CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A winning taboo position can have a nonwinning child

Statement refuted

In a game with terminal taboos, a position from which a player can force a terminal taboo for the opponent need not have only such winning children. In particular, deleting all positions from which either player can force an opponent taboo need not leave a prefix-closed tree. Here infinite play does not count as successful terminal reachability for either player.

Facts & Assumptions

[F1]

A terminal taboo loses for its named player, while nonterminal moves and fixed-history parity are as in Game trees with terminal taboos.

Counterexample

Given: The tree $T = {\emptyset, (0)} \cup {(1)^{⌢} 0^{n} : n \in N}$ . Its only terminal node is $(0)$ ; declare it taboo for II. The root is I-to-move. For definiteness the infinite payoff is empty; terminal reachability ignores that payoff.

1.1

Prefixes of $(1)^{⌢} 0^{n}$ are the root or words $(1)^{⌢} 0^{m}$ with $m \leq n$ , all listed in $T$ . The only other nonempty word is $(0)$ , whose sole proper prefix is the root. Thus $T$ is a nonempty tree. The node $(0)$ is terminal and every node on the other ray has the unique child obtained by appending $0$ , verifying the asserted terminal partition.

givenF1

2.1

At the root I can choose $0$ and reach the II taboo immediately. Below the child $(1)$ every legal continuation is forced and the only maximal continuation is the infinite sequence $(1, 0, 0, \dots)$ . No continuation from that child reaches a terminal node. Neither player can therefore force an opponent terminal taboo there, although I can do so at its parent.

F1step 1.1

3.1

The proposed deletion removes the root by step 2.1 and retains $(1)$ by the same step. A set retaining $(1)$ but omitting its empty prefix is not a tree. This is the required witness against both the child assertion and the resulting deletion rule. QED.

step 1.1step 2.1

Remarks

This witness corrects the downward-closure assertion in the proof of Buffard–Levrel–Mayo Lemma 1 (arXiv v1). It does not refute a reduction that keeps only nodes all of whose prefixes avoid the two reachability-winning sets. It is an AI-generated counterexample and is not a dependency supplier.

Depends on

Game trees with terminal taboos

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Lemma 1 proof, downward-closure assertion (claim refuted by the displayed local tree) (standard reference, not scraped)