Question 27307: Every Legal Split of a^p b^p Is Defeated by Pumping Down

Original question and givens

Use positive pumping length p=7 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=7 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 = [7]

derived = [a^7b^7, 14, ALL x>=0,y>=1,x+y<=p]

Pumping ORIGINAL remaining a after i0 split0,1 := sub(7, 1)

Pumping ORIGINAL remaining a after i0 split0,1 = 6

source roles = [7, 0, 1]

derived = [6, 13, [0, 1, 6, a^6b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,2 := sub(7, 2)

Pumping ORIGINAL remaining a after i0 split0,2 = 5

source roles = [7, 0, 2]

derived = [5, 12, [0, 2, 5, a^5b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,3 := sub(7, 3)

Pumping ORIGINAL remaining a after i0 split0,3 = 4

source roles = [7, 0, 3]

derived = [4, 11, [0, 3, 4, a^4b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,4 := sub(7, 4)

Pumping ORIGINAL remaining a after i0 split0,4 = 3

source roles = [7, 0, 4]

derived = [3, 10, [0, 4, 3, a^3b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,5 := sub(7, 5)

Pumping ORIGINAL remaining a after i0 split0,5 = 2

source roles = [7, 0, 5]

derived = [2, 9, [0, 5, 2, a^2b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,6 := sub(7, 6)

Pumping ORIGINAL remaining a after i0 split0,6 = 1

source roles = [7, 0, 6]

derived = [1, 8, [0, 6, 1, a^1b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,7 := sub(7, 7)

Pumping ORIGINAL remaining a after i0 split0,7 = 0

source roles = [7, 0, 7]

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

Pumping ORIGINAL remaining a after i0 split1,1 := sub(7, 1)

Pumping ORIGINAL remaining a after i0 split1,1 = 6

source roles = [7, 1, 1]

derived = [5, 12, [1, 1, 5, a^6b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,2 := sub(7, 2)

Pumping ORIGINAL remaining a after i0 split1,2 = 5

source roles = [7, 1, 2]

derived = [4, 11, [1, 2, 4, a^5b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,3 := sub(7, 3)

Pumping ORIGINAL remaining a after i0 split1,3 = 4

source roles = [7, 1, 3]

derived = [3, 10, [1, 3, 3, a^4b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,4 := sub(7, 4)

Pumping ORIGINAL remaining a after i0 split1,4 = 3

source roles = [7, 1, 4]

derived = [2, 9, [1, 4, 2, a^3b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,5 := sub(7, 5)

Pumping ORIGINAL remaining a after i0 split1,5 = 2

source roles = [7, 1, 5]

derived = [1, 8, [1, 5, 1, a^2b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,6 := sub(7, 6)

Pumping ORIGINAL remaining a after i0 split1,6 = 1

source roles = [7, 1, 6]

derived = [0, 7, [1, 6, 0, a^1b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,1 := sub(7, 1)

Pumping ORIGINAL remaining a after i0 split2,1 = 6

source roles = [7, 2, 1]

derived = [4, 11, [2, 1, 4, a^6b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,2 := sub(7, 2)

Pumping ORIGINAL remaining a after i0 split2,2 = 5

source roles = [7, 2, 2]

derived = [3, 10, [2, 2, 3, a^5b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,3 := sub(7, 3)

Pumping ORIGINAL remaining a after i0 split2,3 = 4

source roles = [7, 2, 3]

derived = [2, 9, [2, 3, 2, a^4b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,4 := sub(7, 4)

Pumping ORIGINAL remaining a after i0 split2,4 = 3

source roles = [7, 2, 4]

derived = [1, 8, [2, 4, 1, a^3b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,5 := sub(7, 5)

Pumping ORIGINAL remaining a after i0 split2,5 = 2

source roles = [7, 2, 5]

derived = [0, 7, [2, 5, 0, a^2b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,1 := sub(7, 1)

Pumping ORIGINAL remaining a after i0 split3,1 = 6

source roles = [7, 3, 1]

derived = [3, 10, [3, 1, 3, a^6b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,2 := sub(7, 2)

Pumping ORIGINAL remaining a after i0 split3,2 = 5

source roles = [7, 3, 2]

derived = [2, 9, [3, 2, 2, a^5b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,3 := sub(7, 3)

Pumping ORIGINAL remaining a after i0 split3,3 = 4

source roles = [7, 3, 3]

derived = [1, 8, [3, 3, 1, a^4b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,4 := sub(7, 4)

Pumping ORIGINAL remaining a after i0 split3,4 = 3

source roles = [7, 3, 4]

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

Pumping ORIGINAL remaining a after i0 split4,1 := sub(7, 1)

Pumping ORIGINAL remaining a after i0 split4,1 = 6

source roles = [7, 4, 1]

derived = [2, 9, [4, 1, 2, a^6b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split4,2 := sub(7, 2)

Pumping ORIGINAL remaining a after i0 split4,2 = 5

source roles = [7, 4, 2]

derived = [1, 8, [4, 2, 1, a^5b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split4,3 := sub(7, 3)

Pumping ORIGINAL remaining a after i0 split4,3 = 4

source roles = [7, 4, 3]

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

Pumping ORIGINAL remaining a after i0 split5,1 := sub(7, 1)

Pumping ORIGINAL remaining a after i0 split5,1 = 6

source roles = [7, 5, 1]

derived = [1, 8, [5, 1, 1, a^6b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split5,2 := sub(7, 2)

Pumping ORIGINAL remaining a after i0 split5,2 = 5

source roles = [7, 5, 2]

derived = [0, 7, [5, 2, 0, a^5b^7, No], Yes]

Pumping ORIGINAL remaining a after i0 split6,1 := sub(7, 1)

Pumping ORIGINAL remaining a after i0 split6,1 = 6

source roles = [7, 6, 1]

derived = [0, 7, [6, 1, 0, a^6b^7, No], Yes]

Pumping ORIGINAL triangular numerator := mul(7, 8)

Pumping ORIGINAL triangular numerator = 56

Pumping ORIGINAL triangular all-splits count := div(56, 2)

Pumping ORIGINAL triangular all-splits count = 28

source roles = [7, [[0, 1, 6, a^6b^7, No], [0, 2, 5, a^5b^7, No], [0, 3, 4, a^4b^7, No], [0, 4, 3, a^3b^7, No], [0, 5, 2, a^2b^7, No], [0, 6, 1, a^1b^7, No], [0, 7, 0, a^0b^7, No], [1, 1, 5, a^6b^7, No], [1, 2, 4, a^5b^7, No], [1, 3, 3, a^4b^7, No], [1, 4, 2, a^3b^7, No], [1, 5, 1, a^2b^7, No], [1, 6, 0, a^1b^7, No], [2, 1, 4, a^6b^7, No], [2, 2, 3, a^5b^7, No], [2, 3, 2, a^4b^7, No], [2, 4, 1, a^3b^7, No], [2, 5, 0, a^2b^7, No], [3, 1, 3, a^6b^7, No], [3, 2, 2, a^5b^7, No], [3, 3, 1, a^4b^7, No], [3, 4, 0, a^3b^7, No], [4, 1, 2, a^6b^7, No], [4, 2, 1, a^5b^7, No], [4, 3, 0, a^4b^7, No], [5, 1, 1, a^6b^7, No], [5, 2, 0, a^5b^7, No], [6, 1, 0, a^6b^7, No]]]

derived = [28, 28, Yes]

Reverse checks

• S061 · 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.
• S060 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular all-splits count with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular numerator with exact rational arithmetic, not rounded decimals.
• S058 · 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.
• S057 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split6,1 with exact rational arithmetic, not rounded decimals.
• S056 · 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.
• S055 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,2 with exact rational arithmetic, not rounded decimals.
• S054 · 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.
• S053 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,1 with exact rational arithmetic, not rounded decimals.
• S052 · 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.
• S051 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,3 with exact rational arithmetic, not rounded decimals.
• S050 · 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.
• S049 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,2 with exact rational arithmetic, not rounded decimals.
• S048 · 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.
• S047 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,1 with exact rational arithmetic, not rounded decimals.
• S046 · 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.
• S045 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,4 with exact rational arithmetic, not rounded decimals.
• S044 · 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.
• S043 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,3 with exact rational arithmetic, not rounded decimals.
• S042 · 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.
• S041 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,2 with exact rational arithmetic, not rounded decimals.
• S040 · 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.
• S039 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,1 with exact rational arithmetic, not rounded decimals.
• S038 · 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.
• S037 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,5 with exact rational arithmetic, not rounded decimals.
• S036 · 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.
• S035 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,4 with exact rational arithmetic, not rounded decimals.
• S034 · 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.
• S033 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,3 with exact rational arithmetic, not rounded decimals.
• S032 · 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.
• S031 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,2 with exact rational arithmetic, not rounded decimals.
• S030 · 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.
• S029 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split2,1 with exact rational arithmetic, not rounded decimals.
• S028 · 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.
• S027 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,6 with exact rational arithmetic, not rounded decimals.
• S026 · 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.
• S025 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,5 with exact rational arithmetic, not rounded decimals.
• S024 · 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.
• S023 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split1,4 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 split1,3 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 split1,2 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 split1,1 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 split0,7 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 split0,6 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 split0,5 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

(7, a^7b^7, [[0, 1, 6, a^6b^7, No], [0, 2, 5, a^5b^7, No], [0, 3, 4, a^4b^7, No], [0, 4, 3, a^3b^7, No], [0, 5, 2, a^2b^7, No], [0, 6, 1, a^1b^7, No], [0, 7, 0, a^0b^7, No], [1, 1, 5, a^6b^7, No], [1, 2, 4, a^5b^7, No], [1, 3, 3, a^4b^7, No], [1, 4, 2, a^3b^7, No], [1, 5, 1, a^2b^7, No], [1, 6, 0, a^1b^7, No], [2, 1, 4, a^6b^7, No], [2, 2, 3, a^5b^7, No], [2, 3, 2, a^4b^7, No], [2, 4, 1, a^3b^7, No], [2, 5, 0, a^2b^7, No], [3, 1, 3, a^6b^7, No], [3, 2, 2, a^5b^7, No], [3, 3, 1, a^4b^7, No], [3, 4, 0, a^3b^7, No], [4, 1, 2, a^6b^7, No], [4, 2, 1, a^5b^7, No], [4, 3, 0, a^4b^7, No], [5, 1, 1, a^6b^7, No], [5, 2, 0, a^5b^7, No], [6, 1, 0, a^6b^7, No]], 28, 28, 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.