InCover_of_mem
plain-language theorem explainer
Membership of a vertex v in the list S directly implies that S covers v under the InCover predicate. Complexity theorists analyzing vertex cover instances in the Recognition framework would cite this for foundational set properties. The proof reduces immediately via a simpa tactic that unfolds the definition of InCover.
Claim. Let $S$ be a list of natural numbers and $v$ a natural number. If $v$ belongs to $S$, then the predicate InCover$(S, v)$ holds, where InCover$(S, v)$ means precisely that $v$ is an element of the list $S$.
background
The module treats complexity pairs as functions of input size. InCover is defined directly as membership: InCover$(S, v)$ holds exactly when $v$ is in the list $S$. This supplies the basic predicate for vertex cover constructions alongside siblings such as InCover_cons and Covers. The result depends on the InCover definition and on the ILG action $S[g, ψ; C_lag, α] := S_EH[g] + S_ψ[g,ψ]$ from the relativity module, though the latter is not invoked in the proof.
proof idea
The proof is a one-line wrapper that applies simpa with the InCover definition to reduce the goal to the supplied membership hypothesis.
why it matters
This lemma supplies basic infrastructure for vertex cover arguments inside the complexity module. It supports later definitions such as HasCover and Covers, though no downstream uses are recorded. It aligns with the framework's use of discrete structures without invoking the forcing chain T0-T8, the Recognition Composition Law, or the phi-ladder.
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