The Tseitin clauses for an AND gate define the gate

Identifier: F:tseitin-and-gate
Proof route: smt-term-level
External status: proved
Axiom footprint: axeyum-ir.bool-evaluator, classical-two-valued-bool-semantics

Recorded description

For all propositions a, b, t: the conjunction of the three clauses ((not t) or a), ((not t) or b) and (t or (not a) or (not b)) is equivalent to t if and only if (a and b).

Formal statement

(assert (= (and (or (not t) a) (or (not t) b) (or t (not a) (not b))) (= t (and a b))))

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

tseitin-and-gate-term-level

Kind: exhaustive-enumeration
Status: checked

Supports: For all propositions a, b, t: the conjunction of the three clauses ((not t) or a), ((not t) or b) and (t or (not a) or (not b)) is equivalent to t if and only if (a and b).

Checker command

test "$(cargo run -q -p axeyum-bench --example smtcomp_cli -- --evidence artifacts/facts/smt2/neg-tseitin-and-gate.smt2 2>/dev/null | tail -1)" = unsat

Evidence notes

Ran 2026-08-14: the file asserts the NEGATION of formal.statement and the solver reported `unsat` with `kind=unsat-term-level certified=1 arena=ok`. `unsat-term-level` is exhaustive evaluation of the 8 Boolean assignments by the axeyum-ir evaluator alone -- it trusts neither the bit-blaster, the CNF encoder, nor the SAT solver. `arena=ok` means Evidence::check re-ran that enumeration against a FRESH PARSE of the file, independent of anything the producing solve held in memory. Independently cross-checked: `z3 -smt2` on the same file also reports `unsat`.

Provenance

{
  "date": "2026-08-14",
  "established_by": "facts-logic lane, axeyum SAT/SMT front door",
  "source": "authored from the S:logic-and-proof strand of the math-education concept graph; statement written here, not copied",
  "prior_art": [
    {
      "citation": "G. S. Tseitin (1968), 'On the complexity of derivation in propositional calculus', Studies in Constructive Mathematics and Mathematical Logic Part II",
      "establishes": "the linear-size clausal encoding of a formula by gate definitions, of which this AND gate is one case",
      "attribution": "standard textbook attribution; this lane did not consult the primary source, and records the citation as the received one"
    }
  ]
}