LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Ph quantifier block closure

Statement

For each fixed $k \geq 0$ , both $Σ_{k}^{p}$ and $Π_{k}^{p}$ are closed under polynomial-time many-one preimages and polynomially many uniformly indexed conjunctions and disjunctions. Precisely, if the language of pairs $(x, i)$ lies in the class and $0 \leq i < q (∣ x ∣)$ for a polynomial-time computable polynomial bound $q$ , then both “every such $i$ ” and “some such $i$ ” define languages in that class. Complements interchange $Σ_{k}^{p}$ and $Π_{k}^{p}$ , and each is contained in both classes at level $k + 1$ . These assertions also hold with one fixed base oracle.

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F1]

The levels are defined by a fixed number of alternating polynomial-length string blocks over a uniform polynomial-time predicate. (Polynomial hierarchy levels).

Proof

1.1

The bounded-block definition supplies one uniform predicate for $(x, i)$ , not a separately chosen machine for each index. A polynomial-time preimage first computes its image; witness lengths can be padded to a polynomial bound in $∣ x ∣$ . At level zero all these operations are deterministic polynomial-time loops.

2.1

For positive $k$ , conjunctions use a tuple of independent variables at each quantifier block and the conjunction of the matrix tests. The identities $(Q a U (a)) \land (Q b V (b)) ⟺ Q (a, b) (U (a) \land V (b))$ hold for either quantifier over nonempty domains. One verifies the universal identity by fixing an arbitrary value of the other variable, and the existential identity by combining witnesses. Repeated application proves both directions of the tuple construction. A tuple contains only polynomially many polynomial-length strings.

step 1.1algebra

3.1

Disjunction follows by complementing a conjunction of complements; this simultaneously proves closure for both starting polarities. Negating a quantified predicate flips every quantifier and negates its deterministic matrix. A zero-term conjunction is true and a zero-term disjunction is false, both deterministic. Length-zero blocks still have one value.

step 2.1algebra

4.1

Append an unused last block for containment in the next level of the same polarity; prepend an unused opposite block for the other polarity. Both preserve truth because binary-string domains are nonempty. The proof copies calls to a fixed base oracle verbatim, so all constructions relativize.

F1step 3.1∎

Depends on

Polynomial hierarchy levels

Used by

Ph complete language forces collapse Corollary
Collapse at one level propagates Lemma
Ph adaptive oracle transcript normal form Lemma
Ph containments and polynomial space Proposition

Dependency tree · two levels

2 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

Goldreich, Computational Complexity lecture notes; Lecture9 §9.1, Proposition9.1.1 p104 and proof of Theorem9.9 pp105–107. (standard reference, not scraped)