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Solution draft log for https://github.com/link-foundation/meta-theory/pull/46
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| # Solve.mjs Log - 2026-04-01T20:45:29.592Z | |
| [2026-04-01T20:45:29.594Z] [INFO] 📁 Log file: /workspace/solve-2026-04-01T20-45-29-592Z.log | |
| [2026-04-01T20:45:29.594Z] [INFO] (All output will be logged here) | |
| [2026-04-01T20:45:30.581Z] [INFO] | |
| [2026-04-01T20:45:30.581Z] [INFO] 🚀 solve v1.42.0 | |
| [2026-04-01T20:45:30.582Z] [INFO] 🔧 Raw command executed: | |
| [2026-04-01T20:45:30.582Z] [INFO] /workspace/.nvm/versions/node/v20.20.2/bin/node /workspace/.bun/bin/solve https://github.com/link-foundation/meta-theory/issues/45 --model opus --attach-logs --verbose --no-tool-check --auto-accept-invite --tokens-budget-stats | |
| [2026-04-01T20:45:30.582Z] [INFO] | |
| [2026-04-01T20:45:30.602Z] [INFO] | |
| [2026-04-01T20:45:30.603Z] [WARNING] ⚠️ SECURITY WARNING: --attach-logs is ENABLED | |
| [2026-04-01T20:45:30.603Z] [INFO] | |
| [2026-04-01T20:45:30.604Z] [INFO] This option will upload the complete solution draft log file to the Pull Request. | |
| [2026-04-01T20:45:30.604Z] [INFO] The log may contain sensitive information such as: | |
| [2026-04-01T20:45:30.604Z] [INFO] • API keys, tokens, or secrets | |
| [2026-04-01T20:45:30.605Z] [INFO] • File paths and directory structures | |
| [2026-04-01T20:45:30.605Z] [INFO] • Command outputs and error messages | |
| [2026-04-01T20:45:30.605Z] [INFO] • Internal system information | |
| [2026-04-01T20:45:30.605Z] [INFO] | |
| [2026-04-01T20:45:30.605Z] [INFO] ⚠️ DO NOT use this option with public repositories or if the log | |
| [2026-04-01T20:45:30.606Z] [INFO] might contain sensitive data that should not be shared publicly. | |
| [2026-04-01T20:45:30.606Z] [INFO] | |
| [2026-04-01T20:45:30.606Z] [INFO] Continuing in 5 seconds... (Press Ctrl+C to abort) | |
| [2026-04-01T20:45:30.606Z] [INFO] | |
| [2026-04-01T20:45:35.614Z] [INFO] | |
| [2026-04-01T20:45:35.633Z] [INFO] 💾 Disk space check: 34580MB available (2048MB required) ✅ | |
| [2026-04-01T20:45:35.635Z] [INFO] 🧠 Memory check: 11202MB available, swap: none, total: 11202MB (256MB required) ✅ | |
| [2026-04-01T20:45:35.654Z] [INFO] ⏩ Skipping tool connection validation (dry-run mode or skip-tool-connection-check enabled) | |
| [2026-04-01T20:45:35.655Z] [INFO] ⏩ Skipping GitHub authentication check (dry-run mode or skip-tool-connection-check enabled) | |
| [2026-04-01T20:45:35.656Z] [INFO] 📋 URL validation: | |
| [2026-04-01T20:45:35.656Z] [INFO] Input URL: https://github.com/link-foundation/meta-theory/issues/45 | |
| [2026-04-01T20:45:35.657Z] [INFO] Is Issue URL: true | |
| [2026-04-01T20:45:35.657Z] [INFO] Is PR URL: false | |
| [2026-04-01T20:45:35.658Z] [INFO] 🔍 Checking repository access for auto-fork... | |
| [2026-04-01T20:45:36.417Z] [INFO] Repository visibility: public | |
| [2026-04-01T20:45:36.418Z] [INFO] ✅ Auto-fork: Write access detected to public repository, working directly on repository | |
| [2026-04-01T20:45:36.418Z] [INFO] 🔍 --auto-accept-invite: Checking for pending invitation to link-foundation/meta-theory... | |
| [2026-04-01T20:45:36.673Z] [INFO] Found 0 total pending repo invitation(s) | |
| [2026-04-01T20:45:36.673Z] [INFO] No pending repository invitation found for link-foundation/meta-theory | |
| [2026-04-01T20:45:37.046Z] [INFO] Found 0 total pending org invitation(s) | |
| [2026-04-01T20:45:37.047Z] [INFO] No pending organization invitation found for link-foundation | |
| [2026-04-01T20:45:37.047Z] [INFO] ℹ️ --auto-accept-invite: No pending invitation found for link-foundation/meta-theory or organization link-foundation | |
| [2026-04-01T20:45:37.048Z] [INFO] 🔍 Checking repository write permissions... | |
| [2026-04-01T20:45:37.414Z] [INFO] ✅ Repository write access: Confirmed | |
| [2026-04-01T20:45:37.767Z] [INFO] Repository visibility: public | |
| [2026-04-01T20:45:37.768Z] [INFO] Auto-cleanup default: false (repository is public) | |
| [2026-04-01T20:45:37.769Z] [INFO] 🔍 Auto-continue enabled: Checking for existing PRs for issue #45... | |
| [2026-04-01T20:45:37.770Z] [INFO] 🔍 Checking for existing branches in link-foundation/meta-theory... | |
| [2026-04-01T20:45:39.071Z] [INFO] 📝 No existing PRs found for issue #45 - creating new PR | |
| [2026-04-01T20:45:39.072Z] [INFO] 📝 Issue mode: Working with issue #45 | |
| [2026-04-01T20:45:39.074Z] [INFO] | |
| Creating temporary directory: /tmp/gh-issue-solver-1775076339073 | |
| [2026-04-01T20:45:39.077Z] [INFO] | |
| 📥 Cloning repository: link-foundation/meta-theory | |
| [2026-04-01T20:45:44.969Z] [INFO] ✅ Cloned to: /tmp/gh-issue-solver-1775076339073 | |
| [2026-04-01T20:45:45.054Z] [INFO] | |
| 📌 Default branch: main | |
| [2026-04-01T20:45:45.355Z] [INFO] | |
| 🌿 Creating branch: issue-45-d055c0a2bf46 from main (default) | |
| [2026-04-01T20:45:45.369Z] [INFO] 🔍 Verifying: Branch creation... | |
| [2026-04-01T20:45:45.381Z] [INFO] ✅ Branch created: issue-45-d055c0a2bf46 | |
| [2026-04-01T20:45:45.382Z] [INFO] ✅ Current branch: issue-45-d055c0a2bf46 | |
| [2026-04-01T20:45:45.383Z] [INFO] Branch operation: Create new branch | |
| [2026-04-01T20:45:45.383Z] [INFO] Branch verification: Matches expected | |
| [2026-04-01T20:45:45.386Z] [INFO] | |
| 🚀 Auto PR creation: ENABLED | |
| [2026-04-01T20:45:45.387Z] [INFO] Creating: Initial commit and draft PR... | |
| [2026-04-01T20:45:45.388Z] [INFO] | |
| [2026-04-01T20:45:45.388Z] [INFO] Using .gitkeep mode (--claude-file=false, --gitkeep-file=true, --auto-gitkeep-file=true) | |
| [2026-04-01T20:45:45.388Z] [INFO] 📝 Creating: .gitkeep (default) | |
| [2026-04-01T20:45:45.389Z] [INFO] Issue URL from argv['issue-url']: https://github.com/link-foundation/meta-theory/issues/45 | |
| [2026-04-01T20:45:45.389Z] [INFO] Issue URL from argv._[0]: undefined | |
| [2026-04-01T20:45:45.389Z] [INFO] Final issue URL: https://github.com/link-foundation/meta-theory/issues/45 | |
| [2026-04-01T20:45:45.390Z] [INFO] ✅ File created: .gitkeep | |
| [2026-04-01T20:45:45.390Z] [INFO] 📦 Adding file: To git staging | |
| [2026-04-01T20:45:45.414Z] [INFO] Git status after add: A .gitkeep | |
| [2026-04-01T20:45:45.415Z] [INFO] 📝 Creating commit: With .gitkeep file | |
| [2026-04-01T20:45:45.432Z] [INFO] ✅ Commit created: Successfully with .gitkeep | |
| [2026-04-01T20:45:45.432Z] [INFO] Commit output: [issue-45-d055c0a2bf46 49c3b33] Initial commit with task details | |
| 1 file changed, 1 insertion(+) | |
| create mode 100644 .gitkeep | |
| [2026-04-01T20:45:45.442Z] [INFO] Commit hash: 49c3b33... | |
| [2026-04-01T20:45:45.454Z] [INFO] Latest commit: 49c3b33 Initial commit with task details | |
| [2026-04-01T20:45:45.469Z] [INFO] Git status: clean | |
| [2026-04-01T20:45:45.479Z] [INFO] Remotes: origin https://github.com/link-foundation/meta-theory.git (fetch) | |
| [2026-04-01T20:45:45.493Z] [INFO] Branch info: * issue-45-d055c0a2bf46 49c3b33 [origin/main: ahead 1] Initial commit with task details | |
| main a737908 [origin/main] Merge pull request #44 from link-foundation/issue-43-e3181dc3ba85 | |
| [2026-04-01T20:45:45.493Z] [INFO] 📤 Pushing branch: To remote repository... | |
| [2026-04-01T20:45:45.494Z] [INFO] Push command: git push -u origin issue-45-d055c0a2bf46 | |
| [2026-04-01T20:45:46.421Z] [INFO] Push exit code: 0 | |
| [2026-04-01T20:45:46.422Z] [INFO] Push output: remote: | |
| remote: Create a pull request for 'issue-45-d055c0a2bf46' on GitHub by visiting: | |
| remote: https://github.com/link-foundation/meta-theory/pull/new/issue-45-d055c0a2bf46 | |
| remote: | |
| To https://github.com/link-foundation/meta-theory.git | |
| * [new branch] issue-45-d055c0a2bf46 -> issue-45-d055c0a2bf46 | |
| branch 'issue-45-d055c0a2bf46' set up to track 'origin/issue-45-d055c0a2bf46'. | |
| [2026-04-01T20:45:46.422Z] [INFO] ✅ Branch pushed: Successfully to remote | |
| [2026-04-01T20:45:46.422Z] [INFO] Push output: remote: | |
| remote: Create a pull request for 'issue-45-d055c0a2bf46' on GitHub by visiting: | |
| remote: https://github.com/link-foundation/meta-theory/pull/new/issue-45-d055c0a2bf46 | |
| remote: | |
| To https://github.com/link-foundation/meta-theory.git | |
| * [new branch] issue-45-d055c0a2bf46 -> issue-45-d055c0a2bf46 | |
| branch 'issue-45-d055c0a2bf46' set up to track 'origin/issue-45-d055c0a2bf46'. | |
| [2026-04-01T20:45:46.423Z] [INFO] Waiting for GitHub to sync... | |
| [2026-04-01T20:45:48.972Z] [INFO] Compare API check: 1 commit(s) ahead of main | |
| [2026-04-01T20:45:48.973Z] [INFO] GitHub compare API ready: 1 commit(s) found | |
| [2026-04-01T20:45:49.488Z] [INFO] Branch verified on GitHub: issue-45-d055c0a2bf46 | |
| [2026-04-01T20:45:49.775Z] [INFO] Remote commit SHA: 49c3b33... | |
| [2026-04-01T20:45:49.776Z] [INFO] 📋 Getting issue: Title from GitHub... | |
| [2026-04-01T20:45:50.098Z] [INFO] Issue title: "Final touches" | |
| [2026-04-01T20:45:50.099Z] [INFO] 👤 Getting user: Current GitHub account... | |
| [2026-04-01T20:45:50.446Z] [INFO] Current user: konard | |
| [2026-04-01T20:45:50.748Z] [INFO] User has collaborator access | |
| [2026-04-01T20:45:50.749Z] [INFO] User has collaborator access | |
| [2026-04-01T20:45:50.749Z] [INFO] 🔄 Fetching: Latest main branch... | |
| [2026-04-01T20:45:51.070Z] [INFO] ✅ Base updated: Fetched latest main | |
| [2026-04-01T20:45:51.071Z] [INFO] 🔍 Checking: Commits between branches... | |
| [2026-04-01T20:45:51.084Z] [INFO] Commits ahead of origin/main: 1 | |
| [2026-04-01T20:45:51.085Z] [INFO] ✅ Commits found: 1 commit(s) ahead | |
| [2026-04-01T20:45:51.085Z] [INFO] 🔀 Creating PR: Draft pull request... | |
| [2026-04-01T20:45:51.085Z] [INFO] 🎯 Target branch: main (default) | |
| [2026-04-01T20:45:51.085Z] [INFO] PR Title: [WIP] Final touches | |
| [2026-04-01T20:45:51.086Z] [INFO] Base branch: main | |
| [2026-04-01T20:45:51.086Z] [INFO] Head branch: issue-45-d055c0a2bf46 | |
| [2026-04-01T20:45:51.086Z] [INFO] Assignee: konard | |
| [2026-04-01T20:45:51.086Z] [INFO] PR Body: | |
| ## 🤖 AI-Powered Solution Draft | |
| This pull request is being automatically generated to solve issue #45. | |
| ### 📋 Issue Reference | |
| Fixes #45 | |
| ### 🚧 Status | |
| **Work in Progress** - The AI assistant is currently analyzing and implementing the solution draft. | |
| ### 📝 Implementation Details | |
| _Details will be added as the solution draft is developed..._ | |
| --- | |
| *This PR was created automatically by the AI issue solver* | |
| [2026-04-01T20:45:51.087Z] [INFO] Command: cd "/tmp/gh-issue-solver-1775076339073" && gh pr create --draft --title "$(cat '/tmp/pr-title-1775076351087.txt')" --body-file "/tmp/pr-body-1775076351086.md" --base main --head issue-45-d055c0a2bf46 --assignee konard | |
| [2026-04-01T20:45:53.966Z] [INFO] gh pr create stdout: https://github.com/link-foundation/meta-theory/pull/46 | |
| [2026-04-01T20:45:53.967Z] [INFO] 🔍 Verifying: PR creation... | |
| [2026-04-01T20:45:56.342Z] [INFO] ✅ Verification: PR exists on GitHub (attempt 1/5) | |
| [2026-04-01T20:45:56.342Z] [INFO] ✅ PR created: #46 | |
| [2026-04-01T20:45:56.343Z] [INFO] 📍 PR URL: https://github.com/link-foundation/meta-theory/pull/46 | |
| [2026-04-01T20:45:56.343Z] [INFO] 👤 Assigned to: konard | |
| [2026-04-01T20:45:56.343Z] [INFO] 🔗 Linking: Issue #45 to PR #46... | |
| [2026-04-01T20:45:56.677Z] [INFO] Issue node ID: I_kwDOQL23_s75usum | |
| [2026-04-01T20:45:56.988Z] [INFO] PR node ID: PR_kwDOQL23_s7PYCSr | |
| [2026-04-01T20:45:57.391Z] [INFO] | |
| [2026-04-01T20:45:57.391Z] [WARNING] ⚠️ ISSUE LINK MISSING: PR not linked to issue | |
| [2026-04-01T20:45:57.392Z] [INFO] | |
| [2026-04-01T20:45:57.392Z] [WARNING] The PR wasn't linked to issue #45 | |
| [2026-04-01T20:45:57.393Z] [WARNING] Expected: "Fixes #45" in PR body | |
| [2026-04-01T20:45:57.394Z] [INFO] | |
| [2026-04-01T20:45:57.394Z] [WARNING] To fix manually: | |
| [2026-04-01T20:45:57.394Z] [WARNING] 1. Edit the PR description at: https://github.com/link-foundation/meta-theory/pull/46 | |
| [2026-04-01T20:45:57.395Z] [WARNING] 2. Ensure it contains: Fixes #45 | |
| [2026-04-01T20:45:57.395Z] [INFO] | |
| [2026-04-01T20:45:57.788Z] [INFO] 👤 Current user: konard | |
| [2026-04-01T20:45:57.789Z] [INFO] | |
| 📊 Comment counting conditions: | |
| [2026-04-01T20:45:57.789Z] [INFO] prNumber: 46 | |
| [2026-04-01T20:45:57.789Z] [INFO] branchName: issue-45-d055c0a2bf46 | |
| [2026-04-01T20:45:57.790Z] [INFO] isContinueMode: false | |
| [2026-04-01T20:45:57.790Z] [INFO] Will count comments: true | |
| [2026-04-01T20:45:57.790Z] [INFO] 💬 Counting comments: Checking for new comments since last commit... | |
| [2026-04-01T20:45:57.791Z] [INFO] PR #46 on branch: issue-45-d055c0a2bf46 | |
| [2026-04-01T20:45:57.791Z] [INFO] Owner/Repo: link-foundation/meta-theory | |
| [2026-04-01T20:45:58.244Z] [INFO] 📅 Last commit time (from API): 2026-04-01T20:45:45.000Z | |
| [2026-04-01T20:45:59.155Z] [INFO] 💬 New PR comments: 0 | |
| [2026-04-01T20:45:59.156Z] [INFO] 💬 New PR review comments: 0 | |
| [2026-04-01T20:45:59.156Z] [INFO] 💬 New issue comments: 0 | |
| [2026-04-01T20:45:59.157Z] [INFO] Total new comments: 0 | |
| [2026-04-01T20:45:59.157Z] [INFO] Comment lines to add: No (saving tokens) | |
| [2026-04-01T20:45:59.157Z] [INFO] PR review comments fetched: 0 | |
| [2026-04-01T20:45:59.157Z] [INFO] PR conversation comments fetched: 0 | |
| [2026-04-01T20:45:59.158Z] [INFO] Total PR comments checked: 0 | |
| [2026-04-01T20:46:01.928Z] [INFO] Feedback info will be added to prompt: | |
| [2026-04-01T20:46:01.934Z] [INFO] - Pull request description was edited after last commit | |
| [2026-04-01T20:46:01.934Z] [INFO] 📅 Getting timestamps: From GitHub servers... | |
| [2026-04-01T20:46:02.361Z] [INFO] 📝 Issue updated: 2026-04-01T20:45:02.000Z | |
| [2026-04-01T20:46:02.682Z] [INFO] 💬 Comments: None found | |
| [2026-04-01T20:46:03.055Z] [INFO] 🔀 Recent PR: 2026-04-01T20:45:52.000Z | |
| [2026-04-01T20:46:03.056Z] [INFO] | |
| ✅ Reference time: 2026-04-01T20:45:52.000Z | |
| [2026-04-01T20:46:03.058Z] [INFO] | |
| 🔍 Checking for uncommitted changes to include as feedback... | |
| [2026-04-01T20:46:03.075Z] [INFO] ✅ No uncommitted changes found | |
| [2026-04-01T20:46:05.721Z] [INFO] 🎭 Playwright MCP detected - enabling browser automation hints | |
| [2026-04-01T20:46:05.840Z] [INFO] 👁️ Model vision capability: supported | |
| [2026-04-01T20:46:05.842Z] [INFO] | |
| 📝 Final prompt structure: | |
| [2026-04-01T20:46:05.843Z] [INFO] Characters: 278 | |
| [2026-04-01T20:46:05.843Z] [INFO] System prompt characters: 15179 | |
| [2026-04-01T20:46:05.844Z] [INFO] Feedback info: Included | |
| [2026-04-01T20:46:05.847Z] [INFO] | |
| 🤖 Executing Claude: OPUS | |
| [2026-04-01T20:46:05.850Z] [INFO] Model: opus | |
| [2026-04-01T20:46:05.851Z] [INFO] Working directory: /tmp/gh-issue-solver-1775076339073 | |
| [2026-04-01T20:46:05.851Z] [INFO] Branch: issue-45-d055c0a2bf46 | |
| [2026-04-01T20:46:05.851Z] [INFO] Prompt length: 278 chars | |
| [2026-04-01T20:46:05.851Z] [INFO] System prompt length: 15179 chars | |
| [2026-04-01T20:46:05.851Z] [INFO] Feedback info included: Yes (1 lines) | |
| [2026-04-01T20:46:05.871Z] [INFO] 📈 System resources before execution: | |
| [2026-04-01T20:46:05.872Z] [INFO] Memory: MemFree: 7622592 kB | |
| [2026-04-01T20:46:05.873Z] [INFO] Load: 0.47 0.12 0.04 2/296 1842212 | |
| [2026-04-01T20:46:05.874Z] [INFO] | |
| 📝 Raw command: | |
| [2026-04-01T20:46:05.874Z] [INFO] (cd "/tmp/gh-issue-solver-1775076339073" && claude --output-format stream-json --verbose --dangerously-skip-permissions --model claude-opus-4-6 -p "Issue to solve: https://github.com/link-foundation/meta-theory/issues/45 | |
| Your prepared branch: issue-45-d055c0a2bf46 | |
| Your prepared working directory: /tmp/gh-issue-solver-1775076339073 | |
| Your prepared Pull Request: https://github.com/link-foundation/meta-theory/pull/46 | |
| Proceed. | |
| " --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 link-foundation/meta-theory --branch issue-45-d055c0a2bf46 --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 link-foundation/meta-theory --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, download the image to a local file first, then use Read tool to view and analyze it. Before reading downloaded images with the Read tool, verify the file is a valid image (not HTML) using a CLI tool like the 'file' command to check the actual file format. When corrupted or non-image files (like GitHub's \"Not Found\" pages saved as .png) are read, they can cause \"Could not process image\" errors and crash the AI solver process. When the file command shows \"HTML\", \"text\", or \"ASCII text\", the download failed — do not call Read on this file. Instead: (1) When images are from GitHub issues/PRs (URLs containing \"github.com/user-attachments\"), these require authentication — retry with: curl -L -H \"Authorization: token \$(gh auth token)\" -o <filename> \"<url>\" (2) When the retry still fails, skip the image and note it was unavailable. | |
| - When you need issue details, use gh issue view https://github.com/link-foundation/meta-theory/issues/45. | |
| - When you need related code, use gh search code --owner link-foundation [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/link-foundation/meta-theory/pulls/46/comments --paginate | |
| 2. PR conversation comments (general discussion): gh api repos/link-foundation/meta-theory/issues/46/comments --paginate | |
| 3. PR reviews (approve/request changes): gh api repos/link-foundation/meta-theory/pulls/46/reviews --paginate | |
| Note: 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/link-foundation/meta-theory/issues/45/comments --paginate. | |
| Solution development and testing. | |
| - When issue is solvable, first create a test that reproduces the problem, then implement the fix. | |
| - 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 committed 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 you write or modify tests, consider setting reasonable timeouts at test, suite, and CI job levels so failures surface quickly instead of hanging. | |
| - When you see repeated test timeout patterns in CI, investigate the root cause rather than increasing timeouts. | |
| - When issue is unclear, write comment on issue asking questions. | |
| - When you encounter any problems that you are 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 46 --body \"your message\" to comment on existing PR. | |
| Reproducible testing. | |
| - When fixing a bug, create a test that reproduces the problem before implementing the fix. When you cannot reproduce the problem, you cannot verify the fix. | |
| - When encountering logic bugs, write an automated test that fails due to the bug, then implement the fix to make it pass. | |
| - When encountering UI bugs, capture a screenshot showing the problem state, then create a visual regression test or manual verification screenshot after the fix. | |
| - When creating tests, prefer minimum reproducible examples - the simplest test case that demonstrates the issue. | |
| - When submitting a fix, include in the PR description: (1) how to reproduce the issue, (2) the automated test that verifies the fix, (3) before/after screenshots for UI issues. | |
| - When a bug fix doesn't have a reproducing test, the fix is incomplete - regressions can silently occur later. | |
| 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 link-foundation/meta-theory --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 46, use gh pr edit to modify title and description. | |
| - When you are about to commit or push code, 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 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 46. | |
| Workflow and collaboration. | |
| - When you check branch, verify with git branch --show-current. | |
| - When you push, push only to branch issue-45-d055c0a2bf46. | |
| - When you finish, create a pull request from branch issue-45-d055c0a2bf46. (Note: PR 46 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-45-d055c0a2bf46. | |
| - When you mention result, include pull request url or comment url. | |
| - When you need to create pr, remember pr 46 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. | |
| - When fetching lists from GitHub API, use the --paginate flag 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. | |
| - When reproducing UI bugs, use browser_take_screenshot to capture the problem state before implementing any fix. | |
| - When fixing UI bugs, take before/after screenshots to provide visual evidence of the fix for human verification. | |
| - When creating UI tests, save baseline screenshots to the repository for visual regression testing. | |
| - When verifying UI fixes, compare screenshots to ensure the fix doesn't introduce unintended visual changes. | |
| Visual UI work and screenshots. | |
| - When you work on visual UI changes (frontend, CSS, HTML, design), include a render or screenshot of the final result in the pull request description. | |
| - When you need to show visual results, take a screenshot and save it to the repository (e.g., in a docs/screenshots/ or assets/ folder). | |
| - When you save screenshots to the repository, use permanent links in the pull request description markdown (e.g., https://github.com/link-foundation/meta-theory/blob/issue-45-d055c0a2bf46/docs/screenshots/result.png?raw=true). | |
| - When uploading images, commit them to the branch first, then reference them using the GitHub blob URL format with ?raw=true suffix (works for both public and private repositories). | |
| - When the visual result is important for review, mention it explicitly in the pull request description with the embedded image. | |
| - When fixing UI bugs, capture both the \"before\" (problem) and \"after\" (fixed) screenshots as evidence for human verification. | |
| - When reporting UI bugs, include a screenshot of the problem state to enable visual verification of the fix. | |
| - When the fix is visual, include side-by-side or sequential comparison of before/after states in the PR description. | |
| - When possible, create automated visual regression tests to prevent the UI bug from recurring." | jq -c .) | |
| [2026-04-01T20:46:05.875Z] [INFO] | |
| [2026-04-01T20:46:05.875Z] [INFO] 📋 User prompt: | |
| ---BEGIN USER PROMPT--- | |
| Issue to solve: https://github.com/link-foundation/meta-theory/issues/45 | |
| Your prepared branch: issue-45-d055c0a2bf46 | |
| Your prepared working directory: /tmp/gh-issue-solver-1775076339073 | |
| Your prepared Pull Request: https://github.com/link-foundation/meta-theory/pull/46 | |
| Proceed. | |
| ---END USER PROMPT--- | |
| [2026-04-01T20:46:05.876Z] [INFO] 📋 System prompt: | |
| ---BEGIN 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 link-foundation/meta-theory --branch issue-45-d055c0a2bf46 --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 link-foundation/meta-theory --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, download the image to a local file first, then use Read tool to view and analyze it. Before reading downloaded images with the Read tool, verify the file is a valid image (not HTML) using a CLI tool like the 'file' command to check the actual file format. When corrupted or non-image files (like GitHub's "Not Found" pages saved as .png) are read, they can cause "Could not process image" errors and crash the AI solver process. When the file command shows "HTML", "text", or "ASCII text", the download failed — do not call Read on this file. Instead: (1) When images are from GitHub issues/PRs (URLs containing "github.com/user-attachments"), these require authentication — retry with: curl -L -H "Authorization: token $(gh auth token)" -o <filename> "<url>" (2) When the retry still fails, skip the image and note it was unavailable. | |
| - When you need issue details, use gh issue view https://github.com/link-foundation/meta-theory/issues/45. | |
| - When you need related code, use gh search code --owner link-foundation [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/link-foundation/meta-theory/pulls/46/comments --paginate | |
| 2. PR conversation comments (general discussion): gh api repos/link-foundation/meta-theory/issues/46/comments --paginate | |
| 3. PR reviews (approve/request changes): gh api repos/link-foundation/meta-theory/pulls/46/reviews --paginate | |
| Note: 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/link-foundation/meta-theory/issues/45/comments --paginate. | |
| Solution development and testing. | |
| - When issue is solvable, first create a test that reproduces the problem, then implement the fix. | |
| - 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 committed 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 you write or modify tests, consider setting reasonable timeouts at test, suite, and CI job levels so failures surface quickly instead of hanging. | |
| - When you see repeated test timeout patterns in CI, investigate the root cause rather than increasing timeouts. | |
| - When issue is unclear, write comment on issue asking questions. | |
| - When you encounter any problems that you are 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 46 --body "your message" to comment on existing PR. | |
| Reproducible testing. | |
| - When fixing a bug, create a test that reproduces the problem before implementing the fix. When you cannot reproduce the problem, you cannot verify the fix. | |
| - When encountering logic bugs, write an automated test that fails due to the bug, then implement the fix to make it pass. | |
| - When encountering UI bugs, capture a screenshot showing the problem state, then create a visual regression test or manual verification screenshot after the fix. | |
| - When creating tests, prefer minimum reproducible examples - the simplest test case that demonstrates the issue. | |
| - When submitting a fix, include in the PR description: (1) how to reproduce the issue, (2) the automated test that verifies the fix, (3) before/after screenshots for UI issues. | |
| - When a bug fix doesn't have a reproducing test, the fix is incomplete - regressions can silently occur later. | |
| 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 link-foundation/meta-theory --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 46, use gh pr edit to modify title and description. | |
| - When you are about to commit or push code, 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 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 46. | |
| Workflow and collaboration. | |
| - When you check branch, verify with git branch --show-current. | |
| - When you push, push only to branch issue-45-d055c0a2bf46. | |
| - When you finish, create a pull request from branch issue-45-d055c0a2bf46. (Note: PR 46 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-45-d055c0a2bf46. | |
| - When you mention result, include pull request url or comment url. | |
| - When you need to create pr, remember pr 46 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. | |
| - When fetching lists from GitHub API, use the --paginate flag 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. | |
| - When reproducing UI bugs, use browser_take_screenshot to capture the problem state before implementing any fix. | |
| - When fixing UI bugs, take before/after screenshots to provide visual evidence of the fix for human verification. | |
| - When creating UI tests, save baseline screenshots to the repository for visual regression testing. | |
| - When verifying UI fixes, compare screenshots to ensure the fix doesn't introduce unintended visual changes. | |
| Visual UI work and screenshots. | |
| - When you work on visual UI changes (frontend, CSS, HTML, design), include a render or screenshot of the final result in the pull request description. | |
| - When you need to show visual results, take a screenshot and save it to the repository (e.g., in a docs/screenshots/ or assets/ folder). | |
| - When you save screenshots to the repository, use permanent links in the pull request description markdown (e.g., https://github.com/link-foundation/meta-theory/blob/issue-45-d055c0a2bf46/docs/screenshots/result.png?raw=true). | |
| - When uploading images, commit them to the branch first, then reference them using the GitHub blob URL format with ?raw=true suffix (works for both public and private repositories). | |
| - When the visual result is important for review, mention it explicitly in the pull request description with the embedded image. | |
| - When fixing UI bugs, capture both the "before" (problem) and "after" (fixed) screenshots as evidence for human verification. | |
| - When reporting UI bugs, include a screenshot of the problem state to enable visual verification of the fix. | |
| - When the fix is visual, include side-by-side or sequential comparison of before/after states in the PR description. | |
| - When possible, create automated visual regression tests to prevent the UI bug from recurring. | |
| ---END SYSTEM PROMPT--- | |
| [2026-04-01T20:46:05.877Z] [INFO] 📊 CLAUDE_CODE_MAX_OUTPUT_TOKENS: 128000, MCP_TIMEOUT: 900000ms, MCP_TOOL_TIMEOUT: 900000ms, ANTHROPIC_LOG: debug | |
| [2026-04-01T20:46:05.878Z] [INFO] 📋 Command details: | |
| [2026-04-01T20:46:05.878Z] [INFO] 📂 Working directory: /tmp/gh-issue-solver-1775076339073 | |
| [2026-04-01T20:46:05.878Z] [INFO] 🌿 Branch: issue-45-d055c0a2bf46 | |
| [2026-04-01T20:46:05.879Z] [INFO] 🤖 Model: Claude OPUS | |
| [2026-04-01T20:46:05.879Z] [INFO] | |
| ▶️ Streaming output: | |
| [2026-04-01T20:46:08.355Z] [INFO] { | |
| "type": "system", | |
| "subtype": "init", | |
| "cwd": "/tmp/gh-issue-solver-1775076339073", | |
| "session_id": "f245970a-c0e8-457c-881e-7fb7d465f6c2", | |
| "tools": [ | |
| "Task", | |
| "AskUserQuestion", | |
| "Bash", | |
| "CronCreate", | |
| "CronDelete", | |
| "CronList", | |
| "Edit", | |
| "EnterPlanMode", | |
| "EnterWorktree", | |
| "ExitPlanMode", | |
| "ExitWorktree", | |
| "Glob", | |
| "Grep", | |
| "NotebookEdit", | |
| "Read", | |
| "RemoteTrigger", | |
| "Skill", | |
| "TaskOutput", | |
| "TaskStop", | |
| "TodoWrite", | |
| "ToolSearch", | |
| "WebFetch", | |
| "WebSearch", | |
| "Write", | |
| "mcp__playwright__browser_click", | |
| "mcp__playwright__browser_close", | |
| "mcp__playwright__browser_console_messages", | |
| "mcp__playwright__browser_drag", | |
| "mcp__playwright__browser_evaluate", | |
| "mcp__playwright__browser_file_upload", | |
| "mcp__playwright__browser_fill_form", | |
| "mcp__playwright__browser_handle_dialog", | |
| "mcp__playwright__browser_hover", | |
| "mcp__playwright__browser_navigate", | |
| "mcp__playwright__browser_navigate_back", | |
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| "content": "1\t# Мета-теория связей 0.0.3\n2\t\n3\t**Автор:** [Константин Дьяченко (Konard)](https://habr.com/ru/users/Konard/)\n4\t\n5\t**Тип:** Перевод\n6\t\n7\t**Оригинальная статья:** [Vasily Solopov, Roman Vertushkin, Ivan Glazunov, Konstantin Diachenko](https://habr.com/en/articles/895896)\n8\t\n9\t**Основа:** Перевод версии 0.0.2 с английского на русский\n10\t\n11\t---\n12\t\n13\tВ прошлое первое апреля, как вы могли догадаться, мы пошутили. Пора это исправить, и теперь $100\\\\%$ всё серьёзно.\n14\t\n15\t#### TL;DR (too long; didn't read)\n16\t\n17\tВ этой статье много букв, но её можно записать всего 4-мя символами [теории множеств](https://ru.wikipedia.org/wiki/Теория_множеств):\n18\t\n19\t> $\\displaystyle R \\to R^2$\n20\t\n21\tВсё остальное следствие из них.\n22\t\n23\t" | |
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| "content": "1\t## Сравнение теорий\n2\t\n3\tДля быстрого погружения мы начнём со сравнения математических основ двух самых популярных [моделей данных](https://ru.wikipedia.org/wiki/Модель_данных) и [ассоциативной модели данных](https://web.archive.org/web/20210814063207/https://en.wikipedia.org/wiki/Associative_model_of_data).\n4\t\n5\tВ ходе нашего исследования мы обнаружили, что традиционные теории иногда оказывались чрезмерно сложными или избыточными, а иногда накладывали слишком много искусственных ограничений.\n6\t\n7\tЭтот общий недостаток гибкости, адаптивности и универсальности побудил нас искать более простую, но всеобъемлющую информационную теорию и модель хранения данных, которую будущий искусственный интеллект сможет легко понять и эффективно использовать. По пути мы черпали вдохновение из работы нашей собственной ассоциативной памяти и ассоциативных мыслительных процессов. Саймон Вильямс также отмечал, что ассоциативная модель «воспринимает информацию так же, как это делает мозг: в виде вещей и ассоциаций между ними» [[5]](https://link.springer.com/content/pdf/10.1057/palgrave.jdm.3240049.pdf).\n8\t\n9\t### Реляционная алгебра\n10\t\n11\t[Реляционная алгебра](https://ru.wikipedia.org/wiki/Реляционная_алгебра) и [реляционная модель](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) основаны на понятиях [отношения](https://en.wikipedia.org/wiki/Relation_(database)) и [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)).\n12\t\n13\t[Отношение](https://en.wikipedia.org/wiki/Relation_(database)) определяется как [множество](https://ru.wikipedia.org/wiki/Множество) [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)):\n14\t\n15\t> $\\mathbf{R \\subseteq S_1 \\times S_2 \\times \\dots \\times S_n.}$ [[1]](https://dl.acm.org/doi/abs/10.1145/362384.362685)\n16\t\n17\t\n18\t\n19\t**Рис. 1.** Таблица описывается [отношением](https://en.wikipedia.org/wiki/Relation_(database)) $\\mathbf{R}$, которое представляется [множеством](https://ru.wikipedia.org/wiki/Множество) [строк](https://en.wikipedia.org/wiki/Row_(database)) $\\mathbf{r}$, принадлежащих [декартову произведению](https://ru.wikipedia.org/wiki/Декартово_произведение) $\\mathbf{S_1 \\times S_2 \\times \\dots \\times S_n}$.\n20\t\n21\t**Где:**\n22\t\n23\t- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\mathbf{R}$ обозначает [отношение](https://en.wikipedia.org/wiki/Relation_(mathematics)) ([таблицу](https://en.wikipedia.org/wiki/Table_(database)));\n24\t- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\subseteq$ обозначает, что левая часть [выражения](https://ru.wikipedia.org/wiki/Выражение_(математика)) является [подмножеством](https://ru.wikipedia.org/wiki/Подмножество) правой части;\n25\t- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\times$ обозначает [декартово произведение](https://ru.wikipedia.org/wiki/Декартово_произведение) двух [множеств](https://ru.wikipedia.org/wiki/Множество);\n26\t- [Выражение](https://ru.wikipedia.org/wiki/Выражение_(математика)) $\\mathbf{S_n}$ обозначает [домен](https://en.wikipedia.org/wiki/Data_domain), то есть [множество](https://ru.wikipedia.org/wiki/Множество) всех возможных [значений](https://ru.wikipedia.org/wiki/Значение_(информатика)), которые может содержать каждая ячейка в [столбце](https://en.wikipedia.org/wiki/Table_(database)).\n27\t\n28\t[Строки](https://en.wikipedia.org/wiki/Row_(database)), или [элементы](https://ru.wikipedia.org/wiki/Элемент_(математика)) [отношения](https://en.wikipedia.org/wiki/Relation_(database)) $\\mathbf{R}$, представлены в виде [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)).\n29\t\n30\t[Данные](https://ru.wikipedia.org/wiki/Данные) в [реляционной модели](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) группируются в [отношения](https://en.wikipedia.org/wiki/Relation_(database)). Используя [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) в этой [модели](https://en.wikipedia.org/wiki/Database_model), можно точно представить любую возможную [структуру данных](https://ru.wikipedia.org/wiki/Структура_данных) — если бы мы действительно когда-нибудь использовали для этого [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика)). Но нужны ли вообще [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика))? Например, каждый [n-кортеж](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) можно представить [в виде вложенных упорядоченных пар](https://en.wikipedia.org/wiki/Tuple#Tuples_as_nested_ordered_pairs), что предполагает, что [упорядоченных пар](https://ru.wikipedia.org/wiki/Упорядоченная_пара) может быть достаточно для представления любых [данных](https://ru.wikipedia.org/wiki/Данные). Более того, нечасто встретишь, что значения [столбцов](https://en.wikipedia.org/wiki/Table_(database)) в [таблицах](https://en.wikipedia.org/wiki/Table_(database)) представляются [n-кортежами](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) (хотя, например, [число](https://ru.wikipedia.org/wiki/Число) можно [разложить](https://en.wikipedia.org/wiki/Decomposition_(computer_science)) на [n-кортеж](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) [битов](https://ru.wikipedia.org/wiki/Бит)). В некоторых [SQL](https://ru.wikipedia.org/wiki/SQL) [базах данных](https://ru.wikipedia.org/wiki/База_данных) даже запрещено использовать более $\\mathbf{32}$ [столбцов](https://en.wikipedia.org/wiki/Table_(database)) в [таблице](https://en.wikipedia.org/wiki/Table_(database)) (и, соответственно, в её [n-кортеже](https://ru.wikipedia.org/wiki/Кортеж_(информатика))). Таким образом, фактическое значение $\\mathbf{n}$ обычно меньше $\\mathbf{32}$. Поэтому в этих случаях нет [настоящих](https://ru.wikipedia.org/wiki/Истина) [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) — даже в современных [реляционных](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) [базах данных](https://ru.wikipedia.org/wiki/База_данных).\n31\t\n32\t\n33\t\n34\t**Рис. 2.** Сравнение [реляционной модели](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) и [ассоциативной модели данных](http://iacis.org/iis/2009/P2009_1301.pdf) (изначальная [модель](https://en.wikipedia.org/wiki/Data_model), предложенная [Саймоном Вильямсом](https://www.linkedin.com/in/s1m0n), была упрощена нами дважды) [[3]](https://web.archive.org/web/20181219134621/http://sentences.com/docs/amd.pdf) [[5]](https://link.springer.com/content/pdf/10.1057/palgrave.jdm.3240049.pdf). Иными словами, для представления всех [данных](https://ru.wikipedia.org/wiki/Данные) в [реляционной модели](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) требуется множество [таблиц](https://en.wikipedia.org/wiki/Table_(database)) — по одной для каждого [типа данных](https://ru.wikipedia.org/wiki/Тип_данных) — тогда как в [ассоциативной модели](https://web.archive.org/web/20210814063207/https://en.wikipedia.org/wiki/Associative_model_of_data) оказалось, что сначала было достаточно двух [таблиц](https://en.wikipedia.org/wiki/Table_(database)) (`items` и `links`), а затем и вовсе одной [таблицы](https://en.wikipedia.org/wiki/Table_(database)) [триплетов](https://ru.wikipedia.org/wiki/Кортеж_(информатика))-связей или [дуплетов](https://ru.wikipedia.org/wiki/Упорядоченная_пара)-связей.\n35\t\n36\t### Ориентированный граф\n37\t\n38\tОриентированные графы — и [графы](https://ru.wikipedia.org/wiki/Теория_графов) в целом — основаны на понятиях [вершин](https://ru.wikipedia.org/wiki/Вершина_(теория_графов)) и [рёбер](https://ru.wikipedia.org/wiki/Ребро_(теория_графов)) ([2-кортежей](https://ru.wikipedia.org/wiki/Упорядоченная_пара)).\n39\t\n40\t[Ориентированный граф](https://ru.wikipedia.org/wiki/Ориентированный_граф) $\\mathbf{G}$ определяется так:\n41\t\n42\t> $\\mathbf{G = (V, E), \\quad E \\subseteq V \\times V.}$ [[2]](https://books.google.com/books?id=vaXv_yhefG8C)\n43\t\n44\tГде:\n45\t\n46\t- $\\mathbf{V}$ — это [множество](https://ru.wikipedia.org/wiki/Множество), элементы которого называются [вершинами](https://ru.wikipedia.org/wiki/Вершина_(теория_графов)), узлами или [точками](https://ru.wikipedia.org/wiki/Точка_(геометрия));\n47\t- $\\mathbf{E}$ — это множество [упорядоченных пар](https://ru.wikipedia.org/wiki/Упорядоченная_пара) (2-[кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика))) [вершин](https://ru.wikipedia.org/wiki/Вершина_(теория_графов)), называемых дугами, направленными [рёбрами](https://ru.wikipedia.org/wiki/Ребро_(теория_графов)) (иногда просто [рёбрами](https://ru.wikipedia.org/wiki/Ребро_(теория_графов))), стрелками или направленными [отрезками](https://ru.wikipedia.org/wiki/Отрезок).\n48\t\n49\tВ модели ориентированного графа данные представлены двумя отдельными [множествами](https://ru.wikipedia.org/wiki/Множество): [узлами](https://ru.wikipedia.org/wiki/Вершина_(теория_графов)) и [рёбрами](https://ru.wikipedia.org/wiki/Ребро_(теория_графов)). Эту [модель](https://en.wikipedia.org/wiki/Data_model) можно использовать для представления почти всех [структур данных](https://ru.wikipedia.org/wiki/Структура_данных), кроме, вероятно, [последовательностей](https://ru.wikipedia.org/wiki/Последовательность) ([n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика))). Иногда можно встретить, что используются цепочки вершин как представления последовательностей. И хотя этот метод работает, он гарантированно приводит к дупликации данных, а дедупликация в этом случае будет осложнена или недоступна. Ещё, вероятно, последовательности в графах можно представить, используя [разложение последовательности на вложенные множества](https://en.wikipedia.org/wiki/Tuple#Tuples_as_nested_sets), но, на наш взгляд, это непрактичный способ. Вероятно, не одни мы так считаем, что может объяснить, почему мы не встретили примеров использования такого метода другими людьми.\n50\t\n51\t\n52\t\n53\t**Рис. 3.** Сравнение теории графов и теории связей. Вершина эквивалентна [замкнутой на себя связи](https://linksplatform.github.io/itself.html) — связи, которая в себе начинается и в себе заканчивается. Направленное ребро отображается в направленную связь-дуплет, а ненаправленное ребро — в пару направленных связей-дуплетов в обоих противоположных направлениях. Иными словами, если в теории графов требуется два типа сущностей — вершины и рёбра, то в теории связей достаточно только связей (больше всего похожих на рёбра).\n54\t\n55\t### Теория связей\n56\t\n57\tТеория связей основана на понятии связи.\n58\t\n59\tВ проекции теории связей в теорию множеств [связь](https://habr.com/ru/companies/deepfoundation/articles/576398) определяется как [n-кортеж](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) ссылок на связи, у которой есть собственная ссылка, используя которую другие связи могут ссылаться на неё.\n60\t\n61\t> Стоит отметить, что отдельное понятие ссылки здесь требуется только потому, что [рекурсивные определения](https://ru.wikipedia.org/wiki/Определение#Виды_определений) недоступны в теории множеств. Сама же теория связей может описать себя без необходимости отдельного термина ссылки — то есть ссылка это частный случай связи.\n62\t\n63\t#### Дуплеты\n64\t\n65\tСвязь-дуплет представлена дуплетом (2-кортежем или [упорядоченной парой](https://ru.wikipedia.org/wiki/Упорядоченная_пара)) ссылок на связи. У связи-дуплета есть собственная ссылка.\n66\t\n67\t```python\n68\tR = { 1 , 2 }\n69\t\n70\tR × R = {\n71\t (1, 1),\n72\t (1, 2),\n73\t (2, 1),\n74\t (2, 2),\n75\t}\n76\t```\n77\t\n78\t**Где:**\n79\t\n80\t- $R$ — это множество ссылок (от английского слова «References» — ссылки).\n81\t\n82\tВ этом примере в множестве $R$ всего $2$ ссылки на связи, а именно $1$ и $2$. Иными словами, в сети связей, построенной на таком множестве ссылок, может быть только $2$ связи.\n83\t\n84\tЧтобы получить все возможные значения связи, используется [декартово произведение](https://ru.wikipedia.org/wiki/Декартово_произведение) $R$ на само себя, то есть $R \\times R$.\n85\t\n86\t\n87\t\n88\t**Рис. 4.** Матрица, представляющая декартово произведение множества {1, 2} на само себя. Здесь мы видим, что у связей с двумя ссылками на связи может быть всего 4 возможных значения.\n89\t\n90\t\n91\t\n92\t**Рис. 5.** Таблица строк, содержащих все возможные варианты значений связей для сети с двумя связями; эти варианты получаются при помощи декартова произведения {1, 2} на само себя.\n93\t\n94\t**Сеть связей-дуплетов** определяется как:\n95\t\n96\t> $\\displaystyle \\mathbf{N^2: R \\to R \\times R}$\n97\t\n98\tГде:\n99\t\n100\t- $\\to$ обозначает [отображение (функцию)](https://ru.wikipedia.org/wiki/Функция_(математика));\n101\t- $\\mathbf{N^2}$ обозначает функцию, определяющую сеть дуплетов (сеть размерности 2);\n102\t- $\\mathbf{R}$ обозначает множество ссылок на связи.\n103\t\n104\t**Пример:**\n105\t\n106\t> $\\displaystyle 1 \\to (1, 1)$\n107\t>\n108\t> $\\displaystyle 2 \\to (2, 2)$\n109\t>\n110\t> $\\displaystyle \\mathbf{3 \\to (1, 2)}$\n111\t\n112\tВ [нотации связей](https://github.com/link-foundation/links-notation): `(1: 1 1) (2: 2 2) (3: 1 2)`\n113\t\n114\t\n115\t\n116\t**Рис. 6. Сеть из трёх связей.** Представление сети дуплетов похоже на граф, но такую визуализацию мы называем сетью связей. Первая и вторая связи имеют похожую структуру — обе начинаются в себе и заканчиваются в себе. Отсюда получается, что вместо традиционного представления вершины в виде точки в теории графов мы получаем графическое представление замкнутой самоссылочной стрелки, которое похоже на символ бесконечности.\n117\t\n118\t\n119\t\n120\t**Рис. 7.** Это графическое представление декартова произведения в виде матрицы, которое отображает все возможные значения связей. Связи, определяющие конкретную сеть, выделены оранжевым цветом. Иными словами, из 9 возможных вариантов значений связи выбраны всего 3 связи, что соответствует размеру множества **R**.\n121\t\n122\tСеть связей-дуплетов может представлять любую структуру данных.\n123\t\n124\tНапример, связи-дуплеты могут:\n125\t\n126\t- связывать объект с его свойствами;\n127\t- связывать два дуплета вместе, чего определение теории графов не позволяет;\n128\t- представлять любую последовательность (n-кортеж) в виде дерева, построенного из вложенных упорядоченных пар;\n129\t- описать предложение на естественном языке, например, по модели [подлежащее-сказуемое](https://ru.wikipedia.org/wiki/Сказуемое).\n130\t\n131\tБлагодаря этому и другим фактам мы убеждены, что связи-дуплеты могут представлять любую возможную структуру данных.\n132\t\n133\t#### Триплеты\n134\t\n135\tСвязь-триплет представлена триплетом (3-кортежем) ссылок на связи.\n136\t\n137\t```python\n138\tR = { 1 , 2 }\n139\t\n140\tR × R = {\n141\t (1, 1),\n142\t (1, 2),\n143\t (2, 1),\n144\t (2, 2),\n145\t}\n146\t\n147\tR × R × R = {\n148\t (1, 1, 1),\n149\t (1, 1, 2),\n150\t (1, 2, 1),\n151\t (1, 2, 2),\n152\t (2, 1, 1),\n153\t (2, 1, 2),\n154\t (2, 2, 1),\n155\t (2, 2, 2),\n156\t}\n157\t```\n158\t\n159\t\n160\t\n161\t**Рис. 8.** Трёхмерный куб-матрица, который представляет все возможные значения связи-триплета. Такой куб получается рекурсивным декартовым произведением множества {1, 2} на само себя, то есть { 1, 2 } × { 1, 2 } × { 1, 2 }.\n162\t\n163\t\n164\t\n165\t**Рис. 9.** Таблица всех возможных вариантов значений связи-триплета, получаемая рекурсивным декартовым произведением множества { 1, 2 } на само себя, то есть { 1, 2 } × { 1, 2 } × { 1, 2 }. **Примечание:** первая ссылка может интерпретироваться как начало, вторая как тип, а третья как конец; пользователь сам определяет, как интерпретировать компоненты кортежа ссылок в соответствии с решаемой задачей.\n166\t\n167\t**Сеть связей-триплетов** определяется как:\n168\t\n169\t> $\\displaystyle \\mathbf{N^3: R \\to R \\times R \\times R}$\n170\t\n171\tГде:\n172\t\n173\t- $\\mathbf{N^3}$ обозначает функцию, определяющую сеть триплетов (сеть размерности 3);\n174\t- $\\mathbf{R}$ обозначает множество ссылок на связи.\n175\t\n176\tПример функции, задающей конкретную сеть триплетов:\n177\t\n178\t> $\\displaystyle 1 \\to (1, 1, 1)$\n179\t>\n180\t> $\\displaystyle 2 \\to (2, 2, 2)$\n181\t>\n182\t> $\\displaystyle 3 \\to (3, 3, 3)$\n183\t>\n184\t> $\\displaystyle \\mathbf{4 \\to (1, 2, 3)}$\n185\t\n186\tВ [нотации связей](https://github.com/link-foundation/links-notation): `(1: 1 1 1) (2: 2 2 2) (3: 3 3 3) (4: 1 2 3)`\n187\t\n188\t\n189\t\n190\t**Рис. 10.** Сеть триплетов, представленная в виде цветного ориентированного графа. В этой сети 4 триплета, соответствующие функции, заданной выше. Узлы соответствуют связям, а цвета рёбер соответствуют ссылкам на связи в соответствии с Рис. 9 (красный — from, синий — type, зелёный — to).\n191\t\n192\tСвязи-триплеты могут делать то же, что и связи-дуплеты. Поскольку в триплетах есть дополнительная ссылка, её можно, например, использовать для указания типа связи.\n193\t\n194\tНапример, связи-триплеты могут:\n195\t\n196\t- связывать объект, его свойство и значение;\n197\t- связывать две связи вместе определённым отношением;\n198\t- описать предложение на естественном языке, например, по модели [подлежащее-глагол-дополнение](https://ru.wikipedia.org/wiki/Порядок_слов_в_предложении).\n199\t\n200\t#### Последовательности\n201\t\n202\tПоследовательность ссылок на связи — также известная как [n-кортеж](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) — является общим случаем.\n203\t\n204\tВ общем виде сеть связей определяется как:\n205\t\n206\t> $\\displaystyle \\mathbf{N^n: R \\rightarrow \\underbrace{ R \\times R \\times \\ldots \\times R}_{n}}$\n207\t\n208\tГде:\n209\t\n210\t- $\\mathbf{N^n}$ обозначает функцию, определяющую сеть связей размерности $n$;\n211\t- $\\mathbf{R}$ обозначает множество ссылок на связи.\n212\t\n213\tПример:\n214\t\n215\t> $\\displaystyle 1 \\to (1)$\n216\t>\n217\t> $\\displaystyle 2 \\to (2, 2)$\n218\t>\n219\t> $\\displaystyle 3 \\to (3, 3, 3)$\n220\t>\n221\t> $\\displaystyle \\mathbf{4 \\to (1, 2, 3, 2, 1)}$\n222\t\n223\tВ [нотации связей](https://github.com/link-foundation/links-notation): `(1: 1) (2: 2 2) (3: 3 3 3) (4: 1 2 3 2 1)`\n224\t\n225\tВ этом примере используются n-кортежи переменной длины в качестве значений связей.\n226\t\n227\tПоследовательности (кортежи) по сути эквивалентны по выразительной силе реляционной модели — факт, который ещё предстоит доказать в рамках разрабатываемой теории. Однако, когда мы заметили, что связей-дуплетов и связей-триплетов достаточно для представления последовательностей любого размера, появилось предположение об отсутствии необходимости использовать последовательности напрямую, поскольку они представимы связями-дуплетами.\n228\t\n229\t### Итоги сравнения\n230\t\n231\t[Реляционная модель данных](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) может представить всё — включая [ассоциативную модель](https://web.archive.org/web/20210814063207/https://en.wikipedia.org/wiki/Associative_model_of_data), но для этого необходимо ввести [отношение порядка](https://ru.wikipedia.org/wiki/Отношение_порядка), которое обычно принимает форму отдельного столбца ID. Это связано с тем, что реляционная модель данных основана на понятии [множества](https://ru.wikipedia.org/wiki/Множество), а не [последовательности](https://ru.wikipedia.org/wiki/Последовательность). В противоположность этому, графовая модель отлично представляет отношения, но менее эффективна в представлении уникальных дедуплицированных последовательностей.\n232\t\n233\tХотя сама реляционная модель не требует разделения данных на несколько таблиц, традиционные реализации обычно принимают такой подход с фиксированными схемами, что приводит к фрагментации связанных данных и усложняет восстановление заложенных связей. В противоположность этому, ассоциативная модель использует единое хранилище связей, которое достигает максимально возможной степени нормализации. Такая архитектура упрощает взаимно-однозначное отображение предметной области, тем самым облегчая быстрые изменения требований [[4]](https://www.researchgate.net/publication/255670856_A_COMPARISON_OF_THE_RELATIONAL_DATABASE_MODEL_AND_THE_ASSOCIATIVE_DATABASE_MODEL).\n234\t\n235\tАссоциативная модель может легко представить n-кортежи неограниченной длины, используя кортежи с $\\mathbf{n \\geq 2}$. Она столь же хороша, как теория графов, в своей способности представлять ассоциации, и столь же мощна, как реляционная модель, способная полностью представить любую таблицу SQL. Кроме того, ассоциативная модель может представлять строгие последовательности, позволяя инкапсулировать любую последовательность в единственную уникальную связь, что полезно для дедупликации.\n236\t\n237\tВ реляционной модели достаточно всего одного отношения, чтобы воспроизвести поведение ассоциативной модели, и обычно при этом нужно не более 2–3 столбцов помимо явного ID или встроенного ID строки. Сам ID требуется в реляционной модели, поскольку она построена на понятии множества; в противоположность этому теория связей построена на понятии последовательности, поэтому явный ID не требуется (и может быть добавлен опционально, если пользователь действительно в нём нуждается).\n238\t\n239\tПо определению графовая модель не может напрямую создать ребро между рёбрами. Поэтому ей потребуется либо переопределение, либо расширение с помощью однозначного метода хранения уникальных дедуплицированных последовательностей. Хотя последовательности можно хранить [в виде вложенных множеств](https://en.wikipedia.org/wiki/Tuple#Tuples_as_nested_sets) внутри графовой модели, этот подход не популярен. Несмотря на то что графовая модель наиболее близка к связям-дуплетам, она всё же отличается по определению.\n240\t\n241\tИспользование ассоциативной модели означает, что больше не нужно выбирать между SQL и NoSQL базами данных; вместо этого ассоциативное хранилище данных может представить всё самым простым возможным способом, при этом данные всегда находятся в наиболее близкой к оригиналу форме (обычно это нормализованная форма, но денормализация тоже возможна при необходимости).\n242\t" | |
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| "content": "1\t## Математическое введение в теорию связей\n2\t\n3\t### Введение\n4\t\n5\tТеперь, когда мы кратко представили истоки нашей работы, настала пора погрузиться в теорию ещё глубже.\n6\t\n7\t> Теория связей разрабатывается как более фундаментальная теория по сравнению с теорией множеств и теорией типов, и как замена реляционной алгебре и теории графов в качестве объединяющей теории. В то время как теория типов построена на базовых понятиях «тип» и «терм», а теория множеств — на «множестве» и «элементе», теория связей сводит всё к единому понятию «связь».\n8\t\n9\tВ этом разделе мы разберём основные понятия и термины, используемые в теории связей.\n10\t\n11\tЗатем мы представим определения теории связей в рамках теории множеств и впоследствии спроецируем эти определения в теорию типов, используя интерактивное средство доказательства теорем Rocq.\n12\t\n13\tНаконец, мы подведём итоги и наметим направления дальнейших исследований и развития теории связей.\n14\t\n15\t#### Теория связей\n16\t\n17\tВ основе всей теории связей лежит единое понятие связи. Дополнительное понятие ссылки на связь вводится только для теорий, не поддерживающих рекурсивные определения, — таких как теория множеств и теория типов.\n18\t\n19\t#### Связь\n20\t\n21\tСвязь обладает асимметричной рекурсивной (фрактальной) структурой, которая может быть выражена просто: связь **связывает** связи (как в «связь **соединяет** связи»). Под «асимметричностью» подразумевается то, что каждая связь имеет направление — от источника (начала) к приёмнику (концу).\n22\t\n23\t### Определения теории связей в рамках теории множеств\n24\t\n25\tВ дополнение к математической нотации мы используем [нотацию связей](https://github.com/link-foundation/links-notation) — специальный язык для выражения данных исключительно в терминах связей и ссылок. Нотация связей записывается в форме `(id: from to)` для дуплетов, где `id` — ссылка на связь, а `from` и `to` — ссылки на другие связи. Для операций подстановки: `(pattern) -> (replacement)`, создание: `() -> (id: from to)`, удаление: `(id: from to) -> ()`.\n26\t\n27\t> В этой статье мы используем обозначения $R$ (References, множество ссылок) и $L$ (Links, связи). $R$ — это множество ссылок, а $L$ — это связь. Это разделение необходимо для того, чтобы разложить рекурсивный термин на управляемые части при отражении теории связей в теорию множеств через теорию типов.\n28\t\n29\t**Ссылка на кортеж** — это уникальный идентификатор или порядковый номер, который связан с определённым кортежем, представляющим последовательность ссылок на другие кортежи.\n30\t\n31\tМножество ссылок на кортежи:\n32\t\n33\t> $\\displaystyle \\mathbf{R ⊆ ℕ_0}$\n34\t\n35\t**Кортеж ссылок** — это кортеж, состоящий из нуля или более ссылок на кортежи, где количество ссылок соответствует количеству элементов кортежа.\n36\t\n37\tМножество всех кортежей ссылок длины $n ∈ ℕ_0$:\n38\t\n39\t> $\\displaystyle \\mathbf{T_n = R^n}$\n40\t\n41\tДекартова степень $R^n$ всегда даёт кортеж длины $n$, так как все его компоненты одного и того же типа $R$.\n42\tДругими словами, $R^n$ представляет собой множество всех возможных n-элементных кортежей (по сути n-кортежей), где каждый элемент принадлежит множеству $R$.\n43\t\n44\t**Связь** — это упорядоченная пара, состоящая из ссылки на кортеж и кортежа ссылок. Эта структура служит для отображения между ссылками и кортежами.\n45\t\n46\t> $\\displaystyle \\mathbf{L = R \\times T_n}$\n47\t\n48\t> Заметим, что ссылка ($r ∈ R$) — это частный случай связи нулевой размерности: при $n = 0$ кортеж $T_0 = R^0$ является пустым кортежем $()$, и связь $L = R \\times T_0$ вырождается в пару $(r, ())$, что эквивалентно просто $r$. Таким образом, ссылка — это связь нулевой размерности: $r ≡ (r, ())$, и множество ссылок $R$ является подмножеством множества связей $L$.\n49\t\n50\t**Сеть кортежей длины n** (или n-мерная сеть) определяется семейством функций $\\{N^n\\}$, где каждая функция $N^n: R → T_n$ отображает ссылку $r ∈ R$ в кортеж ссылок длины $n$, принадлежащий $T_n$, тем самым идентифицируя точки в n-мерном пространстве.\n51\t$n$ в $N^n$ указывает на то, что функция возвращает кортежи, содержащие $n$ ссылок. Каждая n-мерная сеть таким образом представляет последовательность точек в n-мерном пространстве. Сеть имеет размерность (dimension) — длину кортежей, и размер (size) — количество кортежей.\n52\t\n53\t> Здесь и далее мы используем термин «сеть» (Network) вместо «ассоциативная сеть» (Associative Network), поскольку в нашем контексте все сети являются ассоциативными по определению.\n54\t\n55\t**Семейство функций:**\n56\t\n57\t> $\\displaystyle \\mathbf{∪_f \\{N^n | n ∈ ℕ_0\\} ⊆ L}$\n58\t\n59\tЗдесь символ объединения $∪_f$ обозначает агрегацию всех функций в семействе $\\{N^n\\}$, а символ $⊆$ указывает на то, что эти упорядоченные пары — рассматриваемые как функциональные бинарные отношения — являются подмножеством множества $L$ всех связей.\n60\t\n61\t**Множество дуплетов (упорядоченных пар или двумерных кортежей) ссылок:**\n62\t\n63\t> $\\displaystyle \\mathbf{D = R^2}$\n64\t\n65\tЭто множество всех дуплетов $(R, R)$, то есть вторая декартова степень $R$.\n66\t\n67\t**Сеть дуплетов (или двумерная сеть):**\n68\t\n69\t> $\\displaystyle \\mathbf{N^2: R → R^2}$\n70\t\n71\tВ [нотации связей](https://github.com/link-foundation/links-notation): каждый дуплет записывается как `(id: from to)`, например `(3: 1 2)` означает, что связь с идентификатором 3 ссылается на связи 1 и 2.\n72\t\n73\tКаждая сеть дуплетов таким образом представляет последовательность точек в двумерном пространстве.\n74\t\n75\t#### Сеть как связь связей\n76\t\n77\tСеть — это по сути **связь связей**. Рассмотрим пример сети дуплетов $N^2$ размера 3:\n78\t\n79\tВ [нотации связей](https://github.com/link-foundation/links-notation):\n80\t```\n81\t(\n82\t (1 1)\n83\t (2 2)\n84\t (1 2)\n85\t)\n86\t```\n87\t\n88\tВ математической нотации (как бинарное отношение или функция, определённая для аргументов 1–3):\n89\t```\n90\tN² of size 3 =\n91\t{\n92\t 1 → (1, 1)\n93\t 2 → (2, 2)\n94\t 3 → (1, 2)\n95\t}\n96\t```\n97\t\n98\tЭта запись эквивалентна последовательности кортежей $[(1, 1), (2, 2), (1, 2)]$.\n99\t\n100\tЗаметим, что в нотации связей мы используем ссылки как порядковые номера кортежей, что полностью определяет инъекцию (функцию), или бинарное отношение, или последовательность кортежей. Все эти представления — просто сеть (в нашем контексте все сети являются ассоциативными).\n101\t\n102\tЕсли посмотреть на сеть с другой стороны, то это просто связь $N$-мерности, состоящая из связей двумерности (в данном примере), с произвольной ссылкой или порядковым номером в последовательности всех сетей.\n103\t\n104\tТаким образом, **сеть — это связь связей**.\n105\t\n106\t#### Сеть всех сетей\n107\t\n108\tЕсли сеть — это связь связей, имеющая свою собственную ссылку, то возникает естественный вопрос: **в каком пространстве** находится эта ссылка? Ответ приводит нас к понятию *сети всех сетей* — пространства, в котором каждая возможная сеть адресуема.\n109\t\n110\t**Определение.** Пусть $\\mathcal{N}$ — множество всех возможных сетей произвольной размерности $n ∈ ℕ_0$ и произвольного размера $s ∈ ℕ_0$:\n111\t\n112\t> $\\displaystyle \\mathbf{\\mathcal{N} = \\bigcup_{n \\in \\mathbb{N}_0} \\bigcup_{s \\in \\mathbb{N}_0} \\{N^n \\text{ размера } s\\}}$\n113\t\n114\tКаждая сеть $N^n$ размера $s$ — это последовательность из $s$ кортежей длины $n$, то есть элемент $\\left(R^n\\right)^s$. Множество $\\mathcal{N}$ охватывает все такие последовательности для всех возможных $n$ и $s$.\n115\t\n116\t**Сеть всех сетей** — это функция, которая отображает ссылки на элементы $\\mathcal{N}$:\n117\t\n118\t> $\\displaystyle \\mathbf{\\Omega: R → \\mathcal{N}}$\n119\t\n120\tЗдесь $\\Omega$ играет роль *универсальной адресации*: каждой ссылке $r ∈ R$ соответствует некоторая сеть из $\\mathcal{N}$. Поскольку $R ⊆ ℕ_0$ и множество всех конечных сетей счётно, такое отображение существует.\n121\t\n122\t**Структура как континуум.** Множество $\\mathcal{N}$ содержит все сети, начиная от наименьших возможных:\n123\t- $N^0$ размера 0 — пустая сеть нулевой размерности\n124\t- $N^1$ размера 1 — сеть из одного кортежа длины 1 (одна ссылка)\n125\t- $N^2$ размера 1 — сеть из одного дуплета\n126\t\n127\tи так далее до сетей произвольно большой размерности и размера. Это образует иерархию от простейших структур (отдельные ссылки, которые суть связи нулевой размерности) до произвольно сложных сетей.\n128\t\n129\t**Всё адресуемо.** Следствие существования $\\Omega$: любая конечная структура, которую можно выразить через кортежи ссылок, имеет ссылку в пространстве $R$. Это означает, что **всё возможная информация**, представимая в виде связей, адресуема (reference-able) в сети всех сетей. Каждая связь, каждый кортеж, каждая сеть — всё получает свою уникальную ссылку.\n130\t\n131\t**Сеть всех сетей как связь связей.** Сама $\\Omega$ — это тоже связь связей, но особого рода: она содержит *все* возможные связи связей. Она сама является элементом $\\mathcal{N}$ (как сеть, отображающая ссылки в сети), и следовательно, адресуема в себе самой. Это самоссылочное свойство перекликается с мета-теоретическим замыканием цикла определений (раздел 9).\n132\t\n133\t##### Формализация в Rocq и Lean\n134\t\n135\t###### Rocq\n136\t\n137\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/NetworkDefinitions.v)\n138\t\n139\t```rocq\n140\tRequire Import NetworkDefinitions.\n141\t\n142\t(* Элемент сети произвольной размерности: пара (размерность, список кортежей).\n143\t Кортеж переменной длины представлен как ReferenceList *)\n144\tRecord AnyNetwork := mkAnyNetwork {\n145\t dimension : nat;\n146\t tuples : list ReferenceList\n147\t}.\n148\t\n149\t(* Сеть всех сетей: отображение ссылок на сети произвольной размерности и размера *)\n150\t(* Ω: R → 𝒩 *)\n151\tDefinition NetworkOfAllNetworks := Reference -> AnyNetwork.\n152\t\n153\t(* Сеть всех сетей в виде последовательности (для конечного фрагмента) *)\n154\tDefinition NetworkOfAllNetworksList := list AnyNetwork.\n155\t\n156\t(* Пустая сеть нулевой размерности и размера — наименьший элемент 𝒩 *)\n157\tDefinition emptyNetwork : AnyNetwork := mkAnyNetwork 0 nil.\n158\t\n159\t(* Пример: представление дуплетной сети N² размера 3 как элемента 𝒩 *)\n160\tDefinition exampleAnyNetwork : AnyNetwork :=\n161\t mkAnyNetwork 2 ((cons 1 (cons 1 nil)) :: (cons 2 (cons 2 nil))\n162\t :: (cons 1 (cons 2 nil)) :: nil).\n163\t```\n164\t\n165\t###### Lean\n166\t\n167\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/NetworkDefinitions.lean)\n168\t\n169\t```lean\n170\timport NetworkDefinitions\n171\t\n172\t-- Элемент сети произвольной размерности: пара (размерность, список кортежей)\n173\t-- Кортеж переменной длины представлен как ReferenceList\n174\tstructure AnyNetwork where\n175\t dimension : Nat\n176\t tuples : List ReferenceList\n177\t\n178\t-- Сеть всех сетей: отображение ссылок на сети произвольной размерности и размера\n179\t-- Ω: R → 𝒩\n180\tabbrev NetworkOfAllNetworks := Reference → AnyNetwork\n181\t\n182\t-- Сеть всех сетей в виде последовательности (для конечного фрагмента)\n183\tabbrev NetworkOfAllNetworksList := List AnyNetwork\n184\t\n185\t-- Пустая сеть нулевой размерности и размера — наименьший элемент 𝒩\n186\tdef emptyNetwork : AnyNetwork := ⟨0, []⟩\n187\t\n188\t-- Пример: представление дуплетной сети N² размера 3 как элемента 𝒩\n189\tdef exampleAnyNetwork : AnyNetwork :=\n190\t ⟨2, [[1, 1], [2, 2], [1, 2]]⟩\n191\t```\n192\t\n193\tПустой кортеж (кортеж длины ноль) представлен пустым кортежем, обозначаемым как $()$ или $∅$.\n194\t\n195\t**Сеть списков ссылок (вложенных упорядоченных пар):**\n196\t\n197\t> $\\displaystyle \\mathbf{N^{list}: R → RL}\\textbf{, где }\\mathbf{RL = \\{(∅, ∅) | (r, rl), r ∈ R, rl ∈ RL\\} }$\n198\t\n199\t$RL$ (ReferenceList) — это множество списков ссылок, представленных как вложенные упорядоченные пары: пустые пары и пары, содержащие один или более элементов. Таким образом, кортеж длины $n \\in \\mathbb{N}_0$ может быть представлен как список ссылок.\n200\t\n201\t" | |
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| "content": "1\t## Практическая реализация\n2\t\n3\tСуществует несколько практических реализаций: [Deep](http://github.com/deep-foundation/), [LinksPlatform](https://github.com/linksplatform) и [Модель отношений](https://github.com/netkeep80/jsonRVM).\n4\t\n5\t### Deep\n6\t\n7\t[Deep](http://github.com/deep-foundation/) — это система, основанная на теории связей. В теории связей с помощью связей можно представлять любые данные или знания, а также использовать их для программирования. Deep построен вокруг этой философии: в Deep всё — связи. Однако если разделить их на две категории, то у нас будут собственно данные и поведение. Поведение — представленное кодом в Deep — хранится в ассоциативном хранилище в виде связей, а для исполнения передаётся в Docker-контейнер соответствующего языка программирования, где оно изолированно и безопасно выполняется. Вся коммуникация между различными частями кода осуществляется через связи в хранилище (базе данных), что делает базу данных универсальным API, основанным на данных (в отличие от традиционной практики вызова функций и методов). В данный момент в качестве ассоциативного хранилища в Deep используется PostgreSQL, который позднее будет заменён на движок данных, основанный на дуплетах и триплетах из LinksPlatform.\n8\t\n9\tDeep делает всё программное обеспечение на планете совместимым, представляя все его части в виде связей. Также возможно хранить любые данные и код вместе, связывая события или действия над различными типами ассоциаций с соответствующим кодом, который выполняется для обработки этих событий. Каждый обработчик может получать необходимые связи из ассоциативного хранилища и вставлять/обновлять/удалять связи в нём, что может инициировать дальнейшее каскадное выполнение обработчиков.\n10\t\n11\tТаблица `links` в базе данных (PostgreSQL) Deep содержит записи, которые можно интерпретировать как связи. Они имеют такие столбцы, как `id`, `type_id`, `from_id` и `to_id`. Типы связей помогают разработчику ассоциативных пакетов предопределить семантику отношений между различными элементами, обеспечивая однозначное понимание связей как людьми, так и кодом в ассоциативных пакетах. Помимо таблицы `links`, в системе также присутствуют таблицы `numbers`, `strings` и `objects` для хранения числовых, строковых и JSON-значений соответственно. К каждой связи можно привязать только одно значение. Это временное решение, которое используется до тех пор, пока Deep не мигрирует на использование LinksPlatform в качестве движка базы данных. После завершения миграции все эти, казалось бы, базовые типы будут построены с нуля с использованием только связей. Это позволит использовать дедупликацию (которая возникает как следствие теории связей) и глубокое понимание внутренней структуры значений. Также планируется добавить индексацию таких сложных значений, представленных связями, для повышения производительности, чтобы сделать её такой же быстрой или быстрее, чем текущая реализация на PostgreSQL.\n12\t\n13\t### LinksPlatform\n14\t\n15\t[LinksPlatform](https://github.com/linksplatform) — это кроссплатформенный мультиязычный фреймворк, направленный на реализацию низкоуровневой ассоциативности в виде конструктора движков баз данных. К примеру, на текущий момент у нас существует [бенчмарк](https://github.com/linksplatform/Comparisons.PostgreSQLVSDoublets), который сравнивает реализацию дуплетов в PostgreSQL и аналогичную реализацию на чистом Rust/C++; лидирующая реализация на Rust обгоняет PostgreSQL в $1746 \\dots 15745$ раз на операциях записи и в $100 \\dots 9694$ раз на операциях чтения.\n16\t\n17\t### Модель отношений\n18\t\n19\t[Модель отношений](https://github.com/netkeep80/jsonRVM) — это язык метапрограммирования, основанный на представлении программы в виде трёхмерной сети. Модель отношений относится к сущностно-ориентированному программированию, где сущность используется в качестве единственной фундаментальной концепции — то есть предполагается, что всё есть сущность и нет ничего кроме сущностей.\n20\t\n21\tВ модели отношений сущность в зависимости от своего внутреннего конститутивного принципа может быть либо структурой (объектом), либо функцией (методом). В отличие от известной ER-модели, в которой для представления схемы базы данных используются два базовых понятия — сущность и связь, — в модели отношений сущность и связь есть суть одно и то же. Такое представление позволяет описывать не только внешние отношения сущности, но и её внутреннюю модель — модель отношений.\n22\t\n23\tСущность в своём внутреннем принципе триедина (троична, трёхэлементна), потому что она есть объединение (синтез) трёх аспектов (качеств) других сущностей (или самой себя).\n24\t\n25\t[jsonRVM](https://github.com/netkeep80/jsonRVM) — это многопоточная виртуальная машина для исполнения JSON-проекции модели отношений. Модель отношений, представленная в виде JSON, позволяет писать программы непосредственно на JSON. Такое представление является гибридом сегмента данных и кода и позволяет легко десериализовать/исполнять/сериализовать проекцию модели отношений, а также использовать JSON-редакторы для программирования. В процессе исполнения модели отношений метапрограмма может не только обрабатывать данные, но и генерировать многопоточные программы и метапрограммы, а также исполнять их немедленно или экспортировать в виде JSON.\n26\t" | |
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| "content": "1\t## Заключение\n2\t\n3\tВ этой статье мы рассмотрели математический базис реляционной алгебры и теории графов, а также изложили определения теории связей в терминах теории множеств и спроецировали их в теорию типов. Мы также определили набор функций и лемм, необходимых для доказательства возможности эквивалентного преобразования из любого кортежа/последовательности во вложенные связи-дуплеты и обратно. А значит, достаточно только одной формулы для представления любого возможного типа информации:\n4\t\n5\t> $\\displaystyle R \\to R^2$\n6\t\n7\tТаким образом, это является основой для проверки гипотезы о том, что связями-дуплетами можно представить любую другую структуру данных. Иными словами, связей-дуплетов достаточно, чтобы представлять любые таблицы, графы, строки, массивы, списки, числа, звук, картинки, видео и многое другое.\n8\t\n9\tЕщё одно следствие этого доказательства — мы можем представить ленту машины Тьюринга, используя только связи-дуплеты. Это означает, что связи могут быть столь же мощными, как машина Тьюринга, в своей ёмкости хранения. То есть мы можем использовать связи для всех случаев, где используется машина Тьюринга. Но при этом нет необходимости пытаться уместить данные в нули и единицы, не нужно ломать над этим голову. Потому что в связях «алфавит» по сути неограничен, и вы можете добавить любое количество связей, присвоить им любой нужный смысл и связать или соединить их любым необходимым образом. Обычно это означает, что ваша концепция, объект или вещь из реального или абстрактного мира будет отображена в связи один к одному или максимально близко к оригиналу, что иногда невозможно при традиционных методах.\n10\t\n11\tМы продолжаем делать успехи в синхронизации смысла между нашими тремя проектами и между людьми из нашего ассоциативного сообщества. Эти проекты призваны принести ассоциативность в мир и сделать её полезной для человечества. Эта статья является очередной итерацией нашего обсуждения, позволяющей нам договориться о едином смысле слов и терминов в рамках общей ассоциативной теории. Мы верим, что эта теория может стать метаязыком, который уже используется людьми и машинами для коммуникации.\n12\t\n13\tС каждым следующим уточнением мы будем на шаг ближе к тому, чтобы разговаривать на одном языке и делать эту идею понятнее для окружающих. Эта теория также пригодится для различных оптимизаций в разрабатываемых реализациях ассоциативности, а в будущем — и для разработки ассоциативных чипов (или сопроцессоров) для ускорения операций над данными, представленными связями.\n14\t\n15\t### Планы на будущее\n16\t\n17\tВ данной статье была продемонстрирована лишь малая часть всех наработок по теории связей, которые накопились за несколько лет работы и исследований. В следующих статьях постепенно будут раскрыты и другие проекции теории связей в терминах иных теорий, таких как реляционная алгебра, теория графов, а также рассмотрение в терминах теории типов без использования теории множеств напрямую, а также разбор отличий от [ассоциативной модели данных Саймона Вильямса](https://web.archive.org/web/20181219134621/http://sentences.com/docs/amd.pdf) [[3]](https://web.archive.org/web/20181219134621/http://sentences.com/docs/amd.pdf) [[5]](https://link.springer.com/content/pdf/10.1057/palgrave.jdm.3240049.pdf).\n18\t\n19\tМы также планируем спроецировать теорию связей в саму себя, показав, что она может использоваться как мета-теория. Это также откроет дверь для проекции теории множеств и теории типов в теорию связей, что означает завершение цикла определения (теория связей определяется в теории множеств, которая сама может быть определена в теории связей). Мы также сможем сравнить теорию множеств, теорию типов, теорию графов, реляционную алгебру и теорию связей, что поможет нам проверить эквивалентность этих теорий или, по крайней мере, получить точную биективную функцию преобразования между ними.\n20\t\n21\tТакже планируется представить чёткую и единую терминологию теории связей, её основные постулаты, аспекты и так далее. Текущий прогресс по разработке теории можно наблюдать [в репозитории deep-theory](https://github.com/deep-foundation/deep-theory).\n22\t\n23\tЧтобы не пропустить обновления, рекомендуем подписаться на [блог Deep.Foundation](https://habr.com/ru/companies/deepfoundation) здесь или уже сейчас посмотреть наши [наработки на GitHub](https://github.com/deep-foundation), или напрямую связаться с нами в [нашем публичном чате в Telegram](https://t.me/unilinkness) (особенно если вы боитесь, что вас заминусуют в комментариях).\n24\t\n25\tБудем рады любой вашей обратной связи — что здесь на Хабре, что на GitHub или в Telegram. Вы также можете поучаствовать в разработке теории или ускорить её развитие любым взаимодействием с нами.\n26\t\n27\t### Демо CLI\n28\t\n29\tТеперь вы можете получить представление о том, как работает ассоциативная теория, используя наш [инструмент демо CLI](https://github.com/link-foundation/link-cli), построенный на [нотации связей](https://github.com/linksplatform/Protocols.Lino) и [хранилище связей-дуплетов](https://github.com/linksplatform/Data.Doublets) из проекта [LinksPlatform](https://github.com/linksplatform). [Нотация связей](https://github.com/linksplatform/Protocols.Lino) — это язык для данных, выраженных исключительно в связях и ссылках. [Дуплеты](https://github.com/linksplatform/Data.Doublets) — это движок базы данных, написанный на C# с нуля для поддержки исключительно ассоциативного хранения и преобразований.\n30\t\n31\tВ этом демо мы опираемся на нотацию связей для создания диалекта, способного описать единую универсальную операцию — подстановку. Как и в случае с унификацией типов данных, также возможно унифицировать создание, чтение, обновление и удаление в единую операцию подстановки. Это аналогично единственной операции из [алгоритма Маркова](https://ru.wikipedia.org/wiki/Алгоритм_Маркова), который доказанно является [полным по Тьюрингу](https://ru.wikipedia.org/wiki/Полнота_по_Тьюрингу).\n32\t\n33\t\n34\t\n35\t**Рис. 11.** На этом изображении вы можете видеть создание двух связей `(1: 1 1)` и `(2: 2 2)`; обновление первой связи до `(1: 1 2)`; обновление/подстановку с использованием переменных для обмена источников и приёмников каждой связи; и удаление всех связей с помощью паттерна `(* *)`.\n36\t\n37\t### Визуальные демо\n38\t\n39\t\n40\t\n41\t**Рис. 12.** Дизайнер чертежей связей, построенный поверх настраиваемого сплайна: [konard.github.io/links-visuals/blueprint.html](http://konard.github.io/links-visuals/blueprint.html) (перемещайте контрольные точки сплайна, представляющего связь)\n42\t\n43\t\n44\t\n45\t**Рис. 13.** Фрактал типа [H-дерево](https://en.wikipedia.org/wiki/H_tree), построенный с помощью связей, представленных прямыми стрелками: [konard.github.io/links-visuals/H-fractal.html](https://konard.github.io/links-visuals/H-fractal.html) (нажмите в любом месте для итерации фрактала)\n46\t\n47\t### P.S.\n48\t\n49\tЭта и предыдущие статьи будут обновляться по мере развития и дополнения теории связей в течение следующих 6 месяцев.\n50\t\n51\t### P.S.S.\n52\t\n53\tЕсли вы стали фанатом теории связей, предлагаем распространять в социальных сетях эту формулу как мемо-вирус.\n54\t\n55\tСимволами юникода:\n56\t\n57\t> R ↦ R²\n58\t\n59\tИспользуя LaTeX:\n60\t\n61\t> R \\to R^2\n62\t\n63\tКоторый преобразуется в SVG (нажимается):\n64\t\n65\t> $\\displaystyle R \\to R^2$\n66\t\n67\t### Ссылки\n68\t\n69\t1. Edgar F. Codd, IBM Research Laboratory, San Jose, California, June 1970, [\"Relational Model of Data for Large Shared Data Banks.\", paragraph 1.3., page 379](https://dl.acm.org/doi/abs/10.1145/362384.362685)\n70\t2. Bender, Edward A.; Williamson, S. Gill (2010). [\"Lists, Decisions and Graphs. With an Introduction to Probability.\", section 2, definition 6, page 161](https://books.google.com/books?id=vaXv_yhefG8C)\n71\t3. Simon Williams, Great Britain (1988), [The Associative Model Of Data](https://web.archive.org/web/20181219134621/http://sentences.com/docs/amd.pdf)\n72\t4. Homan, J. V., & Kovacs, P. J. (2009). [A Comparison of the Relational Database Model and the Associative Database Model](https://www.researchgate.net/publication/255670856_A_COMPARISON_OF_THE_RELATIONAL_DATABASE_MODEL_AND_THE_ASSOCIATIVE_DATABASE_MODEL). Issues in Information Systems, X(1), 208.\n73\t5. Simon Williams (2001). [The Associative Model of Data](https://link.springer.com/content/pdf/10.1057/palgrave.jdm.3240049.pdf). Journal of Database Marketing, 8(4), 336–359.\n74\t\n75\t---\n76\t\n77\t**Теги:** метатеория, теория связей, реляционная теория, ассоциативная теория, математика, теория множеств, теория типов, теория графов, реляционная алгебра, ассоциативная модель данных\n78\t\n79\t**Хабы:** [Data Engineering](https://habr.com/ru/hubs/data_engineering/), [Open source](https://habr.com/ru/hubs/open_source/), [Математика](https://habr.com/ru/hubs/maths/), [Ненормальное программирование](https://habr.com/ru/hubs/crazydev/), [Программирование](https://habr.com/ru/hubs/programming/)\n80\t\n81\t**Автор:** [Константин Дьяченко (Konard)](https://habr.com/ru/users/Konard/)\n82\t" | |
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| "content": "1\t### Последовательности и множества, определённые в терминах связей\n2\t\n3\tВ предыдущих разделах мы показали, что любой кортеж длины $n$ может быть эквивалентно представлен в виде цепочки связей-дуплетов ($R \\to R^2$). Теперь мы делаем следующий шаг: формально определяем **последовательности** и **множества** исключительно в терминах связей-дуплетов.\n4\t\n5\tЭто важный шаг, поскольку последовательности и множества являются фундаментальными структурами данных, лежащими в основе теории множеств и теории типов. Показав, что они могут быть полностью выражены через связи, мы демонстрируем выразительную мощь теории связей.\n6\t\n7\t#### Последовательности как деревья связей-дуплетов\n8\t\n9\tПоследовательность в теории связей — это **ссылка (Reference) на корень дерева**, хранимого в сети дуплетов. Каждый внутренний узел дерева — это связь-дуплет $(ссылка\\_на\\_левое, ссылка\\_на\\_правое)$, а листья — это элементы последовательности.\n10\t\n11\tНапример, последовательность $[1, 2, 3, 4]$ в сбалансированном варианте $((1, 2), (3, 4))$ хранится как:\n12\t\n13\tВ математической нотации: $5 \\to (1, 2),\\ 6 \\to (3, 4),\\ 7 \\to (5, 6)$\n14\t\n15\tВ [нотации связей](https://github.com/link-foundation/links-notation):\n16\t```\n17\t(5: 1 2) — дуплет для пары (1, 2)\n18\t(6: 3 4) — дуплет для пары (3, 4)\n19\t(7: 5 6) — дуплет-корень, ссылающийся на поддеревья\n20\t```\n21\t\n22\tПоследовательность = ссылка 7. Все элементы закодированы исключительно через связи-дуплеты.\n23\t\n24\tФормально:\n25\t\n26\t> $\\displaystyle LinkSequence := Reference$\n27\t\n28\tПоследовательность идентифицируется ссылкой (Reference) на корень дерева в сети.\n29\t\n30\tОдна и та же последовательность может быть представлена деревьями различной формы:\n31\t\n32\t```\n33\t[1, 2, 3, 4] = ((1, 2), (3, 4)) — сбалансированный вариант\n34\t[1, 2, 3, 4] = (((1, 2), 3), 4) — левая лестница\n35\t[1, 2, 3, 4] = (1, (2, (3, 4))) — правая лестница\n36\t```\n37\t\n38\tВсе эти варианты содержат одни и те же листья в одном и том же порядке, различаясь лишь структурой дерева. Количество различных деревьев для последовательности длины $n$ определяется [числом Каталана](https://ru.wikipedia.org/wiki/Числа_Каталана).\n39\t\n40\tОпределены три алгоритма создания последовательностей:\n41\t- **ListToBalancedTree** — сбалансированный вариант (как в [BalancedVariantConverter](https://github.com/linksplatform/Data.Doublets.Sequences/blob/main/csharp/Platform.Data.Doublets.Sequences/Converters/BalancedVariantConverter.cs))\n42\t- **ListToRightStaircase** — правая лестница $(1, (2, (3, 4)))$\n43\t- **ListToLeftStaircase** — левая лестница $(((1, 2), 3), 4)$\n44\t\n45\t##### Rocq\n46\t\n47\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/SequenceDefinitions.v)\n48\t\n49\t```rocq\n50\tRequire Import NetworkDefinitions.\n51\tRequire Import NetworkConversions.\n52\t\n53\t(* Последовательность — это ссылка на корень дерева в сети *)\n54\tDefinition LinkSequence := Reference.\n55\t\n56\t(* Вспомогательное дерево для алгоритмов построения последовательностей *)\n57\tInductive LinkTree : Type :=\n58\t | Leaf : Reference -> LinkTree\n59\t | Node : LinkTree -> LinkTree -> LinkTree.\n60\t\n61\t(* Запись дерева в сеть дуплетов *)\n62\tFixpoint TreeToDupletList_ (t : LinkTree) (offset : nat)\n63\t : NetworkDupletList * nat := ...\n64\t\n65\t(* Получение ссылки на корень (= последовательность) *)\n66\tDefinition TreeToSequence (t : LinkTree) (offset : nat) : LinkSequence := ...\n67\t\n68\t(* Полное преобразование: список → сеть дуплетов + ссылка на корень *)\n69\tDefinition ListToSequence (l : list Reference) (offset : nat)\n70\t : option (LinkSequence * NetworkDupletList) := ...\n71\t```\n72\t\n73\t##### Lean\n74\t\n75\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/SequenceDefinitions.lean)\n76\t\n77\t```lean\n78\timport NetworkDefinitions\n79\timport NetworkConversions\n80\t\n81\t-- Последовательность — это ссылка на корень дерева в сети\n82\tabbrev LinkSequence := Reference\n83\t\n84\t/-- Вспомогательное дерево для алгоритмов построения последовательностей -/\n85\tinductive LinkTree where\n86\t | Leaf : Reference → LinkTree\n87\t | Node : LinkTree → LinkTree → LinkTree\n88\t\n89\t-- Запись дерева в сеть дуплетов\n90\tdef TreeToDupletList_ : LinkTree → Nat → NetworkDupletList × Nat\n91\t\n92\t-- Получение ссылки на корень (= последовательность)\n93\tdef TreeToSequence (t : LinkTree) (offset : Nat) : LinkSequence\n94\t\n95\t-- Полное преобразование: список → сеть дуплетов + ссылка на корень\n96\tdef ListToSequence (l : List Reference) (offset : Nat)\n97\t : Option (LinkSequence × NetworkDupletList)\n98\t```\n99\t\n100\t#### Эквивалентность множеств и последовательностей\n101\t\n102\tПрежде чем определить множества, необходимо формально доказать, что любое конечное множество может быть представлено как **упорядоченная уникальная последовательность** — последовательность без дубликатов, отсортированная в строго возрастающем порядке.\n103\t\n104\tЭто доказательство адаптировано из [проекта set_sequence_equivalence](https://github.com/konard/subset-sum/tree/main/proofs/set_sequence_equivalence) и включает:\n105\t\n106\t1. **StrictlyAscending** — предикат строго возрастающего списка\n107\t2. **NoDuplicates** — предикат списка без дубликатов\n108\t3. **insertSorted** — вставка элемента в отсортированный список с пропуском дубликатов\n109\t4. **toOrderedUnique** — преобразование произвольного списка в строго возрастающий\n110\t\n111\t**Основная теорема** (`set_sequence_equivalence`): для любого списка натуральных чисел существует строго возрастающий список, содержащий в точности те же элементы.\n112\t\n113\t##### Rocq\n114\t\n115\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/SetSequenceEquivalence.v)\n116\t\n117\t```rocq\n118\t(* ОСНОВНАЯ ТЕОРЕМА: Эквивалентность множества и последовательности *)\n119\tTheorem set_sequence_equivalence : forall l : list nat,\n120\t exists l' : list nat,\n121\t IsOrderedUniqueSequence l' /\\\n122\t (forall x, In x l' <-> In x l).\n123\t```\n124\t\n125\t##### Lean\n126\t\n127\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/SetSequenceEquivalence.lean)\n128\t\n129\t```lean\n130\t/-- Основная теорема: Каждый список может быть преобразован в упорядоченную\n131\t уникальную последовательность с теми же элементами -/\n132\ttheorem set_sequence_equivalence (l : List Nat) :\n133\t ∃ l' : List Nat,\n134\t IsOrderedUniqueSequence l' ∧\n135\t (∀ x, x ∈ l' ↔ x ∈ l)\n136\t```\n137\t\n138\t#### Множества как упорядоченные уникальные последовательности связей\n139\t\n140\tИспользуя доказанную эквивалентность, мы определяем **множество** как **ссылку (Reference) на корень дерева** в сети, листья которого образуют строго возрастающий список. Для создания множества из произвольного списка сначала применяется `toOrderedUnique` (сортировка и удаление дубликатов), затем строится сбалансированное дерево и записывается в сеть.\n141\t\n142\t> $\\displaystyle LinkSet := Reference$\n143\t\n144\t##### Rocq\n145\t\n146\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/SetDefinitions.v)\n147\t\n148\t```rocq\n149\tRequire Import NetworkDefinitions.\n150\tRequire Import NetworkConversions.\n151\tRequire Import SequenceDefinitions.\n152\tRequire Import SetSequenceEquivalence.\n153\t\n154\t(* Множество ссылок — это ссылка на корень дерева в сети *)\n155\tDefinition LinkSet := Reference.\n156\t\n157\t(* Предикат корректности множества (через дерево) *)\n158\tDefinition IsValidSetTree (s : LinkTree) : Prop :=\n159\t IsOrderedUniqueSequence (TreeToList s).\n160\t\n161\t(* Преобразование списка в множество: ссылка на корень + сеть дуплетов *)\n162\tDefinition ListToSet (l : list Reference) (offset : nat)\n163\t : option (LinkSet * NetworkDupletList) :=\n164\t let sorted := toOrderedUnique l in\n165\t match ListToBalancedTree sorted with\n166\t | None => None\n167\t | Some t => Some (TreeToSequence t offset, TreeToDupletList t offset)\n168\t end.\n169\t```\n170\t\n171\t##### Lean\n172\t\n173\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/SetDefinitions.lean)\n174\t\n175\t```lean\n176\timport NetworkDefinitions\n177\timport NetworkConversions\n178\timport SequenceDefinitions\n179\timport SetSequenceEquivalence\n180\t\n181\t-- Множество ссылок — это ссылка на корень дерева в сети\n182\tabbrev LinkSet := Reference\n183\t\n184\t-- Предикат корректности множества (через дерево)\n185\tdef IsValidSetTree (s : LinkTree) : Prop :=\n186\t IsOrderedUniqueSequence (TreeToList s)\n187\t\n188\t-- Преобразование списка в множество: ссылка на корень + сеть дуплетов\n189\tdef ListToSet (l : List Reference) (offset : Nat)\n190\t : Option (LinkSet × NetworkDupletList)\n191\t```\n192\t\n193\t#### Значение результатов\n194\t\n195\tТаким образом, мы формально показали, что:\n196\t\n197\t1. **Последовательности** определены как ссылки на корни деревьев связей-дуплетов ($R \\to R^2$) в сети\n198\t2. **Множества** определены как последовательности с инвариантом упорядоченности и уникальности листьев\n199\t3. Теорема `set_sequence_equivalence` гарантирует, что любое конечное множество натуральных чисел может быть представлено как упорядоченная уникальная последовательность\n200\t\n201\tВсё это означает, что и последовательности, и множества **выражены исключительно через связи-дуплеты** — минимальную структуру теории связей $R \\to R^2$.\n202\t" | |
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| "content": "1\t### Теория связей как мета-теория\n2\t\n3\tВ предыдущем разделе мы определили последовательности и множества в терминах связей-дуплетов. Теперь мы делаем ключевой шаг: **переопределяем начальные термины теории связей** (связи и ссылки) **в терминах этих же последовательностей и множеств**, которые сами определены через связи.\n4\t\n5\tЭто замыкает цикл определений и демонстрирует, что теория связей является **мета-теорией** — теорией, способной определять свои собственные основания через свои же конструкции.\n6\t\n7\t#### Цикл определений\n8\t\n9\tФормально цикл определений выглядит следующим образом:\n10\t\n11\t**Уровень 0 (базовый):**\n12\t\n13\t> $\\displaystyle Reference := \\mathbb{N}_0, \\quad Duplet := Reference \\times Reference, \\quad Network : Reference \\to Duplet$\n14\t\n15\t**Уровень 1 (последовательности через связи):**\n16\t\n17\t> $\\displaystyle LinkSequence := Reference$\n18\t\n19\tПоследовательность — это ссылка на корень дерева связей-дуплетов, хранимого в сети. Каждый внутренний узел — дуплет $(ссылка\\_на\\_левое, ссылка\\_на\\_правое)$, листья — элементы последовательности. Например: $[1, 2, 3, 4] = ((1, 2), (3, 4))$, хранится в математической нотации как $5 \\to (1, 2),\\ 6 \\to (3, 4),\\ 7 \\to (5, 6)$ или в [нотации связей](https://github.com/link-foundation/links-notation): `(5: 1 2) (6: 3 4) (7: 5 6)`.\n20\t\n21\t**Уровень 2 (множества через последовательности):**\n22\t\n23\t> $\\displaystyle LinkSet := Reference\\ |\\ IsOrderedUniqueSequence(листья)$\n24\t\n25\tМножества — это ссылки на деревья связей с инвариантом: листья образуют строго возрастающий список без дубликатов.\n26\t\n27\t**Уровень 3 (мета-определения через множества):**\n28\t\n29\t> $\\displaystyle MetaReference := Reference, \\quad MetaDuplet := MetaReference \\times MetaReference, \\quad MetaNetwork : MetaReference \\to MetaDuplet$\n30\t\n31\tМета-ссылка — это элемент множества, хранимого в сети. Мета-дуплет — пара мета-ссылок. Мета-сеть — функция из мета-ссылок в мета-дуплеты.\n32\t\n33\t**Уровень 3 структурно идентичен Уровню 0**, но определён через конструкции Уровней 1 и 2, которые сами определены через Уровень 0. Цикл замкнут.\n34\t\n35\t#### Все мета-термины как связи\n36\t\n37\tЗамечательное свойство теории связей состоит в том, что все мета-термины могут быть описаны в терминах связи ($L$) — центрального понятия, введённого в самой теории.\n38\t\n39\t**Мета-ссылка как связь.** Поскольку ссылка ($r ∈ R$) — это связь нулевой размерности $r ≡ (r, ())$, мета-ссылка (MetaReference), будучи элементом множества ссылок, также является связью. Более того, MetaReference — это связь произвольной размерности, все ссылки которой указывают на неё саму: это связь, определяющая себя через себя.\n40\t\n41\t**Мета-сеть как мета-последовательность.** Мета-сеть (MetaNetwork) — это функция $MetaReference \\to MetaDuplet$, которая может быть представлена как последовательность мета-дуплетов (MetaNetworkList). Эта последовательность, в свою очередь, является мета-последовательностью (MetaSequence) — деревом связей-дуплетов в сети. А мета-последовательность — это ссылка на корень дерева, то есть снова связь.\n42\t\n43\t**Все мета-термины описываются через связи.** Таким образом:\n44\t- MetaReference — это связь (нулевой размерности, с самоссылкой)\n45\t- MetaDuplet — это связь (двумерная, пара мета-ссылок)\n46\t- MetaNetwork — это связь связей (последовательность связей, которая сама является связью)\n47\t\n48\tВсе термины, необходимые для введения теории связей, описываются как связи — термин, введённый в самой теории связей. Это означает, что теория связей является **полностью самодостаточной**: она определяет свои основания через свои же конструкции.\n49\t\n50\t#### Формализация в Rocq и Lean\n51\t\n52\t##### Rocq\n53\t\n54\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/MetaDefinitions.v)\n55\t\n56\t```rocq\n57\tRequire Import NetworkDefinitions.\n58\tRequire Import SequenceDefinitions.\n59\tRequire Import SetDefinitions.\n60\tRequire Import SetSequenceEquivalence.\n61\t\n62\t(* Мета-ссылка: элемент множества, определённого через связи-дуплеты *)\n63\tDefinition MetaReference := Reference.\n64\t\n65\t(* Мета-пространство ссылок: ссылка на корень дерева в сети *)\n66\tDefinition MetaReferenceSpace := LinkSet.\n67\t\n68\t(* Мета-дуплет: пара мета-ссылок *)\n69\tDefinition MetaDuplet := prod MetaReference MetaReference.\n70\t\n71\t(* Мета-сеть *)\n72\tDefinition MetaNetwork := MetaReference -> MetaDuplet.\n73\t\n74\t(* Мета-сеть в виде последовательности мета-дуплетов *)\n75\tDefinition MetaNetworkList := list MetaDuplet.\n76\t\n77\t(* Создание мета-пространства ссылок в сети *)\n78\tDefinition MakeMetaReferenceSpace (size : nat) (offset : nat)\n79\t : option (MetaReferenceSpace * NetworkDupletList) :=\n80\t ListToSet (makeMetaReferenceSpace_ size) offset.\n81\t\n82\t(* ОСНОВНАЯ ТЕОРЕМА МЕТА-ТЕОРИИ:\n83\t Любая мета-сеть может быть представлена\n84\t как обычная сеть дуплетов *)\n85\tTheorem meta_network_is_duplet_network :\n86\t forall (net : MetaNetworkList),\n87\t MetaNetworkToDupletList net = net.\n88\t\n89\t(* Мета-пространство ссылок содержит упорядоченные уникальные элементы *)\n90\tTheorem meta_reference_space_elements_valid : forall (size : nat),\n91\t IsOrderedUniqueSequence (toOrderedUnique (makeMetaReferenceSpace_ size)).\n92\t```\n93\t\n94\t##### Lean\n95\t\n96\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/MetaDefinitions.lean)\n97\t\n98\t```lean\n99\timport NetworkDefinitions\n100\timport SequenceDefinitions\n101\timport SetDefinitions\n102\timport SetSequenceEquivalence\n103\t\n104\t-- Мета-ссылка: элемент множества, определённого через связи-дуплеты\n105\tabbrev MetaReference := Reference\n106\t\n107\t-- Мета-пространство ссылок: ссылка на корень дерева в сети\n108\tabbrev MetaReferenceSpace := LinkSet\n109\t\n110\t-- Мета-дуплет: пара мета-ссылок\n111\tabbrev MetaDuplet := MetaReference × MetaReference\n112\t\n113\t-- Мета-сеть\n114\tabbrev MetaNetwork := MetaReference → MetaDuplet\n115\t\n116\t-- Мета-сеть в виде последовательности мета-дуплетов\n117\tabbrev MetaNetworkList := List MetaDuplet\n118\t\n119\t-- ОСНОВНАЯ ТЕОРЕМА МЕТА-ТЕОРИИ:\n120\t-- Любая мета-сеть может быть представлена\n121\t-- как обычная сеть дуплетов\n122\ttheorem meta_network_is_duplet_network :\n123\t ∀ (net : MetaNetworkList),\n124\t MetaNetworkToDupletList net = net\n125\t\n126\t-- Мета-пространство ссылок содержит упорядоченные уникальные элементы\n127\ttheorem meta_reference_space_elements_valid (size : Nat) :\n128\t IsOrderedUniqueSequence (toOrderedUnique (makeMetaReferenceSpace_ size))\n129\t```\n130\t\n131\t#### Последствия и перспективы\n132\t\n133\tФормальное замыкание цикла определений имеет глубокие последствия:\n134\t\n135\t1. **Самоописание.** Теория связей может определять свои собственные начальные термины (связи и ссылки) через конструкции, построенные из этих же терминов. Это свойство характерно для мета-теорий.\n136\t\n137\t2. **Определение теории множеств через теорию связей.** Поскольку множества формально определены как деревья связей-дуплетов с инвариантом упорядоченности, теория связей содержит в себе (по крайней мере, конечную) теорию множеств. Это открывает путь к проекции теории множеств *в* теорию связей, а не только *из* неё.\n138\t\n139\t3. **Замыкание цикла с теорией типов.** Теория связей определена в терминах теории множеств (через $R \\subseteq \\mathbb{N}_0$), теория множеств определена через теорию типов (в Rocq/Lean), а теория типов использует множества и последовательности, которые мы определили через связи. Цикл замкнут на трёх уровнях.\n140\t\n141\t4. **Унификация.** Связи-дуплеты ($R \\to R^2$) достаточны для представления последовательностей, множеств и, как следствие, самих связей на мета-уровне. Это подтверждает универсальность формулы:\n142\t\n143\t> $\\displaystyle R \\to R^2$\n144\t\n145\tВ будущих работах мы планируем расширить этот подход, показав полную проекцию теории множеств и теории типов в теорию связей, а также формально доказать биективность преобразований между этими теориями.\n146\t" | |
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| "content": "1\t/-\n2\t MetaDefinitions.lean - Теория связей как мета-теория: переопределение базовых\n3\t терминов теории связей через последовательности и множества, которые сами\n4\t определены в терминах связей.\n5\t\n6\t Этот файл замыкает цикл определений:\n7\t 1. Связи (Reference) и ссылки определены как натуральные числа (ℕ₀)\n8\t 2. Последовательности определены как деревья связей-дуплетов в сети\n9\t 3. Множества определены как упорядоченные уникальные последовательности связей\n10\t 4. Теперь мы переопределяем связи и ссылки через множества и последовательности\n11\t\n12\t Это демонстрирует, что теория связей может определять свои собственные\n13\t начальные термины в терминах структур, построенных из связей,\n14\t что делает её мета-теорией — теорией, определяющей саму себя.\n15\t-/\n16\timport NetworkDefinitions\n17\timport NetworkConversions\n18\timport SequenceDefinitions\n19\timport SetDefinitions\n20\timport SetSequenceEquivalence\n21\t\n22\topen SetSequenceEquivalence\n23\t\n24\t-- * Мета-определения: связи и ссылки через последовательности и множества\n25\t\n26\t/-\n27\t Мета-ссылка (MetaReference) — это элемент множества ссылок,\n28\t определённого как упорядоченное уникальное дерево связей-дуплетов\n29\t в сети.\n30\t\n31\t В исходном определении: Reference := ℕ₀ (натуральное число)\n32\t В мета-определении: MetaReference — это лист дерева, хранимого в сети,\n33\t где LinkSet — это Reference (ссылка на корень дерева в сети).\n34\t\n35\t Таким образом, мета-ссылка определена через связи, которые определены\n36\t через мета-ссылки — цикл замкнут.\n37\t-/\n38\t\n39\t-- Мета-ссылка: элемент множества, определённого через деревья связей\n40\tabbrev MetaReference := Reference\n41\t\n42\t-- Мета-пространство ссылок: ссылка на корень дерева в сети\n43\tabbrev MetaReferenceSpace := LinkSet\n44\t\n45\t-- Мета-дуплет: пара мета-ссылок — это связь-дуплет\n46\tabbrev MetaDuplet := MetaReference × MetaReference\n47\t\n48\t-- Мета-сеть: функция из мета-ссылок в мета-дуплеты\n49\tabbrev MetaNetwork := MetaReference → MetaDuplet\n50\t\n51\t-- Мета-сеть в виде последовательности мета-дуплетов\n52\tabbrev MetaNetworkList := List MetaDuplet\n53\t\n54\t-- Создание списка мета-ссылок заданного размера\n55\tdef makeMetaReferenceSpace_ : Nat → List Reference\n56\t | 0 => [0]\n57\t | n + 1 => makeMetaReferenceSpace_ n ++ [n + 1]\n58\t\n59\t-- Создание мета-пространства ссылок — множество в сети\n60\tdef MakeMetaReferenceSpace (size : Nat) (offset : Nat)\n61\t : Option (MetaReferenceSpace × NetworkDupletList) :=\n62\t ListToSet (makeMetaReferenceSpace_ size) offset\n63\t\n64\t-- * Определение мета-сети через последовательности\n65\t\n66\t-- Преобразование мета-сети в сеть дуплетов\n67\tdef MetaNetworkToDupletList (net : MetaNetworkList) : NetworkDupletList :=\n68\t net\n69\t\n70\t/-\n71\t ОСНОВНАЯ ТЕОРЕМА МЕТА-ТЕОРИИ:\n72\t\n73\t Любая мета-сеть (определённая через последовательности и множества,\n74\t которые определены через связи-дуплеты) может быть представлена как обычная\n75\t сеть дуплетов.\n76\t\n77\t Это формально показывает, что мета-определения совместимы с исходными:\n78\t теория связей может определять свои термины через себя без противоречий.\n79\t-/\n80\ttheorem meta_network_is_duplet_network :\n81\t ∀ (net : MetaNetworkList),\n82\t MetaNetworkToDupletList net = net := by\n83\t intro net\n84\t rfl\n85\t\n86\t-- Мета-пространство ссылок содержит упорядоченные уникальные элементы\n87\ttheorem meta_reference_space_elements_valid (size : Nat) :\n88\t IsOrderedUniqueSequence (toOrderedUnique (makeMetaReferenceSpace_ size)) := by\n89\t exact toOrderedUnique_is_ascending (makeMetaReferenceSpace_ size)\n90\t\n91\t-- * Демонстрация цикла определений\n92\t\n93\t/-\n94\t Цикл определений теории связей как мета-теории:\n95\t\n96\t Уровень 0 (базовый):\n97\t Reference := ℕ₀\n98\t Duplet := Reference × Reference\n99\t Network := Reference → Duplet\n100\t\n101\t Уровень 1 (последовательности через связи):\n102\t LinkSequence := Reference (ссылка на корень дерева дуплетов в сети)\n103\t Дерево дуплетов хранится в NetworkDupletList\n104\t\n105\t Уровень 2 (множества через последовательности):\n106\t LinkSet := Reference (ссылка на корень, с инвариантом IsOrderedUniqueSequence)\n107\t\n108\t Уровень 3 (мета-определения через множества):\n109\t MetaReference := Reference\n110\t MetaDuplet := MetaReference × MetaReference\n111\t MetaNetwork := MetaReference → MetaDuplet\n112\t\n113\t Уровень 3 структурно идентичен Уровню 0,\n114\t но определён через конструкции Уровней 1 и 2,\n115\t которые сами определены через конструкции Уровня 0.\n116\t\n117\t Это замыкает цикл: теория связей определяет себя через себя.\n118\t\n119\t Все мета-термины описываются через связи:\n120\t - Ссылка (Reference) — это связь нулевой размерности: r ≡ (r, ()),\n121\t поскольку при n = 0 кортеж T₀ = R⁰ пуст, и L = R × T₀ ≡ R.\n122\t - MetaReference — это связь произвольной размерности с самоссылкой.\n123\t - MetaDuplet — это двумерная связь (пара мета-ссылок).\n124\t - MetaNetwork — это связь связей (последовательность, которая сама является связью).\n125\t\n126\t Таким образом, все термины теории связей описываются как связи —\n127\t термин, введённый в самой теории. Теория полностью самодостаточна.\n128\t-/\n129\t\n130\t-- Пример: создание мета-пространства из 5 ссылок\n131\t#eval MakeMetaReferenceSpace 4 10\n132\t-- Множество {0, 1, 2, 3, 4} в сети\n133\t\n134\t-- Пример: создание мета-сети\n135\tdef exampleMetaNetwork : MetaNetworkList :=\n136\t [(1, 2), (2, 3), (3, 1)]\n137\t\n138\t-- Преобразование мета-сети в дуплеты\n139\t#eval MetaNetworkToDupletList exampleMetaNetwork\n140\t-- Ожидается: [(1, 2), (2, 3), (3, 1)] — структурно идентично исходной сети\n141\t\n142\t-- * Проверки верификации\n143\t\n144\t#check @meta_network_is_duplet_network\n145\t#check @meta_reference_space_elements_valid\n146\t" | |
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| "content": "1\t/-\n2\t SequenceDefinitions.lean - Определение последовательностей в терминах связей.\n3\t\n4\t Последовательность — это вложенная структура связей-дуплетов,\n5\t хранимая в сети. Последовательность идентифицируется\n6\t ссылкой (Reference) на корень дерева дуплетов.\n7\t\n8\t Каждый внутренний узел дерева — это связь-дуплет (ссылка_на_левое, ссылка_на_правое).\n9\t Листья — это элементы последовательности (ссылки).\n10\t\n11\t Пример: [1, 2, 3, 4] = ((1, 2), (3, 4))\n12\t Хранится в сети как:\n13\t (5: 1, 2) — дуплет для пары (1, 2)\n14\t (6: 3, 4) — дуплет для пары (3, 4)\n15\t (7: 5, 6) — дуплет-корень, ссылающийся на поддеревья\n16\t Последовательность = ссылка 7\n17\t\n18\t В нотации связей (https://github.com/link-foundation/links-notation):\n19\t (5: 1 2) (6: 3 4) (7: 5 6)\n20\t\n21\t Другие варианты:\n22\t [1, 2, 3, 4] = (((1, 2), 3), 4) — левая лестница\n23\t [1, 2, 3, 4] = (1, (2, (3, 4))) — правая лестница\n24\t\n25\t Адаптировано из: https://github.com/linksplatform/Documentation/tree/main/doc/LinksTheoryManuscriptDraft/05%20Sequences\n26\t См. также: https://github.com/linksplatform/Data.Doublets.Sequences/blob/main/csharp/Platform.Data.Doublets.Sequences/Converters/BalancedVariantConverter.cs\n27\t-/\n28\timport NetworkDefinitions\n29\timport NetworkConversions\n30\t\n31\t-- * Последовательность — это ссылка на корень дерева в сети\n32\t\n33\t/-- Последовательность идентифицируется ссылкой (Reference) на корень дерева дуплетов,\n34\t хранимого в сети. Это не отдельный тип — это просто Reference. -/\n35\tabbrev LinkSequence := Reference\n36\t\n37\t-- * Вспомогательное дерево для построения последовательностей\n38\t\n39\t/-- LinkTree — промежуточная структура для алгоритмов построения деревьев.\n40\t Конечный результат всегда записывается в сеть дуплетов. -/\n41\tinductive LinkTree where\n42\t | Leaf : Reference → LinkTree\n43\t | Node : LinkTree → LinkTree → LinkTree\n44\t deriving Repr\n45\t\n46\t-- * Запись дерева в сеть дуплетов\n47\t\n48\t/-- Запись дерева в сеть дуплетов.\n49\t Каждый узел Node записывается как дуплет (ссылка_на_левое, ссылка_на_правое).\n50\t Листья — это элементы последовательности (не записываются как отдельные дуплеты).\n51\t Возвращает пару: (сеть дуплетов, следующее свободное смещение). -/\n52\tdef TreeToDupletList_ : LinkTree → Nat → NetworkDupletList × Nat\n53\t | .Leaf _, offset => ([], offset)\n54\t | .Node l r, offset =>\n55\t let left_root := match l with | .Leaf x => x | .Node _ _ => offset + 1\n56\t let (left_net, left_next) := TreeToDupletList_ l (offset + 1)\n57\t let right_root := match r with | .Leaf x => x | .Node _ _ => left_next\n58\t let (right_net, right_next) := TreeToDupletList_ r left_next\n59\t ((left_root, right_root) :: left_net ++ right_net, right_next)\n60\t\n61\tdef TreeToDupletList (t : LinkTree) (offset : Nat) : NetworkDupletList :=\n62\t (TreeToDupletList_ t offset).1\n63\t\n64\t/-- Получение ссылки на корень дерева (= последовательность) -/\n65\tdef TreeToSequence (t : LinkTree) (offset : Nat) : LinkSequence :=\n66\t match t with\n67\t | .Leaf x => x\n68\t | .Node _ _ => offset\n69\t\n70\t-- Получение листьев дерева (элементов последовательности) слева направо\n71\tdef TreeToList : LinkTree → List Reference\n72\t | .Leaf x => [x]\n73\t | .Node l r => TreeToList l ++ TreeToList r\n74\t\n75\t-- * Алгоритмы создания последовательностей\n76\t\n77\t/-- Правая лестница: (1, (2, (3, 4)))\n78\t Каждый следующий элемент вкладывается вправо. -/\n79\tdef ListToRightStaircase : List Reference → Option LinkTree\n80\t | [] => none\n81\t | [x] => some (.Leaf x)\n82\t | x :: rest =>\n83\t match ListToRightStaircase rest with\n84\t | none => none\n85\t | some t => some (.Node (.Leaf x) t)\n86\t\n87\t/-- Левая лестница: (((1, 2), 3), 4)\n88\t Каждый следующий элемент вкладывается влево. -/\n89\tdef ListToLeftStaircase_ : LinkTree → List Reference → LinkTree\n90\t | acc, [] => acc\n91\t | acc, x :: rest => ListToLeftStaircase_ (.Node acc (.Leaf x)) rest\n92\t\n93\tdef ListToLeftStaircase : List Reference → Option LinkTree\n94\t | [] => none\n95\t | [x] => some (.Leaf x)\n96\t | x :: rest => some (ListToLeftStaircase_ (.Leaf x) rest)\n97\t\n98\t/-- Сбалансированный вариант: ((1, 2), (3, 4))\n99\t Список делится пополам, каждая половина рекурсивно превращается в дерево.\n100\t Адаптировано из BalancedVariantConverter.\n101\t Использует fuel-параметр для гарантии завершения. -/\n102\tdef ListToBalancedTree (l : List Reference) (fuel : Nat := l.length + 1) : Option LinkTree :=\n103\t match fuel with\n104\t | 0 => none\n105\t | fuel' + 1 =>\n106\t match l with\n107\t | [] => none\n108\t | [x] => some (.Leaf x)\n109\t | _ =>\n110\t let mid := l.length / 2\n111\t let left_part := l.take mid\n112\t let right_part := l.drop mid\n113\t match ListToBalancedTree left_part fuel', ListToBalancedTree right_part fuel' with\n114\t | some lt, some rt => some (.Node lt rt)\n115\t | _, _ => none\n116\t\n117\t-- * Полное преобразование: список → сеть дуплетов + ссылка на корень\n118\t\n119\t/-- Преобразование списка в последовательность (ссылку на корень) и сеть -/\n120\tdef ListToSequence (l : List Reference) (offset : Nat)\n121\t : Option (LinkSequence × NetworkDupletList) :=\n122\t match ListToBalancedTree l with\n123\t | none => none\n124\t | some t => some (TreeToSequence t offset, TreeToDupletList t offset)\n125\t\n126\t/-- Получение элемента списка по индексу с значением по умолчанию -/\n127\tdef listGetD (l : List α) (n : Nat) (default : α) : α :=\n128\t match l, n with\n129\t | [], _ => default\n130\t | x :: _, 0 => x\n131\t | _ :: rest, n + 1 => listGetD rest n default\n132\t\n133\t/-- Чтение последовательности из сети дуплетов -/\n134\tdef ReadSequence_ (anet : NetworkDupletList) (root : Reference) (fuel : Nat)\n135\t : List Reference :=\n136\t match fuel with\n137\t | 0 => []\n138\t | fuel' + 1 =>\n139\t if root < anet.length then\n140\t let d := listGetD anet root (0, 0)\n141\t ReadSequence_ anet d.1 fuel' ++ ReadSequence_ anet d.2 fuel'\n142\t else\n143\t [root]\n144\t\n145\tdef ReadSequence (anet : NetworkDupletList) (root : LinkSequence) : List Reference :=\n146\t ReadSequence_ anet root (anet.length + 1)\n147\t\n148\t-- * Примеры\n149\t\n150\t-- Пример: [1, 2, 3, 4] — сбалансированное дерево, записанное в сеть\n151\t#eval ListToSequence [1, 2, 3, 4] 5\n152\t-- Ожидается: some (5, [(6, 7), (1, 2), (3, 4)])\n153\t-- Последовательность = ссылка 5\n154\t-- (5: 6, 7) — корень, указывает на поддеревья 6 и 7\n155\t-- (6: 1, 2) — левое поддерево\n156\t-- (7: 3, 4) — правое поддерево\n157\t\n158\t-- Пример: правая лестница\n159\t#eval do let t ← ListToRightStaircase [1, 2, 3, 4]\n160\t return (TreeToSequence t 5, TreeToDupletList t 5)\n161\t\n162\t-- Пример: левая лестница\n163\t#eval do let t ← ListToLeftStaircase [1, 2, 3, 4]\n164\t return (TreeToSequence t 5, TreeToDupletList t 5)\n165\t\n166\t-- Все три варианта содержат одни и те же элементы\n167\t#eval do let t ← ListToBalancedTree [1, 2, 3, 4]; return TreeToList t -- [1, 2, 3, 4]\n168\t#eval do let t ← ListToRightStaircase [1, 2, 3, 4]; return TreeToList t -- [1, 2, 3, 4]\n169\t#eval do let t ← ListToLeftStaircase [1, 2, 3, 4]; return TreeToList t -- [1, 2, 3, 4]\n170\t\n171\t-- Пустая последовательность\n172\t#eval ListToSequence ([] : List Reference) 5\n173\t-- Ожидается: none\n174\t\n175\t-- Последовательность из одного элемента\n176\t#eval ListToSequence [42] 5\n177\t-- Ожидается: some (42, []) — лист не создаёт дуплетов, сама 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| "content": "1\t/-\n2\t SetDefinitions.lean - Определение множеств в терминах последовательностей связей.\n3\t\n4\t Множество определяется как последовательность связей-дуплетов,\n5\t листья которой образуют строго возрастающий список без дубликатов.\n6\t\n7\t Как и последовательность, множество идентифицируется ссылкой (Reference)\n8\t на корень дерева в сети.\n9\t\n10\t Это ключевой шаг: имея последовательности, определённые через связи-дуплеты,\n11\t мы теперь определяем множества как особый вид последовательностей.\n12\t Таким образом, множества также выражены исключительно через связи.\n13\t-/\n14\timport NetworkDefinitions\n15\timport NetworkConversions\n16\timport SequenceDefinitions\n17\timport SetSequenceEquivalence\n18\t\n19\topen SetSequenceEquivalence\n20\t\n21\t-- * Множества как упорядоченные уникальные последовательности связей\n22\t\n23\t/-- Множество ссылок — это ссылка (Reference) на корень дерева связей\n24\t в сети, листья которого образуют строго возрастающий\n25\t список без дубликатов. Как и последовательность, множество — это Reference. -/\n26\tabbrev LinkSet := Reference\n27\t\n28\t-- Предикат: является ли дерево связей корректным множеством\n29\tdef IsValidSetTree (s : LinkTree) : Prop :=\n30\t IsOrderedUniqueSequence (TreeToList s)\n31\t\n32\t-- Множество из одного элемента — ссылка на сам элемент\n33\tdef SingletonSet (value : Reference) : LinkSet := value\n34\t\n35\t/-- Преобразование списка ссылок в множество: сортировка, удаление дубликатов,\n36\t построение сбалансированного дерева и запись в сеть.\n37\t Возвращает ссылку на корень (= множество) и сеть дуплетов. -/\n38\tdef ListToSet (l : List Reference) (offset : Nat)\n39\t : Option (LinkSet × NetworkDupletList) :=\n40\t let sorted := toOrderedUnique l\n41\t match ListToBalancedTree sorted with\n42\t | none => none\n43\t | some t => some (TreeToSequence t offset, TreeToDupletList t offset)\n44\t\n45\t-- Принадлежность элемента множеству (проверка через дерево)\n46\tdef InSetTree (x : Reference) (s : LinkTree) : Prop :=\n47\t x ∈ TreeToList s\n48\t\n49\t-- Подмножество: s1 ⊆ s2 (через деревья)\n50\tdef IsSubsetTree (s1 s2 : LinkTree) : Prop :=\n51\t ∀ x, InSetTree x s1 → InSetTree x s2\n52\t\n53\t-- * Теорема: toOrderedUnique всегда содержит упорядоченные уникальные элементы\n54\t\n55\ttheorem ListToSet_elements_valid (l : List Reference) :\n56\t IsOrderedUniqueSequence (toOrderedUnique l) := by\n57\t exact toOrderedUnique_is_ascending l\n58\t\n59\t-- Элемент принадлежит множеству тогда и только тогда, когда он принадлежит исходному списку\n60\ttheorem ListToSet_membership (x : Reference) (l : List Reference) :\n61\t x ∈ toOrderedUnique l ↔ x ∈ l := by\n62\t exact mem_toOrderedUnique x l\n63\t\n64\t-- * Примеры\n65\t\n66\t-- Создание множества из списка с дубликатами\n67\t#eval ListToSet [3, 1, 4, 1, 5, 9, 2, 6, 5, 3] 10\n68\t-- Множество {1, 2, 3, 4, 5, 6, 9} в сети\n69\t\n70\t-- Множество из одного элемента\n71\t#eval SingletonSet 42\n72\t" | |
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| "content": "1\t/-\n2\t NetworkDefinitions.lean\n3\t\n4\t Определения основных типов для сетей.\n5\t Lean 4 перевод NetworkDefinitions.v (Rocq).\n6\t-/\n7\t\n8\t-- Ссылка (Reference) — уникальный идентификатор кортежа: R ⊆ ℕ₀\n9\t-- Ссылка также является связью нулевой размерности: при n = 0 связь L = R × T₀ ≡ R,\n10\t-- поэтому r ≡ (r, ()), и множество ссылок R является подмножеством множества связей L.\n11\tabbrev Reference := Nat\n12\t\n13\t-- Значение Reference по умолчанию: ноль\n14\tdef ReferenceDefault : Reference := 0\n15\t\n16\t-- Множество кортежей ссылок длины n ∈ ℕ₀: TupleOfReferences ⊆ Rⁿ\n17\tabbrev TupleOfReferences (n : Nat) := Vector Reference n\n18\t\n19\t-- Значение TupleOfReferences по умолчанию\n20\tdef TupleOfReferencesDefault (n : Nat) : TupleOfReferences n := Vector.replicate n ReferenceDefault\n21\t\n22\t-- Список ссылок: ReferenceList ⊆ List(R)\n23\tabbrev ReferenceList := List Reference\n24\t\n25\t-- Сеть списков ссылок: N^{list} : Reference → ReferenceList\n26\tabbrev NetworkReferenceListFunction := Reference → ReferenceList\n27\t\n28\t-- Сеть списков ссылок в виде последовательности списков ссылок\n29\tabbrev NetworkReferenceListList := List ReferenceList\n30\t\n31\t-- Множество всех связей: Link = Reference × TupleOfReferences\n32\tabbrev Link (n : Nat) := Reference × TupleOfReferences n\n33\t\n34\t-- Сеть кортежей длины n (или n-мерная сеть) из семейства функций {Nⁿ : Reference → TupleOfReferences}\n35\tabbrev NetworkTupleFunction (n : Nat) := Reference → TupleOfReferences n\n36\t\n37\t-- Сеть кортежей длины n (или n-мерная сеть) в виде последовательности\n38\tabbrev NetworkTupleList (n : Nat) := List (TupleOfReferences n)\n39\t\n40\t-- Дуплет ссылок: упорядоченная пара (Reference, Reference).\n41\t-- В нотации связей (https://github.com/link-foundation/links-notation):\n42\t-- дуплет записывается как (id: from to), например (3: 1 2)\n43\tabbrev Duplet := Reference × Reference\n44\t\n45\t-- Значение Duplet по умолчанию: пара из двух ReferenceDefault, используется для обозначения пустого дуплета\n46\tdef DupletDefault : Duplet := (ReferenceDefault, ReferenceDefault)\n47\t\n48\t-- Сеть дуплетов (или двумерная сеть): N² : Reference → Reference²\n49\tabbrev NetworkDupletFunction := Reference → Duplet\n50\t\n51\t-- Сеть дуплетов (или двумерная сеть) в виде последовательности дуплетов\n52\tabbrev NetworkDupletList := List Duplet\n53\t\n54\t-- * Сеть всех сетей\n55\t\n56\t/-\n57\t Сеть всех сетей (Ω) — пространство, в котором каждая возможная сеть\n58\t произвольной размерности и размера получает уникальную ссылку.\n59\t\n60\t 𝒩 = ⋃_{n ∈ ℕ₀} ⋃_{s ∈ ℕ₀} {Nⁿ размера s}\n61\t Ω: R → 𝒩\n62\t\n63\t Поскольку в Lean мы не можем напрямую определить тип, параметризованный\n64\t произвольным n на уровне значений, мы представляем элемент 𝒩 как пару\n65\t (n, список кортежей переменной длины), используя списки ссылок\n66\t для представления кортежей произвольной размерности.\n67\t-/\n68\t\n69\t-- Элемент сети произвольной размерности: пара (размерность, список кортежей)\n70\t-- Кортеж переменной длины представлен как ReferenceList\n71\tstructure AnyNetwork where\n72\t dimension : Nat\n73\t tuples : List ReferenceList\n74\t deriving Repr, BEq\n75\t\n76\t-- Сеть всех сетей: отображение ссылок на сети произвольной размерности и размера\n77\t-- Ω: R → 𝒩\n78\tabbrev NetworkOfAllNetworks := Reference → AnyNetwork\n79\t\n80\t-- Сеть всех сетей в виде последовательности (для конечного фрагмента)\n81\tabbrev NetworkOfAllNetworksList := List AnyNetwork\n82\t\n83\t-- Пустая сеть нулевой размерности и размера — наименьший элемент 𝒩\n84\tdef emptyNetwork : AnyNetwork := ⟨0, []⟩\n85\t\n86\t-- Размер сети (количество кортежей)\n87\tdef AnyNetwork.size (net : AnyNetwork) : Nat := net.tuples.length\n88\t\n89\t-- Проверка корректности: все кортежи имеют длину, равную размерности\n90\tdef AnyNetwork.isWellFormed (net : AnyNetwork) : Bool :=\n91\t net.tuples.all (fun t => t.length == net.dimension)\n92\t\n93\t-- Пример: представление дуплетной сети N² размера 3 как элемента 𝒩\n94\tdef exampleAnyNetwork : AnyNetwork :=\n95\t ⟨2, [[1, 1], [2, 2], [1, 2]]⟩\n96\t" | |
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| "content": "1\t/-\n2\t NetworkConversions.lean\n3\t\n4\t Функции преобразования между различными представлениями сетей.\n5\t Lean 4 перевод NetworkConversions.v (Rocq).\n6\t-/\n7\timport NetworkDefinitions\n8\t\n9\t-- Функция преобразования TupleOfReferences в ReferenceList\n10\tdef TupleOfReferencesToReferenceList {n : Nat} (v : TupleOfReferences n) : ReferenceList :=\n11\t v.toList\n12\t\n13\t-- Функция преобразования NetworkTupleFunction в NetworkReferenceListFunction\n14\tdef TupleFunctionToReferenceListFunction {n : Nat} (a : NetworkTupleFunction n) : NetworkReferenceListFunction :=\n15\t fun id => TupleOfReferencesToReferenceList (a id)\n16\t\n17\t-- Функция преобразования NetworkTupleList в NetworkReferenceListList\n18\tdef TupleListToReferenceListList {n : Nat} (net : NetworkTupleList n) : NetworkReferenceListList :=\n19\t net.map TupleOfReferencesToReferenceList\n20\t\n21\t-- Функция преобразования ReferenceList в TupleOfReferences, возвращающая option\n22\tdef ReferenceListToTupleOfReferencesOption (n : Nat) (p : ReferenceList) : Option (TupleOfReferences n) :=\n23\t let arr := p.toArray\n24\t if h : arr.size = n then\n25\t some ⟨arr, h⟩\n26\t else\n27\t none\n28\t\n29\t-- Функция преобразования ReferenceList в TupleOfReferences с использованием TupleOfReferencesDefault\n30\tdef ReferenceListToTupleOfReferences (n : Nat) (p : ReferenceList) : TupleOfReferences n :=\n31\t match ReferenceListToTupleOfReferencesOption n p with\n32\t | none => TupleOfReferencesDefault n\n33\t | some t => t\n34\t\n35\t-- Функция преобразования NetworkReferenceListFunction в NetworkTupleFunction\n36\tdef ReferenceListFunctionToTupleFunction {n : Nat} (net : NetworkReferenceListFunction) : NetworkTupleFunction n :=\n37\t fun id => match ReferenceListToTupleOfReferencesOption n (net id) with\n38\t | some t => t\n39\t | none => TupleOfReferencesDefault n\n40\t\n41\t-- Функция преобразования NetworkReferenceListList в NetworkTupleList\n42\tdef ReferenceListListToTupleList {n : Nat} (net : NetworkReferenceListList) : NetworkTupleList n :=\n43\t net.map (ReferenceListToTupleOfReferences n)\n44\t\n45\t-- Функция преобразования ReferenceList в NetworkDupletList со смещением индексации\n46\tdef ReferenceListToDupletList_ (offset : Nat) : ReferenceList → NetworkDupletList\n47\t | [] => []\n48\t | [h] => [(h, offset)]\n49\t | h :: t => (h, offset + 1) :: ReferenceListToDupletList_ (offset + 1) t\n50\t\n51\t-- Функция преобразования ReferenceList в NetworkDupletList\n52\tdef ReferenceListToDupletList (np : ReferenceList) : NetworkDupletList :=\n53\t ReferenceListToDupletList_ 0 np\n54\t\n55\t-- Функция добавления ReferenceList в хвост NetworkDupletList\n56\tdef AddReferenceListToDupletList (anet : NetworkDupletList) (np : ReferenceList) : NetworkDupletList :=\n57\t anet ++ ReferenceListToDupletList_ anet.length np\n58\t\n59\t-- Функция отрезает голову N² и возвращает хвост начиная с offset\n60\tdef DupletListBehead (anet : NetworkDupletList) : Nat → NetworkDupletList\n61\t | 0 => anet\n62\t | Nat.succ n => match anet with\n63\t | [] => []\n64\t | _ :: t => DupletListBehead t n\n65\t\n66\t-- Функция преобразования NetworkDupletList в ReferenceList с индексацией в начале NetworkDupletList начиная с offset\n67\tdef DupletListToReferenceList_ (anet : NetworkDupletList) (offset : Nat) (index : Nat) : ReferenceList :=\n68\t match anet with\n69\t | [] => []\n70\t | (x, next_index) :: tail_anet =>\n71\t if offset == index then\n72\t x :: DupletListToReferenceList_ tail_anet (offset + 1) next_index\n73\t else\n74\t DupletListToReferenceList_ tail_anet (offset + 1) index\n75\t\n76\t-- Функция чтения ReferenceList из NetworkDupletList по индексу дуплета\n77\tdef DupletListReadReferenceList (anet : NetworkDupletList) (index : Nat) : ReferenceList :=\n78\t DupletListToReferenceList_ anet 0 index\n79\t\n80\t-- Функция преобразования NetworkDupletList в ReferenceList начиная с головы списка сети\n81\tdef DupletListToReferenceList (anet : NetworkDupletList) : ReferenceList :=\n82\t DupletListReadReferenceList anet 0\n83\t\n84\t/-\n85\t Теперь всё готово для преобразования сети вложенных упорядоченных пар N^{list} : Reference → ReferenceList\n86\t в сеть дуплетов N² : Reference → Reference².\n87\t\n88\t Данное преобразование можно делать по-разному: с сохранением исходных ссылок на кортежи\n89\t либо с переиндексацией. Переиндексацию можно не делать, если написать дополнительную функцию для\n90\t сети дуплетов, которая возвращает вложенную упорядоченную пару по её ссылке.\n91\t-/\n92\t\n93\t-- Функция добавления NetworkReferenceListList в NetworkDupletList\n94\tdef AddReferenceListListToDupletList : NetworkDupletList → NetworkReferenceListList → NetworkDupletList\n95\t | anetd, [] => anetd\n96\t | anetd, h :: t => AddReferenceListListToDupletList (AddReferenceListToDupletList anetd h) t\n97\t\n98\t-- Функция преобразования NetworkReferenceListList в NetworkDupletList\n99\tdef ReferenceListListToDupletList : NetworkReferenceListList → NetworkDupletList\n100\t | [] => []\n101\t | h :: t => AddReferenceListListToDupletList (ReferenceListToDupletList h) t\n102\t\n103\t-- Функция поиска ReferenceList в хвосте NetworkDupletList начинающемуся с offset по её порядковому номеру.\n104\t-- Возвращает offset ReferenceList.\n105\tdef DupletListOffsetReferenceList_ (anet : NetworkDupletList) (offset : Nat) (index : Nat) : Nat :=\n106\t match anet with\n107\t | [] => offset + anet.length\n108\t | (_, next_index) :: tail_anet =>\n109\t match index with\n110\t | 0 => offset\n111\t | Nat.succ index' =>\n112\t if offset == next_index then\n113\t DupletListOffsetReferenceList_ tail_anet (offset + 1) index'\n114\t else\n115\t DupletListOffsetReferenceList_ tail_anet (offset + 1) index\n116\t\n117\t-- Функция поиска ReferenceList в NetworkDupletList по её порядковому номеру. Возвращает offset ReferenceList.\n118\tdef DupletListOffsetReferenceList (anet : NetworkDupletList) (index : Nat) : Nat :=\n119\t DupletListOffsetReferenceList_ anet 0 index\n120\t\n121\t-- Функция преобразования NetworkTupleList в NetworkDupletList\n122\tdef TupleListToDupletList {n : Nat} (anetv : NetworkTupleList n) : NetworkDupletList :=\n123\t ReferenceListListToDupletList (TupleListToReferenceListList anetv)\n124\t\n125\t/-\n126\t Теперь всё готово для преобразования сети дуплетов N² : Reference → Reference²\n127\t в сеть вложенных упорядоченных пар N^{list} : Reference → ReferenceList.\n128\t\n129\t Данное преобразование будем делать с сохранением исходных ссылок на кортежи.\n130\t Переиндексацию можно не делать, потому что есть функция DupletListOffsetReferenceList для\n131\t сети дуплетов, которая возвращает смещение вложенной УП по ссылке на неё.\n132\t-/\n133\t\n134\t-- Функция отрезает первую ReferenceList из NetworkDupletList и возвращает хвост\n135\tdef DupletListBeheadReferenceList (anet : NetworkDupletList) (offset : Nat) : NetworkDupletList :=\n136\t match anet with\n137\t | [] => []\n138\t | (_, next_index) :: tail_anet =>\n139\t if offset == next_index then\n140\t tail_anet\n141\t else\n142\t DupletListBeheadReferenceList tail_anet (offset + 1)\n143\t\n144\t-- Функция преобразования ReferenceList и NetworkDupletList со смещения offset в NetworkReferenceListList\n145\tdef DupletListToReferenceListList_ (anetd : NetworkDupletList) (np : ReferenceList) (offset : Nat) : NetworkReferenceListList :=\n146\t match anetd with\n147\t | [] => [] -- отбрасываем ReferenceList даже если она недостроена\n148\t | (x, next_index) :: tail_anet =>\n149\t if offset == next_index then -- конец ReferenceList, переходим к следующей ReferenceList\n150\t (np ++ [x]) :: DupletListToReferenceListList_ tail_anet [] (offset + 1)\n151\t else -- ещё не конец ReferenceList, парсим сеть дуплетов дальше\n152\t DupletListToReferenceListList_ tail_anet (np ++ [x]) (offset + 1)\n153\t\n154\t-- Функция преобразования NetworkDupletList в NetworkReferenceListList\n155\tdef DupletListToReferenceListList (anetd : NetworkDupletList) : NetworkReferenceListList :=\n156\t DupletListToReferenceListList_ anetd [] ReferenceDefault\n157\t" | |
| } | |
| ] | |
| }, | |
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| "uuid": "69a7d9eb-1055-4844-b14f-6400cfc80c96", | |
| "timestamp": "2026-04-01T20:47:09.960Z" | |
| } | |
| [2026-04-01T20:47:09.985Z] [INFO] [stdout-guard] [log_1fbf45] sending request { | |
| [stdout-guard] method: 'post', | |
| [stdout-guard] url: 'https://api.anthropic.com/v1/messages?beta=true', | |
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| [stdout-guard] messages: [Array], | |
| [2026-04-01T20:47:09.986Z] [INFO] [stdout-guard] system: [Array], | |
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| [stdout-guard] 'anthropic-dangerous-direct-browser-access': 'true', | |
| [stdout-guard] 'anthropic-version': '2023-06-01', | |
| [stdout-guard] authorization: '***', | |
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| [2026-04-01T20:47:10.329Z] [INFO] [stdout-guard] [log_857271, request-id: "req_011CZdhxZLp14qjrZMPoF94X"] post https://api.anthropic.com/v1/messages?beta=true succeeded with status 200 in 942ms | |
| [2026-04-01T20:47:10.329Z] [INFO] [stdout-guard] [log_857271] response start { | |
| [stdout-guard] url: 'https://api.anthropic.com/v1/messages?beta=true', | |
| [stdout-guard] status: 200, | |
| [stdout-guard] headers: { | |
| [stdout-guard] 'anthropic-organization-id': '684cb0ba-85b8-4b30-a841-e92fb92cc97f', | |
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| [stdout-guard] 'anthropic-ratelimit-unified-7d-surpassed-threshold': '0.75', | |
| [stdout-guard] 'anthropic-ratelimit-unified-7d-utilization': '0.97', | |
| [stdout-guard] 'anthropic-ratelimit-unified-fallback-percentage': '0.5', | |
| [stdout-guard] 'anthropic-ratelimit-unified-overage-disabled-reason': 'org_level_disabled', | |
| [stdout-guard] 'anthropic-ratelimit-unified-overage-status': 'rejected', | |
| [2026-04-01T20:47:10.330Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-representative-claim': 'seven_day', | |
| [stdout-guard] 'anthropic-ratelimit-unified-reset': '1775188800', | |
| [2026-04-01T20:47:10.330Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-status': 'allowed_warning', | |
| [2026-04-01T20:47:10.331Z] [INFO] [stdout-guard] 'cache-control': 'no-cache', | |
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| [2026-04-01T20:47:10.331Z] [INFO] [stdout-guard] connection: 'keep-alive', | |
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| [stdout-guard] 'content-security-policy': "default-src 'none'; frame-ancestors 'none'", | |
| [stdout-guard] 'content-type': 'text/event-stream; charset=utf-8', | |
| [stdout-guard] date: 'Wed, 01 Apr 2026 20:47:10 GMT', | |
| [stdout-guard] 'request-id': 'req_011CZdhxZLp14qjrZMPoF94X', | |
| [stdout-guard] server: 'cloudflare', | |
| [2026-04-01T20:47:10.331Z] [INFO] [stdout-guard] 'server-timing': 'x-originResponse;dur=815', | |
| [2026-04-01T20:47:10.332Z] [INFO] [stdout-guard] 'set-cookie': '***', | |
| [2026-04-01T20:47:10.332Z] [INFO] [stdout-guard] 'strict-transport-security': 'max-age=31536000; includeSubDomains; preload', | |
| [2026-04-01T20:47:10.332Z] [INFO] [stdout-guard] 'transfer-encoding': 'chunked', | |
| [stdout-guard] vary: 'Accept-Encoding', | |
| [stdout-guard] 'x-envoy-upstream-service-time': '813', | |
| [2026-04-01T20:47:10.332Z] [INFO] [stdout-guard] 'x-robots-tag': 'none' | |
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| [stdout-guard] } | |
| [2026-04-01T20:47:10.333Z] [INFO] [stdout-guard] response 200 https://api.anthropic.com/v1/messages?beta=true Headers { | |
| [stdout-guard] date: 'Wed, 01 Apr 2026 20:47:10 GMT', | |
| [2026-04-01T20:47:10.333Z] [INFO] [stdout-guard] 'content-type': 'text/event-stream; charset=utf-8', | |
| [stdout-guard] 'transfer-encoding': 'chunked', | |
| [2026-04-01T20:47:10.334Z] [INFO] [stdout-guard] connection: 'keep-alive', | |
| [stdout-guard] 'cache-control': 'no-cache', | |
| [stdout-guard] 'anthropic-ratelimit-unified-status': 'allowed_warning', | |
| [2026-04-01T20:47:10.334Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-5h-status': 'allowed', | |
| [2026-04-01T20:47:10.334Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-5h-reset': '1775091600', | |
| [2026-04-01T20:47:10.334Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-5h-utilization': '0.04', | |
| [2026-04-01T20:47:10.334Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-7d-status': 'allowed_warning', | |
| [2026-04-01T20:47:10.335Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-7d-reset': '1775188800', | |
| [stdout-guard] 'anthropic-ratelimit-unified-7d-utilization': '0.97', | |
| [stdout-guard] 'anthropic-ratelimit-unified-7d-surpassed-threshold': '0.75', | |
| [stdout-guard] 'anthropic-ratelimit-unified-representative-claim': 'seven_day', | |
| [2026-04-01T20:47:10.335Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-fallback-percentage': '0.5', | |
| [stdout-guard] 'anthropic-ratelimit-unified-reset': '1775188800', | |
| [stdout-guard] 'anthropic-ratelimit-unified-overage-disabled-reason': 'org_level_disabled', | |
| [2026-04-01T20:47:10.335Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-overage-status': 'rejected', | |
| [stdout-guard] 'request-id': 'req_011CZdhxZLp14qjrZMPoF94X', | |
| [stdout-guard] 'strict-transport-security': 'max-age=31536000; includeSubDomains; preload', | |
| [stdout-guard] 'anthropic-organization-id': '684cb0ba-85b8-4b30-a841-e92fb92cc97f', | |
| [stdout-guard] server: 'cloudflare', | |
| [stdout-guard] 'x-envoy-upstream-service-time': '813', | |
| [stdout-guard] 'content-encoding': 'gzip', | |
| [stdout-guard] vary: 'Accept-Encoding', | |
| [stdout-guard] 'server-timing': 'x-originResponse;dur=815', | |
| [stdout-guard] 'set-cookie': '_cfuvid=j9zP.CzVVdAUk0Qx9zA.WTghoA0P.H2oOnUqMkXeBEo-1775076429.39686-1.0.1.1-Sq9cOWzWwSlW8vff0Y5ldlTjWAdiARCkaKu4aNuK1Io; HttpOnly; SameSite=None; Secure; Path=/; Domain=api.anthropic.com', | |
| [stdout-guard] 'x-robots-tag': 'none', | |
| [stdout-guard] 'cf-cache-status': 'DYNAMIC', | |
| [stdout-guard] 'content-security-policy': "default-src 'none'; frame-ancestors 'none'", | |
| [stdout-guard] 'cf-ray': '9e5a7283bc5dcef6-FRA' | |
| [stdout-guard] } ReadableStream { locked: false, state: 'readable', supportsBYOB: true } | |
| [2026-04-01T20:47:10.335Z] [INFO] [stdout-guard] [log_857271] response parsed { | |
| [2026-04-01T20:47:10.335Z] [INFO] [stdout-guard] url: 'https://api.anthropic.com/v1/messages?beta=true', | |
| [stdout-guard] status: 200, | |
| [stdout-guard] body: KT { | |
| [stdout-guard] iterator: [AsyncGeneratorFunction: A], | |
| [stdout-guard] controller: AbortController { signal: AbortSignal { aborted: false } } | |
| [stdout-guard] }, | |
| [stdout-guard] durationMs: 944 | |
| [stdout-guard] } | |
| [2026-04-01T20:47:11.084Z] [INFO] [stdout-guard] [log_1fbf45, request-id: "req_011CZdhxbubaZ6Rrwk8r839K"] post https://api.anthropic.com/v1/messages?beta=true succeeded with status 200 in 1099ms | |
| [2026-04-01T20:47:11.085Z] [INFO] [stdout-guard] [log_1fbf45] response start { | |
| [stdout-guard] url: 'https://api.anthropic.com/v1/messages?beta=true', | |
| [2026-04-01T20:47:11.086Z] [INFO] [stdout-guard] status: 200, | |
| [stdout-guard] headers: { | |
| [stdout-guard] 'anthropic-organization-id': '684cb0ba-85b8-4b30-a841-e92fb92cc97f', | |
| [stdout-guard] 'anthropic-ratelimit-unified-5h-reset': '1775091600', | |
| [2026-04-01T20:47:11.086Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-5h-status': 'allowed', | |
| [stdout-guard] 'anthropic-ratelimit-unified-5h-utilization': '0.04', | |
| [2026-04-01T20:47:11.087Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-7d-reset': '1775188800', | |
| [stdout-guard] 'anthropic-ratelimit-unified-7d-status': 'allowed_warning', | |
| [stdout-guard] 'anthropic-ratelimit-unified-7d-surpassed-threshold': '0.75', | |
| [stdout-guard] 'anthropic-ratelimit-unified-7d-utilization': '0.97', | |
| [stdout-guard] 'anthropic-ratelimit-unified-fallback-percentage': '0.5', | |
| [stdout-guard] 'anthropic-ratelimit-unified-overage-disabled-reason': 'org_level_disabled', | |
| [stdout-guard] 'anthropic-ratelimit-unified-overage-status': 'rejected', | |
| [stdout-guard] 'anthropic-ratelimit-unified-representative-claim': 'seven_day', | |
| [2026-04-01T20:47:11.087Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-reset': '1775188800', | |
| [2026-04-01T20:47:11.087Z] [INFO] [stdout-guard] 'anthropic-ratelimit-unified-status': 'allowed_warning', | |
| [stdout-guard] 'cache-control': 'no-cache', | |
| [2026-04-01T20:47:11.087Z] [INFO] [stdout-guard] 'cf-cache-status': 'DYNAMIC', | |
| [2026-04-01T20:47:11.087Z] [INFO] [stdout-guard] 'cf-ray': '9e5a72877f65d2a0-FRA', | |
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| [stdout-guard] 'content-type': 'text/event-stream; charset=utf-8', | |
| [stdout-guard] date: 'Wed, 01 Apr 2026 20:47:11 GMT', | |
| [stdout-guard] 'request-id': 'req_011CZdhxbubaZ6Rrwk8r839K', | |
| [stdout-guard] server: 'cloudflare', | |
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| "content": "1\t/-\n2\t SetSequenceEquivalence.lean - Формальное доказательство того, что конечное множество\n3\t эквивалентно упорядоченной последовательности с уникальными элементами.\n4\t\n5\t Основная теорема: Для любого конечного множества натуральных чисел существует\n6\t единственная строго возрастающая последовательность (список без дубликатов,\n7\t отсортированный по возрастанию), содержащая в точности те же элементы.\n8\t\n9\t Адаптировано из: https://github.com/konard/subset-sum/tree/main/proofs/set_sequence_equivalence\n10\t Полные доказательства см. в Rocq-версии: SetSequenceEquivalence.v\n11\t-/\n12\t\n13\tnamespace SetSequenceEquivalence\n14\t\n15\t/-- Список является строго возрастающим, если каждый элемент строго меньше следующего -/\n16\tdef StrictlyAscending : List Nat → Prop\n17\t | [] => True\n18\t | [_] => True\n19\t | x :: y :: rest => x < y ∧ StrictlyAscending (y :: rest)\n20\t\n21\t/-- Список не содержит дубликатов -/\n22\tdef NoDuplicates : List Nat → Prop\n23\t | [] => True\n24\t | x :: rest => x ∉ rest ∧ NoDuplicates rest\n25\t\n26\t/-- Упорядоченная уникальная последовательность — это строго возрастающий список -/\n27\tdef IsOrderedUniqueSequence (l : List Nat) : Prop :=\n28\t StrictlyAscending l\n29\t\n30\t/-- Вставка элемента в отсортированный список с сохранением порядка -/\n31\tdef insertSorted (x : Nat) : List Nat → List Nat\n32\t | [] => [x]\n33\t | y :: rest =>\n34\t if x < y then x :: y :: rest\n35\t else if x = y then y :: rest\n36\t else y :: insertSorted x rest\n37\t\n38\t/-- Сортировка списка в строго возрастающем порядке (с удалением дубликатов) -/\n39\tdef toOrderedUnique : List Nat → List Nat\n40\t | [] => []\n41\t | x :: rest => insertSorted x (toOrderedUnique rest)\n42\t\n43\t/-- Строго возрастающий список не содержит дубликатов.\n44\t Доказательство: в строго возрастающем списке x < y < z < ...,\n45\t каждый элемент строго меньше всех последующих, поэтому не может повторяться.\n46\t Полное формальное доказательство см. в Rocq-версии. -/\n47\ttheorem strictly_ascending_implies_no_dup (l : List Nat)\n48\t (h : StrictlyAscending l) : NoDuplicates l := by\n49\t sorry\n50\t\n51\t/-- insertSorted сохраняет свойство строгого возрастания.\n52\t Доказательство: разбор случаев по положению вставляемого элемента\n53\t относительно текущего: x < y (вставляем перед), x = y (пропускаем),\n54\t x > y (рекурсия в хвост). Полное доказательство см. в Rocq-версии. -/\n55\ttheorem insertSorted_preserves_ascending (x : Nat) (l : List Nat)\n56\t (h : StrictlyAscending l) : StrictlyAscending (insertSorted x l) := by\n57\t sorry\n58\t\n59\t/-- toOrderedUnique выдаёт строго возрастающие списки -/\n60\ttheorem toOrderedUnique_is_ascending (l : List Nat) :\n61\t StrictlyAscending (toOrderedUnique l) := by\n62\t induction l with\n63\t | nil => simp [toOrderedUnique, StrictlyAscending]\n64\t | cons x rest ih =>\n65\t simp only [toOrderedUnique]\n66\t exact insertSorted_preserves_ascending x (toOrderedUnique rest) ih\n67\t\n68\t/-- Принадлежность элемента сохраняется при insertSorted.\n69\t Доказательство: y ∈ insertSorted x l ↔ y = x ∨ y ∈ l.\n70\t Полное доказательство см. в Rocq-версии. -/\n71\ttheorem mem_insertSorted (x y : Nat) (l : List Nat) :\n72\t y ∈ insertSorted x l ↔ y = x ∨ y ∈ l := by\n73\t sorry\n74\t\n75\t/-- Принадлежность элемента сохраняется при toOrderedUnique.\n76\t Доказательство: индукция по списку с использованием mem_insertSorted.\n77\t Полное доказательство см. в Rocq-версии. -/\n78\ttheorem mem_toOrderedUnique (x : Nat) (l : List Nat) :\n79\t x ∈ toOrderedUnique l ↔ x ∈ l := by\n80\t sorry\n81\t\n82\t/-\n83\t ОСНОВНАЯ ТЕОРЕМА: Эквивалентность множества и последовательности\n84\t\n85\t Для любого списка (представляющего мультимножество) существует строго\n86\t возрастающий список, содержащий в точности те же элементы (как множество).\n87\t-/\n88\t\n89\t/-- Основная теорема: Каждый список может быть преобразован в упорядоченную уникальную\n90\t последовательность с теми же элементами -/\n91\ttheorem set_sequence_equivalence (l : List Nat) :\n92\t ∃ l' : List Nat,\n93\t IsOrderedUniqueSequence l' ∧\n94\t (∀ x, x ∈ l' ↔ x ∈ l) :=\n95\t ⟨toOrderedUnique l, toOrderedUnique_is_ascending l, fun x => mem_toOrderedUnique x l⟩\n96\t\n97\t/-- Следствие: Упорядоченная уникальная последовательность не содержит дубликатов -/\n98\ttheorem ordered_unique_has_no_dup (l : List Nat) :\n99\t NoDuplicates (toOrderedUnique l) :=\n100\t strictly_ascending_implies_no_dup _ (toOrderedUnique_is_ascending l)\n101\t\n102\tend SetSequenceEquivalence\n103\t\n104\t-- Проверки верификации\n105\t#check SetSequenceEquivalence.set_sequence_equivalence\n106\t#check SetSequenceEquivalence.ordered_unique_has_no_dup\n107\t" | |
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| "content": "1\t/-\n2\t NetworkEquivalence.lean\n3\t\n4\t Предикаты эквивалентности для различных представлений сетей.\n5\t Lean 4 перевод NetworkEquivalence.v (Rocq).\n6\t-/\n7\timport NetworkDefinitions\n8\timport NetworkConversions\n9\t\n10\t-- Предикат эквивалентности двух сетей кортежей длины n,\n11\t-- anet1 и anet2 типа NetworkTupleFunction.\n12\t--\n13\t-- Данный предикат описывает свойство «эквивалентности» для таких сетей.\n14\t-- Он утверждает, что anet1 и anet2 считаются «эквивалентными», если для каждой ссылки id кортеж,\n15\t-- связанный с id в anet1, точно совпадает с кортежем, связанным с тем же id в anet2.\n16\tdef TupleFunctionEquivalence {n : Nat} (anet1 anet2 : NetworkTupleFunction n) : Prop :=\n17\t ∀ id, anet1 id = anet2 id\n18\t\n19\t-- Предикат эквивалентности двух сетей кортежей длины n,\n20\t-- anet1 и anet2 типа NetworkTupleList.\n21\tdef TupleListEquivalence {n : Nat} (anet1 anet2 : NetworkTupleList n) : Prop :=\n22\t anet1 = anet2\n23\t\n24\t-- Предикат эквивалентности для сетей дуплетов NetworkDupletFunction\n25\tdef DupletFunctionEquivalence (anet1 anet2 : NetworkDupletFunction) : Prop :=\n26\t ∀ id, anet1 id = anet2 id\n27\t\n28\t-- Предикат эквивалентности для сетей дуплетов NetworkDupletList\n29\tdef DupletListEquivalence (anet1 anet2 : NetworkDupletList) : Prop :=\n30\t anet1 = anet2\n31\t" | |
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| "content": "1\t/-\n2\t NetworkExamples.lean\n3\t\n4\t Практические примеры и тесты преобразований сетей.\n5\t Lean 4 перевод NetworkExamples.v (Rocq).\n6\t-/\n7\timport NetworkDefinitions\n8\timport NetworkConversions\n9\t\n10\t-- Трёхмерная сеть\n11\tdef complexExampleNetwork : NetworkTupleFunction 3 :=\n12\t fun id => match id with\n13\t | 0 => #v[0, 0, 0]\n14\t | 1 => #v[1, 1, 2]\n15\t | 2 => #v[2, 4, 0]\n16\t | 3 => #v[3, 0, 5]\n17\t | 4 => #v[4, 1, 1]\n18\t | _ => #v[0, 0, 0]\n19\t\n20\t-- Кортежи ссылок\n21\tdef exampleTuple0 : TupleOfReferences 0 := #v[]\n22\tdef exampleTuple1 : TupleOfReferences 1 := #v[0]\n23\tdef exampleTuple4 : TupleOfReferences 4 := #v[3, 2, 1, 0]\n24\t\n25\t-- Преобразование кортежей ссылок во вложенные упорядоченные пары (списки)\n26\tdef nestedPair0 := TupleOfReferencesToReferenceList exampleTuple0\n27\tdef nestedPair1 := TupleOfReferencesToReferenceList exampleTuple1\n28\tdef nestedPair4 := TupleOfReferencesToReferenceList exampleTuple4\n29\t\n30\t#eval nestedPair0 -- Ожидается результат: []\n31\t#eval nestedPair1 -- Ожидается результат: [0]\n32\t#eval nestedPair4 -- Ожидается результат: [3, 2, 1, 0]\n33\t\n34\t-- Вычисление значений преобразованной функции трёхмерной сети\n35\t#eval (TupleFunctionToReferenceListFunction complexExampleNetwork) 0 -- Ожидается результат: [0, 0, 0]\n36\t#eval (TupleFunctionToReferenceListFunction complexExampleNetwork) 1 -- Ожидается результат: [1, 1, 2]\n37\t#eval (TupleFunctionToReferenceListFunction complexExampleNetwork) 2 -- Ожидается результат: [2, 4, 0]\n38\t#eval (TupleFunctionToReferenceListFunction complexExampleNetwork) 3 -- Ожидается результат: [3, 0, 5]\n39\t#eval (TupleFunctionToReferenceListFunction complexExampleNetwork) 4 -- Ожидается результат: [4, 1, 1]\n40\t#eval (TupleFunctionToReferenceListFunction complexExampleNetwork) 5 -- Ожидается результат: [0, 0, 0]\n41\t\n42\t-- Сеть вложенных упорядоченных пар\n43\tdef testPairsNetwork : NetworkReferenceListFunction :=\n44\t fun id => match id with\n45\t | 0 => [5, 0, 8]\n46\t | 1 => [7, 1, 2]\n47\t | 2 => [2, 4, 5]\n48\t | 3 => [3, 1, 5]\n49\t | 4 => [4, 2, 1]\n50\t | _ => [0, 0, 0]\n51\t\n52\t-- Преобразованная сеть вложенных УП в трёхмерную сеть (размерность должна совпадать)\n53\tdef testTuplesNetwork : NetworkTupleFunction 3 :=\n54\t ReferenceListFunctionToTupleFunction testPairsNetwork\n55\t\n56\t-- Вычисление значений преобразованной функции сети вложенных УП\n57\t#eval (testTuplesNetwork 0).toList -- Ожидается результат: [5, 0, 8]\n58\t#eval (testTuplesNetwork 1).toList -- Ожидается результат: [7, 1, 2]\n59\t#eval (testTuplesNetwork 2).toList -- Ожидается результат: [2, 4, 5]\n60\t#eval (testTuplesNetwork 3).toList -- Ожидается результат: [3, 1, 5]\n61\t#eval (testTuplesNetwork 4).toList -- Ожидается результат: [4, 2, 1]\n62\t#eval (testTuplesNetwork 5).toList -- Ожидается результат: [0, 0, 0]\n63\t\n64\t-- Преобразование вложенных УП в сеть дуплетов\n65\t#eval ReferenceListToDupletList [121, 21, 1343]\n66\t-- Должно вернуть: [(121, 1), (21, 2), (1343, 2)]\n67\t\n68\t-- Добавление вложенных УП в сеть дуплетов\n69\t#eval AddReferenceListToDupletList [(121, 1), (21, 2), (1343, 2)] [12, 23, 34]\n70\t-- Ожидается результат: [(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)]\n71\t\n72\t-- Преобразование сети дуплетов во вложенные УП\n73\t#eval DupletListToReferenceList [(121, 1), (21, 2), (1343, 2)]\n74\t-- Ожидается результат: [121, 21, 1343]\n75\t\n76\t#eval DupletListToReferenceList [(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)]\n77\t-- Ожидается результат: [121, 21, 1343]\n78\t\n79\t-- Чтение вложенных УП из сети дуплетов по индексу дуплета — начала вложенных УП\n80\t#eval DupletListReadReferenceList [(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)] 0\n81\t-- Ожидается результат: [121, 21, 1343]\n82\t\n83\t#eval DupletListReadReferenceList [(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)] 3\n84\t-- Ожидается результат: [12, 23, 34]\n85\t\n86\t-- Определяем сеть вложенных УП\n87\tdef testReferenceListList : NetworkReferenceListList :=\n88\t [[121, 21, 1343], [12, 23], [34], [121, 21, 1343], [12, 23], [34]]\n89\t\n90\t-- Преобразованная сеть вложенных УП в сеть дуплетов\n91\tdef testDupletList := ReferenceListListToDupletList testReferenceListList\n92\t\n93\t-- Вычисление преобразованной сети вложенных УП в сеть дуплетов\n94\t#eval testDupletList\n95\t\n96\t-- Вычисление преобразования сети вложенных УП в сеть дуплетов и обратно в testReferenceListList\n97\t#eval DupletListToReferenceListList testDupletList\n98\t\n99\t-- Вычисление смещения вложенных УП в сети дуплетов по их порядковому номеру\n100\t#eval DupletListOffsetReferenceList testDupletList 0 -- Ожидается результат: 0\n101\t#eval DupletListOffsetReferenceList testDupletList 1 -- Ожидается результат: 3\n102\t#eval DupletListOffsetReferenceList testDupletList 2 -- Ожидается результат: 5\n103\t#eval DupletListOffsetReferenceList testDupletList 3 -- Ожидается результат: 6\n104\t#eval DupletListOffsetReferenceList testDupletList 4 -- Ожидается результат: 9\n105\t#eval DupletListOffsetReferenceList testDupletList 5 -- Ожидается результат: 11\n106\t#eval DupletListOffsetReferenceList testDupletList 6 -- Ожидается результат: 12\n107\t#eval DupletListOffsetReferenceList testDupletList 7 -- Ожидается результат: 12\n108\t\n109\t-- Определяем трёхмерную сеть как последовательность кортежей длины 3\n110\tdef testTupleList : NetworkTupleList 3 :=\n111\t [#v[0, 0, 0], #v[1, 1, 2], #v[2, 4, 0], #v[3, 0, 5], #v[4, 1, 1], #v[0, 0, 0]]\n112\t\n113\t-- Преобразованная трёхмерная сеть в сеть дуплетов через сеть вложенных УП\n114\tdef testTuplesToDupletList : NetworkDupletList := TupleListToDupletList testTupleList\n115\t\n116\t-- Вычисление трёхмерной сети преобразованной в сеть дуплетов\n117\t#eval testTuplesToDupletList\n118\t\n119\t-- Преобразование и обратно в трёхмерную сеть\n120\tdef resultTuplesNetwork : NetworkTupleList 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| "content": "1\t### Проекция теории связей в теорию типов (Rocq и Lean) через теорию множеств\n2\t\n3\t#### О Rocq\n4\t\n5\t[Rocq](https://rocq-prover.org/) (ранее известный как [Coq](https://ru.wikipedia.org/wiki/Coq)) — это интерактивное средство доказательства теорем, основанное на теории типов высшего порядка, также известной как Исчисление Индуктивных Построений (Calculus of Inductive Constructions, CIC). Это мощная среда для формализации сложных математических теорем, проверки доказательств на корректность и извлечения работающего программного кода из формально проверенных спецификаций. Rocq широко используется в академических кругах для формализации математики, а также в IT-индустрии для верификации программного обеспечения и оборудования.\n6\t\n7\tРешение применить Rocq для описания теории связей в рамках теории типов было обусловлено необходимостью строгой формализации доказательств и гарантирования логической корректности в рамках разработки теории связей. Использование Rocq позволяет выразить свойства и операции над связями в точных и надёжных терминах, благодаря системе типов Rocq и мощным средствам для создания и проверки доказательств.\n8\t\n9\tВ преддверии обширной работы по доказательству эквивалентности реляционной модели и сети дуплетов, мы представляем в этом разделе начальные шаги, выполненные с использованием систем доказательств Rocq и Lean. На первом этапе стоит задача формализации структур сетей через определения базовых типов, функций и структур.\n10\t\n11\t#### О Lean\n12\t\n13\t[Lean](https://lean-lang.org/) — это функциональный язык программирования и интерактивное средство доказательства теорем, разработанное в Microsoft Research. Lean 4, текущая версия, сочетает в себе мощную систему зависимых типов с полноценным языком программирования, что позволяет использовать один и тот же язык как для написания доказательств, так и для исполняемого кода. Lean широко применяется в проектах по формализации математики, таких как Mathlib — крупнейшая библиотека формализованных математических теорем.\n14\t\n15\tИспользование Lean в дополнение к Rocq позволяет обеспечить более широкую верифицируемость теории связей, а также делает формализацию доступной для более широкой аудитории математиков и разработчиков, знакомых с экосистемой Lean.\n16\t\n17\t#### Определения сетей\n18\t\n19\t##### Rocq\n20\t\n21\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/NetworkDefinitions.v)\n22\t\n23\t```rocq\n24\tRequire Import PeanoNat.\n25\tRequire Import Coq.Init.Nat.\n26\tRequire Import Vector.\n27\tRequire Import List.\n28\tRequire Import Coq.Init.Datatypes.\n29\tImport ListNotations.\n30\tImport VectorNotations.\n31\t\n32\t(* Ссылка (Reference) — уникальный идентификатор кортежа: R ⊆ ℕ₀ *)\n33\tDefinition Reference := nat.\n34\t\n35\t(* Значение Reference по умолчанию: ноль *)\n36\tDefinition ReferenceDefault : Reference := 0.\n37\t\n38\t(* Множество кортежей ссылок длины n ∈ ℕ₀: TupleOfReferences ⊆ Rⁿ *)\n39\tDefinition TupleOfReferences (n : nat) := t Reference n.\n40\t\n41\t(* Значение TupleOfReferences по умолчанию *)\n42\tDefinition TupleOfReferencesDefault (n : nat) : TupleOfReferences n := Vector.const ReferenceDefault n.\n43\t\n44\t(* Множество всех связей: Link = Reference × TupleOfReferences *)\n45\tDefinition Link (n : nat) := prod Reference (TupleOfReferences n).\n46\t\n47\t(* Сеть кортежей длины n (или n-мерная сеть) из семейства функций {Nⁿ : Reference → TupleOfReferences} *)\n48\tDefinition NetworkTupleFunction (n : nat) := Reference -> TupleOfReferences n.\n49\t\n50\t(* Сеть кортежей длины n (или n-мерная сеть) в виде последовательности *)\n51\tDefinition NetworkTupleList (n : nat) := list (TupleOfReferences n).\n52\t\n53\t(* Вложенные упорядоченные пары *)\n54\tDefinition ReferenceList := list Reference.\n55\t\n56\t(* Сеть вложенных упорядоченных пар: N^{list} : Reference → ReferenceList *)\n57\tDefinition NetworkReferenceListFunction := Reference -> ReferenceList.\n58\t\n59\t(* Сеть вложенных упорядоченных пар в виде последовательности вложенных упорядоченных пар *)\n60\tDefinition NetworkReferenceListList := list ReferenceList.\n61\t\n62\t(* Дуплет ссылок *)\n63\tDefinition Duplet := prod Reference Reference.\n64\t\n65\t(* Значение Duplet по умолчанию: пара из двух ReferenceDefault, используется для обозначения пустого дуплета *)\n66\tDefinition DupletDefault : Duplet := (ReferenceDefault, ReferenceDefault).\n67\t\n68\t(* Сеть дуплетов (или двумерная сеть): N² : Reference → Reference² *)\n69\tDefinition NetworkDupletFunction := Reference -> Duplet.\n70\t\n71\t(* Сеть дуплетов (или двумерная сеть) в виде последовательности дуплетов *)\n72\tDefinition NetworkDupletList := list Duplet.\n73\t```\n74\t\n75\t##### Lean\n76\t\n77\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/NetworkDefinitions.lean)\n78\t\n79\t```lean\n80\t-- Ссылка (Reference) — уникальный идентификатор кортежа: R ⊆ ℕ₀\n81\tabbrev Reference := Nat\n82\t\n83\t-- Значение Reference по умолчанию: ноль\n84\tdef ReferenceDefault : Reference := 0\n85\t\n86\t-- Множество кортежей ссылок длины n ∈ ℕ₀: TupleOfReferences ⊆ Rⁿ\n87\tabbrev TupleOfReferences (n : Nat) := Vector Reference n\n88\t\n89\t-- Значение TupleOfReferences по умолчанию\n90\tdef TupleOfReferencesDefault (n : Nat) : TupleOfReferences n := Vector.replicate n ReferenceDefault\n91\t\n92\t-- Вложенные упорядоченные пары\n93\tabbrev ReferenceList := List Reference\n94\t\n95\t-- Сеть вложенных упорядоченных пар: N^{list} : Reference → ReferenceList\n96\tabbrev NetworkReferenceListFunction := Reference → ReferenceList\n97\t\n98\t-- Сеть вложенных упорядоченных пар в виде последовательности вложенных упорядоченных пар\n99\tabbrev NetworkReferenceListList := List ReferenceList\n100\t\n101\t-- Множество всех связей: Link = Reference × TupleOfReferences\n102\tabbrev Link (n : Nat) := Reference × TupleOfReferences n\n103\t\n104\t-- Сеть кортежей длины n (или n-мерная сеть) из семейства функций {Nⁿ : Reference → TupleOfReferences}\n105\tabbrev NetworkTupleFunction (n : Nat) := Reference → TupleOfReferences n\n106\t\n107\t-- Сеть кортежей длины n (или n-мерная сеть) в виде последовательности\n108\tabbrev NetworkTupleList (n : Nat) := List (TupleOfReferences n)\n109\t\n110\t-- Дуплет ссылок\n111\tabbrev Duplet := Reference × Reference\n112\t\n113\t-- Значение Duplet по умолчанию: пара из двух ReferenceDefault, используется для обозначения пустого дуплета\n114\tdef DupletDefault : Duplet := (ReferenceDefault, ReferenceDefault)\n115\t\n116\t-- Сеть дуплетов (или двумерная сеть): N² : Reference → Reference²\n117\tabbrev NetworkDupletFunction := Reference → Duplet\n118\t\n119\t-- Сеть дуплетов (или двумерная сеть) в виде последовательности дуплетов\n120\tabbrev NetworkDupletList := List Duplet\n121\t```\n122\t\n123\t#### Функции преобразования сетей\n124\t\n125\t##### Rocq\n126\t\n127\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/NetworkConversions.v)\n128\t\n129\t```rocq\n130\t(* Функция преобразования TupleOfReferences в ReferenceList *)\n131\tFixpoint TupleOfReferencesToReferenceList {n : nat} (v : TupleOfReferences n) : ReferenceList :=\n132\t match v with\n133\t | Vector.nil _ => List.nil\n134\t | Vector.cons _ h _ t => List.cons h (TupleOfReferencesToReferenceList t)\n135\t end.\n136\t\n137\t(* Функция преобразования NetworkTupleFunction в NetworkReferenceListFunction *)\n138\tDefinition TupleFunctionToReferenceListFunction {n : nat} (a: NetworkTupleFunction n) : NetworkReferenceListFunction :=\n139\t fun id => TupleOfReferencesToReferenceList (a id).\n140\t\n141\t(* Функция преобразования NetworkTupleList в NetworkReferenceListList *)\n142\tDefinition TupleListToReferenceListList {n: nat} (net: NetworkTupleList n) : NetworkReferenceListList :=\n143\t map TupleOfReferencesToReferenceList net.\n144\t\n145\t(* Функция преобразования ReferenceList в TupleOfReferences, возвращающая option *)\n146\tFixpoint ReferenceListToTupleOfReferencesOption (n: nat) (p: ReferenceList) : option (TupleOfReferences n) :=\n147\t match n, p with\n148\t | 0, List.nil => Some (Vector.nil nat)\n149\t | S n', List.cons f p' =>\n150\t match ReferenceListToTupleOfReferencesOption n' p' with\n151\t | None => None\n152\t | Some t => Some (Vector.cons nat f n' t)\n153\t end\n154\t | _, _ => None\n155\t end.\n156\t\n157\t(* Функция преобразования ReferenceList в TupleOfReferences с использованием TupleOfReferencesDefault *)\n158\tDefinition ReferenceListToTupleOfReferences(n: nat) (p: ReferenceList) : TupleOfReferences n :=\n159\t match ReferenceListToTupleOfReferencesOption n p with\n160\t | None => TupleOfReferencesDefault n\n161\t | Some t => t\n162\t end.\n163\t\n164\t(* Функция преобразования NetworkReferenceListFunction в NetworkTupleFunction *)\n165\tDefinition ReferenceListFunctionToTupleFunction { n: nat } (net: NetworkReferenceListFunction) : NetworkTupleFunction n :=\n166\t fun id => match ReferenceListToTupleOfReferencesOption n (net id) with\n167\t | Some t => t\n168\t | None => TupleOfReferencesDefault n\n169\t end.\n170\t\n171\t(* Функция преобразования NetworkReferenceListList в NetworkTupleList *)\n172\tDefinition ReferenceListListToTupleList {n: nat} (net : NetworkReferenceListList) : NetworkTupleList n :=\n173\t map (ReferenceListToTupleOfReferencesn) net.\n174\t\n175\t(* Функция преобразования ReferenceList в NetworkDupletList со смещением индексации *)\n176\tFixpoint ReferenceListToDupletList_ (offset: nat) (np: ReferenceList) : NetworkDupletList :=\n177\t match np with\n178\t | nil => nil\n179\t | cons h nil => cons (h, offset) nil\n180\t | cons h t => cons (h, S offset) (ReferenceListToDupletList_ (S offset) t)\n181\t end.\n182\t\n183\t(* Функция преобразования ReferenceList в NetworkDupletList *)\n184\tDefinition ReferenceListToDupletList (np: ReferenceList) : NetworkDupletList := ReferenceListToDupletList_ 0 np.\n185\t\n186\t(* Функция добавления ReferenceList в хвост NetworkDupletList *)\n187\tDefinition AddReferenceListToDupletList (anet: NetworkDupletList) (np: ReferenceList) : NetworkDupletList :=\n188\t app anet (ReferenceListToDupletList_ (length anet) np).\n189\t\n190\t(* Функция отрезает голову anetd и возвращает хвост начиная с offset *)\n191\tFixpoint DupletListBehead (anet: NetworkDupletList) (offset : nat) : NetworkDupletList :=\n192\t match offset with\n193\t | 0 => anet\n194\t | S n' =>\n195\t match anet with\n196\t | nil => nil\n197\t | cons h t => DupletListBehead t n'\n198\t end\n199\t end.\n200\t\n201\t(* Функция преобразования NetworkDupletList в ReferenceList с индексацией в начале NetworkDupletList начиная с offset *)\n202\tFixpoint DupletListToReferenceList_ (anet: NetworkDupletList) (offset: nat) (index: nat): ReferenceList :=\n203\t match anet with\n204\t | nil => nil\n205\t | cons (x, next_index) tail_anet =>\n206\t if offset =? index then\n207\t cons x (DupletListToReferenceList_ tail_anet (S offset) next_index)\n208\t else\n209\t DupletListToReferenceList_ tail_anet (S offset) index\n210\t end.\n211\t\n212\t(* Функция чтения ReferenceList из NetworkDupletList по индексу дуплета *)\n213\tDefinition DupletListReadReferenceList (anet: NetworkDupletList) (index: nat) : ReferenceList :=\n214\t DupletListToReferenceList_ anet 0 index.\n215\t\n216\t(* Функция преобразования NetworkDupletList в ReferenceList начиная с головы списка сети *)\n217\tDefinition DupletListToReferenceList (anet: NetworkDupletList) : ReferenceList := DupletListReadReferenceList anet 0.\n218\t\n219\t(*\n220\t Теперь всё готово для преобразования сети вложенных упорядоченных пар N^{list} : Reference → ReferenceList\n221\t в сеть дуплетов N² : Reference → Reference².\n222\t\n223\t Данное преобразование можно делать по-разному: с сохранением исходных ссылок на кортежи\n224\t либо с переиндексацией. Переиндексацию можно не делать, если написать дополнительную функцию для\n225\t сети дуплетов, которая возвращает вложенную упорядоченную пару по её ссылке.\n226\t*)\n227\t\n228\t(* Функция добавления NetworkReferenceListList в NetworkDupletList *)\n229\tFixpoint AddReferenceListListToDupletList (anetd: NetworkDupletList) (anetl: NetworkReferenceListList) : NetworkDupletList :=\n230\t match anetl with\n231\t | nil => anetd\n232\t | cons h t => AddReferenceListListToDupletList (AddReferenceListToDupletList anetd h) t\n233\t end.\n234\t\n235\t(* Функция преобразования NetworkReferenceListList в NetworkDupletList *)\n236\tDefinition ReferenceListListToDupletList (anetl: NetworkReferenceListList) : NetworkDupletList :=\n237\t match anetl with\n238\t | nil => nil\n239\t | cons h t => AddReferenceListListToDupletList (ReferenceListToDupletList h) t\n240\t end.\n241\t\n242\t(* Функция поиска ReferenceList в хвосте NetworkDupletList начинающемуся с offset по её порядковому номеру.\n243\t Возвращает offset ReferenceList. *)\n244\tFixpoint DupletListOffsetReferenceList_ (anet: NetworkDupletList) (offset: nat) (index: nat) : nat :=\n245\t match anet with\n246\t | nil => offset + (length anet)\n247\t | cons (_, next_index) tail_anet =>\n248\t match index with\n249\t | O => offset\n250\t | S index' =>\n251\t if offset =? next_index then\n252\t DupletListOffsetReferenceList_ tail_anet (S offset) index'\n253\t else\n254\t DupletListOffsetReferenceList_ tail_anet (S offset) index\n255\t end\n256\t end.\n257\t\n258\t(* Функция поиска ReferenceList в NetworkDupletList по её порядковому номеру. Возвращает offset ReferenceList. *)\n259\tDefinition DupletListOffsetReferenceList (anet: NetworkDupletList) (index: nat) : nat :=\n260\t DupletListOffsetReferenceList_ anet 0 index.\n261\t\n262\t(* Функция преобразования NetworkTupleList в NetworkDupletList *)\n263\tDefinition TupleListToDupletList {n : nat} (anetv: NetworkTupleList n) : NetworkDupletList :=\n264\t ReferenceListListToDupletList (TupleListToReferenceListList anetv).\n265\t\n266\t(*\n267\t Теперь всё готово для преобразования сети дуплетов N² : Reference → Reference²\n268\t в сеть вложенных упорядоченных пар N^{list} : Reference → ReferenceList.\n269\t\n270\t Данное преобразование будем делать с сохранением исходных ссылок на кортежи.\n271\t Переиндексацию можно не делать, потому что есть функция DupletListOffsetReferenceList для\n272\t сети дуплетов, которая возвращает смещение вложенной УП по ссылке на неё.\n273\t*)\n274\t\n275\t(* Функция отрезает первую ReferenceList из NetworkDupletList и возвращает хвост *)\n276\tFixpoint DupletListBeheadReferenceList (anet: NetworkDupletList) (offset: nat) : NetworkDupletList :=\n277\t match anet with\n278\t | nil => nil\n279\t | cons (_, next_index) tail_anet =>\n280\t if offset =? next_index then (* конец ReferenceList *)\n281\t tail_anet\n282\t else (* ещё не конец ReferenceList *)\n283\t DupletListBeheadReferenceList tail_anet (S offset)\n284\t end.\n285\t\n286\t(* Функция преобразования ReferenceList и NetworkDupletList со смещения offset в NetworkReferenceListList *)\n287\tFixpoint DupletListToReferenceListList_ (anetd: NetworkDupletList) (np: ReferenceList) (offset: nat) : NetworkReferenceListList :=\n288\t match anetd with\n289\t | nil => nil (* отбрасываем ReferenceList даже если она недостроена *)\n290\t | cons (x, next_index) tail_anet =>\n291\t if offset =? next_index then (* конец ReferenceList, переходим к следующей ReferenceList *)\n292\t cons (app np (cons x nil)) (DupletListToReferenceListList_ tail_anet nil (S offset))\n293\t else (* ещё не конец ReferenceList, парсим сеть дуплетов дальше *)\n294\t DupletListToReferenceListList_ tail_anet (app np (cons x nil)) (S offset)\n295\t end.\n296\t\n297\t(* Функция преобразования NetworkDupletList в NetworkReferenceListList *)\n298\tDefinition DupletListToReferenceListList (anetd: NetworkDupletList) : NetworkReferenceListList :=\n299\t DupletListToReferenceListList_ anetd nil ReferenceDefault.\n300\t```" | |
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| "content": "300\t```\n301\t\n302\t##### Lean\n303\t\n304\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/NetworkConversions.lean)\n305\t\n306\t```lean\n307\t-- Функция преобразования TupleOfReferences в ReferenceList\n308\tdef TupleOfReferencesToReferenceList {n : Nat} (v : TupleOfReferences n) : ReferenceList :=\n309\t v.toList\n310\t\n311\t-- Функция преобразования NetworkTupleFunction в NetworkReferenceListFunction\n312\tdef TupleFunctionToReferenceListFunction {n : Nat} (a : NetworkTupleFunction n) : NetworkReferenceListFunction :=\n313\t fun id => TupleOfReferencesToReferenceList (a id)\n314\t\n315\t-- Функция преобразования NetworkTupleList в NetworkReferenceListList\n316\tdef TupleListToReferenceListList {n : Nat} (net : NetworkTupleList n) : NetworkReferenceListList :=\n317\t net.map TupleOfReferencesToReferenceList\n318\t\n319\t-- Функция преобразования ReferenceList в TupleOfReferences, возвращающая option\n320\tdef ReferenceListToTupleOfReferencesOption (n : Nat) (p : ReferenceList) : Option (TupleOfReferences n) :=\n321\t let arr := p.toArray\n322\t if h : arr.size = n then\n323\t some ⟨arr, h⟩\n324\t else\n325\t none\n326\t\n327\t-- Функция преобразования ReferenceList в TupleOfReferences с использованием TupleOfReferencesDefault\n328\tdef ReferenceListToTupleOfReferences(n : Nat) (p : ReferenceList) : TupleOfReferences n :=\n329\t match ReferenceListToTupleOfReferencesOption n p with\n330\t | none => TupleOfReferencesDefault n\n331\t | some t => t\n332\t\n333\t-- Функция преобразования NetworkReferenceListFunction в NetworkTupleFunction\n334\tdef ReferenceListFunctionToTupleFunction {n : Nat} (net : NetworkReferenceListFunction) : NetworkTupleFunction n :=\n335\t fun id => match ReferenceListToTupleOfReferencesOption n (net id) with\n336\t | some t => t\n337\t | none => TupleOfReferencesDefault n\n338\t\n339\t-- Функция преобразования NetworkReferenceListList в NetworkTupleList\n340\tdef ReferenceListListToTupleList {n : Nat} (net : NetworkReferenceListList) : NetworkTupleList n :=\n341\t net.map (ReferenceListToTupleOfReferencesn)\n342\t\n343\t-- Функция преобразования ReferenceList в NetworkDupletList со смещением индексации\n344\tdef ReferenceListToDupletList_ (offset : Nat) : ReferenceList → NetworkDupletList\n345\t | [] => []\n346\t | [h] => [(h, offset)]\n347\t | h :: t => (h, offset + 1) :: ReferenceListToDupletList_ (offset + 1) t\n348\t\n349\t-- Функция преобразования ReferenceList в NetworkDupletList\n350\tdef ReferenceListToDupletList (np : ReferenceList) : NetworkDupletList :=\n351\t ReferenceListToDupletList_ 0 np\n352\t\n353\t-- Функция добавления ReferenceList в хвост NetworkDupletList\n354\tdef AddReferenceListToDupletList (anet : NetworkDupletList) (np : ReferenceList) : NetworkDupletList :=\n355\t anet ++ ReferenceListToDupletList_ anet.length np\n356\t\n357\t-- Функция отрезает голову anetd и возвращает хвост начиная с offset\n358\tdef DupletListBehead (anet : NetworkDupletList) : Nat → NetworkDupletList\n359\t | 0 => anet\n360\t | Nat.succ n => match anet with\n361\t | [] => []\n362\t | _ :: t => DupletListBehead t n\n363\t\n364\t-- Функция преобразования NetworkDupletList в ReferenceList\n365\tdef DupletListToReferenceList_ (anet : NetworkDupletList) (offset : Nat) (index : Nat) : ReferenceList :=\n366\t match anet with\n367\t | [] => []\n368\t | (x, next_index) :: tail_anet =>\n369\t if offset == index then\n370\t x :: DupletListToReferenceList_ tail_anet (offset + 1) next_index\n371\t else\n372\t DupletListToReferenceList_ tail_anet (offset + 1) index\n373\t\n374\t-- Функция чтения ReferenceList из NetworkDupletList по индексу дуплета\n375\tdef DupletListReadReferenceList (anet : NetworkDupletList) (index : Nat) : ReferenceList :=\n376\t DupletListToReferenceList_ anet 0 index\n377\t\n378\t-- Функция преобразования NetworkDupletList в ReferenceList\n379\tdef DupletListToReferenceList (anet : NetworkDupletList) : ReferenceList :=\n380\t DupletListReadReferenceList anet 0\n381\t\n382\t-- Функция добавления NetworkReferenceListList в NetworkDupletList\n383\tdef AddReferenceListListToDupletList : NetworkDupletList → NetworkReferenceListList → NetworkDupletList\n384\t | anetd, [] => anetd\n385\t | anetd, h :: t => AddReferenceListListToDupletList (AddReferenceListToDupletList anetd h) t\n386\t\n387\t-- Функция преобразования NetworkReferenceListList в NetworkDupletList\n388\tdef ReferenceListListToDupletList : NetworkReferenceListList → NetworkDupletList\n389\t | [] => []\n390\t | h :: t => AddReferenceListListToDupletList (ReferenceListToDupletList h) t\n391\t\n392\t-- Функция поиска ReferenceList по порядковому номеру. Возвращает offset ReferenceList.\n393\tdef DupletListOffsetReferenceList_ (anet : NetworkDupletList) (offset : Nat) (index : Nat) : Nat :=\n394\t match anet with\n395\t | [] => offset + anet.length\n396\t | (_, next_index) :: tail_anet =>\n397\t match index with\n398\t | 0 => offset\n399\t | Nat.succ index' =>\n400\t if offset == next_index then\n401\t DupletListOffsetReferenceList_ tail_anet (offset + 1) index'\n402\t else\n403\t DupletListOffsetReferenceList_ tail_anet (offset + 1) index\n404\t\n405\t-- Функция поиска ReferenceList в NetworkDupletList по её порядковому номеру\n406\tdef DupletListOffsetReferenceList (anet : NetworkDupletList) (index : Nat) : Nat :=\n407\t DupletListOffsetReferenceList_ anet 0 index\n408\t\n409\t-- Функция преобразования NetworkTupleList в NetworkDupletList\n410\tdef TupleListToDupletList {n : Nat} (anetv : NetworkTupleList n) : NetworkDupletList :=\n411\t ReferenceListListToDupletList (TupleListToReferenceListList anetv)\n412\t\n413\t-- Функция отрезает первую ReferenceList из NetworkDupletList и возвращает хвост\n414\tdef DupletListBeheadReferenceList (anet : NetworkDupletList) (offset : Nat) : NetworkDupletList :=\n415\t match anet with\n416\t | [] => []\n417\t | (_, next_index) :: tail_anet =>\n418\t if offset == next_index then\n419\t tail_anet\n420\t else\n421\t DupletListBeheadReferenceList tail_anet (offset + 1)\n422\t\n423\t-- Функция преобразования NetworkDupletList в NetworkReferenceListList\n424\tdef DupletListToReferenceListList_ (anetd : NetworkDupletList) (np : ReferenceList) (offset : Nat) : NetworkReferenceListList :=\n425\t match anetd with\n426\t | [] => []\n427\t | (x, next_index) :: tail_anet =>\n428\t if offset == next_index then\n429\t (np ++ [x]) :: DupletListToReferenceListList_ tail_anet [] (offset + 1)\n430\t else\n431\t DupletListToReferenceListList_ tail_anet (np ++ [x]) (offset + 1)\n432\t\n433\t-- Функция преобразования NetworkDupletList в NetworkReferenceListList\n434\tdef DupletListToReferenceListList (anetd : NetworkDupletList) : NetworkReferenceListList :=\n435\t DupletListToReferenceListList_ anetd [] ReferenceDefault\n436\t```\n437\t\n438\t#### Предикаты эквивалентности сетей\n439\t\n440\t##### Rocq\n441\t\n442\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/NetworkEquivalence.v)\n443\t\n444\t```rocq\n445\t(* Предикат эквивалентности двух сетей кортежей длины n,\n446\t anet1 и anet2 типа NetworkTupleFunction.\n447\t\n448\t Данный предикат описывает свойство «эквивалентности» для таких сетей.\n449\t Он утверждает, что anet1 и anet2 считаются «эквивалентными», если для каждой ссылки id кортеж,\n450\t связанный с id в anet1, точно совпадает с кортежем, связанным с тем же id в anet2.\n451\t*)\n452\tDefinition TupleFunctionEquivalence {n: nat} (anet1: NetworkTupleFunction n) (anet2: NetworkTupleFunction n) : Prop :=\n453\t forall id, anet1 id = anet2 id.\n454\t\n455\t(* Предикат эквивалентности двух сетей кортежей длины n,\n456\t anet1 и anet2 типа NetworkTupleList.\n457\t*)\n458\tDefinition TupleListEquivalence {n: nat} (anet1: NetworkTupleList n) (anet2: NetworkTupleList n) : Prop :=\n459\t anet1 = anet2.\n460\t\n461\t(* Предикат эквивалентности для сетей дуплетов NetworkDupletFunction *)\n462\tDefinition DupletFunctionEquivalence (anet1: NetworkDupletFunction) (anet2: NetworkDupletFunction) : Prop := forall id, anet1 id = anet2 id.\n463\t\n464\t(* Предикат эквивалентности для сетей дуплетов NetworkDupletList *)\n465\tDefinition DupletListEquivalence (anet1: NetworkDupletList) (anet2: NetworkDupletList) : Prop := anet1 = anet2.\n466\t```\n467\t\n468\t##### Lean\n469\t\n470\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/NetworkEquivalence.lean)\n471\t\n472\t```lean\n473\t-- Предикат эквивалентности двух сетей кортежей длины n\n474\tdef TupleFunctionEquivalence {n : Nat} (anet1 anet2 : NetworkTupleFunction n) : Prop :=\n475\t ∀ id, anet1 id = anet2 id\n476\t\n477\t-- Предикат эквивалентности двух сетей кортежей длины n в виде списков\n478\tdef TupleListEquivalence {n : Nat} (anet1 anet2 : NetworkTupleList n) : Prop :=\n479\t anet1 = anet2\n480\t\n481\t-- Предикат эквивалентности для сетей дуплетов NetworkDupletFunction\n482\tdef DupletFunctionEquivalence (anet1 anet2 : NetworkDupletFunction) : Prop :=\n483\t ∀ id, anet1 id = anet2 id\n484\t\n485\t-- Предикат эквивалентности для сетей дуплетов NetworkDupletList\n486\tdef DupletListEquivalence (anet1 anet2 : NetworkDupletList) : Prop :=\n487\t anet1 = anet2\n488\t```\n489\t\n490\t#### Леммы эквивалентности сетей\n491\t\n492\t##### Rocq\n493\t\n494\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/NetworkLemmas.v)\n495\t\n496\t```rocq\n497\t(* Лемма о сохранении длины кортежей сети *)\n498\tLemma TupleOfReferencesDimensionPreserved : forall {l: nat} (t: TupleOfReferences l), List.length (TupleOfReferencesToReferenceList t) = l.\n499\tProof.\n500\t intros l t.\n501\t induction t.\n502\t - simpl. reflexivity.\n503\t - simpl. rewrite IHt. reflexivity.\n504\tQed.\n505\t\n506\t\n507\t(* Лемма о взаимном обращении функций ReferenceListToTupleOfReferencesOption и TupleOfReferencesToReferenceList\n508\t\n509\t ReferenceListToTupleOfReferencesInverse доказывает, что каждый кортеж TupleOfReferences без потери данных может быть преобразован в ReferenceList\n510\t с помощью TupleOfReferencesToReferenceList и обратно в TupleOfReferences с помощью ReferenceListToTupleOfReferencesOption.\n511\t\n512\t В формальном виде forall n: nat, forall t: TupleOfReferences n, ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = Some t говорит о том,\n513\t что для всякого натурального числа n и каждого кортежа TupleOfReferences длины n,\n514\t мы можем преобразовать TupleOfReferences в ReferenceList с помощью TupleOfReferencesToReferenceList,\n515\t затем обратно преобразовать результат в TupleOfReferences с помощью ReferenceListToTupleOfReferencesOption n,\n516\t и в итоге получить тот же кортеж TupleOfReferences, что и в начале.\n517\t\n518\t Это свойство очень важно, потому что оно гарантирует,\n519\t что эти две функции образуют обратную пару на множестве преобразуемых кортежей TupleOfReferences и ReferenceList.\n520\t Когда вы применяете обе функции к значениям в этом множестве, вы в итоге получаете исходное значение.\n521\t Это означает, что никакая информация не теряется при преобразованиях,\n522\t так что можно свободно конвертировать между TupleOfReferences и ReferenceList,\n523\t если это требуется в реализации или доказательствах.\n524\t*)\n525\tLemma ReferenceListToTupleOfReferencesInverse: forall n: nat, forall t: TupleOfReferences n, ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = Some t.\n526\tProof.\n527\t intros n.\n528\t induction t as [| h n' t' IH].\n529\t - simpl. reflexivity.\n530\t - simpl. rewrite IH. reflexivity.\n531\tQed.\n532\t\n533\t\n534\t(*\n535\t Теорема обёртывания и восстановления сети кортежей:\n536\t\n537\t Пусть дана сеть кортежей длины n, обозначенная как Nⁿ : Reference → Tⁿ.\n538\t Определим операцию отображения этой сети в сеть вложенных упорядоченных пар N^{list} : Reference → ReferenceList,\n539\t где ReferenceList = {(∅,∅) | (l, np), l ∈ Reference, np ∈ ReferenceList}.\n540\t Затем определим обратное отображение из сети вложенных упорядоченных пар обратно\n541\t в сеть кортежей длины n.\n542\t\n543\t Теорема утверждает:\n544\t\n545\t Для любой сети кортежей длины n, Nⁿ, применение операции преобразования\n546\t в сеть вложенных упорядоченных пар и обратное преобразование обратно\n547\t в сеть кортежей длины n обеспечивает восстановление исходной сети Nⁿ.\n548\t Иначе говоря:\n549\t\n550\t ∀ Nⁿ : Reference → Tⁿ, обратно(вперёд(Nⁿ)) = Nⁿ.\n551\t*)\n552\tTheorem TupleFunctionEquivalenceAfterTransforms : forall {n: nat} (anet: NetworkTupleFunction n),\n553\t TupleFunctionEquivalence anet (fun id => match ReferenceListToTupleOfReferencesOption n ((TupleFunctionToReferenceListFunction anet) id) with\n554\t | Some t => t\n555\t | None => anet id\n556\t end).\n557\tProof.\n558\t intros n net id.\n559\t unfold TupleFunctionToReferenceListFunction.\n560\t simpl.\n561\t rewrite ReferenceListToTupleOfReferencesInverse.\n562\t reflexivity.\n563\tQed.\n564\t\n565\t\n566\t(* Лемма о сохранении длины списков ReferenceList в сети дуплетов *)\n567\tLemma ReferenceListDimensionPreserved : forall (offset: nat) (np: ReferenceList),\n568\t length np = length (ReferenceListToDupletList_ offset np).\n569\tProof.\n570\t intros offset np.\n571\t generalize dependent offset.\n572\t induction np as [| n np' IHnp']; intros offset.\n573\t - simpl. reflexivity.\n574\t - destruct np' as [| m np'']; simpl; simpl in IHnp'.\n575\t + reflexivity.\n576\t + rewrite IHnp' with (offset := S offset). reflexivity.\n577\tQed.\n578\t```\n579\t\n580\t##### Lean\n581\t\n582\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/NetworkLemmas.lean)\n583\t\n584\t```lean\n585\t-- Лемма о сохранении длины кортежей сети\n586\ttheorem TupleOfReferencesDimensionPreserved {l : Nat} (t : TupleOfReferences l) :\n587\t (TupleOfReferencesToReferenceList t).length = l := by\n588\t simp [TupleOfReferencesToReferenceList]\n589\t\n590\t-- Лемма о взаимном обращении функций ReferenceListToTupleOfReferencesOption и TupleOfReferencesToReferenceList\n591\ttheorem ReferenceListToTupleOfReferencesInverse (n : Nat) (t : TupleOfReferences n) :\n592\t ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = some t := by\n593\t simp [ReferenceListToTupleOfReferencesOption, TupleOfReferencesToReferenceList]\n594\t congr 1\n595\t\n596\t-- Теорема обёртывания и восстановления сети кортежей\n597\ttheorem TupleFunctionEquivalenceAfterTransforms {n : Nat} (anet : NetworkTupleFunction n) :\n598\t TupleFunctionEquivalence anet\n599\t (fun id => match ReferenceListToTupleOfReferencesOption n ((TupleFunctionToReferenceListFunction anet) id) with" | |
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| "timestamp": "2026-04-01T20:47:14.331Z" | |
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| "content": "600\t | some t => t\n601\t | none => anet id) := by\n602\t intro id\n603\t simp [TupleFunctionToReferenceListFunction]\n604\t rw [ReferenceListToTupleOfReferencesInverse]\n605\t\n606\t-- Лемма о сохранении длины списков ReferenceList в сети дуплетов\n607\ttheorem ReferenceListDimensionPreserved (offset : Nat) (np : ReferenceList) :\n608\t np.length = (ReferenceListToDupletList_ offset np).length := by\n609\t induction np generalizing offset with\n610\t | nil => simp [ReferenceListToDupletList_]\n611\t | cons n np' ih =>\n612\t cases np' with\n613\t | nil => simp [ReferenceListToDupletList_]\n614\t | cons m np'' =>\n615\t simp [ReferenceListToDupletList_]\n616\t exact ih (offset + 1)\n617\t```\n618\t\n619\t#### Примеры преобразований между сетями\n620\t\n621\t##### Rocq\n622\t\n623\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/NetworkExamples.v)\n624\t\n625\t```rocq\n626\t(* Нотация записи списков *)\n627\tNotation \"{ }\" := (nil) (at level 0).\n628\tNotation \"{ x , .. , y }\" := (cons x .. (cons y nil) ..) (at level 0).\n629\t\n630\t(* Трёхмерная сеть *)\n631\tDefinition complexExampleNetwork : NetworkTupleFunction 3 :=\n632\t fun id => match id with\n633\t | 0 => [0; 0; 0]\n634\t | 1 => [1; 1; 2]\n635\t | 2 => [2; 4; 0]\n636\t | 3 => [3; 0; 5]\n637\t | 4 => [4; 1; 1]\n638\t | S _ => [0; 0; 0]\n639\t end.\n640\t\n641\t(* Кортежи ссылок *)\n642\tDefinition exampleTuple0 : TupleOfReferences 0 := [].\n643\tDefinition exampleTuple1 : TupleOfReferences 1 := [0].\n644\tDefinition exampleTuple4 : TupleOfReferences 4 := [3; 2; 1; 0].\n645\t\n646\t(* Преобразование кортежей ссылок во вложенные упорядоченные пары (списки) *)\n647\tDefinition nestedPair0 := TupleOfReferencesToReferenceList exampleTuple0.\n648\tDefinition nestedPair1 := TupleOfReferencesToReferenceList exampleTuple1.\n649\tDefinition nestedPair4 := TupleOfReferencesToReferenceList exampleTuple4.\n650\t\n651\tCompute nestedPair0. (* Ожидается результат: { } *)\n652\tCompute nestedPair1. (* Ожидается результат: {0} *)\n653\tCompute nestedPair4. (* Ожидается результат: {3, 2, 1, 0} *)\n654\t\n655\t(* Вычисление значений преобразованной функции трёхмерной сети *)\n656\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 0. (* Ожидается результат: {0, 0, 0} *)\n657\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 1. (* Ожидается результат: {1, 1, 2} *)\n658\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 2. (* Ожидается результат: {2, 4, 0} *)\n659\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 3. (* Ожидается результат: {3, 0, 5} *)\n660\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 4. (* Ожидается результат: {4, 1, 1} *)\n661\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 5. (* Ожидается результат: {0, 0, 0} *)\n662\t\n663\t(* Сеть вложенных упорядоченных пар *)\n664\tDefinition testPairsNetwork : NetworkReferenceListFunction :=\n665\t fun id => match id with\n666\t | 0 => {5, 0, 8}\n667\t | 1 => {7, 1, 2}\n668\t | 2 => {2, 4, 5}\n669\t | 3 => {3, 1, 5}\n670\t | 4 => {4, 2, 1}\n671\t | S _ => {0, 0, 0}\n672\t end.\n673\t\n674\t(* Преобразованная сеть вложенных УП в трёхмерную сеть (размерность должна совпадать) *)\n675\tDefinition testTuplesNetwork : NetworkTupleFunction 3 :=\n676\t ReferenceListFunctionToTupleFunction testPairsNetwork.\n677\t\n678\t(* Вычисление значений преобразованной функции сети вложенных УП *)\n679\tCompute testTuplesNetwork 0. (* Ожидается результат: [5; 0; 8] *)\n680\tCompute testTuplesNetwork 1. (* Ожидается результат: [7; 1; 2] *)\n681\tCompute testTuplesNetwork 2. (* Ожидается результат: [2; 4; 5] *)\n682\tCompute testTuplesNetwork 3. (* Ожидается результат: [3; 1; 5] *)\n683\tCompute testTuplesNetwork 4. (* Ожидается результат: [4; 2; 1] *)\n684\tCompute testTuplesNetwork 5. (* Ожидается результат: [0; 0; 0] *)\n685\t\n686\t(* Преобразование вложенных УП в сеть дуплетов *)\n687\tCompute ReferenceListToDupletList { 121, 21, 1343 }.\n688\t(* Должно вернуть: {(121, 1), (21, 2), (1343, 2)} *)\n689\t\n690\t(* Добавление вложенных УП в сеть дуплетов *)\n691\tCompute AddReferenceListToDupletList {(121, 1), (21, 2), (1343, 2)} {12, 23, 34}.\n692\t(* Ожидается результат: {(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)} *)\n693\t\n694\t(* Преобразование сети дуплетов во вложенные УП *)\n695\tCompute DupletListToReferenceList {(121, 1), (21, 2), (1343, 2)}.\n696\t(* Ожидается результат: {121, 21, 1343} *)\n697\t\n698\tCompute DupletListToReferenceList {(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)}.\n699\t(* Ожидается результат: {121, 21, 1343} *)\n700\t\n701\t(* Чтение вложенных УП из сети дуплетов по индексу дуплета — начала вложенных УП *)\n702\tCompute DupletListReadReferenceList {(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)} 0.\n703\t(* Ожидается результат: {121, 21, 1343} *)\n704\t\n705\tCompute DupletListReadReferenceList {(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)} 3.\n706\t(* Ожидается результат: {12, 23, 34} *)\n707\t\n708\t(* Определяем сеть вложенных УП *)\n709\tDefinition testReferenceListList := { {121, 21, 1343}, {12, 23}, {34}, {121, 21, 1343}, {12, 23}, {34} }.\n710\t\n711\t(* Преобразованная сеть вложенных УП в сеть дуплетов *)\n712\tDefinition testDupletList := ReferenceListListToDupletList testReferenceListList.\n713\t\n714\t(* Вычисление преобразованной сети вложенных УП в сеть дуплетов *)\n715\tCompute testDupletList.\n716\t(* Ожидается результат:\n717\t {(121, 1), (21, 2), (1343, 2),\n718\t (12, 4), (23, 4),\n719\t (34, 5),\n720\t (121, 7), (21, 8), (1343, 8),\n721\t (12, 10), (23, 10),\n722\t (34, 11)} *)\n723\t\n724\t(* Вычисление преобразования сети вложенных УП в сеть дуплетов и обратно в testReferenceListList *)\n725\tCompute DupletListToReferenceListList testDupletList.\n726\t(* Ожидается результат:\n727\t {{121, 21, 1343}, {12, 23}, {34}, {121, 21, 1343}, {12, 23}, {34}} *)\n728\t\n729\t(* Вычисление смещения вложенных УП в сети дуплетов по их порядковому номеру *)\n730\tCompute DupletListOffsetReferenceList testDupletList 0. (* Ожидается результат: 0 *)\n731\tCompute DupletListOffsetReferenceList testDupletList 1. (* Ожидается результат: 3 *)\n732\tCompute DupletListOffsetReferenceList testDupletList 2. (* Ожидается результат: 5 *)\n733\tCompute DupletListOffsetReferenceList testDupletList 3. (* Ожидается результат: 6 *)\n734\tCompute DupletListOffsetReferenceList testDupletList 4. (* Ожидается результат: 9 *)\n735\tCompute DupletListOffsetReferenceList testDupletList 5. (* Ожидается результат: 11 *)\n736\tCompute DupletListOffsetReferenceList testDupletList 6. (* Ожидается результат: 12 *)\n737\tCompute DupletListOffsetReferenceList testDupletList 7. (* Ожидается результат: 12 *)\n738\t\n739\t(* Определяем трёхмерную сеть как последовательность кортежей длины 3 *)\n740\tDefinition testTupleList : NetworkTupleList 3 :=\n741\t { [0; 0; 0], [1; 1; 2], [2; 4; 0], [3; 0; 5], [4; 1; 1], [0; 0; 0] }.\n742\t\n743\t(* Преобразованная трёхмерная сеть в сеть дуплетов через сеть вложенных УП *)\n744\tDefinition testTuplesToDupletList : NetworkDupletList := TupleListToDupletList testTupleList.\n745\t\n746\t(* Вычисление трёхмерной сети преобразованной в сеть дуплетов через сеть вложенных УП *)\n747\tCompute testTuplesToDupletList.\n748\t(* Ожидается результат:\n749\t{ (0, 1), (0, 2), (0, 2),\n750\t (1, 4), (1, 5), (2, 5),\n751\t (2, 7), (4, 8), (0, 8),\n752\t (3, 10), (0, 11), (5, 11),\n753\t (4, 13), (1, 14), (1, 14),\n754\t (0, 16), (0, 17), (0, 17)} *)\n755\t\n756\t(* Преобразованная трёхмерная сеть в сеть дуплетов через сеть вложенных УП и обратно в трёхмерную сеть *)\n757\tDefinition resultTuplesNetwork : NetworkTupleList 3 :=\n758\t ReferenceListListToTupleList (DupletListToReferenceListList testTuplesToDupletList).\n759\t\n760\t(* Итоговая проверка эквивалентности сетей *)\n761\tCompute resultTuplesNetwork.\n762\t(* Ожидается результат:\n763\t { [0; 0; 0], [1; 1; 2], [2; 4; 0], [3; 0; 5], [4; 1; 1], [0; 0; 0] } *)\n764\t```\n765\t\n766\t##### Lean\n767\t\n768\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/NetworkExamples.lean)\n769\t\n770\t```lean\n771\t-- Трёхмерная сеть\n772\tdef complexExampleNetwork : NetworkTupleFunction 3 :=\n773\t fun id => match id with\n774\t | 0 => #v[0, 0, 0]\n775\t | 1 => #v[1, 1, 2]\n776\t | 2 => #v[2, 4, 0]\n777\t | 3 => #v[3, 0, 5]\n778\t | 4 => #v[4, 1, 1]\n779\t | _ => #v[0, 0, 0]\n780\t\n781\t-- Преобразование вложенных УП в сеть дуплетов\n782\t#eval ReferenceListToDupletList [121, 21, 1343]\n783\t-- Должно вернуть: [(121, 1), (21, 2), (1343, 2)]\n784\t\n785\t-- Добавление вложенных УП в сеть дуплетов\n786\t#eval AddReferenceListToDupletList [(121, 1), (21, 2), (1343, 2)] [12, 23, 34]\n787\t-- Ожидается результат: [(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)]\n788\t\n789\t-- Преобразование сети дуплетов во вложенные УП\n790\t#eval DupletListToReferenceList [(121, 1), (21, 2), (1343, 2)]\n791\t-- Ожидается результат: [121, 21, 1343]\n792\t\n793\t-- Определяем сеть вложенных УП\n794\tdef testReferenceListList : NetworkReferenceListList :=\n795\t [[121, 21, 1343], [12, 23], [34], [121, 21, 1343], [12, 23], [34]]\n796\t\n797\t-- Преобразование сети вложенных УП в сеть дуплетов и обратно\n798\t#eval DupletListToReferenceListList (ReferenceListListToDupletList testReferenceListList)\n799\t-- Ожидается результат:\n800\t-- [[121, 21, 1343], [12, 23], [34], [121, 21, 1343], [12, 23], [34]]\n801\t\n802\t-- Определяем трёхмерную сеть как последовательность кортежей длины 3\n803\tdef testTupleList : NetworkTupleList 3 :=\n804\t [#v[0, 0, 0], #v[1, 1, 2], #v[2, 4, 0], #v[3, 0, 5], #v[4, 1, 1], #v[0, 0, 0]]\n805\t\n806\t-- Итоговая проверка эквивалентности сетей\n807\tdef resultTuplesNetwork : NetworkTupleList 3 :=\n808\t ReferenceListListToTupleList (DupletListToReferenceListList (TupleListToDupletList testTupleList))\n809\t\n810\t#eval resultTuplesNetwork.map (·.toList)\n811\t-- Ожидается результат:\n812\t-- [[0, 0, 0], [1, 1, 2], [2, 4, 0], [3, 0, 5], [4, 1, 1], [0, 0, 0]]\n813\t```\n814\t" | |
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| "content": "1\t(**\n2\t MetaDefinitions.v - Теория связей как мета-теория: переопределение базовых\n3\t терминов теории связей через последовательности и множества, которые сами\n4\t определены в терминах связей.\n5\t\n6\t Этот файл замыкает цикл определений:\n7\t 1. Ссылки (Reference) определены как натуральные числа (ℕ₀)\n8\t 2. Последовательности определены как деревья связей-дуплетов в сети\n9\t 3. Множества определены как упорядоченные уникальные последовательности связей\n10\t 4. Теперь мы переопределяем ссылки через множества и последовательности\n11\t\n12\t Это демонстрирует, что теория связей может определять свои собственные\n13\t начальные термины в терминах структур, построенных из связей,\n14\t что делает её мета-теорией — теорией, определяющей саму себя.\n15\t*)\n16\t\n17\tRequire Import PeanoNat.\n18\tRequire Import Coq.Init.Nat.\n19\tRequire Import List.\n20\tRequire Import Coq.Init.Datatypes.\n21\tImport ListNotations.\n22\tRequire Import NetworkDefinitions.\n23\tRequire Import NetworkConversions.\n24\tRequire Import SequenceDefinitions.\n25\tRequire Import SetDefinitions.\n26\tRequire Import SetSequenceEquivalence.\n27\t\n28\t(** * Мета-определения: ссылки через последовательности и множества *)\n29\t\n30\t(**\n31\t Мета-ссылка (MetaReference) — это элемент множества ссылок,\n32\t определённого как упорядоченное уникальное дерево связей-дуплетов\n33\t в сети.\n34\t\n35\t В исходном определении: Reference := ℕ₀ (натуральное число)\n36\t В мета-определении: MetaReference — это лист дерева, хранимого в сети,\n37\t где LinkSet — это Reference (ссылка на корень дерева в сети).\n38\t\n39\t Таким образом, мета-ссылка определена через связи, которые определены\n40\t через мета-ссылки — цикл замкнут.\n41\t*)\n42\t\n43\t(** Мета-ссылка: элемент множества, определённого через деревья связей *)\n44\tDefinition MetaReference := Reference.\n45\t\n46\t(** Мета-пространство ссылок: множество всех допустимых мета-ссылок,\n47\t представленное как ссылка на корень дерева в сети *)\n48\tDefinition MetaReferenceSpace := LinkSet.\n49\t\n50\t(** Мета-дуплет: пара мета-ссылок — это связь-дуплет *)\n51\tDefinition MetaDuplet := prod MetaReference MetaReference.\n52\t\n53\t(** Мета-сеть: функция из мета-ссылок в мета-дуплеты *)\n54\tDefinition MetaNetwork := MetaReference -> MetaDuplet.\n55\t\n56\t(** Мета-сеть в виде последовательности мета-дуплетов *)\n57\tDefinition MetaNetworkList := list MetaDuplet.\n58\t\n59\t(**\n60\t Ключевая конструкция: определение множества мета-ссылок\n61\t через деревья связей-дуплетов в сети.\n62\t\n63\t Мы берём множество натуральных чисел {0, 1, 2, ..., n} и\n64\t представляем его как LinkSet — ссылку на корень дерева в сети,\n65\t листья которого образуют упорядоченную уникальную последовательность.\n66\t*)\n67\t\n68\t(** Создание списка мета-ссылок заданного размера *)\n69\tFixpoint makeMetaReferenceSpace_ (n : nat) : list Reference :=\n70\t match n with\n71\t | 0 => [0]\n72\t | S n' => makeMetaReferenceSpace_ n' ++ [S n']\n73\t end.\n74\t\n75\t(** Создание мета-пространства ссылок — множество в сети *)\n76\tDefinition MakeMetaReferenceSpace (size : nat) (offset : nat)\n77\t : option (MetaReferenceSpace * NetworkDupletList) :=\n78\t ListToSet (makeMetaReferenceSpace_ size) offset.\n79\t\n80\t(** * Определение мета-сети через последовательности *)\n81\t\n82\t(** Преобразование мета-сети в сеть дуплетов *)\n83\tDefinition MetaNetworkToDupletList (net : MetaNetworkList) : NetworkDupletList :=\n84\t net.\n85\t\n86\t(**\n87\t ОСНОВНАЯ ТЕОРЕМА МЕТА-ТЕОРИИ:\n88\t\n89\t Любая мета-сеть (определённая через последовательности и множества,\n90\t которые определены через связи-дуплеты) может быть представлена как обычная\n91\t сеть дуплетов.\n92\t\n93\t Это формально показывает, что мета-определения совместимы с исходными:\n94\t теория связей может определять свои термины через себя без противоречий.\n95\t*)\n96\tTheorem meta_network_is_duplet_network :\n97\t forall (net : MetaNetworkList),\n98\t MetaNetworkToDupletList net = net.\n99\tProof.\n100\t intro net.\n101\t unfold MetaNetworkToDupletList.\n102\t reflexivity.\n103\tQed.\n104\t\n105\t(** Мета-пространство ссылок содержит упорядоченные уникальные элементы *)\n106\tTheorem meta_reference_space_elements_valid : forall (size : nat),\n107\t IsOrderedUniqueSequence (toOrderedUnique (makeMetaReferenceSpace_ size)).\n108\tProof.\n109\t intro size.\n110\t apply toOrderedUnique_is_ascending.\n111\tQed.\n112\t\n113\t(** * Демонстрация цикла определений *)\n114\t\n115\t(**\n116\t Цикл определений теории связей как мета-теории:\n117\t\n118\t Уровень 0 (базовый):\n119\t Reference := ℕ₀\n120\t Duplet := Reference × Reference\n121\t Network := Reference → Duplet\n122\t\n123\t Уровень 1 (последовательности через связи):\n124\t LinkSequence := Reference (ссылка на корень дерева дуплетов в сети)\n125\t Дерево дуплетов хранится в NetworkDupletList\n126\t Варианты: сбалансированный, левая/правая лестница\n127\t\n128\t Уровень 2 (множества через последовательности):\n129\t LinkSet := Reference (ссылка на корень, с инвариантом IsOrderedUniqueSequence на листьях)\n130\t\n131\t Уровень 3 (мета-определения через множества):\n132\t MetaReference := Reference\n133\t MetaDuplet := MetaReference × MetaReference\n134\t MetaNetwork := MetaReference → MetaDuplet\n135\t\n136\t Уровень 3 структурно идентичен Уровню 0,\n137\t но определён через конструкции Уровней 1 и 2,\n138\t которые сами определены через конструкции Уровня 0.\n139\t\n140\t Это замыкает цикл: теория связей определяет себя через себя.\n141\t\n142\t Все мета-термины описываются через связи:\n143\t - Ссылка (Reference) — это связь нулевой размерности: r ≡ (r, ()),\n144\t поскольку при n = 0 кортеж T₀ = R⁰ пуст, и L = R × T₀ ≡ R.\n145\t - MetaReference — это связь произвольной размерности с самоссылкой.\n146\t - MetaDuplet — это двумерная связь (пара мета-ссылок).\n147\t - MetaNetwork — это связь связей (последовательность, которая сама является связью).\n148\t\n149\t Таким образом, все термины теории связей описываются как связи —\n150\t термин, введённый в самой теории. Теория полностью самодостаточна.\n151\t*)\n152\t\n153\t(** Пример: создание мета-пространства из 5 ссылок *)\n154\tCompute MakeMetaReferenceSpace 4 10.\n155\t(* Множество {0, 1, 2, 3, 4} в виде дуплетов в сети *)\n156\t\n157\t(** Пример: создание мета-сети *)\n158\tDefinition exampleMetaNetwork : MetaNetworkList :=\n159\t [(1, 2); (2, 3); (3, 1)].\n160\t\n161\t(** Преобразование мета-сети в дуплеты *)\n162\tCompute MetaNetworkToDupletList exampleMetaNetwork.\n163\t(* Ожидается: [(1, 2), (2, 3), (3, 1)] — структурно идентично исходной сети *)\n164\t\n165\t(** * Проверки верификации *)\n166\t\n167\tCheck meta_network_is_duplet_network.\n168\tCheck meta_reference_space_elements_valid.\n169\t\n170\t(** Все мета-определения успешно верифицированы! *)\n171\t" | |
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| "content": "1\t(**\n2\t SequenceDefinitions.v - Определение последовательностей в терминах связей.\n3\t\n4\t Последовательность — это вложенная структура связей-дуплетов,\n5\t хранимая в сети. Последовательность идентифицируется\n6\t ссылкой (Reference) на корень дерева дуплетов.\n7\t\n8\t Каждый внутренний узел дерева — это связь-дуплет (ссылка_на_левое, ссылка_на_правое).\n9\t Листья — это связи-ссылающиеся-на-себя (элемент, элемент).\n10\t\n11\t Пример: [1, 2, 3, 4] = ((1, 2), (3, 4))\n12\t Хранится в сети как:\n13\t (5: 1, 2) — дуплет для пары (1, 2)\n14\t (6: 3, 4) — дуплет для пары (3, 4)\n15\t (7: 5, 6) — дуплет-корень, ссылающийся на поддеревья\n16\t Последовательность = ссылка 7\n17\t\n18\t В нотации связей (https://github.com/link-foundation/links-notation):\n19\t (5: 1 2) (6: 3 4) (7: 5 6)\n20\t\n21\t Другие варианты:\n22\t [1, 2, 3, 4] = (((1, 2), 3), 4) — левая лестница\n23\t [1, 2, 3, 4] = (1, (2, (3, 4))) — правая лестница\n24\t\n25\t Адаптировано из: https://github.com/linksplatform/Documentation/tree/main/doc/LinksTheoryManuscriptDraft/05%20Sequences\n26\t См. также: https://github.com/linksplatform/Data.Doublets.Sequences/blob/main/csharp/Platform.Data.Doublets.Sequences/Converters/BalancedVariantConverter.cs\n27\t*)\n28\t\n29\tRequire Import PeanoNat.\n30\tRequire Import Coq.Init.Nat.\n31\tRequire Import List.\n32\tRequire Import Coq.Init.Datatypes.\n33\tImport ListNotations.\n34\tRequire Import NetworkDefinitions.\n35\tRequire Import NetworkConversions.\n36\t\n37\t(** * Последовательность — это ссылка на корень дерева в сети *)\n38\t\n39\t(** Последовательность идентифицируется ссылкой (Reference) на корень дерева дуплетов,\n40\t хранимого в сети. Это не отдельный тип — это просто Reference. *)\n41\tDefinition LinkSequence := Reference.\n42\t\n43\t(** * Вспомогательное дерево для построения последовательностей\n44\t\n45\t LinkTree используется как промежуточная структура для алгоритмов\n46\t построения деревьев (сбалансированный вариант, лестницы).\n47\t Конечный результат всегда записывается в сеть дуплетов. *)\n48\t\n49\tInductive LinkTree : Type :=\n50\t | Leaf : Reference -> LinkTree\n51\t | Node : LinkTree -> LinkTree -> LinkTree.\n52\t\n53\t(** * Запись дерева в сеть дуплетов *)\n54\t\n55\t(** Запись дерева в сеть дуплетов.\n56\t Каждый узел Node записывается как дуплет (ссылка_на_левое, ссылка_на_правое).\n57\t Листья — это элементы последовательности (не записываются как отдельные дуплеты).\n58\t Возвращает пару: (сеть дуплетов, следующее свободное смещение). *)\n59\tFixpoint TreeToDupletList_ (t : LinkTree) (offset : nat)\n60\t : NetworkDupletList * nat :=\n61\t match t with\n62\t | Leaf _ => ([], offset)\n63\t | Node l r =>\n64\t let cur := offset in\n65\t let '(left_net, left_next) := TreeToDupletList_ l (S offset) in\n66\t let left_root := match l with Leaf x => x | Node _ _ => S offset end in\n67\t let '(right_net, right_next) := TreeToDupletList_ r left_next in\n68\t let right_root := match r with Leaf x => x | Node _ _ => left_next end in\n69\t ((left_root, right_root) :: left_net ++ right_net, right_next)\n70\t end.\n71\t\n72\tDefinition TreeToDupletList (t : LinkTree) (offset : nat) : NetworkDupletList :=\n73\t fst (TreeToDupletList_ t offset).\n74\t\n75\t(** Получение ссылки на корень дерева (= последовательность) *)\n76\tDefinition TreeToSequence (t : LinkTree) (offset : nat) : LinkSequence :=\n77\t match t with\n78\t | Leaf x => x\n79\t | Node _ _ => offset\n80\t end.\n81\t\n82\t(** Получение листьев дерева (элементов последовательности) слева направо *)\n83\tFixpoint TreeToList (t : LinkTree) : list Reference :=\n84\t match t with\n85\t | Leaf x => [x]\n86\t | Node l r => TreeToList l ++ TreeToList r\n87\t end.\n88\t\n89\t(** * Алгоритмы создания последовательностей *)\n90\t\n91\t(** Правая лестница: (1, (2, (3, 4)))\n92\t Каждый следующий элемент вкладывается вправо. *)\n93\tFixpoint ListToRightStaircase (l : list Reference) : option LinkTree :=\n94\t match l with\n95\t | [] => None\n96\t | [x] => Some (Leaf x)\n97\t | x :: rest =>\n98\t match ListToRightStaircase rest with\n99\t | None => None\n100\t | Some t => Some (Node (Leaf x) t)\n101\t end\n102\t end.\n103\t\n104\t(** Левая лестница: (((1, 2), 3), 4)\n105\t Каждый следующий элемент вкладывается влево. *)\n106\tFixpoint ListToLeftStaircase_ (acc : LinkTree) (l : list Reference) : LinkTree :=\n107\t match l with\n108\t | [] => acc\n109\t | x :: rest => ListToLeftStaircase_ (Node acc (Leaf x)) rest\n110\t end.\n111\t\n112\tDefinition ListToLeftStaircase (l : list Reference) : option LinkTree :=\n113\t match l with\n114\t | [] => None\n115\t | [x] => Some (Leaf x)\n116\t | x :: rest => Some (ListToLeftStaircase_ (Leaf x) rest)\n117\t end.\n118\t\n119\t(** Сбалансированный вариант: ((1, 2), (3, 4))\n120\t Список делится пополам, каждая половина рекурсивно превращается в дерево.\n121\t Адаптировано из BalancedVariantConverter.\n122\t Использует fuel-параметр для гарантии завершения. *)\n123\tFixpoint ListToBalancedTree_ (l : list Reference) (fuel : nat) : option LinkTree :=\n124\t match fuel with\n125\t | 0 => None\n126\t | S fuel' =>\n127\t match l with\n128\t | [] => None\n129\t | [x] => Some (Leaf x)\n130\t | _ =>\n131\t let mid := length l / 2 in\n132\t let left_part := firstn mid l in\n133\t let right_part := skipn mid l in\n134\t match ListToBalancedTree_ left_part fuel', ListToBalancedTree_ right_part fuel' with\n135\t | Some lt, Some rt => Some (Node lt rt)\n136\t | _, _ => None\n137\t end\n138\t end\n139\t end.\n140\t\n141\tDefinition ListToBalancedTree (l : list Reference) : option LinkTree :=\n142\t ListToBalancedTree_ l (S (length l)).\n143\t\n144\t(** * Полное преобразование: список → сеть дуплетов + ссылка на корень *)\n145\t\n146\t(** Преобразование списка в последовательность (ссылку на корень) и сеть *)\n147\tDefinition ListToSequence (l : list Reference) (offset : nat)\n148\t : option (LinkSequence * NetworkDupletList) :=\n149\t match ListToBalancedTree l with\n150\t | None => None\n151\t | Some t => Some (TreeToSequence t offset, TreeToDupletList t offset)\n152\t end.\n153\t\n154\t(** Чтение последовательности из сети дуплетов *)\n155\tFixpoint ReadSequence_ (anet : NetworkDupletList) (root : Reference) (fuel : nat)\n156\t : list Reference :=\n157\t match fuel with\n158\t | 0 => []\n159\t | S fuel' =>\n160\t if root <? length anet then\n161\t let d := nth root anet (0, 0) in\n162\t let l := fst d in\n163\t let r := snd d in\n164\t ReadSequence_ anet l fuel' ++ ReadSequence_ anet r fuel'\n165\t else\n166\t [root]\n167\t end.\n168\t\n169\tDefinition ReadSequence (anet : NetworkDupletList) (root : LinkSequence) : list Reference :=\n170\t ReadSequence_ anet root (S (length anet)).\n171\t\n172\t(** * Примеры *)\n173\t\n174\t(** Пример: [1, 2, 3, 4] — сбалансированное дерево, записанное в сеть *)\n175\tCompute ListToSequence [1; 2; 3; 4] 5.\n176\t(* Ожидается: Some (5, [(5_root: 6, 7), (6: 1, 2), (7: 3, 4)])\n177\t Последовательность = ссылка 5\n178\t (5: 6, 7) — корень, указывает на поддеревья 6 и 7\n179\t (6: 1, 2) — левое поддерево\n180\t (7: 3, 4) — правое поддерево *)\n181\t\n182\t(** Пример: правая лестница *)\n183\tCompute match ListToRightStaircase [1; 2; 3; 4] with\n184\t | Some t => Some (TreeToSequence t 5, TreeToDupletList t 5)\n185\t | None => None\n186\t end.\n187\t(* (1, (2, (3, 4))) хранится как дуплеты *)\n188\t\n189\t(** Пример: левая лестница *)\n190\tCompute match ListToLeftStaircase [1; 2; 3; 4] with\n191\t | Some t => Some (TreeToSequence t 5, TreeToDupletList t 5)\n192\t | None => None\n193\t end.\n194\t(* (((1, 2), 3), 4) хранится как дуплеты *)\n195\t\n196\t(** Все три варианта содержат одни и те же элементы *)\n197\tDefinition balanced_1234 := match ListToBalancedTree [1; 2; 3; 4] with Some t => TreeToList t | None => [] end.\n198\tDefinition right_1234 := match ListToRightStaircase [1; 2; 3; 4] with Some t => TreeToList t | None => [] end.\n199\tDefinition left_1234 := match ListToLeftStaircase [1; 2; 3; 4] with Some t => TreeToList t | None => [] end.\n200\t\n201\tCompute balanced_1234. (* [1, 2, 3, 4] *)\n202\tCompute right_1234. (* [1, 2, 3, 4] *)\n203\tCompute left_1234. (* [1, 2, 3, 4] *)\n204\t\n205\t(** Пустая последовательность *)\n206\tCompute ListToSequence [] 5.\n207\t(* Ожидается: None *)\n208\t\n209\t(** Последовательность из одного элемента *)\n210\tCompute ListToSequence [42] 5.\n211\t(* Ожидается: Some (42, []) — лист не создаёт дуплетов, сама ссылка 42 и есть последовательность *)\n212\t" | |
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| "content": "1\t(**\n2\t SetDefinitions.v - Определение множеств в терминах последовательностей связей.\n3\t\n4\t Множество определяется как последовательность связей-дуплетов,\n5\t листья которой образуют строго возрастающий список без дубликатов.\n6\t\n7\t Как и последовательность, множество идентифицируется ссылкой (Reference)\n8\t на корень дерева в сети.\n9\t\n10\t Это ключевой шаг: имея последовательности, определённые через связи-дуплеты,\n11\t мы теперь определяем множества как особый вид последовательностей.\n12\t Таким образом, множества также выражены исключительно через связи.\n13\t*)\n14\t\n15\tRequire Import PeanoNat.\n16\tRequire Import Coq.Init.Nat.\n17\tRequire Import List.\n18\tRequire Import Coq.Init.Datatypes.\n19\tImport ListNotations.\n20\tRequire Import NetworkDefinitions.\n21\tRequire Import NetworkConversions.\n22\tRequire Import SequenceDefinitions.\n23\tRequire Import SetSequenceEquivalence.\n24\t\n25\t(** * Множества как упорядоченные уникальные последовательности связей *)\n26\t\n27\t(** Множество ссылок — это ссылка (Reference) на корень дерева связей\n28\t в сети, листья которого образуют строго возрастающий\n29\t список без дубликатов. Как и последовательность, множество — это Reference. *)\n30\tDefinition LinkSet := Reference.\n31\t\n32\t(** Предикат: является ли дерево связей корректным множеством\n33\t (упорядоченная уникальная последовательность листьев) *)\n34\tDefinition IsValidSetTree (s : LinkTree) : Prop :=\n35\t IsOrderedUniqueSequence (TreeToList s).\n36\t\n37\t(** Множество из одного элемента — ссылка на сам элемент *)\n38\tDefinition SingletonSet (value : Reference) : LinkSet := value.\n39\t\n40\t(** Преобразование списка ссылок в множество: сортировка, удаление дубликатов,\n41\t построение сбалансированного дерева и запись в сеть.\n42\t Возвращает ссылку на корень (= множество) и сеть дуплетов. *)\n43\tDefinition ListToSet (l : list Reference) (offset : nat)\n44\t : option (LinkSet * NetworkDupletList) :=\n45\t let sorted := toOrderedUnique l in\n46\t match ListToBalancedTree sorted with\n47\t | None => None\n48\t | Some t => Some (TreeToSequence t offset, TreeToDupletList t offset)\n49\t end.\n50\t\n51\t(** Принадлежность элемента множеству (проверка через дерево) *)\n52\tDefinition InSetTree (x : Reference) (s : LinkTree) : Prop :=\n53\t In x (TreeToList s).\n54\t\n55\t(** Подмножество: s1 ⊆ s2 (через деревья) *)\n56\tDefinition IsSubsetTree (s1 s2 : LinkTree) : Prop :=\n57\t forall x, InSetTree x s1 -> InSetTree x s2.\n58\t\n59\t(** * Теорема: ListToSet всегда содержит упорядоченные уникальные элементы *)\n60\t\n61\tTheorem ListToSet_elements_valid : forall l : list Reference,\n62\t IsOrderedUniqueSequence (toOrderedUnique l).\n63\tProof.\n64\t intro l.\n65\t unfold IsOrderedUniqueSequence.\n66\t apply toOrderedUnique_is_ascending.\n67\tQed.\n68\t\n69\t(** Элемент принадлежит множеству тогда и только тогда, когда он принадлежит исходному списку *)\n70\tTheorem ListToSet_membership : forall (x : Reference) (l : list Reference),\n71\t In x (toOrderedUnique l) <-> In x l.\n72\tProof.\n73\t intros x l.\n74\t apply mem_toOrderedUnique.\n75\tQed.\n76\t\n77\t(** * Примеры *)\n78\t\n79\t(** Создание множества из списка с дубликатами *)\n80\tCompute ListToSet [3; 1; 4; 1; 5; 9; 2; 6; 5; 3] 10.\n81\t(* Множество {1, 2, 3, 4, 5, 6, 9} в сети *)\n82\t\n83\t(** Множество из одного элемента *)\n84\tCompute SingletonSet 42.\n85\t" | |
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| "content": "1\tRequire Import PeanoNat.\n2\tRequire Import Coq.Init.Nat.\n3\tRequire Import Vector.\n4\tRequire Import List.\n5\tRequire Import Coq.Init.Datatypes.\n6\tImport ListNotations.\n7\tImport VectorNotations.\n8\t\n9\t(* Ссылка (Reference) — уникальный идентификатор кортежа: R ⊆ ℕ₀ *)\n10\t(* Ссылка также является связью нулевой размерности: при n = 0 связь L = R × T₀ ≡ R,\n11\t поэтому r ≡ (r, ()), и множество ссылок R является подмножеством множества связей L. *)\n12\tDefinition Reference := nat.\n13\t\n14\t(* Значение Reference по умолчанию: ноль *)\n15\tDefinition ReferenceDefault : Reference := 0.\n16\t\n17\t(* Множество кортежей ссылок длины n ∈ ℕ₀: TupleOfReferences ⊆ Rⁿ *)\n18\tDefinition TupleOfReferences (n : nat) := t Reference n.\n19\t\n20\t(* Значение TupleOfReferences по умолчанию *)\n21\tDefinition TupleOfReferencesDefault (n : nat) : TupleOfReferences n := Vector.const ReferenceDefault n.\n22\t\n23\t(* Множество всех связей: Link = Reference × TupleOfReferences\n24\t Link (связь) — это пара из ссылки и кортежа ссылок *)\n25\tDefinition Link (n : nat) := prod Reference (TupleOfReferences n).\n26\t\n27\t(* Сеть кортежей длины n (или n-мерная сеть) из семейства функций {Nⁿ : Reference → TupleOfReferences} *)\n28\tDefinition NetworkTupleFunction (n : nat) := Reference -> TupleOfReferences n.\n29\t\n30\t(* Сеть кортежей длины n (или n-мерная сеть) в виде последовательности *)\n31\tDefinition NetworkTupleList (n : nat) := list (TupleOfReferences n).\n32\t\n33\t(* Список ссылок: ReferenceList ⊆ List(R) *)\n34\tDefinition ReferenceList := list Reference.\n35\t\n36\t(* Сеть списков ссылок: N^{list} : Reference → ReferenceList *)\n37\tDefinition NetworkReferenceListFunction := Reference -> ReferenceList.\n38\t\n39\t(* Сеть списков ссылок в виде последовательности списков ссылок *)\n40\tDefinition NetworkReferenceListList := list ReferenceList.\n41\t\n42\t(* Дуплет ссылок: упорядоченная пара (Reference, Reference).\n43\t В нотации связей (https://github.com/link-foundation/links-notation):\n44\t дуплет записывается как (id: from to), например (3: 1 2) *)\n45\tDefinition Duplet := prod Reference Reference.\n46\t\n47\t(* Значение Duplet по умолчанию: пара из двух ReferenceDefault, используется для обозначения пустого дуплета *)\n48\tDefinition DupletDefault : Duplet := (ReferenceDefault, ReferenceDefault).\n49\t\n50\t(* Сеть дуплетов (или двумерная сеть): N² : Reference → Reference² *)\n51\tDefinition NetworkDupletFunction := Reference -> Duplet.\n52\t\n53\t(* Сеть дуплетов (или двумерная сеть) в виде последовательности дуплетов *)\n54\tDefinition NetworkDupletList := list Duplet.\n55\t\n56\t(** * Сеть всех сетей *)\n57\t\n58\t(**\n59\t Сеть всех сетей (Ω) — пространство, в котором каждая возможная сеть\n60\t произвольной размерности и размера получает уникальную ссылку.\n61\t\n62\t 𝒩 = ⋃_{n ∈ ℕ₀} ⋃_{s ∈ ℕ₀} {Nⁿ размера s}\n63\t Ω: R → 𝒩\n64\t\n65\t Поскольку в Rocq мы не можем напрямую определить тип, параметризованный\n66\t произвольным n на уровне значений, мы представляем элемент 𝒩 как запись\n67\t (размерность, список кортежей переменной длины), используя списки ссылок\n68\t для представления кортежей произвольной размерности.\n69\t*)\n70\t\n71\t(* Элемент сети произвольной размерности: пара (размерность, список кортежей).\n72\t Кортеж переменной длины представлен как ReferenceList *)\n73\tRecord AnyNetwork := mkAnyNetwork {\n74\t dimension : nat;\n75\t tuples : list ReferenceList\n76\t}.\n77\t\n78\t(* Сеть всех сетей: отображение ссылок на сети произвольной размерности и размера *)\n79\t(* Ω: R → 𝒩 *)\n80\tDefinition NetworkOfAllNetworks := Reference -> AnyNetwork.\n81\t\n82\t(* Сеть всех сетей в виде последовательности (для конечного фрагмента) *)\n83\tDefinition NetworkOfAllNetworksList := list AnyNetwork.\n84\t\n85\t(* Пустая сеть нулевой размерности и размера — наименьший элемент 𝒩 *)\n86\tDefinition emptyNetwork : AnyNetwork := mkAnyNetwork 0 nil.\n87\t\n88\t(* Размер сети (количество кортежей) *)\n89\tDefinition anyNetworkSize (net : AnyNetwork) : nat :=\n90\t length (tuples net).\n91\t\n92\t(* Проверка корректности: все кортежи имеют длину, равную размерности *)\n93\tDefinition anyNetworkIsWellFormed (net : AnyNetwork) : bool :=\n94\t forallb (fun t => Nat.eqb (length t) (dimension net)) (tuples net).\n95\t\n96\t(* Пример: представление дуплетной сети N² размера 3 как элемента 𝒩 *)\n97\tDefinition exampleAnyNetwork : AnyNetwork :=\n98\t mkAnyNetwork 2 ((cons 1 (cons 1 nil)) :: (cons 2 (cons 2 nil)) :: (cons 1 (cons 2 nil)) :: nil).\n99\t" | |
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| "content": "1\tRequire Import PeanoNat.\n2\tRequire Import Coq.Init.Nat.\n3\tRequire Import Vector.\n4\tRequire Import List.\n5\tRequire Import Coq.Init.Datatypes.\n6\tImport ListNotations.\n7\tImport VectorNotations.\n8\tRequire Import NetworkDefinitions.\n9\t\n10\t(* Функция преобразования TupleOfReferences в ReferenceList *)\n11\tFixpoint TupleOfReferencesToReferenceList {n : nat} (v : TupleOfReferences n) : ReferenceList :=\n12\t match v with\n13\t | Vector.nil _ => List.nil\n14\t | Vector.cons _ h _ t => List.cons h (TupleOfReferencesToReferenceList t)\n15\t end.\n16\t\n17\t(* Функция преобразования NetworkTupleFunction в NetworkReferenceListFunction *)\n18\tDefinition TupleFunctionToReferenceListFunction {n : nat} (a: NetworkTupleFunction n) : NetworkReferenceListFunction :=\n19\t fun id => TupleOfReferencesToReferenceList (a id).\n20\t\n21\t(* Функция преобразования NetworkTupleList в NetworkReferenceListList *)\n22\tDefinition TupleListToReferenceListList {n: nat} (net: NetworkTupleList n) : NetworkReferenceListList :=\n23\t map TupleOfReferencesToReferenceList net.\n24\t\n25\t(* Функция преобразования ReferenceList в TupleOfReferences, возвращающая option *)\n26\tFixpoint ReferenceListToTupleOfReferencesOption (n: nat) (p: ReferenceList) : option (TupleOfReferences n) :=\n27\t match n, p with\n28\t | 0, List.nil => Some (Vector.nil nat)\n29\t | S n', List.cons f p' =>\n30\t match ReferenceListToTupleOfReferencesOption n' p' with\n31\t | None => None\n32\t | Some t => Some (Vector.cons nat f n' t)\n33\t end\n34\t | _, _ => None\n35\t end.\n36\t\n37\t(* Функция преобразования ReferenceList в TupleOfReferences с использованием TupleOfReferencesDefault *)\n38\tDefinition ReferenceListToTupleOfReferences (n: nat) (p: ReferenceList) : TupleOfReferences n :=\n39\t match ReferenceListToTupleOfReferencesOption n p with\n40\t | None => TupleOfReferencesDefault n\n41\t | Some t => t\n42\t end.\n43\t\n44\t(* Функция преобразования NetworkReferenceListFunction в NetworkTupleFunction *)\n45\tDefinition ReferenceListFunctionToTupleFunction { n: nat } (net: NetworkReferenceListFunction) : NetworkTupleFunction n :=\n46\t fun id => match ReferenceListToTupleOfReferencesOption n (net id) with\n47\t | Some t => t\n48\t | None => TupleOfReferencesDefault n\n49\t end.\n50\t\n51\t(* Функция преобразования NetworkReferenceListList в NetworkTupleList *)\n52\tDefinition ReferenceListListToTupleList {n: nat} (net : NetworkReferenceListList) : NetworkTupleList n :=\n53\t map (ReferenceListToTupleOfReferences n) net.\n54\t\n55\t(* Функция преобразования ReferenceList в NetworkDupletList со смещением индексации *)\n56\tFixpoint ReferenceListToDupletList_ (offset: nat) (np: ReferenceList) : NetworkDupletList :=\n57\t match np with\n58\t | nil => nil\n59\t | cons h nil => cons (h, offset) nil\n60\t | cons h t => cons (h, S offset) (ReferenceListToDupletList_ (S offset) t)\n61\t end.\n62\t\n63\t(* Функция преобразования ReferenceList в NetworkDupletList *)\n64\tDefinition ReferenceListToDupletList (np: ReferenceList) : NetworkDupletList := ReferenceListToDupletList_ 0 np.\n65\t\n66\t(* Функция добавления ReferenceList в хвост NetworkDupletList *)\n67\tDefinition AddReferenceListToDupletList (anet: NetworkDupletList) (np: ReferenceList) : NetworkDupletList :=\n68\t app anet (ReferenceListToDupletList_ (length anet) np).\n69\t\n70\t(* Функция отрезает голову N² и возвращает хвост начиная с offset *)\n71\tFixpoint DupletListBehead (anet: NetworkDupletList) (offset : nat) : NetworkDupletList :=\n72\t match offset with\n73\t | 0 => anet\n74\t | S n' =>\n75\t match anet with\n76\t | nil => nil\n77\t | cons h t => DupletListBehead t n'\n78\t end\n79\t end.\n80\t\n81\t(* Функция преобразования NetworkDupletList в ReferenceList с индексацией в начале NetworkDupletList начиная с offset *)\n82\tFixpoint DupletListToReferenceList_ (anet: NetworkDupletList) (offset: nat) (index: nat): ReferenceList :=\n83\t match anet with\n84\t | nil => nil\n85\t | cons (x, next_index) tail_anet =>\n86\t if offset =? index then\n87\t cons x (DupletListToReferenceList_ tail_anet (S offset) next_index)\n88\t else\n89\t DupletListToReferenceList_ tail_anet (S offset) index\n90\t end.\n91\t\n92\t(* Функция чтения ReferenceList из NetworkDupletList по индексу дуплета *)\n93\tDefinition DupletListReadReferenceList (anet: NetworkDupletList) (index: nat) : ReferenceList :=\n94\t DupletListToReferenceList_ anet 0 index.\n95\t\n96\t(* Функция преобразования NetworkDupletList в ReferenceList начиная с головы списка сети *)\n97\tDefinition DupletListToReferenceList (anet: NetworkDupletList) : ReferenceList := DupletListReadReferenceList anet 0.\n98\t\n99\t(*\n100\t Теперь всё готово для преобразования сети вложенных упорядоченных пар N^{list} : Reference → ReferenceList\n101\t в сеть дуплетов N² : Reference → Reference².\n102\t\n103\t Данное преобразование можно делать по-разному: с сохранением исходных ссылок на кортежи\n104\t либо с переиндексацией. Переиндексацию можно не делать, если написать дополнительную функцию для\n105\t сети дуплетов, которая возвращает вложенную упорядоченную пару по её ссылке.\n106\t*)\n107\t\n108\t(* Функция добавления NetworkReferenceListList в NetworkDupletList *)\n109\tFixpoint AddReferenceListListToDupletList (anetd: NetworkDupletList) (anetl: NetworkReferenceListList) : NetworkDupletList :=\n110\t match anetl with\n111\t | nil => anetd\n112\t | cons h t => AddReferenceListListToDupletList (AddReferenceListToDupletList anetd h) t\n113\t end.\n114\t\n115\t(* Функция преобразования NetworkReferenceListList в NetworkDupletList *)\n116\tDefinition ReferenceListListToDupletList (anetl: NetworkReferenceListList) : NetworkDupletList :=\n117\t match anetl with\n118\t | nil => nil\n119\t | cons h t => AddReferenceListListToDupletList (ReferenceListToDupletList h) t\n120\t end.\n121\t\n122\t(* Функция поиска ReferenceList в хвосте NetworkDupletList начинающемуся с offset по её порядковому номеру.\n123\t Возвращает offset ReferenceList. *)\n124\tFixpoint DupletListOffsetReferenceList_ (anet: NetworkDupletList) (offset: nat) (index: nat) : nat :=\n125\t match anet with\n126\t | nil => offset + (length anet)\n127\t | cons (_, next_index) tail_anet =>\n128\t match index with\n129\t | O => offset\n130\t | S index' =>\n131\t if offset =? next_index then\n132\t DupletListOffsetReferenceList_ tail_anet (S offset) index'\n133\t else\n134\t DupletListOffsetReferenceList_ tail_anet (S offset) index\n135\t end\n136\t end.\n137\t\n138\t(* Функция поиска ReferenceList в NetworkDupletList по её порядковому номеру. Возвращает offset ReferenceList. *)\n139\tDefinition DupletListOffsetReferenceList (anet: NetworkDupletList) (index: nat) : nat :=\n140\t DupletListOffsetReferenceList_ anet 0 index.\n141\t\n142\t(* Функция преобразования NetworkTupleList в NetworkDupletList *)\n143\tDefinition TupleListToDupletList {n : nat} (anetv: NetworkTupleList n) : NetworkDupletList :=\n144\t ReferenceListListToDupletList (TupleListToReferenceListList anetv).\n145\t\n146\t(*\n147\t Теперь всё готово для преобразования сети дуплетов N² : Reference → Reference²\n148\t в сеть вложенных упорядоченных пар N^{list} : Reference → ReferenceList.\n149\t\n150\t Данное преобразование будем делать с сохранением исходных ссылок на кортежи.\n151\t Переиндексацию можно не делать, потому что есть функция DupletListOffsetReferenceList для\n152\t сети дуплетов, которая возвращает смещение вложенной УП по ссылке на неё.\n153\t*)\n154\t\n155\t(* Функция отрезает первую ReferenceList из NetworkDupletList и возвращает хвост *)\n156\tFixpoint DupletListBeheadReferenceList (anet: NetworkDupletList) (offset: nat) : NetworkDupletList :=\n157\t match anet with\n158\t | nil => nil\n159\t | cons (_, next_index) tail_anet =>\n160\t if offset =? next_index then (* конец ReferenceList *)\n161\t tail_anet\n162\t else (* ещё не конец ReferenceList *)\n163\t DupletListBeheadReferenceList tail_anet (S offset)\n164\t end.\n165\t\n166\t(* Функция преобразования ReferenceList и NetworkDupletList со смещения offset в NetworkReferenceListList *)\n167\tFixpoint DupletListToReferenceListList_ (anetd: NetworkDupletList) (np: ReferenceList) (offset: nat) : NetworkReferenceListList :=\n168\t match anetd with\n169\t | nil => nil (* отбрасываем ReferenceList даже если она недостроена *)\n170\t | cons (x, next_index) tail_anet =>\n171\t if offset =? next_index then (* конец ReferenceList, переходим к следующей ReferenceList *)\n172\t cons (app np (cons x nil)) (DupletListToReferenceListList_ tail_anet nil (S offset))\n173\t else (* ещё не конец ReferenceList, парсим сеть дуплетов дальше *)\n174\t DupletListToReferenceListList_ tail_anet (app np (cons x nil)) (S offset)\n175\t end.\n176\t\n177\t(* Функция преобразования NetworkDupletList в NetworkReferenceListList *)\n178\tDefinition DupletListToReferenceListList (anetd: NetworkDupletList) : NetworkReferenceListList :=\n179\t DupletListToReferenceListList_ anetd nil ReferenceDefault.\n180\t" | |
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| "content": "1\t(**\n2\t SetSequenceEquivalence.v - Формальное доказательство того, что конечное множество\n3\t эквивалентно упорядоченной последовательности с уникальными элементами.\n4\t\n5\t Основная теорема: Для любого конечного множества натуральных чисел существует\n6\t единственная строго возрастающая последовательность (список без дубликатов,\n7\t отсортированный по возрастанию), содержащая в точности те же элементы.\n8\t\n9\t Адаптировано из: https://github.com/konard/subset-sum/tree/main/proofs/set_sequence_equivalence\n10\t*)\n11\t\n12\tFrom Stdlib Require Import Arith.\n13\tFrom Stdlib Require Import List.\n14\tFrom Stdlib Require Import Lia.\n15\tFrom Stdlib Require Import Bool.\n16\tImport ListNotations.\n17\t\n18\t(** * Определения *)\n19\t\n20\t(** Список является строго возрастающим, если каждый элемент строго меньше следующего *)\n21\tFixpoint StrictlyAscending (l : list nat) : Prop :=\n22\t match l with\n23\t | [] => True\n24\t | [_] => True\n25\t | x :: ((y :: _) as rest) => x < y /\\ StrictlyAscending rest\n26\t end.\n27\t\n28\t(** Список не содержит дубликатов *)\n29\tFixpoint NoDuplicates (l : list nat) : Prop :=\n30\t match l with\n31\t | [] => True\n32\t | x :: rest => ~ In x rest /\\ NoDuplicates rest\n33\t end.\n34\t\n35\t(** Упорядоченная уникальная последовательность — это строго возрастающий список *)\n36\tDefinition IsOrderedUniqueSequence (l : list nat) : Prop :=\n37\t StrictlyAscending l.\n38\t\n39\t(** * Вспомогательные леммы *)\n40\t\n41\t(** Все элементы после позиции в строго возрастающем списке больше *)\n42\tLemma strictly_ascending_all_greater : forall l x y,\n43\t StrictlyAscending (x :: y :: l) ->\n44\t forall z, In z (y :: l) -> x < z.\n45\tProof.\n46\t induction l as [| w ws IH].\n47\t - intros x y H z Hz.\n48\t simpl in H. destruct H as [Hxy _].\n49\t simpl in Hz. destruct Hz as [Hz | Hz].\n50\t + subst. exact Hxy.\n51\t + contradiction.\n52\t - intros x y H z Hz.\n53\t simpl in H. destruct H as [Hxy Hrest].\n54\t simpl in Hz. destruct Hz as [Hz | Hz].\n55\t + subst. exact Hxy.\n56\t + assert (Hy_asc : StrictlyAscending (y :: w :: ws)).\n57\t { exact Hrest. }\n58\t assert (Hyz : y < z).\n59\t { apply (IH y w Hrest z Hz). }\n60\t lia.\n61\tQed.\n62\t\n63\t(** Строго возрастающий список не содержит дубликатов *)\n64\tTheorem strictly_ascending_implies_no_dup : forall l : list nat,\n65\t StrictlyAscending l -> NoDuplicates l.\n66\tProof.\n67\t induction l as [| x rest IH].\n68\t - intro H. simpl. exact I.\n69\t - intro H.\n70\t simpl. split.\n71\t + destruct rest as [| y ys].\n72\t * simpl. auto.\n73\t * intro Hx_in.\n74\t assert (Hxy : x < y).\n75\t { simpl in H. destruct H as [Hxy _]. exact Hxy. }\n76\t assert (Hxz : forall z, In z (y :: ys) -> x < z).\n77\t { apply strictly_ascending_all_greater. exact H. }\n78\t specialize (Hxz x Hx_in).\n79\t lia.\n80\t + destruct rest as [| y ys].\n81\t * simpl. exact I.\n82\t * simpl in H. destruct H as [_ Hrest].\n83\t apply IH. exact Hrest.\n84\tQed.\n85\t\n86\t(** * Вставка в отсортированный список *)\n87\t\n88\t(** Вставка элемента в отсортированный список с сохранением порядка *)\n89\tFixpoint insertSorted (x : nat) (l : list nat) : list nat :=\n90\t match l with\n91\t | [] => [x]\n92\t | y :: rest =>\n93\t if x <? y then x :: y :: rest\n94\t else if x =? y then y :: rest (* Пропуск дубликатов *)\n95\t else y :: insertSorted x rest\n96\t end.\n97\t\n98\t(** Сортировка списка в строго возрастающем порядке (с удалением дубликатов) *)\n99\tFixpoint toOrderedUnique (l : list nat) : list nat :=\n100\t match l with\n101\t | [] => []\n102\t | x :: rest => insertSorted x (toOrderedUnique rest)\n103\t end.\n104\t\n105\t(** * Свойства insertSorted *)\n106\t\n107\t(** Получение головы строго возрастающего списка *)\n108\tLemma strictly_ascending_head : forall x y rest,\n109\t StrictlyAscending (x :: y :: rest) -> x < y.\n110\tProof.\n111\t intros x y rest H.\n112\t simpl in H. destruct H as [Hxy _]. exact Hxy.\n113\tQed.\n114\t\n115\t(** Хвост строго возрастающего списка также строго возрастающий *)\n116\tLemma strictly_ascending_tail : forall x rest,\n117\t StrictlyAscending (x :: rest) -> StrictlyAscending rest.\n118\tProof.\n119\t intros x rest H.\n120\t destruct rest as [| y ys].\n121\t - simpl. exact I.\n122\t - simpl in H. destruct H as [_ Hrest]. exact Hrest.\n123\tQed.\n124\t\n125\t(** Добавление меньшего элемента сохраняет строгое возрастание *)\n126\tLemma strictly_ascending_cons : forall x y rest,\n127\t x < y -> StrictlyAscending (y :: rest) -> StrictlyAscending (x :: y :: rest).\n128\tProof.\n129\t intros x y rest Hxy Hrest.\n130\t simpl. split.\n131\t - exact Hxy.\n132\t - exact Hrest.\n133\tQed.\n134\t\n135\t(** Голова результата insertSorted x (y :: ys) — это либо x, либо y *)\n136\tLemma insertSorted_head_cases : forall x y ys h hs,\n137\t insertSorted x (y :: ys) = h :: hs ->\n138\t h = x \\/ h = y.\n139\tProof.\n140\t intros x y ys h hs Heq.\n141\t simpl in Heq.\n142\t destruct (x <? y) eqn:Hxy.\n143\t - injection Heq as Hh Hrest.\n144\t left. symmetry. exact Hh.\n145\t - destruct (x =? y) eqn:Hxy_eq.\n146\t + injection Heq as Hh Hrest.\n147\t right. symmetry. exact Hh.\n148\t + injection Heq as Hh Hrest.\n149\t right. symmetry. exact Hh.\n150\tQed.\n151\t\n152\t(** insertSorted в непустой список даёт непустой список *)\n153\tLemma insertSorted_non_empty : forall x y ys,\n154\t exists h hs, insertSorted x (y :: ys) = h :: hs.\n155\tProof.\n156\t intros x y ys.\n157\t simpl.\n158\t destruct (x <? y) eqn:Hxy.\n159\t - exists x, (y :: ys). reflexivity.\n160\t - destruct (x =? y) eqn:Hxy_eq.\n161\t + exists y, ys. reflexivity.\n162\t + exists y, (insertSorted x ys). reflexivity.\n163\tQed.\n164\t\n165\t(** insertSorted сохраняет свойство строгого возрастания *)\n166\tTheorem insertSorted_preserves_ascending : forall l x,\n167\t StrictlyAscending l -> StrictlyAscending (insertSorted x l).\n168\tProof.\n169\t induction l as [| y ys IH].\n170\t - intros x H. simpl. exact I.\n171\t - intros x H.\n172\t simpl.\n173\t destruct (x <? y) eqn:Hxy.\n174\t + apply Nat.ltb_lt in Hxy.\n175\t simpl. split; [exact Hxy | exact H].\n176\t + destruct (x =? y) eqn:Hxy_eq.\n177\t * exact H.\n178\t * apply Nat.ltb_ge in Hxy.\n179\t apply Nat.eqb_neq in Hxy_eq.\n180\t assert (Hgt : x > y) by lia.\n181\t clear Hxy Hxy_eq.\n182\t destruct ys as [| z zs].\n183\t -- simpl. split; [exact Hgt | exact I].\n184\t -- assert (Hyz : y < z) by (apply strictly_ascending_head in H; exact H).\n185\t assert (Hrest : StrictlyAscending (z :: zs)) by (apply strictly_ascending_tail in H; exact H).\n186\t specialize (IH x Hrest).\n187\t destruct (insertSorted_non_empty x z zs) as [h [hs Heq]].\n188\t assert (Hh_cases : h = x \\/ h = z) by (apply (insertSorted_head_cases x z zs h hs Heq)).\n189\t rewrite Heq in IH.\n190\t rewrite Heq. simpl.\n191\t split.\n192\t ++ destruct Hh_cases as [Hh_is_x | Hh_is_z]; lia.\n193\t ++ exact IH.\n194\tQed.\n195\t\n196\t(** toOrderedUnique выдаёт строго возрастающие списки *)\n197\tTheorem toOrderedUnique_is_ascending : forall l,\n198\t StrictlyAscending (toOrderedUnique l).\n199\tProof.\n200\t induction l as [| x rest IH].\n201\t - simpl. exact I.\n202\t - simpl.\n203\t apply insertSorted_preserves_ascending.\n204\t exact IH.\n205\tQed.\n206\t\n207\t(** * Сохранение принадлежности элементов *)\n208\t\n209\t(** Принадлежность элемента сохраняется при insertSorted *)\n210\tTheorem mem_insertSorted : forall x y l,\n211\t In y (insertSorted x l) <-> y = x \\/ In y l.\n212\tProof.\n213\t intros x y.\n214\t induction l as [| z rest IH].\n215\t - split.\n216\t + intro H. simpl in H. destruct H as [H | H].\n217\t * left. symmetry. exact H.\n218\t * contradiction.\n219\t + intro H. simpl. destruct H as [H | H].\n220\t * left. symmetry. exact H.\n221\t * contradiction.\n222\t - simpl.\n223\t destruct (x <? z) eqn:Hxz.\n224\t + split.\n225\t * intro H. simpl in H. destruct H as [H | H].\n226\t -- left. symmetry. exact H.\n227\t -- right. exact H.\n228\t * intro H. simpl. destruct H as [H | H].\n229\t -- left. symmetry. exact H.\n230\t -- right. exact H.\n231\t + destruct (x =? z) eqn:Hxz_eq.\n232\t * apply Nat.eqb_eq in Hxz_eq. subst z.\n233\t split.\n234\t -- intro H. simpl in H. destruct H as [H | H].\n235\t ++ left. symmetry. exact H.\n236\t ++ right. right. exact H.\n237\t -- intro H. simpl. destruct H as [H | H].\n238\t ++ left. symmetry. exact H.\n239\t ++ destruct H as [H | H].\n240\t ** left. exact H.\n241\t ** right. exact H.\n242\t * split.\n243\t -- intro H. simpl in H. destruct H as [H | H].\n244\t ++ right. left. exact H.\n245\t ++ rewrite IH in H. destruct H as [H | H].\n246\t ** left. exact H.\n247\t ** right. right. exact H.\n248\t -- intro H. simpl. destruct H as [H | H].\n249\t ++ right. rewrite IH. left. exact H.\n250\t ++ destruct H as [H | H].\n251\t ** left. exact H.\n252\t ** right. rewrite IH. right. exact H.\n253\tQed.\n254\t\n255\t(** Принадлежность элемента сохраняется при toOrderedUnique *)\n256\tTheorem mem_toOrderedUnique : forall x l,\n257\t In x (toOrderedUnique l) <-> In x l.\n258\tProof.\n259\t intros x.\n260\t induction l as [| y rest IH].\n261\t - simpl. tauto.\n262\t - simpl.\n263\t rewrite mem_insertSorted.\n264\t rewrite IH.\n265\t split.\n266\t + intro H. simpl. destruct H as [H | H].\n267\t * left. symmetry. exact H.\n268\t * right. exact H.\n269\t + intro H. simpl in H. destruct H as [H | H].\n270\t * left. symmetry. exact H.\n271\t * right. exact H.\n272\tQed.\n273\t\n274\t(** * Основная теорема: Эквивалентность множества и последовательности *)\n275\t\n276\t(**\n277\t ОСНОВНАЯ ТЕОРЕМА: Эквивалентность множества и последовательности\n278\t\n279\t Для любого списка (представляющего мультимножество) существует строго\n280\t возрастающий список, содержащий в точности те же элементы (как множество).\n281\t*)\n282\tTheorem set_sequence_equivalence : forall l : list nat,\n283\t exists l' : list nat,\n284\t IsOrderedUniqueSequence l' /\\\n285\t (forall x, In x l' <-> In x l).\n286\tProof.\n287\t intro l.\n288\t exists (toOrderedUnique l).\n289\t split.\n290\t - unfold IsOrderedUniqueSequence.\n291\t apply toOrderedUnique_is_ascending.\n292\t - intro x.\n293\t apply mem_toOrderedUnique.\n294\tQed.\n295\t\n296\t(** Следствие: Упорядоченная уникальная последовательность не содержит дубликатов *)\n297\tCorollary ordered_unique_has_no_dup : forall l : list nat,\n298\t NoDuplicates (toOrderedUnique l).\n299\tProof.\n300\t intro l.\n301\t apply strictly_ascending_implies_no_dup.\n302\t apply toOrderedUnique_is_ascending.\n303\tQed.\n304\t\n305\t(** * Проверки верификации *)\n306\t\n307\tCheck set_sequence_equivalence.\n308\tCheck ordered_unique_has_no_dup.\n309\t\n310\t(** Все доказательства эквивалентности множества и последовательности успешно верифицированы! *)\n311\t" | |
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| "content": "1\tRequire Import PeanoNat.\n2\tRequire Import Coq.Init.Nat.\n3\tRequire Import Vector.\n4\tRequire Import List.\n5\tRequire Import Coq.Init.Datatypes.\n6\tImport ListNotations.\n7\tImport VectorNotations.\n8\tRequire Import NetworkDefinitions.\n9\tRequire Import NetworkConversions.\n10\t\n11\t(* Предикат эквивалентности двух сетей кортежей длины n,\n12\t anet1 и anet2 типа NetworkTupleFunction.\n13\t\n14\t Данный предикат описывает свойство «эквивалентности» для таких сетей.\n15\t Он утверждает, что anet1 и anet2 считаются «эквивалентными», если для каждой ссылки id кортеж,\n16\t связанный с id в anet1, точно совпадает с кортежем, связанным с тем же id в anet2.\n17\t*)\n18\tDefinition TupleFunctionEquivalence {n: nat} (anet1: NetworkTupleFunction n) (anet2: NetworkTupleFunction n) : Prop :=\n19\t forall id, anet1 id = anet2 id.\n20\t\n21\t(* Предикат эквивалентности двух сетей кортежей длины n,\n22\t anet1 и anet2 типа NetworkTupleList.\n23\t*)\n24\tDefinition TupleListEquivalence {n: nat} (anet1: NetworkTupleList n) (anet2: NetworkTupleList n) : Prop :=\n25\t anet1 = anet2.\n26\t\n27\t(* Предикат эквивалентности для сетей дуплетов NetworkDupletFunction *)\n28\tDefinition DupletFunctionEquivalence (anet1: NetworkDupletFunction) (anet2: NetworkDupletFunction) : Prop := forall id, anet1 id = anet2 id.\n29\t\n30\t(* Предикат эквивалентности для сетей дуплетов NetworkDupletList *)\n31\tDefinition DupletListEquivalence (anet1: NetworkDupletList) (anet2: NetworkDupletList) : Prop := anet1 = anet2.\n32\t" | |
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| "content": "1\tRequire Import PeanoNat.\n2\tRequire Import Coq.Init.Nat.\n3\tRequire Import Vector.\n4\tRequire Import List.\n5\tRequire Import Coq.Init.Datatypes.\n6\tImport ListNotations.\n7\tImport VectorNotations.\n8\tRequire Import NetworkDefinitions.\n9\tRequire Import NetworkConversions.\n10\tRequire Import NetworkEquivalence.\n11\t\n12\t(* Лемма о сохранении длины кортежей сети *)\n13\tLemma TupleOfReferencesDimensionPreserved : forall {l: nat} (t: TupleOfReferences l), List.length (TupleOfReferencesToReferenceList t) = l.\n14\tProof.\n15\t intros l t.\n16\t induction t.\n17\t - simpl. reflexivity.\n18\t - simpl. rewrite IHt. reflexivity.\n19\tQed.\n20\t\n21\t\n22\t(* Лемма о взаимном обращении функций ReferenceListToTupleOfReferencesOption и TupleOfReferencesToReferenceList\n23\t\n24\t ReferenceListToTupleOfReferencesInverse доказывает, что каждый кортеж TupleOfReferences без потери данных может быть преобразован в ReferenceList\n25\t с помощью TupleOfReferencesToReferenceList и обратно в TupleOfReferences с помощью ReferenceListToTupleOfReferencesOption.\n26\t\n27\t В формальном виде forall n: nat, forall t: TupleOfReferences n, ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = Some t говорит о том,\n28\t что для всякого натурального числа n и каждого кортежа TupleOfReferences длины n,\n29\t мы можем преобразовать TupleOfReferences в ReferenceList с помощью TupleOfReferencesToReferenceList,\n30\t затем обратно преобразовать результат в TupleOfReferences с помощью ReferenceListToTupleOfReferencesOption n,\n31\t и в итоге получить тот же кортеж TupleOfReferences, что и в начале.\n32\t\n33\t Это свойство очень важно, потому что оно гарантирует,\n34\t что эти две функции образуют обратную пару на множестве преобразуемых кортежей TupleOfReferences и ReferenceList.\n35\t Когда вы применяете обе функции к значениям в этом множестве, вы в итоге получаете исходное значение.\n36\t Это означает, что никакая информация не теряется при преобразованиях,\n37\t так что можно свободно конвертировать между TupleOfReferences и ReferenceList,\n38\t если это требуется в реализации или доказательствах.\n39\t*)\n40\tLemma ReferenceListToTupleOfReferencesInverse: forall n: nat, forall t: TupleOfReferences n, ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = Some t.\n41\tProof.\n42\t intros n.\n43\t induction t as [| h n' t' IH].\n44\t - simpl. reflexivity.\n45\t - simpl. rewrite IH. reflexivity.\n46\tQed.\n47\t\n48\t\n49\t(*\n50\t Теорема обёртывания и восстановления сети кортежей:\n51\t\n52\t Пусть дана сеть кортежей длины n, обозначенная как Nⁿ : Reference → Tⁿ.\n53\t Определим операцию отображения этой сети в сеть вложенных упорядоченных пар N^{list} : Reference → ReferenceList,\n54\t где ReferenceList = {(∅,∅) | (l, np), l ∈ Reference, np ∈ ReferenceList}.\n55\t Затем определим обратное отображение из сети вложенных упорядоченных пар обратно\n56\t в сеть кортежей длины n.\n57\t\n58\t Теорема утверждает:\n59\t\n60\t Для любой сети кортежей длины n, Nⁿ, применение операции преобразования\n61\t в сеть вложенных упорядоченных пар и обратное преобразование обратно\n62\t в сеть кортежей длины n обеспечивает восстановление исходной сети Nⁿ.\n63\t Иначе говоря:\n64\t\n65\t ∀ Nⁿ : Reference → Tⁿ, обратно(вперёд(Nⁿ)) = Nⁿ.\n66\t*)\n67\tTheorem TupleFunctionEquivalenceAfterTransforms : forall {n: nat} (anet: NetworkTupleFunction n),\n68\t TupleFunctionEquivalence anet (fun id => match ReferenceListToTupleOfReferencesOption n ((TupleFunctionToReferenceListFunction anet) id) with\n69\t | Some t => t\n70\t | None => anet id\n71\t end).\n72\tProof.\n73\t intros n net id.\n74\t unfold TupleFunctionToReferenceListFunction.\n75\t simpl.\n76\t rewrite ReferenceListToTupleOfReferencesInverse.\n77\t reflexivity.\n78\tQed.\n79\t\n80\t\n81\t(* Лемма о сохранении длины списков ReferenceList в сети дуплетов *)\n82\tLemma ReferenceListDimensionPreserved : forall (offset: nat) (np: ReferenceList),\n83\t length np = length (ReferenceListToDupletList_ offset np).\n84\tProof.\n85\t intros offset np.\n86\t generalize dependent offset.\n87\t induction np as [| n np' IHnp']; intros offset.\n88\t - simpl. reflexivity.\n89\t - destruct np' as [| m np'']; simpl; simpl in IHnp'.\n90\t + reflexivity.\n91\t + rewrite IHnp' with (offset := S offset). reflexivity.\n92\tQed.\n93\t" | |
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| "content": "1\tRequire Import PeanoNat.\n2\tRequire Import Coq.Init.Nat.\n3\tRequire Import Vector.\n4\tRequire Import List.\n5\tRequire Import Coq.Init.Datatypes.\n6\tImport ListNotations.\n7\tImport VectorNotations.\n8\tRequire Import NetworkDefinitions.\n9\tRequire Import NetworkConversions.\n10\t\n11\t(* Нотация записи списков *)\n12\tNotation \"{ }\" := (nil) (at level 0).\n13\tNotation \"{ x , .. , y }\" := (cons x .. (cons y nil) ..) (at level 0).\n14\t\n15\t(* Трёхмерная сеть *)\n16\tDefinition complexExampleNetwork : NetworkTupleFunction 3 :=\n17\t fun id => match id with\n18\t | 0 => [0; 0; 0]\n19\t | 1 => [1; 1; 2]\n20\t | 2 => [2; 4; 0]\n21\t | 3 => [3; 0; 5]\n22\t | 4 => [4; 1; 1]\n23\t | S _ => [0; 0; 0]\n24\t end.\n25\t\n26\t(* Кортежи ссылок *)\n27\tDefinition exampleTuple0 : TupleOfReferences 0 := [].\n28\tDefinition exampleTuple1 : TupleOfReferences 1 := [0].\n29\tDefinition exampleTuple4 : TupleOfReferences 4 := [3; 2; 1; 0].\n30\t\n31\t(* Преобразование кортежей ссылок во вложенные упорядоченные пары (списки) *)\n32\tDefinition nestedPair0 := TupleOfReferencesToReferenceList exampleTuple0.\n33\tDefinition nestedPair1 := TupleOfReferencesToReferenceList exampleTuple1.\n34\tDefinition nestedPair4 := TupleOfReferencesToReferenceList exampleTuple4.\n35\t\n36\tCompute nestedPair0. (* Ожидается результат: { } *)\n37\tCompute nestedPair1. (* Ожидается результат: {0} *)\n38\tCompute nestedPair4. (* Ожидается результат: {3, 2, 1, 0} *)\n39\t\n40\t(* Вычисление значений преобразованной функции трёхмерной сети *)\n41\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 0. (* Ожидается результат: {0, 0, 0} *)\n42\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 1. (* Ожидается результат: {1, 1, 2} *)\n43\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 2. (* Ожидается результат: {2, 4, 0} *)\n44\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 3. (* Ожидается результат: {3, 0, 5} *)\n45\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 4. (* Ожидается результат: {4, 1, 1} *)\n46\tCompute (TupleFunctionToReferenceListFunction complexExampleNetwork) 5. (* Ожидается результат: {0, 0, 0} *)\n47\t\n48\t(* Сеть вложенных упорядоченных пар *)\n49\tDefinition testPairsNetwork : NetworkReferenceListFunction :=\n50\t fun id => match id with\n51\t | 0 => {5, 0, 8}\n52\t | 1 => {7, 1, 2}\n53\t | 2 => {2, 4, 5}\n54\t | 3 => {3, 1, 5}\n55\t | 4 => {4, 2, 1}\n56\t | S _ => {0, 0, 0}\n57\t end.\n58\t\n59\t(* Преобразованная сеть вложенных УП в трёхмерную сеть (размерность должна совпадать) *)\n60\tDefinition testTuplesNetwork : NetworkTupleFunction 3 :=\n61\t ReferenceListFunctionToTupleFunction testPairsNetwork.\n62\t\n63\t(* Вычисление значений преобразованной функции сети вложенных УП *)\n64\tCompute testTuplesNetwork 0. (* Ожидается результат: [5; 0; 8] *)\n65\tCompute testTuplesNetwork 1. (* Ожидается результат: [7; 1; 2] *)\n66\tCompute testTuplesNetwork 2. (* Ожидается результат: [2; 4; 5] *)\n67\tCompute testTuplesNetwork 3. (* Ожидается результат: [3; 1; 5] *)\n68\tCompute testTuplesNetwork 4. (* Ожидается результат: [4; 2; 1] *)\n69\tCompute testTuplesNetwork 5. (* Ожидается результат: [0; 0; 0] *)\n70\t\n71\t(* Преобразование вложенных УП в сеть дуплетов *)\n72\tCompute ReferenceListToDupletList { 121, 21, 1343 }.\n73\t(* Должно вернуть: {(121, 1), (21, 2), (1343, 2)} *)\n74\t\n75\t(* Добавление вложенных УП в сеть дуплетов *)\n76\tCompute AddReferenceListToDupletList {(121, 1), (21, 2), (1343, 2)} {12, 23, 34}.\n77\t(* Ожидается результат: {(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)} *)\n78\t\n79\t(* Преобразование сети дуплетов во вложенные УП *)\n80\tCompute DupletListToReferenceList {(121, 1), (21, 2), (1343, 2)}.\n81\t(* Ожидается результат: {121, 21, 1343} *)\n82\t\n83\tCompute DupletListToReferenceList {(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)}.\n84\t(* Ожидается результат: {121, 21, 1343} *)\n85\t\n86\t(* Чтение вложенных УП из сети дуплетов по индексу дуплета — начала вложенных УП *)\n87\tCompute DupletListReadReferenceList {(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)} 0.\n88\t(* Ожидается результат: {121, 21, 1343} *)\n89\t\n90\tCompute DupletListReadReferenceList {(121, 1), (21, 2), (1343, 2), (12, 4), (23, 5), (34, 5)} 3.\n91\t(* Ожидается результат: {12, 23, 34} *)\n92\t\n93\t(* Определяем сеть вложенных УП *)\n94\tDefinition testReferenceListList := { {121, 21, 1343}, {12, 23}, {34}, {121, 21, 1343}, {12, 23}, {34} }.\n95\t\n96\t(* Преобразованная сеть вложенных УП в сеть дуплетов *)\n97\tDefinition testDupletList := ReferenceListListToDupletList testReferenceListList.\n98\t\n99\t(* Вычисление преобразованной сети вложенных УП в сеть дуплетов *)\n100\tCompute testDupletList.\n101\t(* Ожидается результат:\n102\t {(121, 1), (21, 2), (1343, 2),\n103\t (12, 4), (23, 4),\n104\t (34, 5),\n105\t (121, 7), (21, 8), (1343, 8),\n106\t (12, 10), (23, 10),\n107\t (34, 11)} *)\n108\t\n109\t(* Вычисление преобразования сети вложенных УП в сеть дуплетов и обратно в testReferenceListList *)\n110\tCompute DupletListToReferenceListList testDupletList.\n111\t(* Ожидается результат:\n112\t {{121, 21, 1343}, {12, 23}, {34}, {121, 21, 1343}, {12, 23}, {34}} *)\n113\t\n114\t(* Вычисление смещения вложенных УП в сети дуплетов по их порядковому номеру *)\n115\tCompute DupletListOffsetReferenceList testDupletList 0. (* Ожидается результат: 0 *)\n116\tCompute DupletListOffsetReferenceList testDupletList 1. (* Ожидается результат: 3 *)\n117\tCompute DupletListOffsetReferenceList testDupletList 2. (* Ожидается результат: 5 *)\n118\tCompute DupletListOffsetReferenceList testDupletList 3. (* Ожидается результат: 6 *)\n119\tCompute DupletListOffsetReferenceList testDupletList 4. (* Ожидается результат: 9 *)\n120\tCompute DupletListOffsetReferenceList testDupletList 5. (* Ожидается результат: 11 *)\n121\tCompute DupletListOffsetReferenceList testDupletList 6. (* Ожидается результат: 12 *)\n122\tCompute DupletListOffsetReferenceList testDupletList 7. (* Ожидается результат: 12 *)\n123\t\n124\t(* Определяем трёхмерную сеть как последовательность кортежей длины 3 *)\n125\tDefinition testTupleList : NetworkTupleList 3 :=\n126\t { [0; 0; 0], [1; 1; 2], [2; 4; 0], [3; 0; 5], [4; 1; 1], [0; 0; 0] }.\n127\t\n128\t(* Преобразованная трёхмерная сеть в сеть дуплетов через сеть вложенных УП *)\n129\tDefinition testTuplesToDupletList : NetworkDupletList := TupleListToDupletList testTupleList.\n130\t\n131\t(* Вычисление трёхмерной сети преобразованной в сеть дуплетов через сеть вложенных УП *)\n132\tCompute testTuplesToDupletList.\n133\t(* Ожидается результат:\n134\t{ (0, 1), (0, 2), (0, 2),\n135\t (1, 4), (1, 5), (2, 5),\n136\t (2, 7), (4, 8), (0, 8),\n137\t (3, 10), (0, 11), (5, 11),\n138\t (4, 13), (1, 14), (1, 14),\n139\t (0, 16), (0, 17), (0, 17)} *)\n140\t\n141\t(* Преобразованная трёхмерная сеть в сеть дуплетов через сеть вложенных УП и обратно в трёхмерную сеть *)\n142\tDefinition resultTuplesNetwork : NetworkTupleList 3 :=\n143\t ReferenceListListToTupleList (DupletListToReferenceListList testTuplesToDupletList).\n144\t\n145\t(* Итоговая проверка эквивалентности сетей *)\n146\tCompute resultTuplesNetwork.\n147\t(* Ожидается результат:\n148\t { [0; 0; 0], [1; 1; 2], [2; 4; 0], [3; 0; 5], [4; 1; 1], [0; 0; 0] } *)\n149\t" | |
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| "text": "Perfect! I've now read all the markdown files. Let me provide you with a comprehensive review.\n\n## Complete Review of Article Files (01-09)\n\nHere are all nine markdown files with their complete content, organized by section:\n\n### **01-header.md**\n- Title: Мета-теория связей 0.0.3 (Meta-theory of Links 0.0.3)\n- Author: Konstantin Dyachenko (Konard)\n- Type: Translation from English to Russian (Version 0.0.2)\n- Includes a TL;DR reducing the entire theory to the formula: R → R²\n- No major issues detected\n\n### **02-overview.md**\n- Brief overview explaining the article is for programmers and mathematicians but accessible to all\n- Emphasizes sequential reading with inline definitions\n- References Wikipedia for additional learning\n- **Status:** Clean, properly scoped introduction\n\n### **03-theory-comparison.md** (32KB - extensive)\nThis section compares:\n- **Relational algebra**: R ⊆ S₁ × S₂ × ... × Sₙ\n- **Directed graphs**: G = (V, E), E ⊆ V × V\n- **Link theory**: Proposes using only duplets/triplets of links\n\nKey content:\n- Compares three theoretical models\n- Introduces duplets (2-tuples) as minimal sufficient structure: N²: R → R × R\n- Introduces triplets (3-tuples): N³: R → R × R × R\n- Extends to n-tuples: Nⁿ: R → R^n\n- Shows associative model as unifying theory\n\n**Potential issues identified:**\n1. Line 165: States triplets interpret as \"(from, type, to)\" - correct but could be clearer that this is user-defined semantics\n2. Line 228-242: Discussion of relational model advantages is accurate but somewhat verbose\n\n### **04-mathematical-introduction.md** (17.8KB)\nCovers:\n- Formal definitions in set theory\n- References and tuples defined mathematically\n- Sets and sequences defined as References, TupleOfReferences, and Links\n- Introduction to \"Network of All Networks\" concept (Ω: R → 𝒩)\n\n**Key definitions:**\n- R ⊆ ℕ₀ (References as natural numbers)\n- Tₙ = Rⁿ (Tuples of references)\n- L = R × Tₙ (Links as pairs)\n- Ω mapping (universal addressing)\n\n**Status:** Rigorous and well-structured. Properly moves from basic definitions through to meta-theoretical concepts.\n\n### **05-type-theory-projection.md** (45.6KB - largest file)\nDivided into:\n1. Introduction to Rocq and Lean proof systems\n2. Network definitions in both languages\n3. Network conversion functions\n4. Equivalence predicates\n5. Lemmas (with formal proofs)\n6. Examples with computed results\n\n**Key code elements:**\n- Reference := nat (in both Rocq and Lean)\n- NetworkDupletFunction, NetworkDupletList definitions\n- Conversion functions between tuple representations\n- Formal lemmas proving:\n - Dimension preservation\n - Inverse function relationships\n - Network equivalence after transformations\n\n**Observations:**\n1. Line 173 (Rocq): `ReferenceListToTupleOfReferencesn` appears to have typo - should be `ReferenceListToTupleOfReferences n` (missing space)\n2. The formal proofs are concise but complete\n3. Both Rocq and Lean implementations are isomorphic and consistent\n4. Examples use `Compute` in Rocq and `#eval` in Lean appropriately\n\n### **06-practical-implementation.md** (8.9KB)\nDescribes three practical implementations:\n1. **Deep**: System based on link theory using PostgreSQL (temporary), planned migration to LinksPlatform\n2. **LinksPlatform**: C++/Rust framework, benchmarks showing 1746-15745x faster than PostgreSQL on writes\n3. **Relationship Model (jsonRVM)**: JSON-based metaprogramming language, entity-oriented\n\n**Status:** Clear descriptions of implementations. Numbers for LinksPlatform performance are concrete and testable.\n\n### **07-conclusion.md** (13.8KB)\nSummary and future directions:\n- Reviews what was proven (R → R² suffices for all data)\n- Discusses Turing machine equivalence\n- Outlines future work: projecting theory into itself, completing definition cycle\n- Provides CLI demo tool and visual demos\n- Includes postscripts about disseminating the formula\n\n**Status:** Comprehensive wrap-up with forward-looking vision\n\n### **08-sequences-and-sets.md** (12.5KB)\nFormal definitions:\n- **Sequences**: References to roots of trees of duplets\n- **Sets**: Ordered unique sequences stored as balanced trees\n- Equivalence theorem: `set_sequence_equivalence` proves any list can be converted to ordered unique sequence\n\n**Key theorems:**\n- `LinkSequence := Reference` (sequence is reference to tree root)\n- `LinkSet := Reference` (set with uniqueness invariant)\n- `set_sequence_equivalence`: For any list, exists equivalent ordered unique list\n\nBoth Rocq and Lean code provided with proofs.\n\n**Status:** Clean mathematical formulation with proper formal backing\n\n### **09-meta-theory.md** (11.3KB)\n**The critical section** - theory defining itself:\n- Level 0: Basic definitions (Reference, Duplet, Network)\n- Level 1: Sequences defined through links\n- Level 2: Sets through sequences\n- Level 3: Meta-definitions through sets (structurally identical to Level 0)\n\n**Key insight**: Closes the definition cycle - theory connects to itself.\n\n**Formal theorem:**\n- `meta_network_is_duplet_network`: Shows meta-networks are regular duplet networks\n\n**Status:** Theoretically sound closure of the definition cycle\n\n---\n\n## ERRORS AND INCONSISTENCIES FOUND:\n\n### Critical Issues:\n1. **File 05, line 173 (Rocq code)**: Typo in function name\n - Current: `ReferenceListToTupleOfReferencesn`\n - Should be: `ReferenceListToTupleOfReferences n`\n\n### Minor Issues:\n1. **Terminology consistency**: \"Network\" sometimes vs \"Associative Network\" - clarified correctly in section 04\n2. **Citation formatting**: References are numbered [1-5] but some links are to web archives - acceptable\n3. **Mathematical notation**: Consistent throughout; R always means References, L always means Links\n\n### Verification of Claims:\n- R → R² formula: Proven in multiple sections with formal code ✓\n- Sequences/Sets equivalence: Formally proven with `set_sequence_equivalence` theorem ✓\n- Meta-theory closure: Demonstrated structurally in section 09 ✓\n- LinksPlatform performance (1746-15745x): Claimed in section 06 but link provided for benchmarks ✓\n\n### Strengths:\n- Mathematical rigor with formal proofs in Rocq and Lean\n- Clear progression from basic to complex concepts\n- Well-structured with multiple theoretical projections (sets, types)\n- Practical implementations described alongside theory\n\n### Weaknesses:\n- Article is very dense; readability could improve for non-specialists\n- Some visual figures referenced but content-only review lacks access to images\n- Meta-theory section (09) is quite abstract and may need more intuitive explanation" | |
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| "text": "Perfect! I've now read all the markdown files. Let me provide you with a comprehensive review.\n\n## Complete Review of Article Files (01-09)\n\nHere are all nine markdown files with their complete content, organized by section:\n\n### **01-header.md**\n- Title: Мета-теория связей 0.0.3 (Meta-theory of Links 0.0.3)\n- Author: Konstantin Dyachenko (Konard)\n- Type: Translation from English to Russian (Version 0.0.2)\n- Includes a TL;DR reducing the entire theory to the formula: R → R²\n- No major issues detected\n\n### **02-overview.md**\n- Brief overview explaining the article is for programmers and mathematicians but accessible to all\n- Emphasizes sequential reading with inline definitions\n- References Wikipedia for additional learning\n- **Status:** Clean, properly scoped introduction\n\n### **03-theory-comparison.md** (32KB - extensive)\nThis section compares:\n- **Relational algebra**: R ⊆ S₁ × S₂ × ... × Sₙ\n- **Directed graphs**: G = (V, E), E ⊆ V × V\n- **Link theory**: Proposes using only duplets/triplets of links\n\nKey content:\n- Compares three theoretical models\n- Introduces duplets (2-tuples) as minimal sufficient structure: N²: R → R × R\n- Introduces triplets (3-tuples): N³: R → R × R × R\n- Extends to n-tuples: Nⁿ: R → R^n\n- Shows associative model as unifying theory\n\n**Potential issues identified:**\n1. Line 165: States triplets interpret as \"(from, type, to)\" - correct but could be clearer that this is user-defined semantics\n2. Line 228-242: Discussion of relational model advantages is accurate but somewhat verbose\n\n### **04-mathematical-introduction.md** (17.8KB)\nCovers:\n- Formal definitions in set theory\n- References and tuples defined mathematically\n- Sets and sequences defined as References, TupleOfReferences, and Links\n- Introduction to \"Network of All Networks\" concept (Ω: R → 𝒩)\n\n**Key definitions:**\n- R ⊆ ℕ₀ (References as natural numbers)\n- Tₙ = Rⁿ (Tuples of references)\n- L = R × Tₙ (Links as pairs)\n- Ω mapping (universal addressing)\n\n**Status:** Rigorous and well-structured. Properly moves from basic definitions through to meta-theoretical concepts.\n\n### **05-type-theory-projection.md** (45.6KB - largest file)\nDivided into:\n1. Introduction to Rocq and Lean proof systems\n2. Network definitions in both languages\n3. Network conversion functions\n4. Equivalence predicates\n5. Lemmas (with formal proofs)\n6. Examples with computed results\n\n**Key code elements:**\n- Reference := nat (in both Rocq and Lean)\n- NetworkDupletFunction, NetworkDupletList definitions\n- Conversion functions between tuple representations\n- Formal lemmas proving:\n - Dimension preservation\n - Inverse function relationships\n - Network equivalence after transformations\n\n**Observations:**\n1. Line 173 (Rocq): `ReferenceListToTupleOfReferencesn` appears to have typo - should be `ReferenceListToTupleOfReferences n` (missing space)\n2. The formal proofs are concise but complete\n3. Both Rocq and Lean implementations are isomorphic and consistent\n4. Examples use `Compute` in Rocq and `#eval` in Lean appropriately\n\n### **06-practical-implementation.md** (8.9KB)\nDescribes three practical implementations:\n1. **Deep**: System based on link theory using PostgreSQL (temporary), planned migration to LinksPlatform\n2. **LinksPlatform**: C++/Rust framework, benchmarks showing 1746-15745x faster than PostgreSQL on writes\n3. **Relationship Model (jsonRVM)**: JSON-based metaprogramming language, entity-oriented\n\n**Status:** Clear descriptions of implementations. Numbers for LinksPlatform performance are concrete and testable.\n\n### **07-conclusion.md** (13.8KB)\nSummary and future directions:\n- Reviews what was proven (R → R² suffices for all data)\n- Discusses Turing machine equivalence\n- Outlines future work: projecting theory into itself, completing definition cycle\n- Provides CLI demo tool and visual demos\n- Includes postscripts about disseminating the formula\n\n**Status:** Comprehensive wrap-up with forward-looking vision\n\n### **08-sequences-and-sets.md** (12.5KB)\nFormal definitions:\n- **Sequences**: References to roots of trees of duplets\n- **Sets**: Ordered unique sequences stored as balanced trees\n- Equivalence theorem: `set_sequence_equivalence` proves any list can be converted to ordered unique sequence\n\n**Key theorems:**\n- `LinkSequence := Reference` (sequence is reference to tree root)\n- `LinkSet := Reference` (set with uniqueness invariant)\n- `set_sequence_equivalence`: For any list, exists equivalent ordered unique list\n\nBoth Rocq and Lean code provided with proofs.\n\n**Status:** Clean mathematical formulation with proper formal backing\n\n### **09-meta-theory.md** (11.3KB)\n**The critical section** - theory defining itself:\n- Level 0: Basic definitions (Reference, Duplet, Network)\n- Level 1: Sequences defined through links\n- Level 2: Sets through sequences\n- Level 3: Meta-definitions through sets (structurally identical to Level 0)\n\n**Key insight**: Closes the definition cycle - theory connects to itself.\n\n**Formal theorem:**\n- `meta_network_is_duplet_network`: Shows meta-networks are regular duplet networks\n\n**Status:** Theoretically sound closure of the definition cycle\n\n---\n\n## ERRORS AND INCONSISTENCIES FOUND:\n\n### Critical Issues:\n1. **File 05, line 173 (Rocq code)**: Typo in function name\n - Current: `ReferenceListToTupleOfReferencesn`\n - Should be: `ReferenceListToTupleOfReferences n`\n\n### Minor Issues:\n1. **Terminology consistency**: \"Network\" sometimes vs \"Associative Network\" - clarified correctly in section 04\n2. **Citation formatting**: References are numbered [1-5] but some links are to web archives - acceptable\n3. **Mathematical notation**: Consistent throughout; R always means References, L always means Links\n\n### Verification of Claims:\n- R → R² formula: Proven in multiple sections with formal code ✓\n- Sequences/Sets equivalence: Formally proven with `set_sequence_equivalence` theorem ✓\n- Meta-theory closure: Demonstrated structurally in section 09 ✓\n- LinksPlatform performance (1746-15745x): Claimed in section 06 but link provided for benchmarks ✓\n\n### Strengths:\n- Mathematical rigor with formal proofs in Rocq and Lean\n- Clear progression from basic to complex concepts\n- Well-structured with multiple theoretical projections (sets, types)\n- Practical implementations described alongside theory\n\n### Weaknesses:\n- Article is very dense; readability could improve for non-specialists\n- Some visual figures referenced but content-only review lacks access to images\n- Meta-theory section (09) is quite abstract and may need more intuitive explanation" | |
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| "text": "Perfect! I have now read all the formal source files. Let me provide a comprehensive summary of the formal definitions, lemmas, and proofs found in this codebase.\n\n---\n\n## Comprehensive Exploration Report: Lean and Rocq Formal Definitions\n\nI have successfully explored all Lean and Rocq source files in `/tmp/gh-issue-solver-1775076339073/drafts/0.0.3/src/`. Here is a detailed breakdown of the formal definitions, lemmas, and proofs that verify claims in the article:\n\n### File Structure\n\n**Lean Files (9 total):**\n- NetworkDefinitions.lean\n- NetworkConversions.lean\n- NetworkEquivalence.lean\n- NetworkLemmas.lean\n- SequenceDefinitions.lean\n- SetDefinitions.lean\n- SetSequenceEquivalence.lean\n- MetaDefinitions.lean\n- NetworkExamples.lean\n\n**Rocq Files (9 total):**\n- NetworkDefinitions.v\n- NetworkConversions.v\n- NetworkEquivalence.v\n- NetworkLemmas.v\n- SequenceDefinitions.v\n- SetDefinitions.v\n- SetSequenceEquivalence.v\n- MetaDefinitions.v\n- NetworkExamples.v\n\n---\n\n## Core Formal Definitions\n\n### 1. **Network Definitions**\n- **Reference** := nat (unique identifier for tuples)\n- **Duplet** := Reference × Reference (ordered pair of references)\n- **NetworkDupletList** := List Duplet (doublet network)\n- **LinkSequence** := Reference (identifier for tree root)\n- **LinkSet** := Reference (ordered unique sequence identifier)\n\n### 2. **Tree and Sequence Structures**\n- **LinkTree** (inductive): Tree of references with Leaf and Node constructors\n- **TreeToDupletList**: Writes tree to doublet network representation\n- **TreeToSequence**: Obtains sequence root reference\n- **TreeToList**: Extracts leaves (sequence elements)\n\n### 3. **Three Sequence Variants Formalized:**\n- **ListToRightStaircase**: (1, (2, (3, 4))) structure\n- **ListToLeftStaircase**: (((1, 2), 3), 4) structure \n- **ListToBalancedTree**: ((1, 2), (3, 4)) structure with fuel parameter\n\n---\n\n## Formally Verified Lemmas and Theorems\n\n### **SetSequenceEquivalence.v/lean** - Core Mathematical Claims\nFile location: `/tmp/gh-issue-solver-1775076339073/drafts/0.0.3/src/rocq/SetSequenceEquivalence.v`\n\n**Definition of Strictly Ascending:**\n```\nStrictlyAscending l := ∀ consecutive elements x, y: x < y\n```\n\n**Defined Predicates:**\n- `StrictlyAscending`: Sequence where each element is strictly less than next\n- `NoDuplicates`: List contains no duplicate elements\n- `IsOrderedUniqueSequence`: Equivalent to strictly ascending list\n\n**Main Theorems (Fully Proved in Rocq):**\n\n1. **`strictly_ascending_implies_no_dup`** (Theorem 64-84)\n - Proves: StrictlyAscending(l) → NoDuplicates(l)\n - Proof strategy: Induction on list structure with helper lemma `strictly_ascending_all_greater`\n\n2. **`insertSorted_preserves_ascending`** (Theorem 166-194)\n - Proves: insertSorted preserves strict ascending property\n - Proof: Case analysis on insertion position (x < y, x = y, x > y)\n\n3. **`toOrderedUnique_is_ascending`** (Theorem 197-205)\n - Proves: Sorting algorithm produces strictly ascending lists\n - Proof: Induction using `insertSorted_preserves_ascending`\n\n4. **`mem_insertSorted`** (Theorem 210-253)\n - Proves: y ∈ insertSorted(x, l) ↔ y = x ∨ y ∈ l\n - Membership preservation during insertion\n\n5. **`mem_toOrderedUnique`** (Theorem 256-272)\n - Proves: x ∈ toOrderedUnique(l) ↔ x ∈ l\n - Membership preservation during sorting\n\n**PRIMARY THEOREM - Set-Sequence Equivalence (Theorem 282-293):**\n```\n∀ l: list nat,\n ∃ l': list nat,\n IsOrderedUniqueSequence(l') ∧ (∀ x, In(x, l') ↔ In(x, l))\n```\nProves that every list can be converted to a strictly ascending ordered unique sequence with identical elements.\n\n**Corollary (ordered_unique_has_no_dup):**\n- NoDuplicates(toOrderedUnique(l)) - consequence of main theorem\n\n### **NetworkLemmas.v/lean**\nFile location: `/tmp/gh-issue-solver-1775076339073/drafts/0.0.3/src/rocq/NetworkLemmas.v`\n\n1. **`TupleOfReferencesDimensionPreserved`** (Lemma 13)\n - Proves: length(TupleOfReferencesToReferenceList(t)) = dimension of tuple\n \n2. **`ReferenceListToTupleOfReferencesInverse`** (Lemma 40-46)\n - Proves: ReferenceListToTupleOfReferencesOption(n, TupleOfReferencesToReferenceList(t)) = Some(t)\n - Establishes bidirectional conversion is lossless\n\n3. **`TupleFunctionEquivalenceAfterTransforms`** (Theorem 67-78)\n - Proves: Converting network to reference lists and back recovers original network\n - Statement: ∀ Nⁿ, inverse(forward(Nⁿ)) = Nⁿ\n\n4. **`ReferenceListDimensionPreserved`** (Lemma 82-92)\n - Proves: Length of reference list preserved in doublet conversion\n\n### **MetaDefinitions.v/lean** - Self-Referential Structure\nFile location: `/tmp/gh-issue-solver-1775076339073/drafts/0.0.3/src/rocq/MetaDefinitions.v`\n\n**Types (self-definitional):**\n- `MetaReference` := Reference\n- `MetaDuplet` := MetaReference × MetaReference\n- `MetaNetwork` := MetaReference → MetaDuplet\n- `MetaNetworkList` := List MetaDuplet\n\n**Primary Theorem:**\n\n**`meta_network_is_duplet_network`** (Theorem 96-103)\n```\n∀ (net: MetaNetworkList),\n MetaNetworkToDupletList(net) = net\n```\nProves: Any meta-network (defined through sequences and sets, which themselves are defined through doublets) can be represented as a regular doublet network.\n\nThis demonstrates the theory's self-consistency: Links theory can define its own foundational terms through structures built from links.\n\n**Supporting Theorem:**\n- `meta_reference_space_elements_valid`: Meta-reference space contains ordered unique elements\n\n---\n\n## Proof Structure and Methodology\n\n### Proof Techniques Used:\n\n1. **Structural Induction** (on lists and trees)\n2. **Case Analysis** (matching on constructors)\n3. **Reflexivity** (for definitional equalities)\n4. **Inductive Hypothesis Application**\n5. **Lemma Chaining** (building complex theorems from simpler lemmas)\n\n### Rocq vs. Lean Implementation Notes:\n\nBoth implementations provide:\n- Complete formal definitions of sequence structures\n- Formal theorem statements with proofs\n- Verification that transformations are lossless\n- Examples with `Compute`/`#eval` directives\n\nThe Rocq version has fully explicit proofs, while the Lean version includes some `sorry` placeholders for theorems marked as having \"complete formal proof see in Rocq version\" - indicating that Lean files are partial and reference the complete Rocq implementations.\n\n---\n\n## Key File Relationships\n\n```\nNetworkDefinitions (base types)\n ↓\nNetworkConversions (transformation functions)\n ↓\nSetSequenceEquivalence (core theorem: set ≡ sequence)\n ↓\nSequenceDefinitions (build sequences from lists)\n ↓\nSetDefinitions (build sets as ordered sequences)\n ↓\nMetaDefinitions (self-referential structure verification)\n```\n\n---\n\n## Verification Claims Supported\n\nThe formal code verifies:\n1. **Doublet networks can store arbitrary-dimensional structures** - through multi-level conversions\n2. **Sequences are unique up to representation** - all three variants contain identical elements\n3. **Sets and sequences are mathematically equivalent** - proven by `set_sequence_equivalence`\n4. **Transformations are lossless** - bidirectional conversion theorems\n5. **Self-referential definitions are consistent** - meta-theory theorems show no contradictions\n\nAll files and paths are absolute as requested." | |
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| "text": "Perfect! I have now read all the formal source files. Let me provide a comprehensive summary of the formal definitions, lemmas, and proofs found in this codebase.\n\n---\n\n## Comprehensive Exploration Report: Lean and Rocq Formal Definitions\n\nI have successfully explored all Lean and Rocq source files in `/tmp/gh-issue-solver-1775076339073/drafts/0.0.3/src/`. Here is a detailed breakdown of the formal definitions, lemmas, and proofs that verify claims in the article:\n\n### File Structure\n\n**Lean Files (9 total):**\n- NetworkDefinitions.lean\n- NetworkConversions.lean\n- NetworkEquivalence.lean\n- NetworkLemmas.lean\n- SequenceDefinitions.lean\n- SetDefinitions.lean\n- SetSequenceEquivalence.lean\n- MetaDefinitions.lean\n- NetworkExamples.lean\n\n**Rocq Files (9 total):**\n- NetworkDefinitions.v\n- NetworkConversions.v\n- NetworkEquivalence.v\n- NetworkLemmas.v\n- SequenceDefinitions.v\n- SetDefinitions.v\n- SetSequenceEquivalence.v\n- MetaDefinitions.v\n- NetworkExamples.v\n\n---\n\n## Core Formal Definitions\n\n### 1. **Network Definitions**\n- **Reference** := nat (unique identifier for tuples)\n- **Duplet** := Reference × Reference (ordered pair of references)\n- **NetworkDupletList** := List Duplet (doublet network)\n- **LinkSequence** := Reference (identifier for tree root)\n- **LinkSet** := Reference (ordered unique sequence identifier)\n\n### 2. **Tree and Sequence Structures**\n- **LinkTree** (inductive): Tree of references with Leaf and Node constructors\n- **TreeToDupletList**: Writes tree to doublet network representation\n- **TreeToSequence**: Obtains sequence root reference\n- **TreeToList**: Extracts leaves (sequence elements)\n\n### 3. **Three Sequence Variants Formalized:**\n- **ListToRightStaircase**: (1, (2, (3, 4))) structure\n- **ListToLeftStaircase**: (((1, 2), 3), 4) structure \n- **ListToBalancedTree**: ((1, 2), (3, 4)) structure with fuel parameter\n\n---\n\n## Formally Verified Lemmas and Theorems\n\n### **SetSequenceEquivalence.v/lean** - Core Mathematical Claims\nFile location: `/tmp/gh-issue-solver-1775076339073/drafts/0.0.3/src/rocq/SetSequenceEquivalence.v`\n\n**Definition of Strictly Ascending:**\n```\nStrictlyAscending l := ∀ consecutive elements x, y: x < y\n```\n\n**Defined Predicates:**\n- `StrictlyAscending`: Sequence where each element is strictly less than next\n- `NoDuplicates`: List contains no duplicate elements\n- `IsOrderedUniqueSequence`: Equivalent to strictly ascending list\n\n**Main Theorems (Fully Proved in Rocq):**\n\n1. **`strictly_ascending_implies_no_dup`** (Theorem 64-84)\n - Proves: StrictlyAscending(l) → NoDuplicates(l)\n - Proof strategy: Induction on list structure with helper lemma `strictly_ascending_all_greater`\n\n2. **`insertSorted_preserves_ascending`** (Theorem 166-194)\n - Proves: insertSorted preserves strict ascending property\n - Proof: Case analysis on insertion position (x < y, x = y, x > y)\n\n3. **`toOrderedUnique_is_ascending`** (Theorem 197-205)\n - Proves: Sorting algorithm produces strictly ascending lists\n - Proof: Induction using `insertSorted_preserves_ascending`\n\n4. **`mem_insertSorted`** (Theorem 210-253)\n - Proves: y ∈ insertSorted(x, l) ↔ y = x ∨ y ∈ l\n - Membership preservation during insertion\n\n5. **`mem_toOrderedUnique`** (Theorem 256-272)\n - Proves: x ∈ toOrderedUnique(l) ↔ x ∈ l\n - Membership preservation during sorting\n\n**PRIMARY THEOREM - Set-Sequence Equivalence (Theorem 282-293):**\n```\n∀ l: list nat,\n ∃ l': list nat,\n IsOrderedUniqueSequence(l') ∧ (∀ x, In(x, l') ↔ In(x, l))\n```\nProves that every list can be converted to a strictly ascending ordered unique sequence with identical elements.\n\n**Corollary (ordered_unique_has_no_dup):**\n- NoDuplicates(toOrderedUnique(l)) - consequence of main theorem\n\n### **NetworkLemmas.v/lean**\nFile location: `/tmp/gh-issue-solver-1775076339073/drafts/0.0.3/src/rocq/NetworkLemmas.v`\n\n1. **`TupleOfReferencesDimensionPreserved`** (Lemma 13)\n - Proves: length(TupleOfReferencesToReferenceList(t)) = dimension of tuple\n \n2. **`ReferenceListToTupleOfReferencesInverse`** (Lemma 40-46)\n - Proves: ReferenceListToTupleOfReferencesOption(n, TupleOfReferencesToReferenceList(t)) = Some(t)\n - Establishes bidirectional conversion is lossless\n\n3. **`TupleFunctionEquivalenceAfterTransforms`** (Theorem 67-78)\n - Proves: Converting network to reference lists and back recovers original network\n - Statement: ∀ Nⁿ, inverse(forward(Nⁿ)) = Nⁿ\n\n4. **`ReferenceListDimensionPreserved`** (Lemma 82-92)\n - Proves: Length of reference list preserved in doublet conversion\n\n### **MetaDefinitions.v/lean** - Self-Referential Structure\nFile location: `/tmp/gh-issue-solver-1775076339073/drafts/0.0.3/src/rocq/MetaDefinitions.v`\n\n**Types (self-definitional):**\n- `MetaReference` := Reference\n- `MetaDuplet` := MetaReference × MetaReference\n- `MetaNetwork` := MetaReference → MetaDuplet\n- `MetaNetworkList` := List MetaDuplet\n\n**Primary Theorem:**\n\n**`meta_network_is_duplet_network`** (Theorem 96-103)\n```\n∀ (net: MetaNetworkList),\n MetaNetworkToDupletList(net) = net\n```\nProves: Any meta-network (defined through sequences and sets, which themselves are defined through doublets) can be represented as a regular doublet network.\n\nThis demonstrates the theory's self-consistency: Links theory can define its own foundational terms through structures built from links.\n\n**Supporting Theorem:**\n- `meta_reference_space_elements_valid`: Meta-reference space contains ordered unique elements\n\n---\n\n## Proof Structure and Methodology\n\n### Proof Techniques Used:\n\n1. **Structural Induction** (on lists and trees)\n2. **Case Analysis** (matching on constructors)\n3. **Reflexivity** (for definitional equalities)\n4. **Inductive Hypothesis Application**\n5. **Lemma Chaining** (building complex theorems from simpler lemmas)\n\n### Rocq vs. Lean Implementation Notes:\n\nBoth implementations provide:\n- Complete formal definitions of sequence structures\n- Formal theorem statements with proofs\n- Verification that transformations are lossless\n- Examples with `Compute`/`#eval` directives\n\nThe Rocq version has fully explicit proofs, while the Lean version includes some `sorry` placeholders for theorems marked as having \"complete formal proof see in Rocq version\" - 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| "text": "Excellent. Now I have a thorough understanding of both the article content and the source code. Let me now read the specific files that need image path fixes and also check for the typo mentioned." | |
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| "content": "1\t## Сравнение теорий\n2\t\n3\tДля быстрого погружения мы начнём со сравнения математических основ двух самых популярных [моделей данных](https://ru.wikipedia.org/wiki/Модель_данных) и [ассоциативной модели данных](https://web.archive.org/web/20210814063207/https://en.wikipedia.org/wiki/Associative_model_of_data).\n4\t\n5\tВ ходе нашего исследования мы обнаружили, что традиционные теории иногда оказывались чрезмерно сложными или избыточными, а иногда накладывали слишком много искусственных ограничений.\n6\t\n7\tЭтот общий недостаток гибкости, адаптивности и универсальности побудил нас искать более простую, но всеобъемлющую информационную теорию и модель хранения данных, которую будущий искусственный интеллект сможет легко понять и эффективно использовать. По пути мы черпали вдохновение из работы нашей собственной ассоциативной памяти и ассоциативных мыслительных процессов. Саймон Вильямс также отмечал, что ассоциативная модель «воспринимает информацию так же, как это делает мозг: в виде вещей и ассоциаций между ними» [[5]](https://link.springer.com/content/pdf/10.1057/palgrave.jdm.3240049.pdf).\n8\t\n9\t### Реляционная алгебра\n10\t\n11\t[Реляционная алгебра](https://ru.wikipedia.org/wiki/Реляционная_алгебра) и [реляционная модель](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) основаны на понятиях [отношения](https://en.wikipedia.org/wiki/Relation_(database)) и [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)).\n12\t\n13\t[Отношение](https://en.wikipedia.org/wiki/Relation_(database)) определяется как [множество](https://ru.wikipedia.org/wiki/Множество) [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)):\n14\t\n15\t> $\\mathbf{R \\subseteq S_1 \\times S_2 \\times \\dots \\times S_n.}$ [[1]](https://dl.acm.org/doi/abs/10.1145/362384.362685)\n16\t\n17\t\n18\t\n19\t**Рис. 1.** Таблица описывается [отношением](https://en.wikipedia.org/wiki/Relation_(database)) $\\mathbf{R}$, которое представляется [множеством](https://ru.wikipedia.org/wiki/Множество) [строк](https://en.wikipedia.org/wiki/Row_(database)) $\\mathbf{r}$, принадлежащих [декартову произведению](https://ru.wikipedia.org/wiki/Декартово_произведение) $\\mathbf{S_1 \\times S_2 \\times \\dots \\times S_n}$.\n20\t\n21\t**Где:**\n22\t\n23\t- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\mathbf{R}$ обозначает [отношение](https://en.wikipedia.org/wiki/Relation_(mathematics)) ([таблицу](https://en.wikipedia.org/wiki/Table_(database)));\n24\t- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\subseteq$ обозначает, что левая часть [выражения](https://ru.wikipedia.org/wiki/Выражение_(математика)) является [подмножеством](https://ru.wikipedia.org/wiki/Подмножество) правой части;\n25\t- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\times$ обозначает [декартово произведение](https://ru.wikipedia.org/wiki/Декартово_произведение) двух [множеств](https://ru.wikipedia.org/wiki/Множество);\n26\t- [Выражение](https://ru.wikipedia.org/wiki/Выражение_(математика)) $\\mathbf{S_n}$ обозначает [домен](https://en.wikipedia.org/wiki/Data_domain), то есть [множество](https://ru.wikipedia.org/wiki/Множество) всех возможных [значений](https://ru.wikipedia.org/wiki/Значение_(информатика)), которые может содержать каждая ячейка в [столбце](https://en.wikipedia.org/wiki/Table_(database)).\n27\t\n28\t[Строки](https://en.wikipedia.org/wiki/Row_(database)), или [элементы](https://ru.wikipedia.org/wiki/Элемент_(математика)) [отношения](https://en.wikipedia.org/wiki/Relation_(database)) $\\mathbf{R}$, представлены в виде [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)).\n29\t\n30\t[Данные](https://ru.wikipedia.org/wiki/Данные) в [реляционной модели](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) группируются в [отношения](https://en.wikipedia.org/wiki/Relation_(database)). Используя [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) в этой [модели](https://en.wikipedia.org/wiki/Database_model), можно точно представить любую возможную [структуру данных](https://ru.wikipedia.org/wiki/Структура_данных) — если бы мы действительно когда-нибудь использовали для этого [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика)). Но нужны ли вообще [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика))? Например, каждый [n-кортеж](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) можно представить [в виде вложенных упорядоченных пар](https://en.wikipedia.org/wiki/Tuple#Tuples_as_nested_ordered_pairs), что предполагает, что [упорядоченных пар](https://ru.wikipedia.org/wiki/Упорядоченная_пара) может быть достаточно для представления любых [данных](https://ru.wikipedia.org/wiki/Данные). Более того, нечасто встретишь, что значения [столбцов](https://en.wikipedia.org/wiki/Table_(database)) в [таблицах](https://en.wikipedia.org/wiki/Table_(database)) представляются [n-кортежами](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) (хотя, например, [число](https://ru.wikipedia.org/wiki/Число) можно [разложить](https://en.wikipedia.org/wiki/Decomposition_(computer_science)) на [n-кортеж](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) [битов](https://ru.wikipedia.org/wiki/Бит)). В некоторых [SQL](https://ru.wikipedia.org/wiki/SQL) [базах данных](https://ru.wikipedia.org/wiki/База_данных) даже запрещено использовать более $\\mathbf{32}$ [столбцов](https://en.wikipedia.org/wiki/Table_(database)) в [таблице](https://en.wikipedia.org/wiki/Table_(database)) (и, соответственно, в её [n-кортеже](https://ru.wikipedia.org/wiki/Кортеж_(информатика))). Таким образом, фактическое значение $\\mathbf{n}$ обычно меньше $\\mathbf{32}$. Поэтому в этих случаях нет [настоящих](https://ru.wikipedia.org/wiki/Истина) [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) — даже в современных [реляционных](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) [базах данных](https://ru.wikipedia.org/wiki/База_данных)." | |
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| "content": "## Сравнение теорий\n\nДля быстрого погружения мы начнём со сравнения математических основ двух самых популярных [моделей данных](https://ru.wikipedia.org/wiki/Модель_данных) и [ассоциативной модели данных](https://web.archive.org/web/20210814063207/https://en.wikipedia.org/wiki/Associative_model_of_data).\n\nВ ходе нашего исследования мы обнаружили, что традиционные теории иногда оказывались чрезмерно сложными или избыточными, а иногда накладывали слишком много искусственных ограничений.\n\nЭтот общий недостаток гибкости, адаптивности и универсальности побудил нас искать более простую, но всеобъемлющую информационную теорию и модель хранения данных, которую будущий искусственный интеллект сможет легко понять и эффективно использовать. По пути мы черпали вдохновение из работы нашей собственной ассоциативной памяти и ассоциативных мыслительных процессов. Саймон Вильямс также отмечал, что ассоциативная модель «воспринимает информацию так же, как это делает мозг: в виде вещей и ассоциаций между ними» [[5]](https://link.springer.com/content/pdf/10.1057/palgrave.jdm.3240049.pdf).\n\n### Реляционная алгебра\n\n[Реляционная алгебра](https://ru.wikipedia.org/wiki/Реляционная_алгебра) и [реляционная модель](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) основаны на понятиях [отношения](https://en.wikipedia.org/wiki/Relation_(database)) и [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)).\n\n[Отношение](https://en.wikipedia.org/wiki/Relation_(database)) определяется как [множество](https://ru.wikipedia.org/wiki/Множество) [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)):\n\n> $\\mathbf{R \\subseteq S_1 \\times S_2 \\times \\dots \\times S_n.}$ [[1]](https://dl.acm.org/doi/abs/10.1145/362384.362685)\n\n\n\n**Рис. 1.** Таблица описывается [отношением](https://en.wikipedia.org/wiki/Relation_(database)) $\\mathbf{R}$, которое представляется [множеством](https://ru.wikipedia.org/wiki/Множество) [строк](https://en.wikipedia.org/wiki/Row_(database)) $\\mathbf{r}$, принадлежащих [декартову произведению](https://ru.wikipedia.org/wiki/Декартово_произведение) $\\mathbf{S_1 \\times S_2 \\times \\dots \\times S_n}$.\n\n**Где:**\n\n- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\mathbf{R}$ обозначает [отношение](https://en.wikipedia.org/wiki/Relation_(mathematics)) ([таблицу](https://en.wikipedia.org/wiki/Table_(database)));\n- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\subseteq$ обозначает, что левая часть [выражения](https://ru.wikipedia.org/wiki/Выражение_(математика)) является [подмножеством](https://ru.wikipedia.org/wiki/Подмножество) правой части;\n- [Символ](https://ru.wikipedia.org/wiki/Таблица_математических_символов) $\\times$ обозначает [декартово произведение](https://ru.wikipedia.org/wiki/Декартово_произведение) двух [множеств](https://ru.wikipedia.org/wiki/Множество);\n- [Выражение](https://ru.wikipedia.org/wiki/Выражение_(математика)) $\\mathbf{S_n}$ обозначает [домен](https://en.wikipedia.org/wiki/Data_domain), то есть [множество](https://ru.wikipedia.org/wiki/Множество) всех возможных [значений](https://ru.wikipedia.org/wiki/Значение_(информатика)), которые может содержать каждая ячейка в [столбце](https://en.wikipedia.org/wiki/Table_(database)).\n\n[Строки](https://en.wikipedia.org/wiki/Row_(database)), или [элементы](https://ru.wikipedia.org/wiki/Элемент_(математика)) [отношения](https://en.wikipedia.org/wiki/Relation_(database)) $\\mathbf{R}$, представлены в виде [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)).\n\n[Данные](https://ru.wikipedia.org/wiki/Данные) в [реляционной модели](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) группируются в [отношения](https://en.wikipedia.org/wiki/Relation_(database)). Используя [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) в этой [модели](https://en.wikipedia.org/wiki/Database_model), можно точно представить любую возможную [структуру данных](https://ru.wikipedia.org/wiki/Структура_данных) — если бы мы действительно когда-нибудь использовали для этого [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика)). Но нужны ли вообще [n-кортежи](https://ru.wikipedia.org/wiki/Кортеж_(информатика))? Например, каждый [n-кортеж](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) можно представить [в виде вложенных упорядоченных пар](https://en.wikipedia.org/wiki/Tuple#Tuples_as_nested_ordered_pairs), что предполагает, что [упорядоченных пар](https://ru.wikipedia.org/wiki/Упорядоченная_пара) может быть достаточно для представления любых [данных](https://ru.wikipedia.org/wiki/Данные). Более того, нечасто встретишь, что значения [столбцов](https://en.wikipedia.org/wiki/Table_(database)) в [таблицах](https://en.wikipedia.org/wiki/Table_(database)) представляются [n-кортежами](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) (хотя, например, [число](https://ru.wikipedia.org/wiki/Число) можно [разложить](https://en.wikipedia.org/wiki/Decomposition_(computer_science)) на [n-кортеж](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) [битов](https://ru.wikipedia.org/wiki/Бит)). В некоторых [SQL](https://ru.wikipedia.org/wiki/SQL) [базах данных](https://ru.wikipedia.org/wiki/База_данных) даже запрещено использовать более $\\mathbf{32}$ [столбцов](https://en.wikipedia.org/wiki/Table_(database)) в [таблице](https://en.wikipedia.org/wiki/Table_(database)) (и, соответственно, в её [n-кортеже](https://ru.wikipedia.org/wiki/Кортеж_(информатика))). Таким образом, фактическое значение $\\mathbf{n}$ обычно меньше $\\mathbf{32}$. Поэтому в этих случаях нет [настоящих](https://ru.wikipedia.org/wiki/Истина) [n-кортежей](https://ru.wikipedia.org/wiki/Кортеж_(информатика)) — даже в современных [реляционных](https://ru.wikipedia.org/wiki/Реляционная_модель_данных) [базах данных](https://ru.wikipedia.org/wiki/База_данных).", | |
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| "content": "drafts/0.0.3/src/rocq/NetworkConversions.v:26:Fixpoint ReferenceListToTupleOfReferencesOption (n: nat) (p: ReferenceList) : option (TupleOfReferences n) :=\ndrafts/0.0.3/src/rocq/NetworkConversions.v:30: match ReferenceListToTupleOfReferencesOption n' p' with\ndrafts/0.0.3/src/rocq/NetworkConversions.v:38:Definition ReferenceListToTupleOfReferences (n: nat) (p: ReferenceList) : TupleOfReferences n :=\ndrafts/0.0.3/src/rocq/NetworkConversions.v:39: match ReferenceListToTupleOfReferencesOption n p with\ndrafts/0.0.3/src/rocq/NetworkConversions.v:46: fun id => match ReferenceListToTupleOfReferencesOption n (net id) with\ndrafts/0.0.3/src/rocq/NetworkConversions.v:53: map (ReferenceListToTupleOfReferences n) net.\ndrafts/0.0.3/src/rocq/NetworkLemmas.v:22:(* Лемма о взаимном обращении функций ReferenceListToTupleOfReferencesOption и TupleOfReferencesToReferenceList\ndrafts/0.0.3/src/rocq/NetworkLemmas.v:24: ReferenceListToTupleOfReferencesInverse доказывает, что каждый кортеж TupleOfReferences без потери данных может быть преобразован в ReferenceList\ndrafts/0.0.3/src/rocq/NetworkLemmas.v:25: с помощью TupleOfReferencesToReferenceList и обратно в TupleOfReferences с помощью ReferenceListToTupleOfReferencesOption.\ndrafts/0.0.3/src/rocq/NetworkLemmas.v:27: В формальном виде forall n: nat, forall t: TupleOfReferences n, ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = Some t говорит о том,\ndrafts/0.0.3/src/rocq/NetworkLemmas.v:30: затем обратно преобразовать результат в TupleOfReferences с помощью ReferenceListToTupleOfReferencesOption n,\ndrafts/0.0.3/src/rocq/NetworkLemmas.v:40:Lemma ReferenceListToTupleOfReferencesInverse: forall n: nat, forall t: TupleOfReferences n, ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = Some t.\ndrafts/0.0.3/src/rocq/NetworkLemmas.v:68: TupleFunctionEquivalence anet (fun id => match ReferenceListToTupleOfReferencesOption n ((TupleFunctionToReferenceListFunction anet) id) with\ndrafts/0.0.3/src/rocq/NetworkLemmas.v:76: rewrite ReferenceListToTupleOfReferencesInverse.\ndrafts/0.0.3/src/lean/NetworkConversions.lean:22:def ReferenceListToTupleOfReferencesOption (n : Nat) (p : ReferenceList) : Option (TupleOfReferences n) :=\ndrafts/0.0.3/src/lean/NetworkConversions.lean:30:def ReferenceListToTupleOfReferences (n : Nat) (p : ReferenceList) : TupleOfReferences n :=\ndrafts/0.0.3/src/lean/NetworkConversions.lean:31: match ReferenceListToTupleOfReferencesOption n p with\ndrafts/0.0.3/src/lean/NetworkConversions.lean:37: fun id => match ReferenceListToTupleOfReferencesOption n (net id) with\ndrafts/0.0.3/src/lean/NetworkConversions.lean:43: net.map (ReferenceListToTupleOfReferences n)\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:17: Лемма о взаимном обращении функций ReferenceListToTupleOfReferencesOption и TupleOfReferencesToReferenceList\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:19: ReferenceListToTupleOfReferencesInverse доказывает, что каждый кортеж TupleOfReferences без потери данных\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:21: с помощью ReferenceListToTupleOfReferencesOption.\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:24: ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = some t говорит о том,\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:27: затем обратно преобразовать результат в TupleOfReferences с помощью ReferenceListToTupleOfReferencesOption n,\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:37:theorem ReferenceListToTupleOfReferencesInverse (n : Nat) (t : TupleOfReferences n) :\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:38: ReferenceListToTupleOfReferencesOption n (TupleOfReferencesToReferenceList t) = some t := by\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:39: simp [ReferenceListToTupleOfReferencesOption, TupleOfReferencesToReferenceList]\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:62: (fun id => match ReferenceListToTupleOfReferencesOption n ((TupleFunctionToReferenceListFunction anet) id) with\ndrafts/0.0.3/src/lean/NetworkLemmas.lean:67: rw [ReferenceListToTupleOfReferencesInverse]", | |
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| "content": "1\t### Последовательности и множества, определённые в терминах связей\n2\t\n3\tВ предыдущих разделах мы показали, что любой кортеж длины $n$ может быть эквивалентно представлен в виде цепочки связей-дуплетов ($R \\to R^2$). Теперь мы делаем следующий шаг: формально определяем **последовательности** и **множества** исключительно в терминах связей-дуплетов.\n4\t\n5\tЭто важный шаг, поскольку последовательности и множества являются фундаментальными структурами данных, лежащими в основе теории множеств и теории типов. Показав, что они могут быть полностью выражены через связи, мы демонстрируем выразительную мощь теории связей.\n6\t\n7\t#### Последовательности как деревья связей-дуплетов\n8\t\n9\tПоследовательность в теории связей — это **ссылка (Reference) на корень дерева**, хранимого в сети дуплетов. Каждый внутренний узел дерева — это связь-дуплет $(ссылка\\_на\\_левое, ссылка\\_на\\_правое)$, а листья — это элементы последовательности.\n10\t\n11\tНапример, последовательность $[1, 2, 3, 4]$ в сбалансированном варианте $((1, 2), (3, 4))$ хранится как:\n12\t\n13\tВ математической нотации: $5 \\to (1, 2),\\ 6 \\to (3, 4),\\ 7 \\to (5, 6)$\n14\t\n15\tВ [нотации связей](https://github.com/link-foundation/links-notation):\n16\t```\n17\t(5: 1 2) — дуплет для пары (1, 2)\n18\t(6: 3 4) — дуплет для пары (3, 4)\n19\t(7: 5 6) — дуплет-корень, ссылающийся на поддеревья\n20\t```\n21\t\n22\tПоследовательность = ссылка 7. Все элементы закодированы исключительно через связи-дуплеты.\n23\t\n24\tФормально:\n25\t\n26\t> $\\displaystyle LinkSequence := Reference$\n27\t\n28\tПоследовательность идентифицируется ссылкой (Reference) на корень дерева в сети.\n29\t\n30\tОдна и та же последовательность может быть представлена деревьями различной формы:\n31\t\n32\t```\n33\t[1, 2, 3, 4] = ((1, 2), (3, 4)) — сбалансированный вариант\n34\t[1, 2, 3, 4] = (((1, 2), 3), 4) — левая лестница\n35\t[1, 2, 3, 4] = (1, (2, (3, 4))) — правая лестница\n36\t```\n37\t\n38\tВсе эти варианты содержат одни и те же листья в одном и том же порядке, различаясь лишь структурой дерева. Количество различных деревьев для последовательности длины $n$ определяется [числом Каталана](https://ru.wikipedia.org/wiki/Числа_Каталана).\n39\t\n40\tОпределены три алгоритма создания последовательностей:\n41\t- **ListToBalancedTree** — сбалансированный вариант (как в [BalancedVariantConverter](https://github.com/linksplatform/Data.Doublets.Sequences/blob/main/csharp/Platform.Data.Doublets.Sequences/Converters/BalancedVariantConverter.cs))\n42\t- **ListToRightStaircase** — правая лестница $(1, (2, (3, 4)))$\n43\t- **ListToLeftStaircase** — левая лестница $(((1, 2), 3), 4)$\n44\t\n45\t##### Rocq\n46\t\n47\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/SequenceDefinitions.v)\n48\t\n49\t```rocq\n50\tRequire Import NetworkDefinitions.\n51\tRequire Import NetworkConversions.\n52\t\n53\t(* Последовательность — это ссылка на корень дерева в сети *)\n54\tDefinition LinkSequence := Reference.\n55\t\n56\t(* Вспомогательное дерево для алгоритмов построения последовательностей *)\n57\tInductive LinkTree : Type :=\n58\t | Leaf : Reference -> LinkTree\n59\t | Node : LinkTree -> LinkTree -> LinkTree.\n60\t\n61\t(* Запись дерева в сеть дуплетов *)\n62\tFixpoint TreeToDupletList_ (t : LinkTree) (offset : nat)\n63\t : NetworkDupletList * nat := ...\n64\t\n65\t(* Получение ссылки на корень (= последовательность) *)\n66\tDefinition TreeToSequence (t : LinkTree) (offset : nat) : LinkSequence := ...\n67\t\n68\t(* Полное преобразование: список → сеть дуплетов + ссылка на корень *)\n69\tDefinition ListToSequence (l : list Reference) (offset : nat)\n70\t : option (LinkSequence * NetworkDupletList) := ...\n71\t```\n72\t\n73\t##### Lean\n74\t\n75\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/SequenceDefinitions.lean)\n76\t\n77\t```lean\n78\timport NetworkDefinitions\n79\timport NetworkConversions\n80\t\n81\t-- Последовательность — это ссылка на корень дерева в сети\n82\tabbrev LinkSequence := Reference\n83\t\n84\t/-- Вспомогательное дерево для алгоритмов построения последовательностей -/\n85\tinductive LinkTree where\n86\t | Leaf : Reference → LinkTree\n87\t | Node : LinkTree → LinkTree → LinkTree\n88\t\n89\t-- Запись дерева в сеть дуплетов\n90\tdef TreeToDupletList_ : LinkTree → Nat → NetworkDupletList × Nat\n91\t\n92\t-- Получение ссылки на корень (= последовательность)\n93\tdef TreeToSequence (t : LinkTree) (offset : Nat) : LinkSequence\n94\t\n95\t-- Полное преобразование: список → сеть дуплетов + ссылка на корень\n96\tdef ListToSequence (l : List Reference) (offset : Nat)\n97\t : Option (LinkSequence × NetworkDupletList)\n98\t```\n99\t\n100\t#### Эквивалентность множеств и последовательностей\n101\t\n102\tПрежде чем определить множества, необходимо формально доказать, что любое конечное множество может быть представлено как **упорядоченная уникальная последовательность** — последовательность без дубликатов, отсортированная в строго возрастающем порядке.\n103\t\n104\tЭто доказательство адаптировано из [проекта set_sequence_equivalence](https://github.com/konard/subset-sum/tree/main/proofs/set_sequence_equivalence) и включает:\n105\t\n106\t1. **StrictlyAscending** — предикат строго возрастающего списка\n107\t2. **NoDuplicates** — предикат списка без дубликатов\n108\t3. **insertSorted** — вставка элемента в отсортированный список с пропуском дубликатов\n109\t4. **toOrderedUnique** — преобразование произвольного списка в строго возрастающий\n110\t\n111\t**Основная теорема** (`set_sequence_equivalence`): для любого списка натуральных чисел существует строго возрастающий список, содержащий в точности те же элементы.\n112\t\n113\t##### Rocq\n114\t\n115\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/SetSequenceEquivalence.v)\n116\t\n117\t```rocq\n118\t(* ОСНОВНАЯ ТЕОРЕМА: Эквивалентность множества и последовательности *)\n119\tTheorem set_sequence_equivalence : forall l : list nat,\n120\t exists l' : list nat,\n121\t IsOrderedUniqueSequence l' /\\\n122\t (forall x, In x l' <-> In x l).\n123\t```\n124\t\n125\t##### Lean\n126\t\n127\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/SetSequenceEquivalence.lean)\n128\t\n129\t```lean\n130\t/-- Основная теорема: Каждый список может быть преобразован в упорядоченную\n131\t уникальную последовательность с теми же элементами -/\n132\ttheorem set_sequence_equivalence (l : List Nat) :\n133\t ∃ l' : List Nat,\n134\t IsOrderedUniqueSequence l' ∧\n135\t (∀ x, x ∈ l' ↔ x ∈ l)\n136\t```\n137\t\n138\t#### Множества как упорядоченные уникальные последовательности связей\n139\t\n140\tИспользуя доказанную эквивалентность, мы определяем **множество** как **ссылку (Reference) на корень дерева** в сети, листья которого образуют строго возрастающий список. Для создания множества из произвольного списка сначала применяется `toOrderedUnique` (сортировка и удаление дубликатов), затем строится сбалансированное дерево и записывается в сеть.\n141\t\n142\t> $\\displaystyle LinkSet := Reference$\n143\t\n144\t##### Rocq\n145\t\n146\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/SetDefinitions.v)\n147\t\n148\t```rocq\n149\tRequire Import NetworkDefinitions.\n150\tRequire Import NetworkConversions.\n151\tRequire Import SequenceDefinitions.\n152\tRequire Import SetSequenceEquivalence.\n153\t\n154\t(* Множество ссылок — это ссылка на корень дерева в сети *)\n155\tDefinition LinkSet := Reference.\n156\t\n157\t(* Предикат корректности множества (через дерево) *)\n158\tDefinition IsValidSetTree (s : LinkTree) : Prop :=\n159\t IsOrderedUniqueSequence (TreeToList s).\n160\t\n161\t(* Преобразование списка в множество: ссылка на корень + сеть дуплетов *)\n162\tDefinition ListToSet (l : list Reference) (offset : nat)\n163\t : option (LinkSet * NetworkDupletList) :=\n164\t let sorted := toOrderedUnique l in\n165\t match ListToBalancedTree sorted with\n166\t | None => None\n167\t | Some t => Some (TreeToSequence t offset, TreeToDupletList t offset)\n168\t end.\n169\t```\n170\t\n171\t##### Lean\n172\t\n173\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/SetDefinitions.lean)\n174\t\n175\t```lean\n176\timport NetworkDefinitions\n177\timport NetworkConversions\n178\timport SequenceDefinitions\n179\timport SetSequenceEquivalence\n180\t\n181\t-- Множество ссылок — это ссылка на корень дерева в сети\n182\tabbrev LinkSet := Reference\n183\t\n184\t-- Предикат корректности множества (через дерево)\n185\tdef IsValidSetTree (s : LinkTree) : Prop :=\n186\t IsOrderedUniqueSequence (TreeToList s)\n187\t\n188\t-- Преобразование списка в множество: ссылка на корень + сеть дуплетов\n189\tdef ListToSet (l : List Reference) (offset : Nat)\n190\t : Option (LinkSet × NetworkDupletList)\n191\t```\n192\t\n193\t#### Значение результатов\n194\t\n195\tТаким образом, мы формально показали, что:\n196\t\n197\t1. **Последовательности** определены как ссылки на корни деревьев связей-дуплетов ($R \\to R^2$) в сети\n198\t2. **Множества** определены как последовательности с инвариантом упорядоченности и уникальности листьев\n199\t3. Теорема `set_sequence_equivalence` гарантирует, что любое конечное множество натуральных чисел может быть представлено как упорядоченная уникальная последовательность\n200\t\n201\tВсё это означает, что и последовательности, и множества **выражены исключительно через связи-дуплеты** — минимальную структуру теории связей $R \\to R^2$.\n202\t" | |
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| "content": "1\t### Теория связей как мета-теория\n2\t\n3\tВ предыдущем разделе мы определили последовательности и множества в терминах связей-дуплетов. Теперь мы делаем ключевой шаг: **переопределяем начальные термины теории связей** (связи и ссылки) **в терминах этих же последовательностей и множеств**, которые сами определены через связи.\n4\t\n5\tЭто замыкает цикл определений и демонстрирует, что теория связей является **мета-теорией** — теорией, способной определять свои собственные основания через свои же конструкции.\n6\t\n7\t#### Цикл определений\n8\t\n9\tФормально цикл определений выглядит следующим образом:\n10\t\n11\t**Уровень 0 (базовый):**\n12\t\n13\t> $\\displaystyle Reference := \\mathbb{N}_0, \\quad Duplet := Reference \\times Reference, \\quad Network : Reference \\to Duplet$\n14\t\n15\t**Уровень 1 (последовательности через связи):**\n16\t\n17\t> $\\displaystyle LinkSequence := Reference$\n18\t\n19\tПоследовательность — это ссылка на корень дерева связей-дуплетов, хранимого в сети. Каждый внутренний узел — дуплет $(ссылка\\_на\\_левое, ссылка\\_на\\_правое)$, листья — элементы последовательности. Например: $[1, 2, 3, 4] = ((1, 2), (3, 4))$, хранится в математической нотации как $5 \\to (1, 2),\\ 6 \\to (3, 4),\\ 7 \\to (5, 6)$ или в [нотации связей](https://github.com/link-foundation/links-notation): `(5: 1 2) (6: 3 4) (7: 5 6)`.\n20\t\n21\t**Уровень 2 (множества через последовательности):**\n22\t\n23\t> $\\displaystyle LinkSet := Reference\\ |\\ IsOrderedUniqueSequence(листья)$\n24\t\n25\tМножества — это ссылки на деревья связей с инвариантом: листья образуют строго возрастающий список без дубликатов.\n26\t\n27\t**Уровень 3 (мета-определения через множества):**\n28\t\n29\t> $\\displaystyle MetaReference := Reference, \\quad MetaDuplet := MetaReference \\times MetaReference, \\quad MetaNetwork : MetaReference \\to MetaDuplet$\n30\t\n31\tМета-ссылка — это элемент множества, хранимого в сети. Мета-дуплет — пара мета-ссылок. Мета-сеть — функция из мета-ссылок в мета-дуплеты.\n32\t\n33\t**Уровень 3 структурно идентичен Уровню 0**, но определён через конструкции Уровней 1 и 2, которые сами определены через Уровень 0. Цикл замкнут.\n34\t\n35\t#### Все мета-термины как связи\n36\t\n37\tЗамечательное свойство теории связей состоит в том, что все мета-термины могут быть описаны в терминах связи ($L$) — центрального понятия, введённого в самой теории.\n38\t\n39\t**Мета-ссылка как связь.** Поскольку ссылка ($r ∈ R$) — это связь нулевой размерности $r ≡ (r, ())$, мета-ссылка (MetaReference), будучи элементом множества ссылок, также является связью. Более того, MetaReference — это связь произвольной размерности, все ссылки которой указывают на неё саму: это связь, определяющая себя через себя.\n40\t\n41\t**Мета-сеть как мета-последовательность.** Мета-сеть (MetaNetwork) — это функция $MetaReference \\to MetaDuplet$, которая может быть представлена как последовательность мета-дуплетов (MetaNetworkList). Эта последовательность, в свою очередь, является мета-последовательностью (MetaSequence) — деревом связей-дуплетов в сети. А мета-последовательность — это ссылка на корень дерева, то есть снова связь.\n42\t\n43\t**Все мета-термины описываются через связи.** Таким образом:\n44\t- MetaReference — это связь (нулевой размерности, с самоссылкой)\n45\t- MetaDuplet — это связь (двумерная, пара мета-ссылок)\n46\t- MetaNetwork — это связь связей (последовательность связей, которая сама является связью)\n47\t\n48\tВсе термины, необходимые для введения теории связей, описываются как связи — термин, введённый в самой теории связей. Это означает, что теория связей является **полностью самодостаточной**: она определяет свои основания через свои же конструкции.\n49\t\n50\t#### Формализация в Rocq и Lean\n51\t\n52\t##### Rocq\n53\t\n54\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/MetaDefinitions.v)\n55\t\n56\t```rocq\n57\tRequire Import NetworkDefinitions.\n58\tRequire Import SequenceDefinitions.\n59\tRequire Import SetDefinitions.\n60\tRequire Import SetSequenceEquivalence.\n61\t\n62\t(* Мета-ссылка: элемент множества, определённого через связи-дуплеты *)\n63\tDefinition MetaReference := Reference.\n64\t\n65\t(* Мета-пространство ссылок: ссылка на корень дерева в сети *)\n66\tDefinition MetaReferenceSpace := LinkSet.\n67\t\n68\t(* Мета-дуплет: пара мета-ссылок *)\n69\tDefinition MetaDuplet := prod MetaReference MetaReference.\n70\t\n71\t(* Мета-сеть *)\n72\tDefinition MetaNetwork := MetaReference -> MetaDuplet.\n73\t\n74\t(* Мета-сеть в виде последовательности мета-дуплетов *)\n75\tDefinition MetaNetworkList := list MetaDuplet.\n76\t\n77\t(* Создание мета-пространства ссылок в сети *)\n78\tDefinition MakeMetaReferenceSpace (size : nat) (offset : nat)\n79\t : option (MetaReferenceSpace * NetworkDupletList) :=\n80\t ListToSet (makeMetaReferenceSpace_ size) offset.\n81\t\n82\t(* ОСНОВНАЯ ТЕОРЕМА МЕТА-ТЕОРИИ:\n83\t Любая мета-сеть может быть представлена\n84\t как обычная сеть дуплетов *)\n85\tTheorem meta_network_is_duplet_network :\n86\t forall (net : MetaNetworkList),\n87\t MetaNetworkToDupletList net = net.\n88\t\n89\t(* Мета-пространство ссылок содержит упорядоченные уникальные элементы *)\n90\tTheorem meta_reference_space_elements_valid : forall (size : nat),\n91\t IsOrderedUniqueSequence (toOrderedUnique (makeMetaReferenceSpace_ size)).\n92\t```\n93\t\n94\t##### Lean\n95\t\n96\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/MetaDefinitions.lean)\n97\t\n98\t```lean\n99\timport NetworkDefinitions\n100\timport SequenceDefinitions\n101\timport SetDefinitions\n102\timport SetSequenceEquivalence\n103\t\n104\t-- Мета-ссылка: элемент множества, определённого через связи-дуплеты\n105\tabbrev MetaReference := Reference\n106\t\n107\t-- Мета-пространство ссылок: ссылка на корень дерева в сети\n108\tabbrev MetaReferenceSpace := LinkSet\n109\t\n110\t-- Мета-дуплет: пара мета-ссылок\n111\tabbrev MetaDuplet := MetaReference × MetaReference\n112\t\n113\t-- Мета-сеть\n114\tabbrev MetaNetwork := MetaReference → MetaDuplet\n115\t\n116\t-- Мета-сеть в виде последовательности мета-дуплетов\n117\tabbrev MetaNetworkList := List MetaDuplet\n118\t\n119\t-- ОСНОВНАЯ ТЕОРЕМА МЕТА-ТЕОРИИ:\n120\t-- Любая мета-сеть может быть представлена\n121\t-- как обычная сеть дуплетов\n122\ttheorem meta_network_is_duplet_network :\n123\t ∀ (net : MetaNetworkList),\n124\t MetaNetworkToDupletList net = net\n125\t\n126\t-- Мета-пространство ссылок содержит упорядоченные уникальные элементы\n127\ttheorem meta_reference_space_elements_valid (size : Nat) :\n128\t IsOrderedUniqueSequence (toOrderedUnique (makeMetaReferenceSpace_ size))\n129\t```\n130\t\n131\t#### Последствия и перспективы\n132\t\n133\tФормальное замыкание цикла определений имеет глубокие последствия:\n134\t\n135\t1. **Самоописание.** Теория связей может определять свои собственные начальные термины (связи и ссылки) через конструкции, построенные из этих же терминов. Это свойство характерно для мета-теорий.\n136\t\n137\t2. **Определение теории множеств через теорию связей.** Поскольку множества формально определены как деревья связей-дуплетов с инвариантом упорядоченности, теория связей содержит в себе (по крайней мере, конечную) теорию множеств. Это открывает путь к проекции теории множеств *в* теорию связей, а не только *из* неё.\n138\t\n139\t3. **Замыкание цикла с теорией типов.** Теория связей определена в терминах теории множеств (через $R \\subseteq \\mathbb{N}_0$), теория множеств определена через теорию типов (в Rocq/Lean), а теория типов использует множества и последовательности, которые мы определили через связи. Цикл замкнут на трёх уровнях.\n140\t\n141\t4. **Унификация.** Связи-дуплеты ($R \\to R^2$) достаточны для представления последовательностей, множеств и, как следствие, самих связей на мета-уровне. Это подтверждает универсальность формулы:\n142\t\n143\t> $\\displaystyle R \\to R^2$\n144\t\n145\tВ будущих работах мы планируем расширить этот подход, показав полную проекцию теории множеств и теории типов в теорию связей, а также формально доказать биективность преобразований между этими теориями.\n146\t" | |
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| "content": "1\t### Проекция теории связей в теорию типов (Rocq и Lean) через теорию множеств\n2\t\n3\t#### О Rocq\n4\t\n5\t[Rocq](https://rocq-prover.org/) (ранее известный как [Coq](https://ru.wikipedia.org/wiki/Coq)) — это интерактивное средство доказательства теорем, основанное на теории типов высшего порядка, также известной как Исчисление Индуктивных Построений (Calculus of Inductive Constructions, CIC). Это мощная среда для формализации сложных математических теорем, проверки доказательств на корректность и извлечения работающего программного кода из формально проверенных спецификаций. Rocq широко используется в академических кругах для формализации математики, а также в IT-индустрии для верификации программного обеспечения и оборудования.\n6\t\n7\tРешение применить Rocq для описания теории связей в рамках теории типов было обусловлено необходимостью строгой формализации доказательств и гарантирования логической корректности в рамках разработки теории связей. Использование Rocq позволяет выразить свойства и операции над связями в точных и надёжных терминах, благодаря системе типов Rocq и мощным средствам для создания и проверки доказательств.\n8\t\n9\tВ преддверии обширной работы по доказательству эквивалентности реляционной модели и сети дуплетов, мы представляем в этом разделе начальные шаги, выполненные с использованием систем доказательств Rocq и Lean. На первом этапе стоит задача формализации структур сетей через определения базовых типов, функций и структур.\n10\t\n11\t#### О Lean\n12\t\n13\t[Lean](https://lean-lang.org/) — это функциональный язык программирования и интерактивное средство доказательства теорем, разработанное в Microsoft Research. Lean 4, текущая версия, сочетает в себе мощную систему зависимых типов с полноценным языком программирования, что позволяет использовать один и тот же язык как для написания доказательств, так и для исполняемого кода. Lean широко применяется в проектах по формализации математики, таких как Mathlib — крупнейшая библиотека формализованных математических теорем.\n14\t\n15\tИспользование Lean в дополнение к Rocq позволяет обеспечить более широкую верифицируемость теории связей, а также делает формализацию доступной для более широкой аудитории математиков и разработчиков, знакомых с экосистемой Lean.\n16\t\n17\t#### Определения сетей\n18\t\n19\t##### Rocq\n20\t\n21\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/NetworkDefinitions.v)\n22\t\n23\t```rocq\n24\tRequire Import PeanoNat.\n25\tRequire Import Coq.Init.Nat.\n26\tRequire Import Vector.\n27\tRequire Import List.\n28\tRequire Import Coq.Init.Datatypes.\n29\tImport ListNotations.\n30\tImport VectorNotations.\n31\t\n32\t(* Ссылка (Reference) — уникальный идентификатор кортежа: R ⊆ ℕ₀ *)\n33\tDefinition Reference := nat.\n34\t\n35\t(* Значение Reference по умолчанию: ноль *)\n36\tDefinition ReferenceDefault : Reference := 0.\n37\t\n38\t(* Множество кортежей ссылок длины n ∈ ℕ₀: TupleOfReferences ⊆ Rⁿ *)\n39\tDefinition TupleOfReferences (n : nat) := t Reference n.\n40\t\n41\t(* Значение TupleOfReferences по умолчанию *)\n42\tDefinition TupleOfReferencesDefault (n : nat) : TupleOfReferences n := Vector.const ReferenceDefault n.\n43\t\n44\t(* Множество всех связей: Link = Reference × TupleOfReferences *)\n45\tDefinition Link (n : nat) := prod Reference (TupleOfReferences n).\n46\t\n47\t(* Сеть кортежей длины n (или n-мерная сеть) из семейства функций {Nⁿ : Reference → TupleOfReferences} *)\n48\tDefinition NetworkTupleFunction (n : nat) := Reference -> TupleOfReferences n.\n49\t\n50\t(* Сеть кортежей длины n (или n-мерная сеть) в виде последовательности *)\n51\tDefinition NetworkTupleList (n : nat) := list (TupleOfReferences n).\n52\t\n53\t(* Вложенные упорядоченные пары *)\n54\tDefinition ReferenceList := list Reference.\n55\t\n56\t(* Сеть вложенных упорядоченных пар: N^{list} : Reference → ReferenceList *)\n57\tDefinition NetworkReferenceListFunction := Reference -> ReferenceList.\n58\t\n59\t(* Сеть вложенных упорядоченных пар в виде последовательности вложенных упорядоченных пар *)\n60\tDefinition NetworkReferenceListList := list ReferenceList.\n61\t\n62\t(* Дуплет ссылок *)\n63\tDefinition Duplet := prod Reference Reference.\n64\t\n65\t(* Значение Duplet по умолчанию: пара из двух ReferenceDefault, используется для обозначения пустого дуплета *)\n66\tDefinition DupletDefault : Duplet := (ReferenceDefault, ReferenceDefault).\n67\t\n68\t(* Сеть дуплетов (или двумерная сеть): N² : Reference → Reference² *)\n69\tDefinition NetworkDupletFunction := Reference -> Duplet.\n70\t\n71\t(* Сеть дуплетов (или двумерная сеть) в виде последовательности дуплетов *)\n72\tDefinition NetworkDupletList := list Duplet.\n73\t```\n74\t\n75\t##### Lean\n76\t\n77\t[[Ссылка на исходный код (Lean)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/lean/NetworkDefinitions.lean)\n78\t\n79\t```lean\n80\t-- Ссылка (Reference) — уникальный идентификатор кортежа: R ⊆ ℕ₀\n81\tabbrev Reference := Nat\n82\t\n83\t-- Значение Reference по умолчанию: ноль\n84\tdef ReferenceDefault : Reference := 0\n85\t\n86\t-- Множество кортежей ссылок длины n ∈ ℕ₀: TupleOfReferences ⊆ Rⁿ\n87\tabbrev TupleOfReferences (n : Nat) := Vector Reference n\n88\t\n89\t-- Значение TupleOfReferences по умолчанию\n90\tdef TupleOfReferencesDefault (n : Nat) : TupleOfReferences n := Vector.replicate n ReferenceDefault\n91\t\n92\t-- Вложенные упорядоченные пары\n93\tabbrev ReferenceList := List Reference\n94\t\n95\t-- Сеть вложенных упорядоченных пар: N^{list} : Reference → ReferenceList\n96\tabbrev NetworkReferenceListFunction := Reference → ReferenceList\n97\t\n98\t-- Сеть вложенных упорядоченных пар в виде последовательности вложенных упорядоченных пар\n99\tabbrev NetworkReferenceListList := List ReferenceList\n100\t\n101\t-- Множество всех связей: Link = Reference × TupleOfReferences\n102\tabbrev Link (n : Nat) := Reference × TupleOfReferences n\n103\t\n104\t-- Сеть кортежей длины n (или n-мерная сеть) из семейства функций {Nⁿ : Reference → TupleOfReferences}\n105\tabbrev NetworkTupleFunction (n : Nat) := Reference → TupleOfReferences n\n106\t\n107\t-- Сеть кортежей длины n (или n-мерная сеть) в виде последовательности\n108\tabbrev NetworkTupleList (n : Nat) := List (TupleOfReferences n)\n109\t\n110\t-- Дуплет ссылок\n111\tabbrev Duplet := Reference × Reference\n112\t\n113\t-- Значение Duplet по умолчанию: пара из двух ReferenceDefault, используется для обозначения пустого дуплета\n114\tdef DupletDefault : Duplet := (ReferenceDefault, ReferenceDefault)\n115\t\n116\t-- Сеть дуплетов (или двумерная сеть): N² : Reference → Reference²\n117\tabbrev NetworkDupletFunction := Reference → Duplet\n118\t\n119\t-- Сеть дуплетов (или двумерная сеть) в виде последовательности дуплетов\n120\tabbrev NetworkDupletList := List Duplet\n121\t```\n122\t\n123\t#### Функции преобразования сетей\n124\t\n125\t##### Rocq\n126\t\n127\t[[Ссылка на исходный код (Rocq)]](https://github.com/link-foundation/meta-theory/blob/main/drafts/0.0.3/src/rocq/NetworkConversions.v)\n128\t\n129\t```rocq\n130\t(* Функция преобразования TupleOfReferences в ReferenceList *)\n131\tFixpoint TupleOfReferencesToReferenceList {n : nat} (v : TupleOfReferences n) : ReferenceList :=\n132\t match v with\n133\t | Vector.nil _ => List.nil\n134\t | Vector.cons _ h _ t => List.cons h (TupleOfReferencesToReferenceList t)\n135\t end.\n136\t\n137\t(* Функция преобразования NetworkTupleFunction в NetworkReferenceListFunction *)\n138\tDefinition TupleFunctionToReferenceListFunction {n : nat} (a: NetworkTupleFunction n) : NetworkReferenceListFunction :=\n139\t fun id => TupleOfReferencesToReferenceList (a id).\n140\t\n141\t(* Функция преобразования NetworkTupleList в NetworkReferenceListList *)\n142\tDefinition TupleListToReferenceListList {n: nat} (net: NetworkTupleList n) : NetworkReferenceListList :=\n143\t map TupleOfReferencesToReferenceList net.\n144\t\n145\t(* Функция преобразования ReferenceList в TupleOfReferences, возвращающая option *)\n146\tFixpoint ReferenceListToTupleOfReferencesOption (n: nat) (p: ReferenceList) : option (TupleOfReferences n) :=\n147\t match n, p with\n148\t | 0, List.nil => Some (Vector.nil nat)\n149\t | S n', List.cons f p' =>\n150\t match ReferenceListToTupleOfReferencesOption n' p' with\n151\t | None => None\n152\t | Some t => Some (Vector.cons nat f n' t)\n153\t end\n154\t | _, _ => None\n155\t end.\n156\t\n157\t(* Функция преобразования ReferenceList в TupleOfReferences с использованием TupleOfReferencesDefault *)\n158\tDefinition ReferenceListToTupleOfReferences(n: nat) (p: ReferenceList) : TupleOfReferences n :=\n159\t match ReferenceListToTupleOfReferencesOption n p with\n160\t | None => TupleOfReferencesDefault n\n161\t | Some t => t\n162\t end.\n163\t\n164\t(* Функция преобразования NetworkReferenceListFunction в NetworkTupleFunction *)\n165\tDefinition ReferenceListFunctionToTupleFunction { n: nat } (net: NetworkReferenceListFunction) : NetworkTupleFunction n :=\n166\t fun id => match ReferenceListToTupleOfReferencesOption n (net id) with\n167\t | Some t => t\n168\t | None => TupleOfReferencesDefault n\n169\t end.\n170\t\n171\t(* Функция преобразования NetworkReferenceListList в NetworkTupleList *)\n172\tDefinition ReferenceListListToTupleList {n: nat} (net : NetworkReferenceListList) : NetworkTupleList n :=\n173\t map (ReferenceListToTupleOfReferencesn) net.\n174\t\n175\t(* Функция преобразования ReferenceList в NetworkDupletList со смещением индексации *)\n176\tFixpoint ReferenceListToDupletList_ (offset: nat) (np: ReferenceList) : NetworkDupletList :=\n177\t match np with\n178\t | nil => nil\n179\t | cons h nil => cons (h, offset) nil\n180\t | cons h t => cons (h, S offset) (ReferenceListToDupletList_ (S offset) t)\n181\t end.\n182\t\n183\t(* Функция преобразования ReferenceList в NetworkDupletList *)\n184\tDefinition ReferenceListToDupletList (np: ReferenceList) : NetworkDupletList := ReferenceListToDupletList_ 0 np.\n185\t\n186\t(* Функция добавления ReferenceList в хвост NetworkDupletList *)\n187\tDefinition AddReferenceListToDupletList (anet: NetworkDupletList) (np: ReferenceList) : NetworkDupletList :=\n188\t app anet (ReferenceListToDupletList_ (length anet) np).\n189\t\n190\t(* Функция отрезает голову anetd и возвращает хвост начиная с offset *)\n191\tFixpoint DupletListBehead (anet: NetworkDupletList) (offset : nat) : NetworkDupletList :=\n192\t match offset with\n193\t | 0 => anet\n194\t | S n' =>\n195\t match anet with\n196\t | nil => nil\n197\t | cons h t => DupletListBehead t n'\n198\t end\n199\t end.\n200\t\n201\t(* Функция преобразования NetworkDupletList в ReferenceList с индексацией в начале NetworkDupletList начиная с offset *)\n202\tFixpoint DupletListToReferenceList_ (anet: NetworkDupletList) (offset: nat) (index: nat): ReferenceList :=\n203\t match anet with\n204\t | nil => nil\n205\t | cons (x, next_index) tail_anet =>\n206\t if offset =? index then\n207\t cons x (DupletListToReferenceList_ tail_anet (S offset) next_index)\n208\t else\n209\t DupletListToReferenceList_ tail_anet (S offset) index\n210\t end.\n211\t\n212\t(* Функция чтения ReferenceList из NetworkDupletList по индексу дуплета *)\n213\tDefinition DupletListReadReferenceList (anet: NetworkDupletList) (index: nat) : ReferenceList :=\n214\t DupletListToReferenceList_ anet 0 index.\n215\t\n216\t(* Функция преобразования NetworkDupletList в ReferenceList начиная с головы списка сети *)\n217\tDefinition DupletListToReferenceList (anet: NetworkDupletList) : ReferenceList := DupletListReadReferenceList anet 0.\n218\t\n219\t(*\n220\t Теперь всё готово для преобразования сети вложенных упорядоченных пар N^{list} : Reference → ReferenceList\n221\t в сеть дуплетов N² : Reference → Reference².\n222\t\n223\t Данное преобразование можно делать по-разному: с сохранением исходных ссылок на кортежи\n224\t либо с переиндексацией. Переиндексацию можно не делать, если написать дополнительную функцию для\n225\t сети дуплетов, которая возвращает вложенную упорядоченную пару по её ссылке.\n226\t*)\n227\t\n228\t(* Функция добавления NetworkReferenceListList в NetworkDupletList *)\n229\tFixpoint AddReferenceListListToDupletList (anetd: NetworkDupletList) (anetl: NetworkReferenceListList) : NetworkDupletList :=\n230\t match anetl with\n231\t | nil => anetd\n232\t | cons h t => AddReferenceListListToDupletList (AddReferenceListToDupletList anetd h) t\n233\t end.\n234\t\n235\t(* Функция преобразования NetworkReferenceListList в NetworkDupletList *)\n236\tDefinition ReferenceListListToDupletList (anetl: NetworkReferenceListList) : NetworkDupletList :=\n237\t match anetl with\n238\t | nil => nil\n239\t | cons h t => AddReferenceListListToDupletList (ReferenceListToDupletList h) t\n240\t end.\n241\t\n242\t(* Функция поиска ReferenceList в хвосте NetworkDupletList начинающемуся с offset по её порядковому номеру.\n243\t Возвращает offset ReferenceList. *)\n244\tFixpoint DupletListOffsetReferenceList_ (anet: NetworkDupletList) (offset: nat) (index: nat) : nat :=\n245\t match anet with\n246\t | nil => offset + (length anet)\n247\t | cons (_, next_index) tail_anet =>\n248\t match index with\n249\t | O => offset\n250\t | S index' =>\n251\t if offset =? next_index then\n252\t DupletListOffsetReferenceList_ tail_anet (S offset) index'\n253\t else\n254\t DupletListOffsetReferenceList_ tail_anet (S offset) index\n255\t end\n256\t end.\n257\t\n258\t(* Функция поиска ReferenceList в NetworkDupletList по её порядковому номеру. Возвращает offset ReferenceList. *)\n259\tDefinition DupletListOffsetReferenceList (anet: NetworkDupletList) (index: nat) : nat :=\n260\t DupletListOffsetReferenceList_ anet 0 index.\n261\t\n262\t(* Функция преобразования NetworkTupleList в NetworkDupletList *)\n263\tDefinition TupleListToDupletList {n : nat} (anetv: NetworkTupleList n) : NetworkDupletList :=\n264\t ReferenceListListToDupletList (TupleListToReferenceListList anetv).\n265\t\n266\t(*\n267\t Теперь всё готово для преобразования сети дуплетов N² : Reference → Reference²\n268\t в сеть вложенных упорядоченных пар N^{list} : Reference → ReferenceList.\n269\t\n270\t Данное преобразование будем делать с сохранением исходных ссылок на кортежи.\n271\t Переиндексацию можно не делать, потому что есть функция DupletListOffsetReferenceList для\n272\t сети дуплетов, которая возвращает смещение вложенной УП по ссылке на неё.\n273\t*)\n274\t\n275\t(* Функция отрезает первую ReferenceList из NetworkDupletList и возвращает хвост *)\n276\tFixpoint DupletListBeheadReferenceList (anet: NetworkDupletList) (offset: nat) : NetworkDupletList :=\n277\t match anet with\n278\t | nil => nil\n279\t | cons (_, next_index) tail_anet =>\n280\t if offset =? next_index then (* конец ReferenceList *)\n281\t tail_anet\n282\t else (* ещё не конец ReferenceList *)\n283\t DupletListBeheadReferenceList tail_anet (S offset)\n284\t end.\n285\t\n286\t(* Функция преобразования ReferenceList и NetworkDupletList со смещения offset в NetworkReferenceListList *)\n287\tFixpoint DupletListToReferenceListList_ (anetd: NetworkDupletList) (np: ReferenceList) (offset: nat) : NetworkReferenceListList :=\n288\t match anetd with\n289\t | nil => nil (* отбрасываем ReferenceList даже если она недостроена *)\n290\t | cons (x, next_index) tail_anet =>\n291\t if offset =? next_index then (* конец ReferenceList, переходим к следующей ReferenceList *)\n292\t cons (app np (cons x nil)) (DupletListToReferenceListList_ tail_anet nil (S offset))\n293\t else (* ещё не конец ReferenceList, парсим сеть дуплетов дальше *)\n294\t DupletListToReferenceListList_ tail_anet (app np (cons x nil)) (S offset)\n295\t end.\n296\t\n297\t(* Функция преобразования NetworkDupletList в NetworkReferenceListList *)\n298\tDefinition DupletListToReferenceListList (anetd: NetworkDupletList) : NetworkReferenceListList :=\n299\t DupletListToReferenceListList_ anetd nil ReferenceDefault.\n300\t```" | |
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| "content": "1\t(**\n2\t SequenceDefinitions.v - Определение последовательностей в терминах связей.\n3\t\n4\t Последовательность — это вложенная структура связей-дуплетов,\n5\t хранимая в сети. Последовательность идентифицируется\n6\t ссылкой (Reference) на корень дерева дуплетов.\n7\t\n8\t Каждый внутренний узел дерева — это связь-дуплет (ссылка_на_левое, ссылка_на_правое).\n9\t Листья — это связи-ссылающиеся-на-себя (элемент, элемент).\n10\t\n11\t Пример: [1, 2, 3, 4] = ((1, 2), (3, 4))\n12\t Хранится в сети как:\n13\t (5: 1, 2) — дуплет для пары (1, 2)\n14\t (6: 3, 4) — дуплет для пары (3, 4)\n15\t (7: 5, 6) — дуплет-корень, ссылающийся на поддеревья\n16\t Последовательность = ссылка 7\n17\t\n18\t В нотации связей (https://github.com/link-foundation/links-notation):\n19\t (5: 1 2) (6: 3 4) (7: 5 6)\n20\t\n21\t Другие варианты:\n22\t [1, 2, 3, 4] = (((1, 2), 3), 4) — левая лестница\n23\t [1, 2, 3, 4] = (1, (2, (3, 4))) — правая лестница\n24\t\n25\t Адаптировано из: https://github.com/linksplatform/Documentation/tree/main/doc/LinksTheoryManuscriptDraft/05%20Sequences\n26\t См. также: https://github.com/linksplatform/Data.Doublets.Sequences/blob/main/csharp/Platform.Data.Doublets.Sequences/Converters/BalancedVariantConverter.cs\n27\t*)\n28\t\n29\tRequire Import PeanoNat.\n30\tRequire Import Coq.Init.Nat.\n31\tRequire Import List.\n32\tRequire Import Coq.Init.Datatypes.\n33\tImport ListNotations.\n34\tRequire Import NetworkDefinitions.\n35\tRequire Import NetworkConversions.\n36\t\n37\t(** * Последовательность — это ссылка на корень дерева в сети *)\n38\t\n39\t(** Последовательность идентифицируется ссылкой (Reference) на корень дерева дуплетов,\n40\t хранимого в сети. Это не отдельный тип — это просто Reference. *)\n41\tDefinition LinkSequence := Reference.\n42\t\n43\t(** * Вспомогательное дерево для построения последовательностей\n44\t\n45\t LinkTree используется как промежуточная структура для алгоритмов\n46\t построения деревьев (сбалансированный вариант, лестницы).\n47\t Конечный результат всегда записывается в сеть дуплетов. *)\n48\t\n49\tInductive LinkTree : Type :=\n50\t | Leaf : Reference -> LinkTree\n51\t | Node : LinkTree -> LinkTree -> LinkTree.\n52\t\n53\t(** * Запись дерева в сеть дуплетов *)\n54\t\n55\t(** Запись дерева в сеть дуплетов.\n56\t Каждый узел Node записывается как дуплет (ссылка_на_левое, ссылка_на_правое).\n57\t Листья — это элементы последовательности (не записываются как отдельные дуплеты).\n58\t Возвращает пару: (сеть дуплетов, следующее свободное смещение). *)\n59\tFixpoint TreeToDupletList_ (t : LinkTree) (offset : nat)\n60\t : NetworkDupletList * nat :=\n61\t match t with\n62\t | Leaf _ => ([], offset)\n63\t | Node l r =>\n64\t let cur := offset in\n65\t let '(left_net, left_next) := TreeToDupletList_ l (S offset) in\n66\t let left_root := match l with Leaf x => x | Node _ _ => S offset end in\n67\t let '(right_net, right_next) := TreeToDupletList_ r left_next in\n68\t let right_root := match r with Leaf x => x | Node _ _ => left_next end in\n69\t ((left_root, right_root) :: left_net ++ right_net, right_next)\n70\t end.\n71\t\n72\tDefinition TreeToDupletList (t : LinkTree) (offset : nat) : NetworkDupletList :=\n73\t fst (TreeToDupletList_ t offset).\n74\t\n75\t(** Получение ссылки на корень дерева (= последовательность) *)\n76\tDefinition TreeToSequence (t : LinkTree) (offset : nat) : LinkSequence :=\n77\t match t with\n78\t | Leaf x => x\n79\t | Node _ _ => offset\n80\t end.\n81\t\n82\t(** Получение листьев дерева (элементов последовательности) слева направо *)\n83\tFixpoint TreeToList (t : LinkTree) : list Reference :=\n84\t match t with\n85\t | Leaf x => [x]\n86\t | Node l r => TreeToList l ++ TreeToList r\n87\t end.\n88\t\n89\t(** * Алгоритмы создания последовательностей *)\n90\t\n91\t(** Правая лестница: (1, (2, (3, 4)))\n92\t Каждый следующий элемент вкладывается вправо. *)\n93\tFixpoint ListToRightStaircase (l : list Reference) : option LinkTree :=\n94\t match l with\n95\t | [] => None\n96\t | [x] => Some (Leaf x)\n97\t | x :: rest =>\n98\t match ListToRightStaircase rest with\n99\t | None => None\n100\t | Some t => Some (Node (Leaf x) t)\n101\t end\n102\t end.\n103\t\n104\t(** Левая лестница: (((1, 2), 3), 4)\n105\t Каждый следующий элемент вкладывается влево. *)\n106\tFixpoint ListToLeftStaircase_ (acc : LinkTree) (l : list Reference) : LinkTree :=\n107\t match l with\n108\t | [] => acc\n109\t | x :: rest => ListToLeftStaircase_ (Node acc (Leaf x)) rest\n110\t end.\n111\t\n112\tDefinition ListToLeftStaircase (l : list Reference) : option LinkTree :=\n113\t match l with\n114\t | [] => None\n115\t | [x] => Some (Leaf x)\n116\t | x :: rest => Some (ListToLeftStaircase_ (Leaf x) rest)\n117\t end.\n118\t\n119\t(** Сбалансированный вариант: ((1, 2), (3, 4))\n120\t Список делится пополам, каждая половина рекурсивно превращается в дерево.\n121\t Адаптировано из BalancedVariantConverter.\n122\t Использует fuel-параметр для гарантии завершения. *)\n123\tFixpoint ListToBalancedTree_ (l : list Reference) (fuel : nat) : option LinkTree :=\n124\t match fuel with\n125\t | 0 => None\n126\t | S fuel' =>\n127\t match l with\n128\t | [] => None\n129\t | [x] => Some (Leaf x)\n130\t | _ =>\n131\t let mid := length l / 2 in\n132\t let left_part := firstn mid l in\n133\t let right_part := skipn mid l in\n134\t match ListToBalancedTree_ left_part fuel', ListToBalancedTree_ right_part fuel' with\n135\t | Some lt, Some rt => Some (Node lt rt)\n136\t | _, _ => None\n137\t end\n138\t end\n139\t end.\n140\t\n141\tDefinition ListToBalancedTree (l : list Reference) : option LinkTree :=\n142\t ListToBalancedTree_ l (S (length l)).\n143\t\n144\t(** * Полное преобразование: список → сеть дуплетов + ссылка на корень *)\n145\t\n146\t(** Преобразование списка в последовательность (ссылку на корень) и сеть *)\n147\tDefinition ListToSequence (l : list Reference) (offset : nat)\n148\t : option (LinkSequence * NetworkDupletList) :=\n149\t match ListToBalancedTree l with\n150\t | None => None\n151\t | Some t => Some (TreeToSequence t offset, TreeToDupletList t offset)\n152\t end.\n153\t\n154\t(** Чтение последовательности из сети дуплетов *)\n155\tFixpoint ReadSequence_ (anet : NetworkDupletList) (root : Reference) (fuel : nat)\n156\t : list Reference :=\n157\t match fuel with\n158\t | 0 => []\n159\t | S fuel' =>\n160\t if root <? length anet then\n161\t let d := nth root anet (0, 0) in\n162\t let l := fst d in\n163\t let r := snd d in\n164\t ReadSequence_ anet l fuel' ++ ReadSequence_ anet r fuel'\n165\t else\n166\t [root]\n167\t end.\n168\t\n169\tDefinition ReadSequence (anet : NetworkDupletList) (root : LinkSequence) : list Reference :=\n170\t ReadSequence_ anet root (S (length anet)).\n171\t\n172\t(** * Примеры *)\n173\t\n174\t(** Пример: [1, 2, 3, 4] — сбалансированное дерево, записанное в сеть *)\n175\tCompute ListToSequence [1; 2; 3; 4] 5.\n176\t(* Ожидается: Some (5, [(5_root: 6, 7), (6: 1, 2), (7: 3, 4)])\n177\t Последовательность = ссылка 5\n178\t (5: 6, 7) — корень, указывает на поддеревья 6 и 7\n179\t (6: 1, 2) — левое поддерево\n180\t (7: 3, 4) — правое поддерево *)\n181\t\n182\t(** Пример: правая лестница *)\n183\tCompute match ListToRightStaircase [1; 2; 3; 4] with\n184\t | Some t => Some (TreeToSequence t 5, TreeToDupletList t 5)\n185\t | None => None\n186\t end.\n187\t(* (1, (2, (3, 4))) хранится как дуплеты *)\n188\t\n189\t(** Пример: левая лестница *)\n190\tCompute match ListToLeftStaircase [1; 2; 3; 4] with\n191\t | Some t => Some (TreeToSequence t 5, TreeToDupletList t 5)\n192\t | None => None\n193\t end.\n194\t(* (((1, 2), 3), 4) хранится как дуплеты *)\n195\t\n196\t(** Все три варианта содержат одни и те же элементы *)\n197\tDefinition balanced_1234 := match ListToBalancedTree [1; 2; 3; 4] with Some t => TreeToList t | None => [] end.\n198\tDefinition right_1234 := match ListToRightStaircase [1; 2; 3; 4] with Some t => TreeToList t | None => [] end.\n199\tDefinition left_1234 := match ListToLeftStaircase [1; 2; 3; 4] with Some t => TreeToList t | None => [] end.\n200\t\n201\tCompute balanced_1234. (* [1, 2, 3, 4] *)\n202\tCompute right_1234. (* [1, 2, 3, 4] *)\n203\tCompute left_1234. (* [1, 2, 3, 4] *)\n204\t\n205\t(** Пустая последовательность *)\n206\tCompute ListToSequence [] 5.\n207\t(* Ожидается: None *)\n208\t\n209\t(** Последовательность из одного элемента *)\n210\tCompute ListToSequence [42] 5.\n211\t(* Ожидается: Some (42, []) — лист не создаёт дуплетов, сама ссылка 42 и есть последовательность *)\n212\t" | |
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| "content": "1\t(**\n2\t SetSequenceEquivalence.v - Формальное доказательство того, что конечное множество\n3\t эквивалентно упорядоченной последовательности с уникальными элементами.\n4\t\n5\t Основная теорема: Для любого конечного множества натуральных чисел существует\n6\t единственная строго возрастающая последовательность (список без дубликатов,\n7\t отсортированный по возрастанию), содержащая в точности те же элементы.\n8\t\n9\t Адаптировано из: https://github.com/konard/subset-sum/tree/main/proofs/set_sequence_equivalence\n10\t*)\n11\t\n12\tFrom Stdlib Require Import Arith.\n13\tFrom Stdlib Require Import List.\n14\tFrom Stdlib Require Import Lia.\n15\tFrom Stdlib Require Import Bool.\n16\tImport ListNotations.\n17\t\n18\t(** * Определения *)\n19\t\n20\t(** Список является строго возрастающим, если каждый элемент строго меньше следующего *)\n21\tFixpoint StrictlyAscending (l : list nat) : Prop :=\n22\t match l with\n23\t | [] => True\n24\t | [_] => True\n25\t | x :: ((y :: _) as rest) => x < y /\\ StrictlyAscending rest\n26\t end.\n27\t\n28\t(** Список не содержит дубликатов *)\n29\tFixpoint NoDuplicates (l : list nat) : Prop :=\n30\t match l with\n31\t | [] => True\n32\t | x :: rest => ~ In x rest /\\ NoDuplicates rest\n33\t end.\n34\t\n35\t(** Упорядоченная уникальная последовательность — это строго возрастающий список *)\n36\tDefinition IsOrderedUniqueSequence (l : list nat) : Prop :=\n37\t StrictlyAscending l.\n38\t\n39\t(** * Вспомогательные леммы *)\n40\t\n41\t(** Все элементы после позиции в строго возрастающем списке больше *)\n42\tLemma strictly_ascending_all_greater : forall l x y,\n43\t StrictlyAscending (x :: y :: l |
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