Question 27304: Every Legal Split of a^p b^p Is Defeated by Pumping Down

Original question and givens

Use positive pumping length p=4 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=4 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).

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-anbn-all-splits]

derived = [[p, adversary_word, all_legal_split_rows, split_count, triangular_count, every_split_defeated, pump_down_scope]]

source roles = [4]

derived = [a^4b^4, 8, ALL x>=0,y>=1,x+y<=p]

Pumping ORIGINAL remaining a after i0 split0,1 := sub(4, 1)

Pumping ORIGINAL remaining a after i0 split0,1 = 3

source roles = [4, 0, 1]

derived = [3, 7, [0, 1, 3, a^3b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,2 := sub(4, 2)

Pumping ORIGINAL remaining a after i0 split0,2 = 2

source roles = [4, 0, 2]

derived = [2, 6, [0, 2, 2, a^2b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,3 := sub(4, 3)

Pumping ORIGINAL remaining a after i0 split0,3 = 1

source roles = [4, 0, 3]

derived = [1, 5, [0, 3, 1, a^1b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,4 := sub(4, 4)

Pumping ORIGINAL remaining a after i0 split0,4 = 0

source roles = [4, 0, 4]

derived = [0, 4, [0, 4, 0, a^0b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,1 := sub(4, 1)

Pumping ORIGINAL remaining a after i0 split1,1 = 3

source roles = [4, 1, 1]

derived = [2, 6, [1, 1, 2, a^3b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,2 := sub(4, 2)

Pumping ORIGINAL remaining a after i0 split1,2 = 2

source roles = [4, 1, 2]

derived = [1, 5, [1, 2, 1, a^2b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,3 := sub(4, 3)

Pumping ORIGINAL remaining a after i0 split1,3 = 1

source roles = [4, 1, 3]

derived = [0, 4, [1, 3, 0, a^1b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,1 := sub(4, 1)

Pumping ORIGINAL remaining a after i0 split2,1 = 3

source roles = [4, 2, 1]

derived = [1, 5, [2, 1, 1, a^3b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,2 := sub(4, 2)

Pumping ORIGINAL remaining a after i0 split2,2 = 2

source roles = [4, 2, 2]

derived = [0, 4, [2, 2, 0, a^2b^4, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,1 := sub(4, 1)

Pumping ORIGINAL remaining a after i0 split3,1 = 3

source roles = [4, 3, 1]

derived = [0, 4, [3, 1, 0, a^3b^4, No], Yes]

Pumping ORIGINAL triangular numerator := mul(4, 5)

Pumping ORIGINAL triangular numerator = 20

Pumping ORIGINAL triangular all-splits count := div(20, 2)

Pumping ORIGINAL triangular all-splits count = 10

source roles = [4, [[0, 1, 3, a^3b^4, No], [0, 2, 2, a^2b^4, No], [0, 3, 1, a^1b^4, No], [0, 4, 0, a^0b^4, No], [1, 1, 2, a^3b^4, No], [1, 2, 1, a^2b^4, No], [1, 3, 0, a^1b^4, No], [2, 1, 1, a^3b^4, No], [2, 2, 0, a^2b^4, No], [3, 1, 0, a^3b^4, No]]]

derived = [10, 10, Yes]

Reverse checks

• S025 · 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.
• S024 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular all-splits count with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular numerator with exact rational arithmetic, not rounded decimals.
• S022 · 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.
• S021 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,1 with exact rational arithmetic, not rounded decimals.
• S020 · 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.
• S019 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,2 with exact rational arithmetic, not rounded decimals.
• S018 · 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.
• S017 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,1 with exact rational arithmetic, not rounded decimals.
• S016 · 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.
• S015 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,3 with exact rational arithmetic, not rounded decimals.
• S014 · 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.
• S013 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,2 with exact rational arithmetic, not rounded decimals.
• S012 · 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.
• S011 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,1 with exact rational arithmetic, not rounded decimals.
• S010 · 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.
• S009 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,4 with exact rational arithmetic, not rounded decimals.
• S008 · 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.
• S007 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,3 with exact rational arithmetic, not rounded decimals.
• S006 · 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.
• S005 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split0,2 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.

Result

(4, a^4b^4, [[0, 1, 3, a^3b^4, No], [0, 2, 2, a^2b^4, No], [0, 3, 1, a^1b^4, No], [0, 4, 0, a^0b^4, No], [1, 1, 2, a^3b^4, No], [1, 2, 1, a^2b^4, No], [1, 3, 0, a^1b^4, No], [2, 1, 1, a^3b^4, No], [2, 2, 0, a^2b^4, No], [3, 1, 0, a^3b^4, No]], 10, 10, 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.

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.