{
  "exact_scope": "local inductive certificate obligations, including infinite periodic inclusions",
  "examples": [
    {
      "exact_check": {
        "bound": 0,
        "valid": true
      },
      "instance": {
        "clock": {
          "initial": 0,
          "states": [
            0
          ],
          "table": [
            [
              0,
              "x",
              0,
              0
            ]
          ]
        },
        "rows": [
          {
            "T": 1,
            "below": [
              0
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            "e": 0,
            "p": 1,
            "q": 0,
            "r": 0,
            "residues": []
          }
        ],
        "source": {
          "alphabet": [
            "x"
          ],
          "initial": [
            0
          ],
          "rules": [
            [
              0,
              "x",
              "any",
              -1,
              0
            ],
            [
              0,
              "x",
              "any",
              0,
              0
            ]
          ],
          "states": [
            0
          ]
        }
      },
      "name": "bottom_pop_keep"
    },
    {
      "exact_check": {
        "bound": 1,
        "valid": true
      },
      "instance": {
        "clock": {
          "initial": 0,
          "states": [
            0
          ],
          "table": [
            [
              0,
              "x",
              0,
              0
            ]
          ]
        },
        "rows": [
          {
            "T": 1,
            "below": [
              0
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            "e": 0,
            "p": 1,
            "q": 0,
            "r": 0,
            "residues": []
          },
          {
            "T": 1,
            "below": [
              0
            ],
            "e": 1,
            "p": 1,
            "q": 0,
            "r": 0,
            "residues": []
          }
        ],
        "source": {
          "alphabet": [
            "x"
          ],
          "initial": [
            0
          ],
          "rules": [
            [
              0,
              "x",
              "zero",
              1,
              0
            ],
            [
              0,
              "x",
              "positive",
              0,
              0
            ]
          ],
          "states": [
            0
          ]
        }
      },
      "name": "guarded_push_once"
    },
    {
      "exact_check": {
        "bound": 0,
        "valid": true
      },
      "instance": {
        "clock": {
          "initial": 0,
          "states": [
            0
          ],
          "table": [
            [
              0,
              "a",
              0,
              1
            ],
            [
              0,
              "b",
              0,
              -1
            ],
            [
              0,
              "x",
              0,
              0
            ]
          ]
        },
        "rows": [
          {
            "T": 0,
            "below": [],
            "e": 0,
            "p": 1,
            "q": 0,
            "r": 0,
            "residues": [
              0
            ]
          }
        ],
        "source": {
          "alphabet": [
            "a",
            "b",
            "x"
          ],
          "initial": [
            0
          ],
          "rules": [
            [
              0,
              "a",
              "any",
              1,
              0
            ],
            [
              0,
              "b",
              "positive",
              -1,
              0
            ],
            [
              0,
              "x",
              "any",
              0,
              0
            ],
            [
              0,
              "x",
              "zero",
              -1,
              0
            ]
          ],
          "states": [
            0
          ]
        }
      },
      "name": "dyck_with_bottom_noise"
    },
    {
      "exact_check": {
        "bound": 0,
        "valid": true
      },
      "instance": {
        "clock": {
          "initial": 0,
          "states": [
            0
          ],
          "table": [
            [
              0,
              "a",
              0,
              1
            ]
          ]
        },
        "rows": [
          {
            "T": 0,
            "below": [],
            "e": 0,
            "p": 2,
            "q": 0,
            "r": 0,
            "residues": [
              0
            ]
          },
          {
            "T": 0,
            "below": [],
            "e": 0,
            "p": 2,
            "q": 1,
            "r": 0,
            "residues": [
              1
            ]
          }
        ],
        "source": {
          "alphabet": [
            "a"
          ],
          "initial": [
            0
          ],
          "rules": [
            [
              0,
              "a",
              "any",
              1,
              1
            ],
            [
              1,
              "a",
              "any",
              1,
              0
            ]
          ],
          "states": [
            0,
            1
          ]
        }
      },
      "name": "parity_rows"
    },
    {
      "exact_check": {
        "bound": 1,
        "valid": true
      },
      "instance": {
        "clock": {
          "initial": 0,
          "states": [
            0,
            1
          ],
          "table": [
            [
              0,
              "x",
              1,
              1
            ],
            [
              1,
              "x",
              1,
              0
            ]
          ]
        },
        "rows": [
          {
            "T": 1,
            "below": [
              0
            ],
            "e": 0,
            "p": 1,
            "q": 0,
            "r": 0,
            "residues": []
          },
          {
            "T": 1,
            "below": [
              0
            ],
            "e": -1,
            "p": 1,
            "q": 0,
            "r": 1,
            "residues": []
          }
        ],
        "source": {
          "alphabet": [
            "x"
          ],
          "initial": [
            0
          ],
          "rules": [
            [
              0,
              "x",
              "any",
              -1,
              0
            ],
            [
              0,
              "x",
              "any",
              0,
              0
            ]
          ],
          "states": [
            0
          ]
        }
      },
      "name": "clock_ahead_at_boundary"
    },
    {
      "exact_check": {
        "mode": [
          0,
          0,
          1
        ],
        "reason": "boundary",
        "rule_index": 0,
        "target_mode": [
          0,
          0,
          2
        ],
        "target_z": 0,
        "valid": false,
        "z": 0
      },
      "instance": {
        "clock": {
          "initial": 0,
          "states": [
            0
          ],
          "table": [
            [
              0,
              "x",
              0,
              0
            ]
          ]
        },
        "rows": [
          {
            "T": 1,
            "below": [
              0
            ],
            "e": 0,
            "p": 1,
            "q": 0,
            "r": 0,
            "residues": []
          },
          {
            "T": 1,
            "below": [
              0
            ],
            "e": 1,
            "p": 1,
            "q": 0,
            "r": 0,
            "residues": []
          }
        ],
        "source": {
          "alphabet": [
            "x"
          ],
          "initial": [
            0
          ],
          "rules": [
            [
              0,
              "x",
              "any",
              1,
              0
            ],
            [
              0,
              "x",
              "any",
              0,
              0
            ]
          ],
          "states": [
            0
          ]
        }
      },
      "name": "unbounded_source_truncated_certificate"
    },
    {
      "exact_check": {
        "mode": [
          0,
          0,
          0
        ],
        "reason": "interior",
        "rule_index": 0,
        "target_mode": [
          0,
          0,
          -1
        ],
        "target_z": 0,
        "valid": false,
        "z": 1
      },
      "instance": {
        "clock": {
          "initial": 0,
          "states": [
            0
          ],
          "table": [
            [
              0,
              "x",
              0,
              0
            ]
          ]
        },
        "rows": [
          {
            "T": 0,
            "below": [],
            "e": 0,
            "p": 1,
            "q": 0,
            "r": 0,
            "residues": [
              0
            ]
          }
        ],
        "source": {
          "alphabet": [
            "x"
          ],
          "initial": [
            0
          ],
          "rules": [
            [
              0,
              "x",
              "any",
              -1,
              0
            ],
            [
              0,
              "x",
              "any",
              0,
              0
            ]
          ],
          "states": [
            0
          ]
        }
      },
      "name": "unrestricted_ray_for_bottom_only_source"
    },
    {
      "exact_check": {
        "mode": [
          0,
          0,
          0
        ],
        "reason": "boundary",
        "rule_index": 0,
        "target_mode": [
          1,
          0,
          0
        ],
        "target_z": 1,
        "valid": false,
        "z": 0
      },
      "instance": {
        "clock": {
          "initial": 0,
          "states": [
            0
          ],
          "table": [
            [
              0,
              "a",
              0,
              1
            ]
          ]
        },
        "rows": [
          {
            "T": 0,
            "below": [],
            "e": 0,
            "p": 2,
            "q": 0,
            "r": 0,
            "residues": [
              0
            ]
          },
          {
            "T": 0,
            "below": [],
            "e": 0,
            "p": 2,
            "q": 1,
            "r": 0,
            "residues": [
              0
            ]
          }
        ],
        "source": {
          "alphabet": [
            "a"
          ],
          "initial": [
            0
          ],
          "rules": [
            [
              0,
              "a",
              "any",
              1,
              1
            ],
            [
              1,
              "a",
              "any",
              1,
              0
            ]
          ],
          "states": [
            0,
            1
          ]
        }
      },
      "name": "corrupted_parity_row"
    }
  ],
  "finite_audit": {
    "coordinate_formula_cases": 891,
    "coordinate_ranges": {
      "d_and_c": [
        -1,
        0,
        1
      ],
      "e": [
        -5,
        5
      ],
      "z": [
        0,
        8
      ]
    },
    "inclusion_cases": 28812,
    "periodic_descriptions": 98,
    "periodic_ranges": {
      "oracle_z": [
        1,
        80
      ],
      "period": [
        1,
        3
      ],
      "prefix_and_residue_subsets": "all",
      "threshold": [
        0,
        2
      ]
    },
    "status": "bounded implementation audit passed"
  },
  "not_implemented": [
    "saturation",
    "certificate synthesis",
    "VPA determinization",
    "scaffold generation",
    "proof-assistant verification"
  ],
  "note": "one-counter-certificates.md"
}