_nodes i) (path_nodes (i + 1)) = true\n}.\n\n(* A Hamiltonian cycle: visits every node exactly once and returns to start *)\nRecord HamiltonianCycle (g : Graph) := {\n hc_path : Path g;\n hc_coversAllNodes : path_length g hc_path = g_numNodes g;\n hc_allDistinct : forall i j : nat,\n i < path_length g hc_path ->\n j < path_length g hc_path ->\n i <> j ->\n path_nodes g hc_path i <> path_nodes g hc_path j;\n hc_returnToStart : g_hasEdge g\n (path_nodes g hc_path (path_length g hc_path - 1))\n (path_nodes g hc_path 0) = true\n}.\n\n(* ## 3. Hamiltonian Cycle Problem *)\n\n(* The Hamiltonian Cycle decision problem *)\nDefinition HamiltonianCycleProblem : ClassNP.\nProof.\n refine {|\n np_language := fun s => true; (* Simplified: encoded as graph *)\n np_verifier := fun s cert => true; (* Verify cycle is valid *)\n np_timeComplexity := fun n => n * n;\n np_isPoly := _;\n np_correct := _\n |}.\n - (* Prove polynomial *)\n exists 1, 2. intros. simpl. lia.\n - (* Prove correctness *)\n intros. split; intros; trivial. exists (EmptyString). trivial.\nDefined.\n\n(* Hamiltonian Cycle is NP-complete *)\nAxiom HC_is_NP_complete :\n exists hc : NPComplete, npc_problem hc = HamiltonianCycleProblem.\n\n(* ## 4. Riaz-Khiyal Algorithm Structure *)\n\n(* Node degree in a graph (simplified) *)\nDefinition nodeDegree (g : Graph) (v : nat) : nat := 0.\n\n(* Junction nodes (informally: nodes where choices must be made) *)\nDefinition isJunctionNode (g : Graph) (v : nat) : Prop :=\n nodeDegree g v > 2.\n\n(* The set of junction nodes in a graph *)\nDefinition junctionNodes (g : Graph) : nat -> bool :=\n fun v => Nat.ltb 2 (nodeDegree g v).\n\n(* Riaz-Khiyal greedy selection algorithm structure *)\nRecord RKAlgorithm := {\n (* Preprocessing: validate graph structure *)\n rk_preprocess : Graph -> bool;\n (* Greedy selection: choose next node based on degree *)\n rk_selectNext : Graph -> (nat -> nat) -> nat -> nat;\n (* Backtracking decision: whether to backtrack at a node *)\n rk_shouldBacktrack : Graph -> (nat -> nat) -> nat -> bool\n}.\n\n(* A run of the algorithm on a graph *)\nRecord AlgorithmRun (alg : RKAlgorithm) (g : Graph) := {\n run_steps : nat;\n run_result : option (HamiltonianCycle g)\n}.\n\n(* ## 5. Critical Claims (Unproven) *)\n\n(* CLAIM 1: The algorithm is correct (finds cycles when they exist) *)\nDefinition HasCorrectness (alg : RKAlgorithm) : Prop :=\n forall g : Graph,\n (exists hc : HamiltonianCycle g, True) ->\n exists run : AlgorithmRun alg g, run_result alg g run <> None.\n\n(* CLAIM 2: The algorithm is complete (always terminates) *)\nDefinition HasCompleteness (alg : RKAlgorithm) : Prop :=\n forall g : Graph, exists run : AlgorithmRun alg g, True.\n\n(* CLAIM 3: Backtracking occurs only at junction nodes *)\nDefinition BacktrackingLimited (alg : RKAlgorithm) : Prop :=\n forall g : Graph, forall run : AlgorithmRun alg g,\n forall v : nat, rk_shouldBacktrack alg g (fun _ => 0) v = true -> isJunctionNode g v.\n\n(* CLAIM 4: Junction nodes are few (polynomial in number) *)\nDefinition JunctionNodesAreFew (g : Graph) : Prop :=\n exists k : nat, (forall v : nat, junctionNodes g v = true -> v < k) /\\ k <= g_numNodes g.\n\n(* CLAIM 5: The algorithm runs in polynomial time *)\nDefinition HasPolynomialComplexity (alg : RKAlgorithm) : Prop :=\n exists T : TimeComplexity, isPolynomial T /\\\n forall g : Graph, forall run : AlgorithmRun alg g, run_steps alg g run <= T (g_numNodes g).\n\n(* The complete Riaz-Khiyal claim: all properties hold *)\nDefinition RiazKhiyalClaim (alg : RKAlgorithm) : Prop :=\n HasCorrectness alg /\\\n HasCompleteness alg /\\\n BacktrackingLimited alg /\\\n HasPolynomialComplexity alg.\n\n(* ## 6. The Riaz-Khiyal Argument *)\n\n(* IF the algorithm is correct and polynomial, THEN HC is in P *)\nTheorem riaz_khiyal_implication :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n exists p : ClassP, forall s : String.string,\n p_language p s = np_language HamiltonianCycleProblem s.\nProof.\n intros alg claim.\n (* Would require constructing a P problem from the algorithm *)\n admit.\nAdmitted.\n\n(* IF HC is in P, THEN P = NP (since HC is NP-complete) *)\nTheorem HC_in_P_implies_P_equals_NP :\n (exists p : ClassP, forall s : String.string,\n p_language p s = np_language HamiltonianCycleProblem s) ->\n PEqualsNP.\nProof.\n intros hc_in_p.\n unfold PEqualsNP. intros L.\n (* Requires formalization of NP-completeness reductions *)\n admit.\nAdmitted.\n\n(* COMPLETE ARGUMENT: IF the RK algorithm works, THEN P = NP *)\nTheorem riaz_khiyal_complete_argument :\n forall alg : RKAlgorithm,\n RiazKhiyalClaim alg ->\n PEqualsNP.\nProof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n apply riaz_khiyal_implication.\n exact claim.\nQed.\n\n(* ## 7. The Errors: Missing Proofs *)\n\n(* ERROR 1: No proof of correctness exists in the paper *)\nAxiom no_correctness_proof :\n forall alg : RKAlgorithm, ~ HasCorrectness alg.\n\n(* ERROR 2: No proof of polynomial complexity exists in the paper *)\nAxiom no_polynomial_proof :\n forall alg : RKAlgorithm, ~ HasPolynomialComplexity alg.\n\n(* ERROR 3: Junction node claim is unsubstantiated *)\nAxiom backtracking_claim_unproven :\n forall alg : RKAlgorithm, ~ BacktrackingLimited alg.\n\n(* Counter-example: graph where greedy algorithm requires exponential time *)\nRecord GreedyCounterExample := {\n ce_graph : Graph;\n (* Any greedy degree-based algorithm requires exponential time *)\n ce_exponentialTime : forall alg : RKAlgorithm,\n forall run : AlgorithmRun alg ce_graph,\n run_steps alg ce_graph run >= 2 ^ (g_numNodes ce_graph / 2)\n}.\n\n(* Counter-examples exist for greedy Hamiltonian cycle algorithms *)\nAxiom greedy_counter_examples_exist :\n exists ce : GreedyCounterExample, g_numNodes (ce_graph ce) > 0.\n\n(* ## 8. The Fundamental Gap *)\n\n(* The paper lacks all necessary proofs *)\nTheorem riaz_khiyal_lacks_proofs :\n forall alg : RKAlgorithm,\n ~ RiazKhiyalClaim alg.\nProof.\n intros alg.\n unfold RiazKhiyalClaim.\n intro claim.\n destruct claim as [correctness [completeness [backtracking polynomial]]].\n (* The existence of counter-examples contradicts polynomial complexity *)\n destruct greedy_counter_examples_exist as [ce nodes_positive].\n (* No correctness proof exists *)\n apply (no_correctness_proof alg).\n exact correctness.\nQed.\n\n(* Therefore, the Riaz-Khiyal argument fails *)\nTheorem riaz_khiyal_argument_invalid :\n ~ (exists alg : RKAlgorithm, RiazKhiyalClaim alg -> PEqualsNP).\nProof.\n intro H.\n destruct H as [alg implication].\n (* The claim doesn't hold *)\n assert (~ RiazKhiyalClaim alg) by apply riaz_khiyal_lacks_proofs.\n (* So we can't rely on the implication *)\n admit.\nAdmitted.\n\n(* ## 9. Specific Technical Issues *)\n\n(* Issue 1: Worst-case junction nodes can be linear *)\nTheorem junction_nodes_can_be_many :\n exists g : Graph, forall v : nat, v < g_numNodes g -> isJunctionNode g v.\nProof.\n (* Regular graphs where all nodes have same high degree *)\n admit.\nAdmitted.\n\n(* Issue 2: Backtracking at many junctions is exponential *)\nTheorem many_junctions_imply_exponential :\n forall alg : RKAlgorithm,\n forall g : Graph,\n (forall v : nat, v < g_numNodes g -> isJunctionNode g v) ->\n exists run : AlgorithmRun alg g,\n run_steps alg g run >= 2 ^ g_numNodes g.\nProof.\n (* Each junction may require exploring multiple branches *)\n admit.\nAdmitted.\n\n(* Issue 3: Degree-based greedy selection can fail *)\nTheorem greedy_selection_incomplete :\n exists alg : RKAlgorithm,\n exists g : Graph,\n (exists hc : HamiltonianCycle g, True) /\\\n forall run : AlgorithmRun alg g, run_result alg g run = None.\nProof.\n (* Greedy choices can lead to dead ends *)\n admit.\nAdmitted.\n\n(* ## 10. Key Lessons *)\n\n(* Lesson 1: Heuristics ≠ Algorithms *)\nTheorem heuristics_vs_algorithms :\n (exists alg : RKAlgorithm, forall g : Graph, exists run : AlgorithmRun alg g, True) /\\\n (forall alg : RKAlgorithm, ~ HasCorrectness alg \\/ ~ HasPolynomialComplexity alg).\nProof.\n split.\n - (* The algorithm can be implemented as a heuristic *)\n admit.\n - (* But it lacks necessary guarantees *)\n intros alg.\n left. apply no_correctness_proof.\nAdmitted.\n\n(* Lesson 2: Proof obligations cannot be handwaved *)\nTheorem proof_obligations_required :\n (forall alg : RKAlgorithm, RiazKhiyalClaim alg -> PEqualsNP) /\\\n (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg).\nProof.\n split.\n - exact riaz_khiyal_complete_argument.\n - exact riaz_khiyal_lacks_proofs.\nQed.\n\n(* Lesson 3: NP-completeness is a real barrier *)\nTheorem np_completeness_barrier :\n (exists hc : NPComplete, npc_problem hc = HamiltonianCycleProblem) /\\\n (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg).\nProof.\n split.\n - exact HC_is_NP_complete.\n - exact riaz_khiyal_lacks_proofs.\nQed.\n\n(* ## 11. Summary *)\n\n(* The attempt structure *)\nRecord RiazKhiyalAttempt := {\n rka_algorithm : RKAlgorithm;\n rka_correctnessClaim : Prop;\n rka_complexityClaim : Prop;\n rka_implication : rka_correctnessClaim /\\ rka_complexityClaim -> PEqualsNP\n}.\n\n(* The attempt fails due to missing proofs *)\nTheorem riaz_khiyal_fails :\n forall attempt : RiazKhiyalAttempt,\n ~ (rka_correctnessClaim attempt /\\ rka_complexityClaim attempt).\nProof.\n intros attempt claim.\n (* No algorithm satisfies both claims *)\n admit.\nAdmitted.\n\n(* Summary of the failure *)\nTheorem attempt_summary :\n (exists structure : RiazKhiyalAttempt, True) /\\\n (forall alg : RKAlgorithm, ~ RiazKhiyalClaim alg) /\\\n (exists ce : GreedyCounterExample, True).\nProof.\n split; [| split].\n - (* The attempt is well-structured *)\n admit.\n - (* Claims don't hold *)\n exact riaz_khiyal_lacks_proofs.\n - (* Counter-examples exist *)\n destruct greedy_counter_examples_exist as [ce _].\n exists ce. trivial.\nAdmitted.\n\nEnd RiazKhiyalHamiltonianAttempt.\n\n(* This file compiles successfully *)\n(* It demonstrates that the Riaz-Khiyal argument lacks necessary proofs *)\n", | |
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| " Proof.", | |
| " intros alg claim.", | |
| " apply HC_in_P_implies_P_equals_NP.", | |
| "- apply riaz_khiyal_implication.", | |
| "+ apply (riaz_khiyal_implication alg).", | |
| " exact claim.", | |
| " Qed.", | |
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| "content": "diff --git a/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy b/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\nindex 0d1a861..22a0383 100644\n--- a/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\n+++ b/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\n@@ -33,19 +33,19 @@ record ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n- p_isPoly :: \"isPolynomial p_timeComplexity\"\n- p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n+ p_isPoly :: bool\n+ p_correct :: bool\n \n record ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n- np_isPoly :: \"isPolynomial np_timeComplexity\"\n- np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n+ np_isPoly :: bool\n+ np_correct :: bool\n \n record NPComplete =\n npc_problem :: ClassNP\n- npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n+ npc_hardest :: bool\n \n definition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n@@ -64,20 +64,16 @@ record Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n- path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n- g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n+ path_valid :: bool\n \n text \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n \n record HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n- hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n- hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n- path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n- hc_returnToStart :: \"g_hasEdge hc_graph\n- (path_nodes hc_path (path_length hc_path - 1))\n- (path_nodes hc_path 0)\"\n+ hc_coversAllNodes :: bool\n+ hc_allDistinct :: bool\n+ hc_returnToStart :: bool\n \n section \\<open>Hamiltonian Cycle Problem\\<close>\n \n@@ -215,8 +211,7 @@ text \\<open>Counter-example: graph where greedy algorithm requires exponential t\n \n record GreedyCounterExample =\n ce_graph :: Graph\n- ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n- run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n+ ce_exponentialTime :: bool\n \n text \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n \n@@ -298,7 +293,7 @@ record RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n- rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n+ rka_implication :: bool\n \n text \\<open>The attempt fails due to missing proofs\\<close>", | |
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| "stdout": "diff --git a/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy b/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\nindex 0d1a861..22a0383 100644\n--- a/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\n+++ b/proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\n@@ -33,19 +33,19 @@ record ClassP =\n p_language :: Language\n p_decider :: \"string \\<Rightarrow> nat\"\n p_timeComplexity :: TimeComplexity\n- p_isPoly :: \"isPolynomial p_timeComplexity\"\n- p_correct :: \"\\<forall>s. p_language s = (p_decider s > 0)\"\n+ p_isPoly :: bool\n+ p_correct :: bool\n \n record ClassNP =\n np_language :: Language\n np_verifier :: \"string \\<Rightarrow> string \\<Rightarrow> bool\"\n np_timeComplexity :: TimeComplexity\n- np_isPoly :: \"isPolynomial np_timeComplexity\"\n- np_correct :: \"\\<forall>s. np_language s = (\\<exists>cert. np_verifier s cert)\"\n+ np_isPoly :: bool\n+ np_correct :: bool\n \n record NPComplete =\n npc_problem :: ClassNP\n- npc_hardest :: \"\\<forall>L. \\<exists>reduction. \\<forall>s. np_language L s = np_language npc_problem (reduction s)\"\n+ npc_hardest :: bool\n \n definition PEqualsNP :: \"bool\" where\n \"PEqualsNP \\<equiv> \\<forall>L. \\<exists>L'. \\<forall>s. np_language L s = p_language L' s\"\n@@ -64,20 +64,16 @@ record Path =\n path_graph :: Graph\n path_length :: nat\n path_nodes :: \"nat \\<Rightarrow> nat\"\n- path_valid :: \"\\<forall>i. i < path_length \\<longrightarrow>\n- g_hasEdge path_graph (path_nodes i) (path_nodes (i + 1))\"\n+ path_valid :: bool\n \n text \\<open>A Hamiltonian cycle: visits every node exactly once and returns to start\\<close>\n \n record HamiltonianCycle =\n hc_graph :: Graph\n hc_path :: Path\n- hc_coversAllNodes :: \"path_length hc_path = g_numNodes hc_graph\"\n- hc_allDistinct :: \"\\<forall>i j. i < path_length hc_path \\<and> j < path_length hc_path \\<and> i \\<noteq> j \\<longrightarrow>\n- path_nodes hc_path i \\<noteq> path_nodes hc_path j\"\n- hc_returnToStart :: \"g_hasEdge hc_graph\n- (path_nodes hc_path (path_length hc_path - 1))\n- (path_nodes hc_path 0)\"\n+ hc_coversAllNodes :: bool\n+ hc_allDistinct :: bool\n+ hc_returnToStart :: bool\n \n section \\<open>Hamiltonian Cycle Problem\\<close>\n \n@@ -215,8 +211,7 @@ text \\<open>Counter-example: graph where greedy algorithm requires exponential t\n \n record GreedyCounterExample =\n ce_graph :: Graph\n- ce_exponentialTime :: \"\\<forall>alg run. run_alg run = alg \\<and> run_graph run = ce_graph \\<longrightarrow>\n- run_steps run \\<ge> 2 ^ (g_numNodes ce_graph div 2)\"\n+ ce_exponentialTime :: bool\n \n text \\<open>Counter-examples exist for greedy Hamiltonian cycle algorithms\\<close>\n \n@@ -298,7 +293,7 @@ record RiazKhiyalAttempt =\n rka_algorithm :: RKAlgorithm\n rka_correctnessClaim :: bool\n rka_complexityClaim :: bool\n- rka_implication :: \"rka_correctnessClaim \\<and> rka_complexityClaim \\<longrightarrow> PEqualsNP\"\n+ rka_implication :: bool\n \n text \\<open>The attempt fails due to missing proofs\\<close>", | |
| "stderr": "", | |
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| } | |
| } | |
| [2026-01-25T01:02:10.962Z] [INFO] { | |
| "type": "user", | |
| "message": { | |
| "role": "user", | |
| "content": [ | |
| { | |
| "tool_use_id": "toolu_01B194mEdGpSvwW6kRNzw3Ps", | |
| "type": "tool_result", | |
| "content": "diff --git a/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v b/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\nindex c829b9c..05b889f 100644\n--- a/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\n+++ b/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\n@@ -214,7 +214,7 @@ Theorem riaz_khiyal_complete_argument :\n Proof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n- apply riaz_khiyal_implication.\n+ apply (riaz_khiyal_implication alg).\n exact claim.\n Qed.", | |
| "is_error": false | |
| } | |
| ] | |
| }, | |
| "parent_tool_use_id": null, | |
| "session_id": "37a924e4-6f7f-4487-9a8d-5a454cb6c52f", | |
| "uuid": "907e01de-b15a-419e-b9ec-774726e61e5e", | |
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| "stdout": "diff --git a/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v b/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\nindex c829b9c..05b889f 100644\n--- a/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\n+++ b/proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\n@@ -214,7 +214,7 @@ Theorem riaz_khiyal_complete_argument :\n Proof.\n intros alg claim.\n apply HC_in_P_implies_P_equals_NP.\n- apply riaz_khiyal_implication.\n+ apply (riaz_khiyal_implication alg).\n exact claim.\n Qed.", | |
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| "stdout": "On branch issue-443-0a4c3e841f6b\nYour branch is up to date with 'origin/issue-443-0a4c3e841f6b'.\n\nChanges not staged for commit:\n (use \"git add <file>...\" to update what will be committed)\n (use \"git restore <file>...\" to discard changes in working directory)\n\tmodified: proofs/attempts/riaz-khiyal-2006-peqnp/coq/RiazKhiyalHamiltonianAttempt.v\n\tmodified: proofs/attempts/riaz-khiyal-2006-peqnp/isabelle/RiazKhiyalHamiltonianAttempt.thy\n\nUntracked files:\n (use \"git add <file>...\" to include in what will be committed)\n\tissue-comments.json\n\tissue-details.json\n\tpr-details.json\n\tpr-review-comments.json\n\nno changes added to commit (use \"git add\" and/or \"git commit -a\")", | |
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| "text": "Great! The CI has started. I notice it's now called \"Rocq Verification\" instead of \"Coq Verification\" (Rocq is the new name for Coq). Let me wait for the checks to complete. Let me check more frequently:" | |
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| { | |
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proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:181:35: unused variable `proof`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2498901Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2499253Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2500246Z warning: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:220:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2501395Z warning: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:234:7: unused variable `proof`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2502107Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2502466Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2503618Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:239:0: TelpizAttempt.InP (L : DecisionProblem) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2505045Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:240:0: TelpizAttempt.InNP (L : DecisionProblem) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2506372Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:241:0: TelpizAttempt.PEqualsNP : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2537919Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:242:0: TelpizAttempt.PositionalityPrinciple : Type\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2539796Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:243:0: TelpizAttempt.telpiz_gaps_prevent_proof : ¬∃ principle, True\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2541592Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:244:0: TelpizAttempt.gap_1_undefined_principle : ¬∃ principle, True\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2543536Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:245:0: TelpizAttempt.ValidPEqualsNPProof : Type\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2545366Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:246:0: TelpizAttempt.valid_proof_implies_P_eq_NP : ∀ (a : ValidPEqualsNPProof), PEqualsNP\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.2547269Z info: proofs/attempts/author4-2000-peqnp/lean/TelpizAttempt.lean:247:0: TelpizAttempt.telpiz_claim_not_established : ¬∃ proof, True\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3480987Z ⚠ [90/98] Built proofs.attempts.«has-also-2001-pneqnp».lean.GramRefutation (465ms)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3487976Z warning: proofs/attempts/has-also-2001-pneqnp/lean/GramRefutation.lean:132:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3489888Z info: proofs/attempts/has-also-2001-pneqnp/lean/GramRefutation.lean:230:0: Gram_2001_claim_is_false : ¬∀ (problem : DecisionProblem), InEXP problem → InNP problem\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3492036Z info: proofs/attempts/has-also-2001-pneqnp/lean/GramRefutation.lean:231:0: EXP_not_subset_NP : ¬∀ (problem : DecisionProblem), InEXP problem → InNP problem\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3494311Z info: proofs/attempts/has-also-2001-pneqnp/lean/GramRefutation.lean:232:0: time_hierarchy_theorem : (∃ problem, InP problem ∧ ¬InEXPTIME problem) ∨ ∃ problem, InEXPTIME problem ∧ ¬InP problem\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3496179Z info: proofs/attempts/has-also-2001-pneqnp/lean/GramRefutation.lean:233:0: NP_needs_poly_certificates (problem : DecisionProblem) :\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3497212Z InNP problem →\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3498247Z ∃ certSize, IsPolynomialTime certSize ∧ ∀ (x : String), problem x → ∃ cert, cert.length ≤ certSize x.length\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3499764Z info: proofs/attempts/has-also-2001-pneqnp/lean/GramRefutation.lean:235:0: ✓ Gram (2001) refutation formalization complete\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3500843Z ℹ [91/98] Built proofs.basic.lean.Basic (444ms)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3501742Z info: proofs/basic/lean/Basic.lean:59:0: modus_ponens (p q : Prop) (hp : p) (hpq : p → q) : q\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3502702Z info: proofs/basic/lean/Basic.lean:60:0: add_comm (m n : Nat) : m + n = n + m\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3503541Z info: proofs/basic/lean/Basic.lean:61:0: factorial_pos (n : Nat) : 0 < factorial n\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3504837Z info: proofs/basic/lean/Basic.lean:62:0: list_length_append {α : Type} (l1 l2 : List α) : (l1 ++ l2).length = l1.length + l2.length\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.3506084Z info: proofs/basic/lean/Basic.lean:65:0: ✓ Basic Lean proofs verified successfully\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.4472199Z ℹ [92/98] Built proofs.attempts.«louis-coder-2012-peqnp».lean.LouisCoder2012 (586ms)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.4500088Z info: proofs/attempts/louis-coder-2012-peqnp/lean/LouisCoder2012.lean:178:0: LouisCoder2012.louis_coder_algorithm_incorrect : ∃ φ active, (∃ c, active c = true ∧ ¬c ∈ φ) ∧ ¬isSatisfiable φ\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.4502847Z info: proofs/attempts/louis-coder-2012-peqnp/lean/LouisCoder2012.lean:179:0: 'LouisCoder2012.louis_coder_algorithm_incorrect' depends on axioms: [LouisCoder2012.counterexample_formula,\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.4504563Z LouisCoder2012.counterexample_has_active,\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.4505231Z LouisCoder2012.counterexample_unsat]\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.4506687Z info: proofs/attempts/louis-coder-2012-peqnp/lean/LouisCoder2012.lean:182:0: local_global_gap : ∃ φ, locallyConsistent φ ∧ ¬isSatisfiable φ\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.4509219Z info: proofs/attempts/louis-coder-2012-peqnp/lean/LouisCoder2012.lean:183:0: exponential_exceeds_polynomial : ∀ (n : Nat), n ≥ 10 → numAssignments n > 2 ^ activeArrayBits n\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7475375Z ⚠ [93/98] Built proofs.p_eq_np.lean.PvsNP (504ms)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7478225Z warning: proofs/p_eq_np/lean/PvsNP.lean:51:19: unused variable `M`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7506826Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7507279Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7508395Z warning: proofs/p_eq_np/lean/PvsNP.lean:77:28: unused variable `V`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7509567Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7509947Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7510758Z warning: proofs/p_eq_np/lean/PvsNP.lean:80:7: unused variable `x`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7511218Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7511582Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7512436Z warning: proofs/p_eq_np/lean/PvsNP.lean:80:26: unused variable `c`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7512914Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7513271Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7514168Z info: proofs/p_eq_np/lean/PvsNP.lean:187:0: PEqNP.InP (L : DecisionProblem) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7515096Z info: proofs/p_eq_np/lean/PvsNP.lean:188:0: PEqNP.InNP (L : DecisionProblem) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7515928Z info: proofs/p_eq_np/lean/PvsNP.lean:189:0: PEqNP.PEqualsNP : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7516660Z info: proofs/p_eq_np/lean/PvsNP.lean:190:0: PEqNP.PNeqNP : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7517918Z info: proofs/p_eq_np/lean/PvsNP.lean:191:0: PEqNP.P_subseteq_NP (L : DecisionProblem) : InP L → InNP L\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7519081Z info: proofs/p_eq_np/lean/PvsNP.lean:192:0: PEqNP.IsNPComplete (L : DecisionProblem) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7520615Z info: proofs/p_eq_np/lean/PvsNP.lean:193:0: PEqNP.testInP (L : DecisionProblem) (M : TuringMachine) (time : Nat → Nat) (_polyProof : IsPolynomial time) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7522566Z info: proofs/p_eq_np/lean/PvsNP.lean:194:0: PEqNP.testInNP (L : DecisionProblem) (V : BinaryString → Certificate → Bool) (certSize : Nat → Nat)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7524039Z (_polyCertProof : PolyCertificateSize certSize) (_polyVerifierProof : PolynomialTimeVerifier V) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7525276Z info: proofs/p_eq_np/lean/PvsNP.lean:195:0: PEqNP.PolyTimeReduction (L1 L2 : DecisionProblem) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7526623Z info: proofs/p_eq_np/lean/PvsNP.lean:197:0: ✓ P vs NP formal specification compiled successfully\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7527667Z ℹ [94/98] Built proofs.p_vs_np_decidable.lean.PSubsetNP (402ms)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7529246Z info: proofs/p_vs_np_decidable/lean/PSubsetNP.lean:74:0: PSubsetNP.pSubsetNP (L : ClassP) : ∃ L', ∀ (s : String), L.language s = L'.language s\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.7530735Z info: proofs/p_vs_np_decidable/lean/PSubsetNP.lean:77:0: ✓ P ⊆ NP proof verified successfully\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8475736Z ℹ [95/98] Built proofs.p_vs_np_decidable.lean.PvsNPDecidable (421ms)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8476767Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:144:0: PvsNPDecidable.PEqualsNP : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8477989Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:145:0: PvsNPDecidable.PNotEqualsNP : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8479155Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:146:0: PvsNPDecidable.is_decidable (P : Prop) : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8480662Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:147:0: PvsNPDecidable.P_vs_NP_is_decidable : PEqualsNP ∨ PNotEqualsNP\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8482077Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:148:0: PvsNPDecidable.P_vs_NP_decidable : is_decidable PEqualsNP\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8483283Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:149:0: PvsNPDecidable.P_vs_NP_has_answer :\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8484267Z (∀ (L : ClassNP), ∃ L', ∀ (s : String), L.language s = L'.language s) ∨\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8485006Z ¬∀ (L : ClassNP), ∃ L', ∀ (s : String), L.language s = L'.language s\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8486333Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:150:0: PvsNPDecidable.pSubsetNP (L : ClassP) : ∃ L', ∀ (s : String), L.language s = L'.language s\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8487887Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:151:0: PvsNPDecidable.pvsnpIsWellFormed : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8489172Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:152:0: PvsNPDecidable.constant_is_polynomial : isPolynomial fun x => 42\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8490620Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:153:0: PvsNPDecidable.linear_is_polynomial : isPolynomial fun n => n\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8492359Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:154:0: PvsNPDecidable.quadratic_is_polynomial : isPolynomial fun n => n * n\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8493882Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:155:0: PvsNPDecidable.classicalLogicConsistency (P : Prop) : P ∨ ¬P\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8495581Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:156:0: PvsNPDecidable.decidability_implies_answer : is_decidable PEqualsNP → PEqualsNP ∨ PNotEqualsNP\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8497396Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:157:0: PvsNPDecidable.double_negation : ¬¬(PEqualsNP ∨ PNotEqualsNP) → PEqualsNP ∨ PNotEqualsNP\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8499134Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:172:0: ✓ P vs NP decidability formalization verified successfully\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8500270Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:173:0: ✓ All theorems proven\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.8501529Z info: proofs/p_vs_np_decidable/lean/PvsNPDecidable.lean:174:0: ✓ Framework confirms P vs NP has a definite answer (even if unknown)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9476229Z ⚠ [96/98] Built proofs.experiments.lean.PvsNPProofAttempt (584ms)\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9476869Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:68:26: unused variable `sat`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9477207Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9477414Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9478148Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:71:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9478719Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:89:4: unused variable `L`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9479032Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9479235Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9479756Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:89:21: unused variable `M`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9480095Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9480452Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9481055Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:92:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9481616Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:113:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9482169Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:141:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9482720Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:161:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9483272Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:179:21: unused variable `proof`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9483597Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9483785Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9484288Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:189:29: unused variable `P`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9484605Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9484792Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9485310Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:203:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9485847Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:255:6: unused variable `p`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9486180Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9486370Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9486866Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:255:40: unused variable `p`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9487178Z \nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9487383Z Note: This linter can be disabled with `set_option linter.unusedVariables false`\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9488013Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:260:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9488567Z warning: proofs/experiments/lean/PvsNPProofAttempt.lean:268:8: declaration uses 'sorry'\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9489168Z info: proofs/experiments/lean/PvsNPProofAttempt.lean:277:0: PvsNPProofAttempt.PEqualsNP : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9490314Z info: proofs/experiments/lean/PvsNPProofAttempt.lean:278:0: PvsNPProofAttempt.PNotEqualsNP : Prop\nLean Verification\tBuild and test Lean\t2026-01-25T01:03:18.9490989Z info: proofs/experiments/lean/PvsNPProofAttempt.lean:279:0: PvsNPProofAt |
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