Question 27378: Pumping Quantifiers Place the Word before the Split and the Pump Count after It

Original question and givens

Use positive p=3, word=[a, a, a, b], |x|=2, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).

• G001 — Use positive p=3, word=[a, a, a, b], |x|=2, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-pumping-lemma-regular-languages

Step-by-step derivation

source roles = [pump-quantifier-audit]

derived = [[p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order]]

source roles = [3, [a, a, a, b], 2, 1]

derived = [4, 1, Yes, Yes, Yes, exists p forall MEMBER w exists split forall i>=0]

Reverse checks

• S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
• S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.

Result

(3, [a, a, a, b], 4, 2, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.