Counterexamples and domain eligibility

Mathematical statement

For ∀x∈U P(x), c is a valid counterexample exactly when c∈U and P(c) is false. One valid counterexample refutes the universal statement. Failure to find one is not itself a proof unless the domain is finite and every element has been checked completely.

Representative formula (exact TeX)

P(c)

Assumptions

• The domain and predicate of the universal statement are explicit.
• The least counterexample here means the numerically smallest element of the finite integer failure set; an empty set is explicitly reported as having none.

Symbol scope

• P — predicate of the universal statement under examination
• c — candidate counterexample

Intuition, definition, and misconceptions

First check that c belongs to U, then check that P(c) fails. Even if the same algebraic expression can be evaluated outside U, a failing outside value is not a counterexample to this restricted universal statement.

After restriction to V⊆U, an original counterexample may lie outside V and lose eligibility. This invalidates that witness but does not establish that the new universal statement is true.

A finite integer domain permits enumeration of every failing value, filtering of candidate sets, and selection of the least value. Both the minimum and the report of no counterexample use the full failure set; sampled successes or failures do not settle an infinite-domain claim.

Common errors include using an outside value to refute a restricted universal statement, or taking a bounded search without a counterexample as a universal proof.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Quantifiers and explicit domains — A counterexample refutes a universal statement bound to its domain.
• Mathematical proof — Distinguish a valid refutation, finite enumeration, and general proof obligations.

Verification boundary

The finite witnesses in this pulse can be verified exactly; general counterexample search is not claimed to be decidable or complete.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind witness membership, the negated predicate, and refutation of a restricted universal statement; state existence separately for a finite minimum.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.