lean-bench/docs/reviews/simplex-schemas.html

<!doctype html><html lang="en"><meta charset="utf-8"><title>P0 semantic simplex schemas</title><style>body{font:16px/1.5 system-ui;max-width:1050px;margin:32px auto;padding:16px}pre{white-space:pre-wrap;background:#f5f5f5;padding:16px}section{border-top:1px solid #ccc}</style><h1>65 semantic multi-role relation schemas</h1><p>Schema annotation seeds, not independently validated mathematical statements. A simplex joins the objects, assumptions and conclusion in its slots. References are partial symbol/syntax candidates. This review contains schema annotations and Lean signature text, not textbook passages or full assistant judgment explanations.</p><section id="P01"><h2>P01</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:548ab1a767a685aefa146455:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;field&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;field&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;fact1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;commutative ring&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;commutative&quot;,
          &quot;ring&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;fact2&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;identity nonzero&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;identity&quot;,
          &quot;nonzero&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;nonzero elements are units&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;nonzero&quot;,
          &quot;elements&quot;,
          &quot;are&quot;,
          &quot;units&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:548ab1a767a685aefa146455:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;F&quot;: {
      &quot;name&quot;: &quot;F&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;own&quot;: {
      &quot;name&quot;: &quot;own&quot;,
      &quot;kind&quot;: &quot;own&quot;,
      &quot;type&quot;: &quot;class&quot;,
      &quot;attrs&quot;: {
        &quot;keyword&quot;: &quot;class&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P02"><h2>P02</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:6ea85a17cb5decb0670c4afe:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;whole group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;whole&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;object3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;identity singleton&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;identity&quot;,
          &quot;singleton&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;both sets are subgroups&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;both&quot;,
          &quot;sets&quot;,
          &quot;are&quot;,
          &quot;subgroups&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:6ea85a17cb5decb0670c4afe:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;Subgroup G (Set.univ : Set G) ∧ Subgroup G {1}&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P03"><h2>P03</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:389a27cbae1957943fae2960:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer ring&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;integer&quot;,
          &quot;ring&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;unit&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;unit&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;units equal plus minus one&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;units&quot;,
          &quot;equal&quot;,
          &quot;plus&quot;,
          &quot;minus&quot;,
          &quot;one&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:389a27cbae1957943fae2960:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;premise1&quot;: {
      &quot;name&quot;: &quot;premise1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;(u : ℤ)&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;goal_implication&quot;,
        &quot;classified_by&quot;: &quot;implication&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;premise2&quot;: {
      &quot;name&quot;: &quot;premise2&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;IsUnit u&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;goal_implication&quot;,
        &quot;classified_by&quot;: &quot;implication&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;u = 1 ∨ u = -1&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P04"><h2>P04</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:b63f0ed4d1ea1e11e34e6f8a:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;generated subgroup equals integer powers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;generated&quot;,
          &quot;subgroup&quot;,
          &quot;equals&quot;,
          &quot;integer&quot;,
          &quot;powers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:b63f0ed4d1ea1e11e34e6f8a:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\&lt;a\\&gt; = {a ^ k | k : ℤ}&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;a&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P05"><h2>P05</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:db4553333550bccbd3f0e21d:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;map between groups&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;map&quot;,
          &quot;between&quot;,
          &quot;groups&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;bijective&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;bijective&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;preserves multiplication&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;preserves&quot;,
          &quot;multiplication&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;isomorphic iff such map exists&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;isomorphic&quot;,
          &quot;iff&quot;,
          &quot;such&quot;,
          &quot;map&quot;,
          &quot;exists&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:db4553333550bccbd3f0e21d:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group H&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;own&quot;: {
      &quot;name&quot;: &quot;own&quot;,
      &quot;kind&quot;: &quot;own&quot;,
      &quot;type&quot;: &quot;Prop&quot;,
      &quot;attrs&quot;: {
        &quot;keyword&quot;: &quot;def&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P06"><h2>P06</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:c3e75089fbfe8f386b4a22ec:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;b&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;inverse product reverses inverse factors&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;inverse&quot;,
          &quot;product&quot;,
          &quot;reverses&quot;,
          &quot;inverse&quot;,
          &quot;factors&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:c3e75089fbfe8f386b4a22ec:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;b&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;(a * b)⁻¹ = b⁻¹ * a⁻¹&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;b&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P07"><h2>P07</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:91a6a0c1c21014bf08ffa3b6:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;,
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;p&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;prime factor&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;p&quot;
        ],
        &quot;keywords&quot;: [
          &quot;prime&quot;,
          &quot;factor&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;q&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;prime factor&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;q&quot;
        ],
        &quot;keywords&quot;: [
          &quot;prime&quot;,
          &quot;factor&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;totient function&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;totient&quot;,
          &quot;function&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;totient value can be found&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;totient&quot;,
          &quot;value&quot;,
          &quot;can&quot;,
          &quot;be&quot;,
          &quot;found&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object4&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;recover both prime factors&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;recover&quot;,
          &quot;both&quot;,
          &quot;prime&quot;,
          &quot;factors&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:91a6a0c1c21014bf08ffa3b6:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;p&quot;: {
      &quot;name&quot;: &quot;p&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;q&quot;: {
      &quot;name&quot;: &quot;q&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hn&quot;: {
      &quot;name&quot;: &quot;hn&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;n = p * q&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;,
          &quot;p&quot;,
          &quot;q&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hpq&quot;: {
      &quot;name&quot;: &quot;hpq&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;p.Prime ∧ q.Prime&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;p&quot;,
          &quot;q&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;premise1&quot;: {
      &quot;name&quot;: &quot;premise1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∃ (φ : ℕ → ℕ), (∀ m, φ m = Nat.totient m)&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;goal_implication&quot;,
        &quot;classified_by&quot;: &quot;implication&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;∃ (p&#x27; q&#x27; : ℕ), n = p&#x27; * q&#x27; ∧ p&#x27;.Prime ∧ q&#x27;.Prime&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 1,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P08"><h2>P08</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:675ac9dbf3ec8a0553149a96:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;group isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;map preserves identity and inverses&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;map&quot;,
          &quot;preserves&quot;,
          &quot;identity&quot;,
          &quot;and&quot;,
          &quot;inverses&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:675ac9dbf3ec8a0553149a96:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group H&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;φ (1 : G) = (1 : H) ∧ ∀ g : G, φ g⁻¹ = (φ g)⁻¹&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 1,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P09"><h2>P09</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:8b39fc5b656bac00bc1e863f:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;cyclic group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;cyclic&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;rectangle symmetry group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;rectangle&quot;,
          &quot;symmetry&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object5&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object6&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object7&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;natural number&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;natural&quot;,
          &quot;number&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;nonisomorphism and preservation of powers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;nonisomorphism&quot;,
          &quot;and&quot;,
          &quot;preservation&quot;,
          &quot;of&quot;,
          &quot;powers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P10"><h2>P10</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:3549ad65eee67f7b432a8b70:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;b&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;left right equations uniquely solvable&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;left&quot;,
          &quot;right&quot;,
          &quot;equations&quot;,
          &quot;uniquely&quot;,
          &quot;solvable&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:3549ad65eee67f7b432a8b70:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;b&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\\\(∃! x : G, a * x = b) ∧ (∃! x : G, x * a = b)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 2,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;a&quot;,
          &quot;b&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P11"><h2>P11</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:0b7db1a8100c33a50a08981f:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group size&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;size&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;finite&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;finite&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;nth power equals identity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;nth&quot;,
          &quot;power&quot;,
          &quot;equals&quot;,
          &quot;identity&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:0b7db1a8100c33a50a08981f:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;,
          &quot;Fintype G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;g&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;g ^ Fintype.card G = 1&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;g&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P12"><h2>P12</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:e3d40d93cda13ec59c35d9a5:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;RSA method&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;rsa&quot;,
          &quot;method&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;messages&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;messages&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;design secure authenticated message exchange&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;design&quot;,
          &quot;secure&quot;,
          &quot;authenticated&quot;,
          &quot;message&quot;,
          &quot;exchange&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P13"><h2>P13</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:aaf1250a5104a4f7c1ca0d29:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;Φ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group of units&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;Φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;of&quot;,
          &quot;units&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;relatively prime to modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;relatively&quot;,
          &quot;prime&quot;,
          &quot;to&quot;,
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;totient power congruent to one&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;totient&quot;,
          &quot;power&quot;,
          &quot;congruent&quot;,
          &quot;to&quot;,
          &quot;one&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:aaf1250a5104a4f7c1ca0d29:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;Nat.Coprime a n&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;predicate&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;(a ^ Nat.totient n) ≡ 1 [ZMOD n]&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P14"><h2>P14</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:57966f39253aba58777a4ffd:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;X&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;set&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;X&quot;
        ],
        &quot;keywords&quot;: [
          &quot;set&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;real valued functions&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;real&quot;,
          &quot;valued&quot;,
          &quot;functions&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;ring identity and unit characterization&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;ring&quot;,
          &quot;identity&quot;,
          &quot;and&quot;,
          &quot;unit&quot;,
          &quot;characterization&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:57966f39253aba58777a4ffd:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;X&quot;: {
      &quot;name&quot;: &quot;X&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;Ring (X → ℝ)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;X&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P15"><h2>P15</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:6ef921c49672c73f8f3e0edf:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;finite&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;finite&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;rows columns contain elements once&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;rows&quot;,
          &quot;columns&quot;,
          &quot;contain&quot;,
          &quot;elements&quot;,
          &quot;once&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:6ef921c49672c73f8f3e0edf:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;,
          &quot;Fintype G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;g&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\\\\\n  ∃! (x : G), x * g = h \\\\\\n  ∧ ∃! (y : G), g * y = h&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 2,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;g&quot;,
          &quot;h&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P16"><h2>P16</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:371fe1b32fe04da354ad4478:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;K&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;K&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;object4&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;finite&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;finite&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;K&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;finite&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;K&quot;
        ],
        &quot;keywords&quot;: [
          &quot;finite&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact3&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;isomorphic to product&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;isomorphic&quot;,
          &quot;to&quot;,
          &quot;product&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;element order divides lcm sizes&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;element&quot;,
          &quot;order&quot;,
          &quot;divides&quot;,
          &quot;lcm&quot;,
          &quot;sizes&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:371fe1b32fe04da354ad4478:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Fintype H&quot;,
          &quot;Group H&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;K&quot;: {
      &quot;name&quot;: &quot;K&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Fintype K&quot;,
          &quot;Group K&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;g&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;H × K&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;H&quot;,
          &quot;K&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;OrderOf g ∣ lcm (Fintype.card H) (Fintype.card K)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;H&quot;,
          &quot;K&quot;,
          &quot;g&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P17"><h2>P17</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:a7396b1a5d5ebcd5b48ea7ba:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;real numbers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;real&quot;,
          &quot;numbers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;rational numbers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;rational&quot;,
          &quot;numbers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;object3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;complex numbers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;complex&quot;,
          &quot;numbers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;object4&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;integers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object5&quot;: {
      &quot;name&quot;: &quot;object5&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;real polynomial ring&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;real&quot;,
          &quot;polynomial&quot;,
          &quot;ring&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;three fields two nonfields&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;three&quot;,
          &quot;fields&quot;,
          &quot;two&quot;,
          &quot;nonfields&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P18"><h2>P18</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:59bb865c9bdfb8521e414743:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;complex numbers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;complex&quot;,
          &quot;numbers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;norm one subset&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;norm&quot;,
          &quot;one&quot;,
          &quot;subset&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;object3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;nonzero complex numbers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;nonzero&quot;,
          &quot;complex&quot;,
          &quot;numbers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;unit modulus set is subgroup&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;unit&quot;,
          &quot;modulus&quot;,
          &quot;set&quot;,
          &quot;is&quot;,
          &quot;subgroup&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P19"><h2>P19</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:283d407566264b02fbee6560:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;generated cyclic subgroup&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;generated&quot;,
          &quot;cyclic&quot;,
          &quot;subgroup&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;element order defined by subgroup size&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;element&quot;,
          &quot;order&quot;,
          &quot;defined&quot;,
          &quot;by&quot;,
          &quot;subgroup&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:283d407566264b02fbee6560:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;own&quot;: {
      &quot;name&quot;: &quot;own&quot;,
      &quot;kind&quot;: &quot;own&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;keyword&quot;: &quot;def&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P20"><h2>P20</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:7de228e675b949cd64850c04:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;e&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;identity element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;e&quot;
        ],
        &quot;keywords&quot;: [
          &quot;identity&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;e&#x27;&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;e&#x27;&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;fact1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;left product equals g&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;e&#x27;&quot;,
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;left&quot;,
          &quot;product&quot;,
          &quot;equals&quot;,
          &quot;g&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;,
          &quot;object4&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;element equals group identity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;element&quot;,
          &quot;equals&quot;,
          &quot;group&quot;,
          &quot;identity&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:7de228e675b949cd64850c04:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;e&#x27;&quot;: {
      &quot;name&quot;: &quot;e&#x27;&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;g&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;e&#x27; * g = g&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;e&#x27;&quot;,
          &quot;g&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;e&#x27; = 1&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;e&#x27;&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P21"><h2>P21</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:dc1c450987f3e3a5285260f7:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;group isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;abelian iff target abelian&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;abelian&quot;,
          &quot;iff&quot;,
          &quot;target&quot;,
          &quot;abelian&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:dc1c450987f3e3a5285260f7:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group H&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\\\\\n  Commutative (*) ↔ Commutative ((*) : H → H → H)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: true,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P22"><h2>P22</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:c42dc687ecc042f5b7a00a9d:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;finite group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;finite&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;fact1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;finite&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;finite&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;each row column permutes group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;each&quot;,
          &quot;row&quot;,
          &quot;column&quot;,
          &quot;permutes&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:c42dc687ecc042f5b7a00a9d:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Fintype G&quot;,
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\/\\n  (∀ g : G, ∀ g&#x27; : G, ∃! h : G, g * h = g&#x27;) ∧ \\/\\n  (∀ g : G, ∀ g&#x27; : G, ∃! h : G, h * g = g&#x27;)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 2,
        &quot;forall_count&quot;: 4,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P23"><h2>P23</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:8491370b02906b6efdb3da5e:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;natural modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;natural&quot;,
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;p&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;prime&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;p&quot;
        ],
        &quot;keywords&quot;: [
          &quot;prime&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;q&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;prime&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;q&quot;
        ],
        &quot;keywords&quot;: [
          &quot;prime&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;m&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer exponent&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;m&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;,
          &quot;exponent&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object5&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;product of primes&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;product&quot;,
          &quot;of&quot;,
          &quot;primes&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;m&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;lcm predecessors&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;m&quot;
        ],
        &quot;keywords&quot;: [
          &quot;lcm&quot;,
          &quot;predecessors&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object4&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact3&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;relatively prime to modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;relatively&quot;,
          &quot;prime&quot;,
          &quot;to&quot;,
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object5&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;exponent power congruent to one&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;exponent&quot;,
          &quot;power&quot;,
          &quot;congruent&quot;,
          &quot;to&quot;,
          &quot;one&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:8491370b02906b6efdb3da5e:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;p&quot;: {
      &quot;name&quot;: &quot;p&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;q&quot;: {
      &quot;name&quot;: &quot;q&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hp&quot;: {
      &quot;name&quot;: &quot;hp&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;Nat.Prime p&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;predicate&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;p&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hq&quot;: {
      &quot;name&quot;: &quot;hq&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;Nat.Prime q&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;predicate&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;q&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hn&quot;: {
      &quot;name&quot;: &quot;hn&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;n = p * q&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;,
          &quot;p&quot;,
          &quot;q&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;m&quot;: {
      &quot;name&quot;: &quot;m&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hm&quot;: {
      &quot;name&quot;: &quot;hm&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;m = Nat.lcm (p - 1) (q - 1)&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;m&quot;,
          &quot;p&quot;,
          &quot;q&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;ha&quot;: {
      &quot;name&quot;: &quot;ha&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;Nat.Coprime a n&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;predicate&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;a ^ m ≡ 1 [ZMOD n]&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;m&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P24"><h2>P24</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:437d02fe69d580bde075648c:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;painted square&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;painted&quot;,
          &quot;square&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;color preserving symmetries&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;color&quot;,
          &quot;preserving&quot;,
          &quot;symmetries&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;identify symmetry group size isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;identify&quot;,
          &quot;symmetry&quot;,
          &quot;group&quot;,
          &quot;size&quot;,
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P25"><h2>P25</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:5fe4f3fbbead7e6988f015a9:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;x&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;x&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;y&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;y&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;left and right cancellation implications&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;left&quot;,
          &quot;and&quot;,
          &quot;right&quot;,
          &quot;cancellation&quot;,
          &quot;implications&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:5fe4f3fbbead7e6988f015a9:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;x&quot;: {
      &quot;name&quot;: &quot;x&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;y&quot;: {
      &quot;name&quot;: &quot;y&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;premise1&quot;: {
      &quot;name&quot;: &quot;premise1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;a * x = a * y&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;goal_implication&quot;,
        &quot;classified_by&quot;: &quot;implication&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;x&quot;,
          &quot;y&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;x = y&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;x&quot;,
          &quot;y&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P26"><h2>P26</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:8a55f59982e50e4af47b3d55:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;h&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;m&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;exponent modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;m&quot;
        ],
        &quot;keywords&quot;: [
          &quot;exponent&quot;,
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;congruent one modulo m&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;h&quot;
        ],
        &quot;keywords&quot;: [
          &quot;congruent&quot;,
          &quot;one&quot;,
          &quot;modulo&quot;,
          &quot;m&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;power congruent to original integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;power&quot;,
          &quot;congruent&quot;,
          &quot;to&quot;,
          &quot;original&quot;,
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:8a55f59982e50e4af47b3d55:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;m&quot;: {
      &quot;name&quot;: &quot;m&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hn&quot;: {
      &quot;name&quot;: &quot;hn&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;h % m = 1&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;h&quot;,
          &quot;m&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hm&quot;: {
      &quot;name&quot;: &quot;hm&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;m ∣ n&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;m&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;a ^ h ≡ a [ZMOD n]&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;h&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P27"><h2>P27</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:8d4d64143a8b214e38d7ba1b:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;S_4&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;permutation group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;S_4&quot;
        ],
        &quot;keywords&quot;: [
          &quot;permutation&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;π&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;permutation&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;π&quot;
        ],
        &quot;keywords&quot;: [
          &quot;permutation&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;object3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;stabilizer subgroup&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;stabilizer&quot;,
          &quot;subgroup&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;π&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;fixes fourth point&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;π&quot;
        ],
        &quot;keywords&quot;: [
          &quot;fixes&quot;,
          &quot;fourth&quot;,
          &quot;point&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;stabilizer subgroup isomorphic to S3&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;stabilizer&quot;,
          &quot;subgroup&quot;,
          &quot;isomorphic&quot;,
          &quot;to&quot;,
          &quot;s&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P28"><h2>P28</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:6017927fdafd91ca22ed6ade:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;rational number&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;rational&quot;,
          &quot;number&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;rational number&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;b&quot;
        ],
        &quot;keywords&quot;: [
          &quot;rational&quot;,
          &quot;number&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;object3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;real quadratic subset&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;real&quot;,
          &quot;quadratic&quot;,
          &quot;subset&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;object4&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;conjugation map&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;conjugation&quot;,
          &quot;map&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;subset field conjugation is automorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;subset&quot;,
          &quot;field&quot;,
          &quot;conjugation&quot;,
          &quot;is&quot;,
          &quot;automorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P29"><h2>P29</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:089c452ec2f9b850733a44f6:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object5&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;natural number&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;natural&quot;,
          &quot;number&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;group isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;map preserves powers and identity powers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;map&quot;,
          &quot;preserves&quot;,
          &quot;powers&quot;,
          &quot;and&quot;,
          &quot;identity&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:089c452ec2f9b850733a44f6:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group H&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;g&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\/\\/  φ(g^n) = (φ g)^n ∧ (g^n = 1 → (φ g)^n = 1)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;g&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P30"><h2>P30</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:75009c3d616f02be4d7a5824:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;M&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;set&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;M&quot;
        ],
        &quot;keywords&quot;: [
          &quot;set&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;N&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;subset&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;N&quot;
        ],
        &quot;keywords&quot;: [
          &quot;subset&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;sequence of elements&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;sequence&quot;,
          &quot;of&quot;,
          &quot;elements&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;M&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;associative operation&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;M&quot;
        ],
        &quot;keywords&quot;: [
          &quot;associative&quot;,
          &quot;operation&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;N&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;closed under multiplication&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;N&quot;
        ],
        &quot;keywords&quot;: [
          &quot;closed&quot;,
          &quot;under&quot;,
          &quot;multiplication&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact3&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;positive&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;positive&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact4&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;elements belong to subset&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;elements&quot;,
          &quot;belong&quot;,
          &quot;to&quot;,
          &quot;subset&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object4&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;ordered product belongs to subset&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;ordered&quot;,
          &quot;product&quot;,
          &quot;belongs&quot;,
          &quot;to&quot;,
          &quot;subset&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:75009c3d616f02be4d7a5824:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;M&quot;: {
      &quot;name&quot;: &quot;M&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Semigroup M&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;N&quot;: {
      &quot;name&quot;: &quot;N&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Set M&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;M&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hN&quot;: {
      &quot;name&quot;: &quot;hN&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∀ x y ∈ N, x * y ∈ N&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;N&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hn&quot;: {
      &quot;name&quot;: &quot;hn&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;1 ≤ n&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Fin n → M&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;M&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;ha&quot;: {
      &quot;name&quot;: &quot;ha&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∀ i, a i ∈ N&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;N&quot;,
          &quot;a&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;(∏ i, a i) ∈ N&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;N&quot;,
          &quot;a&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P31"><h2>P31</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:9091f6a277215174e821d373:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;φ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;previously defined ring map&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;φ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;previously&quot;,
          &quot;defined&quot;,
          &quot;ring&quot;,
          &quot;map&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;verify this map preserves ring operations&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;verify&quot;,
          &quot;this&quot;,
          &quot;map&quot;,
          &quot;preserves&quot;,
          &quot;ring&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:9091f6a277215174e821d373:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;R&quot;: {
      &quot;name&quot;: &quot;R&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Ring R&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;S&quot;: {
      &quot;name&quot;: &quot;S&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Ring S&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;IsRingHom φ&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P32"><h2>P32</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:0cd2f0368a0d8234c2593845:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;matrix size&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;matrix&quot;,
          &quot;size&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;real matrix&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;T&quot;
        ],
        &quot;keywords&quot;: [
          &quot;real&quot;,
          &quot;matrix&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;evaluation at matrix is homomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;evaluation&quot;,
          &quot;at&quot;,
          &quot;matrix&quot;,
          &quot;is&quot;,
          &quot;homomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:0cd2f0368a0d8234c2593845:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;T&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Matrix n n ℝ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;RingHom (Polynomial ℝ) (Matrix n n ℝ)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P33"><h2>P33</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:8e55e625fa9b10b7b65f59f4:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;four elements&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;four&quot;,
          &quot;elements&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;nonidentity element squares to identity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;nonidentity&quot;,
          &quot;element&quot;,
          &quot;squares&quot;,
          &quot;to&quot;,
          &quot;identity&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:8e55e625fa9b10b7b65f59f4:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hG&quot;: {
      &quot;name&quot;: &quot;hG&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;Fintype.card G = 4&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;∃ g : G, g ≠ 1 ∧ g * g = 1&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 1,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P34"><h2>P34</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:534c6ffcf56931559e70657b:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;S_3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;symmetric group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;S_3&quot;
        ],
        &quot;keywords&quot;: [
          &quot;symmetric&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;cyclic group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;cyclic&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;nonabelian and abelian hence nonisomorphic&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;nonabelian&quot;,
          &quot;and&quot;,
          &quot;abelian&quot;,
          &quot;hence&quot;,
          &quot;nonisomorphic&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P35"><h2>P35</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:aa16561dceee591e35e42c4d:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;M&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;set&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;M&quot;
        ],
        &quot;keywords&quot;: [
          &quot;set&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;M&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;associative operation&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;M&quot;
        ],
        &quot;keywords&quot;: [
          &quot;associative&quot;,
          &quot;operation&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;positive&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;positive&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;unique coherent associative product family&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;unique&quot;,
          &quot;coherent&quot;,
          &quot;associative&quot;,
          &quot;product&quot;,
          &quot;family&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:aa16561dceee591e35e42c4d:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;M&quot;: {
      &quot;name&quot;: &quot;M&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Semigroup M&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;n ≥ 1&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;premise1&quot;: {
      &quot;name&quot;: &quot;premise1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∃! f : (Fin n → M)&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;goal_implication&quot;,
        &quot;classified_by&quot;: &quot;implication&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;M&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;M, (∀ a : M, f ![a] = a) ∧ (∀ a b : M, f ![a, b] = a * b) ∧ (∀ (k : ℕ) (hk : 1 ≤ k) (hnk : k &lt; n) (a : Fin n → M), f a = f (a ↾ k) * f (a ∘ (Fin.succAbove k).symm))&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 3,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;M&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P36"><h2>P36</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:73ec95bfac2896218d2b1dab:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;C_n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;root of unity group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;C_n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;root&quot;,
          &quot;of&quot;,
          &quot;unity&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;order&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;order&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;ξ&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;complex exponential generator&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;ξ&quot;
        ],
        &quot;keywords&quot;: [
          &quot;complex&quot;,
          &quot;exponential&quot;,
          &quot;generator&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;cyclic with specified order generator&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;cyclic&quot;,
          &quot;with&quot;,
          &quot;specified&quot;,
          &quot;order&quot;,
          &quot;generator&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:73ec95bfac2896218d2b1dab:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;n &gt; 0&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;IsCyclic (ZMod n)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P37"><h2>P37</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:6b5a55ec3bb3c85239919649:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;C_n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;root of unity set&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;C_n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;root&quot;,
          &quot;of&quot;,
          &quot;unity&quot;,
          &quot;set&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;k&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;k&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;k&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;between zero and n minus one&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;k&quot;
        ],
        &quot;keywords&quot;: [
          &quot;between&quot;,
          &quot;zero&quot;,
          &quot;and&quot;,
          &quot;n&quot;,
          &quot;minus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;roots equal specified exponential set&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;roots&quot;,
          &quot;equal&quot;,
          &quot;specified&quot;,
          &quot;exponential&quot;,
          &quot;set&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:6b5a55ec3bb3c85239919649:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;own&quot;: {
      &quot;name&quot;: &quot;own&quot;,
      &quot;kind&quot;: &quot;own&quot;,
      &quot;type&quot;: &quot;Subgroup ℂ&quot;,
      &quot;attrs&quot;: {
        &quot;keyword&quot;: &quot;def&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P38"><h2>P38</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:d6ddc1df3ce6b764f3fb8d73:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;subset&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;subset&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;nonempty&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;nonempty&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;closed under products&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;closed&quot;,
          &quot;under&quot;,
          &quot;products&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact3&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;closed under inverses&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;closed&quot;,
          &quot;under&quot;,
          &quot;inverses&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;conditions imply subgroup and identity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;conditions&quot;,
          &quot;imply&quot;,
          &quot;subgroup&quot;,
          &quot;and&quot;,
          &quot;identity&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:d6ddc1df3ce6b764f3fb8d73:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Set G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h_one&quot;: {
      &quot;name&quot;: &quot;h_one&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∀ h ∈ H, h⁻¹ ∈ H&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h_mul&quot;: {
      &quot;name&quot;: &quot;h_mul&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∀ h1 h2 ∈ H, h1 * h2 ∈ H&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;IsSubgroup H&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P39"><h2>P39</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:ed87589326312890bea42d7c:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;K&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;field&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;K&quot;
        ],
        &quot;keywords&quot;: [
          &quot;field&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;Laurent polynomials&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;laurent&quot;,
          &quot;polynomials&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;ring with identity and units&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;ring&quot;,
          &quot;with&quot;,
          &quot;identity&quot;,
          &quot;and&quot;,
          &quot;units&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:ed87589326312890bea42d7c:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;K&quot;: {
      &quot;name&quot;: &quot;K&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Field K&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;Ring (LaurentSeries K)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;K&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P40"><h2>P40</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:d609c5bb746fc0e05625134a:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;f&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;map&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;f&quot;
        ],
        &quot;keywords&quot;: [
          &quot;map&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;object3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integers&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;integers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;object4&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;residue ring&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;residue&quot;,
          &quot;ring&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;canonical residue map is homomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;canonical&quot;,
          &quot;residue&quot;,
          &quot;map&quot;,
          &quot;is&quot;,
          &quot;homomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:d609c5bb746fc0e05625134a:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;RingHom ℤ (ZMod n)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P41"><h2>P41</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:4405f31187373d5e1de622d2:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;natural number&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;natural&quot;,
          &quot;number&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;S_n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;symmetric group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;S_n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;symmetric&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;factorial cardinality always nonabelian when n≥3&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;factorial&quot;,
          &quot;cardinality&quot;,
          &quot;always&quot;,
          &quot;nonabelian&quot;,
          &quot;when&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:4405f31187373d5e1de622d2:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hn&quot;: {
      &quot;name&quot;: &quot;hn&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;n ≥ 3&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\n  Fintype.card (Symmetric n) = n! ∧ ¬ IsAbelian (Symmetric n)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P42"><h2>P42</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:05a6defa23576b7da41f8fbf:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;subset&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;subset&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;nonempty&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;nonempty&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;subgroup&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;subgroup&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;product closure is necessary&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;product&quot;,
          &quot;closure&quot;,
          &quot;is&quot;,
          &quot;necessary&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:05a6defa23576b7da41f8fbf:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Set G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h_nonempty&quot;: {
      &quot;name&quot;: &quot;h_nonempty&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;H.Nonempty&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h_prod&quot;: {
      &quot;name&quot;: &quot;h_prod&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∀ h1 h2 ∈ H, h1 * h2 ∈ H&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;IsSubgroup H&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P43"><h2>P43</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:47c39e88c29e3974e8864e90:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;fact1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;order two&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;order&quot;,
          &quot;two&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;unique group up to isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;unique&quot;,
          &quot;group&quot;,
          &quot;up&quot;,
          &quot;to&quot;,
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:47c39e88c29e3974e8864e90:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;,
          &quot;Fintype G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group H&quot;,
          &quot;Fintype H&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hG&quot;: {
      &quot;name&quot;: &quot;hG&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;Fintype.card G = 2&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hH&quot;: {
      &quot;name&quot;: &quot;hH&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;Fintype.card H = 2&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;Nonempty (G ≃* H)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P44"><h2>P44</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:fa61f8cb0e88b11e158c4eec:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;b&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;abelian defined by commuting products&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;abelian&quot;,
          &quot;defined&quot;,
          &quot;by&quot;,
          &quot;commuting&quot;,
          &quot;products&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:fa61f8cb0e88b11e158c4eec:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;own&quot;: {
      &quot;name&quot;: &quot;own&quot;,
      &quot;kind&quot;: &quot;own&quot;,
      &quot;type&quot;: &quot;Prop&quot;,
      &quot;attrs&quot;: {
        &quot;keyword&quot;: &quot;class&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P45"><h2>P45</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:3c00c37b39285cbd7d0be732:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;fact1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;order three&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;order&quot;,
          &quot;three&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;unique group up to isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;unique&quot;,
          &quot;group&quot;,
          &quot;up&quot;,
          &quot;to&quot;,
          &quot;isomorphism&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P46"><h2>P46</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:186d922a4121917ac8643316:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;real three variable polynomials&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;real&quot;,
          &quot;three&quot;,
          &quot;variable&quot;,
          &quot;polynomials&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;ring identity and unit characterization&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;ring&quot;,
          &quot;identity&quot;,
          &quot;and&quot;,
          &quot;unit&quot;,
          &quot;characterization&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P47"><h2>P47</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:e845a95d7b5054a0dbdf34eb:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;d&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;d&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;generated subgroup&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;generated&quot;,
          &quot;subgroup&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;integer multiples cyclic opposite generators equal&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;integer&quot;,
          &quot;multiples&quot;,
          &quot;cyclic&quot;,
          &quot;opposite&quot;,
          &quot;generators&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:e845a95d7b5054a0dbdf34eb:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;d&quot;: {
      &quot;name&quot;: &quot;d&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;(Submodule.span ℤ {d}) = {n * d | n : ℤ}&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;d&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P48"><h2>P48</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:e7a3cec23c2ecdc655dd12cf:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;h&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;fact1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;product equals identity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;h&quot;,
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;product&quot;,
          &quot;equals&quot;,
          &quot;identity&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;,
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;left inverse equals inverse element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;left&quot;,
          &quot;inverse&quot;,
          &quot;equals&quot;,
          &quot;inverse&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:e7a3cec23c2ecdc655dd12cf:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h&quot;: {
      &quot;name&quot;: &quot;h&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;g&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hg&quot;: {
      &quot;name&quot;: &quot;hg&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;h * g = 1&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;g&quot;,
          &quot;h&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;h = g⁻¹&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;g&quot;,
          &quot;h&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P49"><h2>P49</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:e966f3b17e32318acd5f2762:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;g&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;inverse of inverse equals element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;inverse&quot;,
          &quot;of&quot;,
          &quot;inverse&quot;,
          &quot;equals&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:e966f3b17e32318acd5f2762:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;g&quot;: {
      &quot;name&quot;: &quot;g&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;g = (g⁻¹)⁻¹&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;g&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P50"><h2>P50</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:2c420bf6ec32069ecd0043dc:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;R&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;set&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;R&quot;
        ],
        &quot;keywords&quot;: [
          &quot;set&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;addition&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;addition&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;object3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;multiplication&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;multiplication&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;R&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;nonempty&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;R&quot;
        ],
        &quot;keywords&quot;: [
          &quot;nonempty&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;ring defined by additive group associativity distributivity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;ring&quot;,
          &quot;defined&quot;,
          &quot;by&quot;,
          &quot;additive&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:2c420bf6ec32069ecd0043dc:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;R&quot;: {
      &quot;name&quot;: &quot;R&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;own&quot;: {
      &quot;name&quot;: &quot;own&quot;,
      &quot;kind&quot;: &quot;own&quot;,
      &quot;type&quot;: &quot;structure&quot;,
      &quot;attrs&quot;: {
        &quot;keyword&quot;: &quot;structure&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P51"><h2>P51</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:25af34567cb2040bf41869e5:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;C_4&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;fourth roots&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;C_4&quot;
        ],
        &quot;keywords&quot;: [
          &quot;fourth&quot;,
          &quot;roots&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;complex multiplication&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;complex&quot;,
          &quot;multiplication&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;specified roots form group with identity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;specified&quot;,
          &quot;roots&quot;,
          &quot;form&quot;,
          &quot;group&quot;,
          &quot;with&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P52"><h2>P52</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:578ae89f353ecbdc0ffb76a2:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;K&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;field&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;K&quot;
        ],
        &quot;keywords&quot;: [
          &quot;field&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;polynomials over field&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;polynomials&quot;,
          &quot;over&quot;,
          &quot;field&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;commutative ring identity units constant&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;commutative&quot;,
          &quot;ring&quot;,
          &quot;identity&quot;,
          &quot;units&quot;,
          &quot;constant&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:578ae89f353ecbdc0ffb76a2:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;K&quot;: {
      &quot;name&quot;: &quot;K&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Field K&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\\\[isCommutativeRing (Polynomial K) ∧ (1 : Polynomial K) = 1 ∧ ∀ p : Polynomial K, IsUnit p → ∃ a : K, p = C a := by \\\\- sorry&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 1,
        &quot;forall_count&quot;: 1,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;K&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P53"><h2>P53</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:1a8d6a5c9aa3cc5000dee474:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;group order is cardinality&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;order&quot;,
          &quot;is&quot;,
          &quot;cardinality&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:1a8d6a5c9aa3cc5000dee474:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;,
          &quot;Fintype G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;own&quot;: {
      &quot;name&quot;: &quot;own&quot;,
      &quot;kind&quot;: &quot;own&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;keyword&quot;: &quot;def&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P54"><h2>P54</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:450c7df89ecd284b0e08504a:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;d&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;residue&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;d&quot;
        ],
        &quot;keywords&quot;: [
          &quot;residue&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;object3&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;generated cyclic subgroup&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;generated&quot;,
          &quot;cyclic&quot;,
          &quot;subgroup&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;subgroup equals integer and residue multiples&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;subgroup&quot;,
          &quot;equals&quot;,
          &quot;integer&quot;,
          &quot;and&quot;,
          &quot;residue&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:450c7df89ecd284b0e08504a:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;d&quot;: {
      &quot;name&quot;: &quot;d&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℕ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hn&quot;: {
      &quot;name&quot;: &quot;hn&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;0 &lt; n&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\\\\\n  (ZMod n).subgroup.closure ({ZMod.mk d} : Set (ZMod n)) = \\\\\\n  {ZMod.mk (k * d % n) | k : ℤ} = \\\\\\n  {ZMod.mk (k * d % n) | k : ZMod n}&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;d&quot;,
          &quot;n&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P55"><h2>P55</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:f908ac15b1bf8228281bcf68:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;first group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;first&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;second group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;second&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;fact1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;groups are isomorphic&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;groups&quot;,
          &quot;are&quot;,
          &quot;isomorphic&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;abelian properties equivalent&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;abelian&quot;,
          &quot;properties&quot;,
          &quot;equivalent&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:f908ac15b1bf8228281bcf68:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group H&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;iso&quot;: {
      &quot;name&quot;: &quot;iso&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;G ≃* H&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;(IsAbelian G ↔ IsAbelian H)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: true,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P56"><h2>P56</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:bcdfa01447e7060e1fd3d935:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;RSA method&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;rsa&quot;,
          &quot;method&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;message&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;message&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;design message authentication method&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;design&quot;,
          &quot;message&quot;,
          &quot;authentication&quot;,
          &quot;method&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P57"><h2>P57</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:f86421f93f1381d2ce728bc1:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;R&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;ring&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;R&quot;
        ],
        &quot;keywords&quot;: [
          &quot;ring&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;S&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;ring&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;S&quot;
        ],
        &quot;keywords&quot;: [
          &quot;ring&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;f&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;map&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;f&quot;
        ],
        &quot;keywords&quot;: [
          &quot;map&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;ring element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;ring&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object5&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;ring element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;b&quot;
        ],
        &quot;keywords&quot;: [
          &quot;ring&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;homomorphism preserves operations bijective means isomorphism&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;homomorphism&quot;,
          &quot;preserves&quot;,
          &quot;operations&quot;,
          &quot;bijective&quot;,
          &quot;means&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:f86421f93f1381d2ce728bc1:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;R&quot;: {
      &quot;name&quot;: &quot;R&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Ring R&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;S&quot;: {
      &quot;name&quot;: &quot;S&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Ring S&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;f&quot;: {
      &quot;name&quot;: &quot;f&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;R → S&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;R&quot;,
          &quot;S&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;Function.IsRingHom f ↔ (∀ a b : R, f (a + b) = f a + f b ∧ f (a * b) = f a * f b)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 1,
        &quot;is_iff&quot;: true,
        &quot;references&quot;: [
          &quot;R&quot;,
          &quot;f&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P58"><h2>P58</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:89fe687db9136379cd276fff:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;cyclic group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;cyclic&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;rectangle symmetry group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;rectangle&quot;,
          &quot;symmetry&quot;,
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;nonisomorphic groups of order four&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;nonisomorphic&quot;,
          &quot;groups&quot;,
          &quot;of&quot;,
          &quot;order&quot;,
          &quot;four&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P59"><h2>P59</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:da65aa56505f6c13e6553d0d:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;subset&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;subset&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;nonempty&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;H&quot;
        ],
        &quot;keywords&quot;: [
          &quot;nonempty&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;subgroup iff group under inherited operation&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;subgroup&quot;,
          &quot;iff&quot;,
          &quot;group&quot;,
          &quot;under&quot;,
          &quot;inherited&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:da65aa56505f6c13e6553d0d:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;H&quot;: {
      &quot;name&quot;: &quot;H&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Set G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h_nonempty&quot;: {
      &quot;name&quot;: &quot;h_nonempty&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;H.Nonempty&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;Subgroup.of G H ↔ IsGroup (H, (*), 1, ⁻¹)&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: true,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;H&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P60"><h2>P60</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:ab2990e9786d5bda5a04e067:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;map&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;T&quot;
        ],
        &quot;keywords&quot;: [
          &quot;map&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;three dimensional space&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;three&quot;,
          &quot;dimensional&quot;,
          &quot;space&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;bijective&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;T&quot;
        ],
        &quot;keywords&quot;: [
          &quot;bijective&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;continuous first partial derivatives&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;T&quot;
        ],
        &quot;keywords&quot;: [
          &quot;continuous&quot;,
          &quot;first&quot;,
          &quot;partial&quot;,
          &quot;derivatives&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;diffeomorphisms form group under composition&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;diffeomorphisms&quot;,
          &quot;form&quot;,
          &quot;group&quot;,
          &quot;under&quot;,
          &quot;composition&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:ab2990e9786d5bda5a04e067:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;U&quot;: {
      &quot;name&quot;: &quot;U&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Set (ℝ × ℝ × ℝ)&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;hU&quot;: {
      &quot;name&quot;: &quot;hU&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;IsOpen U&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;predicate&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;U&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\\\\\n  Group (Subtype (fun T : (ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ) =&gt; \\\\\\n    Bijective T ∧ DifferentiableOn ℝ T U ∧ ContinuousOn (fun x =&gt; deriv (T ∘ (Prod.fst, Prod.snd, Prod.third)) x) U))&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;U&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P61"><h2>P61</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:a892779bfcaca35b91e7d611:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;map&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;T&quot;
        ],
        &quot;keywords&quot;: [
          &quot;map&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;three dimensional space&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;three&quot;,
          &quot;dimensional&quot;,
          &quot;space&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;bijective&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;T&quot;
        ],
        &quot;keywords&quot;: [
          &quot;bijective&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;continuous&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;T&quot;
        ],
        &quot;keywords&quot;: [
          &quot;continuous&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact3&quot;: {
      &quot;name&quot;: &quot;T&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;inverse continuous&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;T&quot;
        ],
        &quot;keywords&quot;: [
          &quot;inverse&quot;,
          &quot;continuous&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;homeomorphisms form group under composition&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;homeomorphisms&quot;,
          &quot;form&quot;,
          &quot;group&quot;,
          &quot;under&quot;,
          &quot;composition&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section><section id="P62"><h2>P62</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:7593062cab7c74cdecec08d0:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;finite order&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;finite&quot;,
          &quot;order&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object1&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;least positive exponent bounded powers generate&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;least&quot;,
          &quot;positive&quot;,
          &quot;exponent&quot;,
          &quot;bounded&quot;,
          &quot;powers&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:7593062cab7c74cdecec08d0:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;ha&quot;: {
      &quot;name&quot;: &quot;ha&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;IsOfFinOrder a&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;predicate&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;a&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;\\/\\/ (orderOf a = (min {n : ℕ | n &gt; 0 ∧ a ^ n = 1})) ∧ (Subgroup.closure ({a} : Set G) = {a ^ k | k : ℕ, k &lt; orderOf a})&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;a&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P63"><h2>P63</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:c8204e9d19a6dd5d730dccc7:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;G&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;e&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;e&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;e&#x27;&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;group element&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;e&#x27;&quot;
        ],
        &quot;keywords&quot;: [
          &quot;group&quot;,
          &quot;element&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;e&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;two sided identity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;e&quot;
        ],
        &quot;keywords&quot;: [
          &quot;two&quot;,
          &quot;sided&quot;,
          &quot;identity&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact2&quot;: {
      &quot;name&quot;: &quot;e&#x27;&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;two sided identity&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;e&#x27;&quot;
        ],
        &quot;keywords&quot;: [
          &quot;two&quot;,
          &quot;sided&quot;,
          &quot;identity&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object3&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;two identities equal&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;two&quot;,
          &quot;identities&quot;,
          &quot;equal&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:c8204e9d19a6dd5d730dccc7:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;G&quot;: {
      &quot;name&quot;: &quot;G&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;Type*&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;implicit&quot;,
        &quot;classified_by&quot;: &quot;sort&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;instances&quot;: [
          &quot;Group G&quot;
        ],
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;e&quot;: {
      &quot;name&quot;: &quot;e&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;e&#x27;&quot;: {
      &quot;name&quot;: &quot;e&#x27;&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;G&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h_e&quot;: {
      &quot;name&quot;: &quot;h_e&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∀ g : G, e * g = g ∧ g * e = g&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;e&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h_e&#x27;&quot;: {
      &quot;name&quot;: &quot;h_e&#x27;&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;∀ g : G, e&#x27; * g = g ∧ g * e&#x27; = g&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;G&quot;,
          &quot;e&#x27;&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;e = e&#x27;&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;e&quot;,
          &quot;e&#x27;&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P64"><h2>P64</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:b40279b634fffd0841ff0f5a:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: false,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;a&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;integer&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;b&quot;
        ],
        &quot;keywords&quot;: [
          &quot;integer&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object3&quot;: {
      &quot;name&quot;: &quot;r&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;exponent&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;r&quot;
        ],
        &quot;keywords&quot;: [
          &quot;exponent&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object4&quot;: {
      &quot;name&quot;: &quot;s&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;exponent&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;s&quot;
        ],
        &quot;keywords&quot;: [
          &quot;exponent&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object5&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;modulus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;n&quot;
        ],
        &quot;keywords&quot;: [
          &quot;modulus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;fact1&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;context&quot;: &quot;congruent to power modulo n&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [
          &quot;b&quot;
        ],
        &quot;keywords&quot;: [
          &quot;congruent&quot;,
          &quot;to&quot;,
          &quot;power&quot;,
          &quot;modulo&quot;,
          &quot;n&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [
          &quot;object2&quot;
        ],
        &quot;reference_status&quot;: &quot;explicit_symbol_candidates&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;ciphertext power returns original residue&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;ciphertext&quot;,
          &quot;power&quot;,
          &quot;returns&quot;,
          &quot;original&quot;,
          &quot;residue&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:b40279b634fffd0841ff0f5a:lean:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;lean&quot;,
    &quot;reference_completeness&quot;: &quot;syntactic_only&quot;
  },
  &quot;roles&quot;: {
    &quot;a&quot;: {
      &quot;name&quot;: &quot;a&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;b&quot;: {
      &quot;name&quot;: &quot;b&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;r&quot;: {
      &quot;name&quot;: &quot;r&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;s&quot;: {
      &quot;name&quot;: &quot;s&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;n&quot;: {
      &quot;name&quot;: &quot;n&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;t&quot;: {
      &quot;name&quot;: &quot;t&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;type&quot;: &quot;ℤ&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;default&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h1&quot;: {
      &quot;name&quot;: &quot;h1&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;b ≡ a^r [ZMOD n]&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;b&quot;,
          &quot;n&quot;,
          &quot;r&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;h2&quot;: {
      &quot;name&quot;: &quot;h2&quot;,
      &quot;kind&quot;: &quot;fact&quot;,
      &quot;type&quot;: &quot;r * s = 1 + n * t&quot;,
      &quot;attrs&quot;: {
        &quot;binder_kind&quot;: &quot;explicit&quot;,
        &quot;classified_by&quot;: &quot;relation&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;,
        &quot;references&quot;: [
          &quot;n&quot;,
          &quot;r&quot;,
          &quot;s&quot;,
          &quot;t&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;
      }
    },
    &quot;goal&quot;: {
      &quot;name&quot;: &quot;goal&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;type&quot;: &quot;b^s ≡ a [ZMOD n]&quot;,
      &quot;attrs&quot;: {
        &quot;exists_count&quot;: 0,
        &quot;forall_count&quot;: 0,
        &quot;is_iff&quot;: false,
        &quot;references&quot;: [
          &quot;a&quot;,
          &quot;b&quot;,
          &quot;n&quot;,
          &quot;s&quot;
        ],
        &quot;reference_status&quot;: &quot;syntactic_candidates&quot;,
        &quot;origin&quot;: &quot;deterministic_signature_projection&quot;
      }
    }
  }
}</pre></section><section id="P65"><h2>P65</h2><h3>source</h3><pre>{
  &quot;name&quot;: &quot;schema:p0-f12735802ba0:f8d2e206681972927560be2c:source:v1&quot;,
  &quot;kind&quot;: &quot;simplex&quot;,
  &quot;attrs&quot;: {
    &quot;side&quot;: &quot;source&quot;,
    &quot;goal_has_exists&quot;: true,
    &quot;reference_completeness&quot;: &quot;partial&quot;
  },
  &quot;roles&quot;: {
    &quot;object1&quot;: {
      &quot;name&quot;: &quot;object1&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;nonsquare rhombus&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;nonsquare&quot;,
          &quot;rhombus&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;object2&quot;: {
      &quot;name&quot;: &quot;object2&quot;,
      &quot;kind&quot;: &quot;object&quot;,
      &quot;context&quot;: &quot;geometric symmetries&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;geometric&quot;,
          &quot;symmetries&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    },
    &quot;goal1&quot;: {
      &quot;name&quot;: &quot;goal1&quot;,
      &quot;kind&quot;: &quot;goal&quot;,
      &quot;context&quot;: &quot;identify symmetry group size and classification&quot;,
      &quot;attrs&quot;: {
        &quot;symbols&quot;: [],
        &quot;keywords&quot;: [
          &quot;identify&quot;,
          &quot;symmetry&quot;,
          &quot;group&quot;,
          &quot;size&quot;,
          &quot;and&quot;
        ],
        &quot;introduced&quot;: &quot;given&quot;,
        &quot;quantifier_status&quot;: &quot;unreviewed&quot;,
        &quot;references&quot;: [],
        &quot;reference_status&quot;: &quot;unknown&quot;,
        &quot;origin&quot;: &quot;frozen_current_chat_annotation&quot;
      }
    }
  }
}</pre><h3>lean</h3><pre>Insufficient projected roles; no object roles were invented.</pre></section></html>