CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-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.

Variable capture changes satisfaction

Statement refuted

False claim: replacing free occurrences by a variable without first avoiding capture always has the intended substitution semantics. In $ϕ = \exists v_{2} (v_{2} \in v_{1})$ , naively substitute $v_{2}$ for $v_{1}$ .

Facts & Assumptions

Given: Work in ZF unless the statement explicitly weakens or supplements it; fix the objects and hypotheses of the statement refuted.

[F1]

There is a canonical total capture-avoiding substitution $ϕ ⟨ t / x ⟩$ on formulas, obtained by renaming obstructing binders using least unused variable indices. It replaces the original free occurrences of $x$ and satisfies $M, s ⊨ ϕ ⟨ t / x ⟩ ⟺ M, s [x : = ⟦ t ⟧_{s}] ⊨ ϕ .$ Canonicity refers to the specified coding and traversal, not to literal invariance under other fresh-variable conventions. (Canonical capture-avoiding substitution)

Counterexample

1.1

In $M = {\emptyset, {\emptyset}}$ with restricted membership, take $s (v_{2}) = {\emptyset}$ . The intended substitution truth, supplied by capture-avoiding substitution, is the truth of $ϕ$ after setting $v_{1}$ to $s (v_{2})$ . It is true, with witness $\emptyset$ .

2.1

Naive replacement gives $\exists v_{2} (v_{2} \in v_{2})$ , false since neither carrier element belongs to itself. Renaming the binder to $v_{3}$ first gives $\exists v_{3} (v_{3} \in v_{2})$ , true at the displayed assignment. Thus the naive operation changes the required truth value.

step 1.1given∎

Depends on

Canonical capture-avoiding substitution

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

Weiss, An Introduction to Set Theory (2014) — chapter 1 worked substitution pp.17–18. (standard reference, not scraped)