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R0053-27301 · q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0051
Use positive pumping length p=1 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope).
G001 — Use positive pumping length p=1 and adversary word a^p b^p. Give (p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_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.
Before
source roles = [pump-anbn-all-splits]
After
derived = [[p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope]]
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 = [1]
After
derived = [a^1b^1, 2, ALL x>=0,y>=1,x+y<=p]
Conditions: none
Under assumed regularity take its pumping length p and choose a^p b^p of length2p>=p.Any legal xy lies in the initial p a-symbols,so y consists of a symbols and is nonempty.Do not preselect a convenient split.
Rule: Pumping_adversary_word_and_ALL_legal_split_domain · Uses: G001, S001
Before
Pumping ORIGINAL remaining a after i0 split0,1 := sub(1, 1)
After
Pumping ORIGINAL remaining a after i0 split0,1 = 0
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [1, 0, 1]
After
derived = [0, 1, [0, 1, 0, a^0b^1, No], Yes]
Conditions: none
The third field p-x-y counts only remaining a symbols in z;full |z| adds all p b-symbols.Pumping i=0 deletes nonempty y,leaving p-y<p a symbols but p b symbols,so this legal split fails membership.
Rule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S003
Before
Pumping ORIGINAL triangular numerator := mul(1, 2)
After
Pumping ORIGINAL triangular numerator = 2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Pumping ORIGINAL triangular all-splits count := div(2, 2)
After
Pumping ORIGINAL triangular all-splits count = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [1, [[0, 1, 0, a^0b^1, No]]]
After
derived = [1, 1, Yes]
Conditions: none
For each x there are p-x choices of y,total p(p+1)/2,so every legal split is included.The general argument repeats for ANY proposed p,with i=0 defeating every split.A table at one fixed p alone is not the general proof.
Rule: Pumping_ALL_splits_and_GENERAL_every_proposed_p_contradiction · Uses: G001, S006
S007 · 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.S006 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular all-splits count with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular numerator with exact rational arithmetic, not rounded decimals.S004 · 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.S003 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,1 with exact rational arithmetic, not rounded decimals.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.(1, a^1b^1, [[0, 1, 0, a^0b^1, No]], 1, 1, Yes, every_legal_split_is_defeated_by_i=0)
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: 08137bddb3e722d3accdf7346ad57d409634a295ef626515d77fa010a1cf916fcandidate record SHA-256: 8220a3ccf65149320a66372bd49de3bad25394b91f72d65736c1df56d81e694bsemantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED