LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Shamir protocol has perfect completeness

Statement

Let $Φ$ be a true closed prenex quantified Boolean formula and let the Shamir protocol of The Shamir interactive protocol for TQBF run on $Φ$ with its honest prover $P_{h}$ . Then $P_{h}$ is a legal prover strategy and the verifier accepts on every random tape, so the protocol has completeness $1$ on $Φ$ . Consequently every true TQBF input has an honest prover that is accepted with probability one.

Facts & Assumptions

Given: A true closed prenex quantified Boolean formula $Φ$ and the Shamir protocol for $Φ$ with its honest prover $P_{h}$ .

[A1]

For every fixed random tape, every verifier test of the run passes and the terminal test accepts (Honest prover maintains the field-value claim).

[A2]

The honest prover's message in a round is determined by the node index and the public current point and claim, both of which are produced by the verifier from the messages exchanged so far; illegal or malformed prover messages are rejected, and an input that is not a well-formed closed prenex formula is rejected immediately (The Shamir interactive protocol for TQBF).

[A3]

$c$ -completeness of a protocol means that on every input of the language an honest prover strategy makes the verifier accept with probability at least $c$ , the probability being over the verifier's coins; the class $I P$ requires some constant-gap protocol with completeness at least $2 / 3$ (The class IP).

Proof

technique · direct

1.1

Fix a random tape. By [A1] every test of the run passes and the terminal test accepts, so this tape leads to acceptance; since the tape was arbitrary, the honest prover is accepted for every one of the $2^{p (∣ x ∣)}$ random tapes of the verifier, hence with probability one on $Φ$ .

A1given

1.2

The honest prover is a legal strategy: in each round its message is a coefficient list of degree at most $D$ as stated in The Shamir interactive protocol for TQBF, it is determined by the public information of the current point and claim, and it never waits on private verifier history; the verifier's own rejection rules for malformed messages therefore never fire.

A2given

2.1

Combining the two steps, on the true input $Φ$ the honest prover makes the verifier accept with probability $1$ , so $Φ$ has completeness $1$ ; in particular $1 \geq 2 / 3$ , and since $Φ$ was an arbitrary true TQBF input the same holds for every true input, which is exactly the statement.

step 1.1step 1.2A3given∎

Depends on

Used by

TQBF has a polynomial-round interactive proof Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Arora and Barak, Computational Complexity: A Modern Approach, §8.5.3, author-hosted draft (standard reference, not scraped)
A. Shen, IP = PSPACE: Simplified Proof, JACM 39(4) 1992, pp. 878–880 (standard reference, not scraped)