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.

Agreement at one assignment is insufficient

Statement refuted

False claim: if two formulas agree at one assignment, they still agree there after the same free-for substitution. Use the two-element equality structure $M = {0, 1}$ with constant $c^{M} = 1$ , distinct variables $x, y$ , and $s (x) = s (y) = 0$ .

Facts & Assumptions

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

[F1]

If $t$ is free for $x$ in $ϕ$ , then for every structure $M$ and assignment $s$ , $M, s ⊨ ϕ [t / x] ⟺ M, s [x : = ⟦ t ⟧_{s}] ⊨ ϕ .$ (Free-for substitution commutes with satisfaction)

Counterexample

1.1

At $s$ , the formulas $x = y$ and $y = y$ are both true. The constant $c$ has no variables, so is free for $x$ in each formula. The free-for identity evaluates substitution by changing $s (x)$ to $1$ .

2.1

After substitution the formulas are $c = y$ and $y = y$ . Their truth values are respectively false ( $1 \neq 0$ ) and true ( $0 = 0$ ). The original agreement at $s$ imposes no agreement at the updated assignment, which is exactly the missing hypothesis.

F1step 1.1∎

Depends on

Free-for substitution commutes with satisfaction

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

Moschovakis, Lecture Notes in Logic (2014) — x1.3.2 p.48, adapted from arithmetic to a finite structure. (standard reference, not scraped)