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R0053-27383 · q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0008
Use positive p=8, word=[a, a, a, a, a, a, a, a, b], |x|=7, |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=8, word=[a, a, a, a, a, a, a, a, b], |x|=7, |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).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.
Before
source roles = [pump-quantifier-audit]
After
derived = [[p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order]]
Conditions: none
List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [8, [a, a, a, a, a, a, a, a, b], 7, 1]
After
derived = [9, 1, Yes, Yes, Yes, exists p forall MEMBER w exists split forall i>=0]
Conditions: none
The word is an array of symbols;length counts tokens,including empty-string tokens.First check legal lengths.The pumping order is p,then every sufficiently long MEMBER word,then a split,then ALL i.A legal split alone proves neither regularity nor all-i membership.
Rule: Pumping_ORIGINAL_token_word_split_lengths_AND_quantifier_order · Uses: G001, S001
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.(8, [a, a, a, a, a, a, a, a, b], 9, 7, 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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: bdc7bb97405723803731b5aa27f7b83c2f8899b5e0ae27c280ab6e963139396fcandidate record SHA-256: 2c4cf4c33c492081c4b05a82c6e52811a5638f1c76b14f0bf624defee2ae6f48semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED