False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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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.
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FALSE: applying mu-minimization to a computable search predicate always yields a total function

Statement

False claim: if $g : N^{2} \to N$ is computable, then the minimization $x \mapsto μ y g (x, y)$ is always total.

Facts & Assumptions

Given: The false claim above.

[A1]

If $g : N^{2} \to N$ is computable, then $x \mapsto μ y g (x, y)$ is total.

[L1]

Unbounded minimization is defined only when some searched value $y$ makes the total function $g (x, y)$ equal to $0$ , by Unbounded minimization and partial recursive functions.

Refutation

technique · direct

1.1

Take the total computable function $g (x, y) = 1$ for all $x, y \in N$ . For this $g$ , there is no $y$ with $g (x, y) = 0$ for any input $x$ .

given

2.1

By [L1], the minimization $μ y g (x, y)$ is therefore undefined for every $x$ . In particular it is not total, contradicting [A1].

A1L1step 1.1∎

Depends on

Unbounded minimization and partial recursive functions

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

Jeremy Avigad and Richard Zach, Recursive Functions (standard reference, not scraped)
Klaus Sutner, Other Models (standard reference, not scraped)