{
  "id": "KL-FCS-041",
  "version": "1.0.0",
  "domain": "Program verification",
  "kind": "hoare",
  "title": "Sum-loop obligations for n ≤ 10",
  "problem": "Verify a loop that increments i and then adds i to total, starting from zero.",
  "specification": {
    "program": "sum-first-n",
    "max_n": 10,
    "sample_n": 5,
    "precondition": "n ≥ 0; i=0; total=0",
    "invariant": "2*total = i*(i+1) and 0 ≤ i ≤ n",
    "variant": "n-i"
  },
  "claim": {
    "postcondition_holds": true,
    "variant_decreases": true
  },
  "witness": {
    "trace": [
      [
        0,
        0
      ],
      [
        1,
        1
      ],
      [
        2,
        3
      ],
      [
        3,
        6
      ],
      [
        4,
        10
      ],
      [
        5,
        15
      ]
    ]
  },
  "verification_scope": "Complete bounded program family · 11 inputs",
  "explanation": "The invariant explains the partial sum at each boundary. The remaining iteration count strictly decreases, and the exit condition turns the invariant into the postcondition.",
  "limitations": "This replay checks n through the stated bound; it does not constitute an unrestricted deductive proof.",
  "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://dafny.org/latest/OnlineTutorial/guide"
  ],
  "license_status": "not-yet-selected",
  "dataset": {
    "family": "program-verification",
    "task": "Verify a loop that increments i and then adds i to total, starting from zero.",
    "input_encoding": "Structured JSON; field meanings are stated in the specification.",
    "coverage": "Complete bounded program family · 11 inputs",
    "acceptance": [
      "Enumerate every admitted bound n.",
      "Start with i=0 and total=0.",
      "Check 2·total=i(i+1) and the decreasing variant n−i.",
      "Replay the sample trace and require total=n(n+1)/2 at exit."
    ],
    "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": "Connect preconditions, invariants, variants, and postconditions in a completely specified bounded program family.",
    "definitions": [
      {
        "term": "Loop invariant",
        "definition": "A property maintained at every loop boundary."
      },
      {
        "term": "Variant",
        "definition": "A value in a well-founded domain that strictly decreases while the loop runs."
      },
      {
        "term": "Postcondition",
        "definition": "The property required when execution exits."
      }
    ],
    "reasoning": [
      "Enumerate every admitted bound n.",
      "Start with i=0 and total=0.",
      "Check 2·total=i(i+1) and the decreasing variant n−i.",
      "Replay the sample trace and require total=n(n+1)/2 at exit."
    ],
    "worked_example": "The invariant explains the partial sum at each boundary. The remaining iteration count strictly decreases, and the exit condition turns the invariant into the postcondition.",
    "complexity": "The family runs Σ n loop iterations for n=0…N, so total replay is O(N²).",
    "common_error": "A postcondition on one execution is weaker than an invariant and a declared input domain.",
    "further_work": "Add inductive proof obligations and externally checked Hoare derivations."
  },
  "related_ids": [
    "KL-FCS-040",
    "KL-FCS-042"
  ]
}