{
  "id": "KL-FCS-035",
  "version": "1.0.0",
  "domain": "Term rewriting",
  "kind": "rewriting",
  "title": "All reduction orders through length 6",
  "problem": "Establish a unique sorted normal form for every bounded binary word.",
  "specification": {
    "max_length": 6,
    "rule": [
      "10",
      "01"
    ],
    "sample_word": "101010"
  },
  "claim": {
    "unique_normal_forms": true
  },
  "witness": {
    "trace": [
      "101010",
      "011010",
      "010110",
      "001110",
      "001101",
      "001011",
      "000111"
    ]
  },
  "verification_scope": "All reduction paths over 127 bounded inputs",
  "explanation": "The checker explores every enabled rewrite, not just the trace shown. Each result preserves the symbol multiset and reaches a block of zeros followed by ones.",
  "limitations": "The mechanical confluence result is restricted to the declared finite word family.",
  "verification_status": "mechanically-checked",
  "review_status": "awaiting-independent-review",
  "provenance": {
    "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
    "external_dataset": null,
    "model_run": null
  },
  "references": [
    "https://isa-afp.org/entries/Abstract-Rewriting.html"
  ],
  "license_status": "not-yet-selected",
  "dataset": {
    "family": "rewriting",
    "task": "Establish a unique sorted normal form for every bounded binary word.",
    "input_encoding": "Structured JSON; field meanings are stated in the specification.",
    "coverage": "All reduction paths over 127 bounded inputs",
    "acceptance": [
      "Enumerate every binary word through the declared length.",
      "Explore all possible one-step adjacent rewrites.",
      "Collect terminal words, memoizing the reduction graph.",
      "Require one normal form per input and replay the supplied concrete trace."
    ],
    "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
    "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
  },
  "lesson": {
    "motivation": "Inspect every allowed reduction order in a finite family, and compare all reachable normal forms.",
    "definitions": [
      {
        "term": "Rewrite rule",
        "definition": "A permitted local replacement; here 10 → 01."
      },
      {
        "term": "Normal form",
        "definition": "A word containing no reducible 10 substring."
      },
      {
        "term": "Confluence on a domain",
        "definition": "Every reduction path from each declared input can reach a common result."
      }
    ],
    "reasoning": [
      "Enumerate every binary word through the declared length.",
      "Explore all possible one-step adjacent rewrites.",
      "Collect terminal words, memoizing the reduction graph.",
      "Require one normal form per input and replay the supplied concrete trace."
    ],
    "worked_example": "The checker explores every enabled rewrite, not just the trace shown. Each result preserves the symbol multiset and reaches a block of zeros followed by ones.",
    "complexity": "Each rewrite decreases the number of inverted 1-before-0 pairs. Exhaustive graph exploration is restricted to the stated length.",
    "common_error": "Two chosen reduction strategies agreeing does not establish that all strategies agree.",
    "further_work": "Publish a general inversion-measure termination argument and a separately checked confluence proof."
  },
  "related_ids": [
    "KL-FCS-034",
    "KL-FCS-036"
  ]
}