Question 27310: Every Legal Split of a^p b^p Is Defeated by Pumping Down

Original question and givens

Use positive pumping length p=10 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=10 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 = [10]

derived = [a^10b^10, 20, ALL x>=0,y>=1,x+y<=p]

Pumping ORIGINAL remaining a after i0 split0,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split0,1 = 9

source roles = [10, 0, 1]

derived = [9, 19, [0, 1, 9, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split0,2 = 8

source roles = [10, 0, 2]

derived = [8, 18, [0, 2, 8, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,3 := sub(10, 3)

Pumping ORIGINAL remaining a after i0 split0,3 = 7

source roles = [10, 0, 3]

derived = [7, 17, [0, 3, 7, a^7b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,4 := sub(10, 4)

Pumping ORIGINAL remaining a after i0 split0,4 = 6

source roles = [10, 0, 4]

derived = [6, 16, [0, 4, 6, a^6b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,5 := sub(10, 5)

Pumping ORIGINAL remaining a after i0 split0,5 = 5

source roles = [10, 0, 5]

derived = [5, 15, [0, 5, 5, a^5b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,6 := sub(10, 6)

Pumping ORIGINAL remaining a after i0 split0,6 = 4

source roles = [10, 0, 6]

derived = [4, 14, [0, 6, 4, a^4b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,7 := sub(10, 7)

Pumping ORIGINAL remaining a after i0 split0,7 = 3

source roles = [10, 0, 7]

derived = [3, 13, [0, 7, 3, a^3b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,8 := sub(10, 8)

Pumping ORIGINAL remaining a after i0 split0,8 = 2

source roles = [10, 0, 8]

derived = [2, 12, [0, 8, 2, a^2b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,9 := sub(10, 9)

Pumping ORIGINAL remaining a after i0 split0,9 = 1

source roles = [10, 0, 9]

derived = [1, 11, [0, 9, 1, a^1b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split0,10 := sub(10, 10)

Pumping ORIGINAL remaining a after i0 split0,10 = 0

source roles = [10, 0, 10]

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

Pumping ORIGINAL remaining a after i0 split1,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split1,1 = 9

source roles = [10, 1, 1]

derived = [8, 18, [1, 1, 8, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split1,2 = 8

source roles = [10, 1, 2]

derived = [7, 17, [1, 2, 7, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,3 := sub(10, 3)

Pumping ORIGINAL remaining a after i0 split1,3 = 7

source roles = [10, 1, 3]

derived = [6, 16, [1, 3, 6, a^7b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,4 := sub(10, 4)

Pumping ORIGINAL remaining a after i0 split1,4 = 6

source roles = [10, 1, 4]

derived = [5, 15, [1, 4, 5, a^6b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,5 := sub(10, 5)

Pumping ORIGINAL remaining a after i0 split1,5 = 5

source roles = [10, 1, 5]

derived = [4, 14, [1, 5, 4, a^5b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,6 := sub(10, 6)

Pumping ORIGINAL remaining a after i0 split1,6 = 4

source roles = [10, 1, 6]

derived = [3, 13, [1, 6, 3, a^4b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,7 := sub(10, 7)

Pumping ORIGINAL remaining a after i0 split1,7 = 3

source roles = [10, 1, 7]

derived = [2, 12, [1, 7, 2, a^3b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,8 := sub(10, 8)

Pumping ORIGINAL remaining a after i0 split1,8 = 2

source roles = [10, 1, 8]

derived = [1, 11, [1, 8, 1, a^2b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split1,9 := sub(10, 9)

Pumping ORIGINAL remaining a after i0 split1,9 = 1

source roles = [10, 1, 9]

derived = [0, 10, [1, 9, 0, a^1b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split2,1 = 9

source roles = [10, 2, 1]

derived = [7, 17, [2, 1, 7, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split2,2 = 8

source roles = [10, 2, 2]

derived = [6, 16, [2, 2, 6, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,3 := sub(10, 3)

Pumping ORIGINAL remaining a after i0 split2,3 = 7

source roles = [10, 2, 3]

derived = [5, 15, [2, 3, 5, a^7b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,4 := sub(10, 4)

Pumping ORIGINAL remaining a after i0 split2,4 = 6

source roles = [10, 2, 4]

derived = [4, 14, [2, 4, 4, a^6b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,5 := sub(10, 5)

Pumping ORIGINAL remaining a after i0 split2,5 = 5

source roles = [10, 2, 5]

derived = [3, 13, [2, 5, 3, a^5b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,6 := sub(10, 6)

Pumping ORIGINAL remaining a after i0 split2,6 = 4

source roles = [10, 2, 6]

derived = [2, 12, [2, 6, 2, a^4b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,7 := sub(10, 7)

Pumping ORIGINAL remaining a after i0 split2,7 = 3

source roles = [10, 2, 7]

derived = [1, 11, [2, 7, 1, a^3b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split2,8 := sub(10, 8)

Pumping ORIGINAL remaining a after i0 split2,8 = 2

source roles = [10, 2, 8]

derived = [0, 10, [2, 8, 0, a^2b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split3,1 = 9

source roles = [10, 3, 1]

derived = [6, 16, [3, 1, 6, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split3,2 = 8

source roles = [10, 3, 2]

derived = [5, 15, [3, 2, 5, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,3 := sub(10, 3)

Pumping ORIGINAL remaining a after i0 split3,3 = 7

source roles = [10, 3, 3]

derived = [4, 14, [3, 3, 4, a^7b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,4 := sub(10, 4)

Pumping ORIGINAL remaining a after i0 split3,4 = 6

source roles = [10, 3, 4]

derived = [3, 13, [3, 4, 3, a^6b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,5 := sub(10, 5)

Pumping ORIGINAL remaining a after i0 split3,5 = 5

source roles = [10, 3, 5]

derived = [2, 12, [3, 5, 2, a^5b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,6 := sub(10, 6)

Pumping ORIGINAL remaining a after i0 split3,6 = 4

source roles = [10, 3, 6]

derived = [1, 11, [3, 6, 1, a^4b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split3,7 := sub(10, 7)

Pumping ORIGINAL remaining a after i0 split3,7 = 3

source roles = [10, 3, 7]

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

Pumping ORIGINAL remaining a after i0 split4,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split4,1 = 9

source roles = [10, 4, 1]

derived = [5, 15, [4, 1, 5, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split4,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split4,2 = 8

source roles = [10, 4, 2]

derived = [4, 14, [4, 2, 4, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split4,3 := sub(10, 3)

Pumping ORIGINAL remaining a after i0 split4,3 = 7

source roles = [10, 4, 3]

derived = [3, 13, [4, 3, 3, a^7b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split4,4 := sub(10, 4)

Pumping ORIGINAL remaining a after i0 split4,4 = 6

source roles = [10, 4, 4]

derived = [2, 12, [4, 4, 2, a^6b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split4,5 := sub(10, 5)

Pumping ORIGINAL remaining a after i0 split4,5 = 5

source roles = [10, 4, 5]

derived = [1, 11, [4, 5, 1, a^5b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split4,6 := sub(10, 6)

Pumping ORIGINAL remaining a after i0 split4,6 = 4

source roles = [10, 4, 6]

derived = [0, 10, [4, 6, 0, a^4b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split5,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split5,1 = 9

source roles = [10, 5, 1]

derived = [4, 14, [5, 1, 4, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split5,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split5,2 = 8

source roles = [10, 5, 2]

derived = [3, 13, [5, 2, 3, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split5,3 := sub(10, 3)

Pumping ORIGINAL remaining a after i0 split5,3 = 7

source roles = [10, 5, 3]

derived = [2, 12, [5, 3, 2, a^7b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split5,4 := sub(10, 4)

Pumping ORIGINAL remaining a after i0 split5,4 = 6

source roles = [10, 5, 4]

derived = [1, 11, [5, 4, 1, a^6b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split5,5 := sub(10, 5)

Pumping ORIGINAL remaining a after i0 split5,5 = 5

source roles = [10, 5, 5]

derived = [0, 10, [5, 5, 0, a^5b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split6,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split6,1 = 9

source roles = [10, 6, 1]

derived = [3, 13, [6, 1, 3, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split6,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split6,2 = 8

source roles = [10, 6, 2]

derived = [2, 12, [6, 2, 2, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split6,3 := sub(10, 3)

Pumping ORIGINAL remaining a after i0 split6,3 = 7

source roles = [10, 6, 3]

derived = [1, 11, [6, 3, 1, a^7b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split6,4 := sub(10, 4)

Pumping ORIGINAL remaining a after i0 split6,4 = 6

source roles = [10, 6, 4]

derived = [0, 10, [6, 4, 0, a^6b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split7,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split7,1 = 9

source roles = [10, 7, 1]

derived = [2, 12, [7, 1, 2, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split7,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split7,2 = 8

source roles = [10, 7, 2]

derived = [1, 11, [7, 2, 1, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split7,3 := sub(10, 3)

Pumping ORIGINAL remaining a after i0 split7,3 = 7

source roles = [10, 7, 3]

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

Pumping ORIGINAL remaining a after i0 split8,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split8,1 = 9

source roles = [10, 8, 1]

derived = [1, 11, [8, 1, 1, a^9b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split8,2 := sub(10, 2)

Pumping ORIGINAL remaining a after i0 split8,2 = 8

source roles = [10, 8, 2]

derived = [0, 10, [8, 2, 0, a^8b^10, No], Yes]

Pumping ORIGINAL remaining a after i0 split9,1 := sub(10, 1)

Pumping ORIGINAL remaining a after i0 split9,1 = 9

source roles = [10, 9, 1]

derived = [0, 10, [9, 1, 0, a^9b^10, No], Yes]

Pumping ORIGINAL triangular numerator := mul(10, 11)

Pumping ORIGINAL triangular numerator = 110

Pumping ORIGINAL triangular all-splits count := div(110, 2)

Pumping ORIGINAL triangular all-splits count = 55

source roles = [10, [[0, 1, 9, a^9b^10, No], [0, 2, 8, a^8b^10, No], [0, 3, 7, a^7b^10, No], [0, 4, 6, a^6b^10, No], [0, 5, 5, a^5b^10, No], [0, 6, 4, a^4b^10, No], [0, 7, 3, a^3b^10, No], [0, 8, 2, a^2b^10, No], [0, 9, 1, a^1b^10, No], [0, 10, 0, a^0b^10, No], [1, 1, 8, a^9b^10, No], [1, 2, 7, a^8b^10, No], [1, 3, 6, a^7b^10, No], [1, 4, 5, a^6b^10, No], [1, 5, 4, a^5b^10, No], [1, 6, 3, a^4b^10, No], [1, 7, 2, a^3b^10, No], [1, 8, 1, a^2b^10, No], [1, 9, 0, a^1b^10, No], [2, 1, 7, a^9b^10, No], [2, 2, 6, a^8b^10, No], [2, 3, 5, a^7b^10, No], [2, 4, 4, a^6b^10, No], [2, 5, 3, a^5b^10, No], [2, 6, 2, a^4b^10, No], [2, 7, 1, a^3b^10, No], [2, 8, 0, a^2b^10, No], [3, 1, 6, a^9b^10, No], [3, 2, 5, a^8b^10, No], [3, 3, 4, a^7b^10, No], [3, 4, 3, a^6b^10, No], [3, 5, 2, a^5b^10, No], [3, 6, 1, a^4b^10, No], [3, 7, 0, a^3b^10, No], [4, 1, 5, a^9b^10, No], [4, 2, 4, a^8b^10, No], [4, 3, 3, a^7b^10, No], [4, 4, 2, a^6b^10, No], [4, 5, 1, a^5b^10, No], [4, 6, 0, a^4b^10, No], [5, 1, 4, a^9b^10, No], [5, 2, 3, a^8b^10, No], [5, 3, 2, a^7b^10, No], [5, 4, 1, a^6b^10, No], [5, 5, 0, a^5b^10, No], [6, 1, 3, a^9b^10, No], [6, 2, 2, a^8b^10, No], [6, 3, 1, a^7b^10, No], [6, 4, 0, a^6b^10, No], [7, 1, 2, a^9b^10, No], [7, 2, 1, a^8b^10, No], [7, 3, 0, a^7b^10, No], [8, 1, 1, a^9b^10, No], [8, 2, 0, a^8b^10, No], [9, 1, 0, a^9b^10, No]]]

derived = [55, 55, Yes]

Reverse checks

• S115 · 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.
• S114 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular all-splits count with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular numerator with exact rational arithmetic, not rounded decimals.
• S112 · 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.
• S111 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split9,1 with exact rational arithmetic, not rounded decimals.
• S110 · 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.
• S109 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split8,2 with exact rational arithmetic, not rounded decimals.
• S108 · 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.
• S107 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split8,1 with exact rational arithmetic, not rounded decimals.
• S106 · 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.
• S105 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split7,3 with exact rational arithmetic, not rounded decimals.
• S104 · 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.
• S103 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split7,2 with exact rational arithmetic, not rounded decimals.
• S102 · 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.
• S101 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split7,1 with exact rational arithmetic, not rounded decimals.
• S100 · 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.
• S099 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split6,4 with exact rational arithmetic, not rounded decimals.
• S098 · 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.
• S097 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split6,3 with exact rational arithmetic, not rounded decimals.
• S096 · 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.
• S095 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split6,2 with exact rational arithmetic, not rounded decimals.
• S094 · 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.
• S093 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split6,1 with exact rational arithmetic, not rounded decimals.
• S092 · 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.
• S091 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,5 with exact rational arithmetic, not rounded decimals.
• S090 · 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.
• S089 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,4 with exact rational arithmetic, not rounded decimals.
• S088 · 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.
• S087 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,3 with exact rational arithmetic, not rounded decimals.
• S086 · 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.
• S085 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,2 with exact rational arithmetic, not rounded decimals.
• S084 · 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.
• S083 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split5,1 with exact rational arithmetic, not rounded decimals.
• S082 · 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.
• S081 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,6 with exact rational arithmetic, not rounded decimals.
• S080 · 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.
• S079 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,5 with exact rational arithmetic, not rounded decimals.
• S078 · 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.
• S077 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,4 with exact rational arithmetic, not rounded decimals.
• S076 · 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.
• S075 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,3 with exact rational arithmetic, not rounded decimals.
• S074 · 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.
• S073 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,2 with exact rational arithmetic, not rounded decimals.
• S072 · 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.
• S071 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split4,1 with exact rational arithmetic, not rounded decimals.
• S070 · 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.
• S069 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,7 with exact rational arithmetic, not rounded decimals.
• S068 · 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.
• S067 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,6 with exact rational arithmetic, not rounded decimals.
• S066 · 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.
• S065 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,5 with exact rational arithmetic, not rounded decimals.
• S064 · 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.
• S063 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,4 with exact rational arithmetic, not rounded decimals.
• S062 · 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.
• S061 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,3 with exact rational arithmetic, not rounded decimals.
• S060 · 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.
• S059 · premise_and_result_audit — Recompute Pumping ORIGINAL remaining a after i0 split3,2 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 split3,1 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 split2,8 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 split2,7 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 split2,6 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 split2,5 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 split2,4 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 split2,3 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 split2,2 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 split2,1 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 split1,9 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 split1,8 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 split1,7 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 split1,6 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 split1,5 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 split1,4 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 split1,3 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 split1,2 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 split1,1 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 split0,10 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 split0,9 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 split0,8 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

(10, a^10b^10, [[0, 1, 9, a^9b^10, No], [0, 2, 8, a^8b^10, No], [0, 3, 7, a^7b^10, No], [0, 4, 6, a^6b^10, No], [0, 5, 5, a^5b^10, No], [0, 6, 4, a^4b^10, No], [0, 7, 3, a^3b^10, No], [0, 8, 2, a^2b^10, No], [0, 9, 1, a^1b^10, No], [0, 10, 0, a^0b^10, No], [1, 1, 8, a^9b^10, No], [1, 2, 7, a^8b^10, No], [1, 3, 6, a^7b^10, No], [1, 4, 5, a^6b^10, No], [1, 5, 4, a^5b^10, No], [1, 6, 3, a^4b^10, No], [1, 7, 2, a^3b^10, No], [1, 8, 1, a^2b^10, No], [1, 9, 0, a^1b^10, No], [2, 1, 7, a^9b^10, No], [2, 2, 6, a^8b^10, No], [2, 3, 5, a^7b^10, No], [2, 4, 4, a^6b^10, No], [2, 5, 3, a^5b^10, No], [2, 6, 2, a^4b^10, No], [2, 7, 1, a^3b^10, No], [2, 8, 0, a^2b^10, No], [3, 1, 6, a^9b^10, No], [3, 2, 5, a^8b^10, No], [3, 3, 4, a^7b^10, No], [3, 4, 3, a^6b^10, No], [3, 5, 2, a^5b^10, No], [3, 6, 1, a^4b^10, No], [3, 7, 0, a^3b^10, No], [4, 1, 5, a^9b^10, No], [4, 2, 4, a^8b^10, No], [4, 3, 3, a^7b^10, No], [4, 4, 2, a^6b^10, No], [4, 5, 1, a^5b^10, No], [4, 6, 0, a^4b^10, No], [5, 1, 4, a^9b^10, No], [5, 2, 3, a^8b^10, No], [5, 3, 2, a^7b^10, No], [5, 4, 1, a^6b^10, No], [5, 5, 0, a^5b^10, No], [6, 1, 3, a^9b^10, No], [6, 2, 2, a^8b^10, No], [6, 3, 1, a^7b^10, No], [6, 4, 0, a^6b^10, No], [7, 1, 2, a^9b^10, No], [7, 2, 1, a^8b^10, No], [7, 3, 0, a^7b^10, No], [8, 1, 1, a^9b^10, No], [8, 2, 0, a^8b^10, No], [9, 1, 0, a^9b^10, No]], 55, 55, 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.