# Question 27303: Every Legal Split of a^p b^p Is Defeated by Pumping Down

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0053-27303 · q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0053



## Original question and givens

Use positive pumping length p=3 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=3 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](https://ocme.evemisslab.com/en/mko/mko-pumping-lemma-regular-languages/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [3]

After

derived = [a^3b^3, 6, 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.

Step identifiers and dependenciesRule: Pumping_adversary_word_and_ALL_legal_split_domain · Uses: G001, S001





S003 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,1 := sub(3, 1)

After

Pumping ORIGINAL remaining a after i0 split0,1 = 2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [3, 0, 1]

After

derived = [2, 5, [0, 1, 2, a^2b^3, 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.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S003





S005 · calculationBefore

Pumping ORIGINAL remaining a after i0 split0,2 := sub(3, 2)

After

Pumping ORIGINAL remaining a after i0 split0,2 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

source roles = [3, 0, 2]

After

derived = [1, 4, [0, 2, 1, a^1b^3, 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.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S005





S007 · calculationBefore

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

After

Pumping ORIGINAL remaining a after i0 split0,3 = 0

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

source roles = [3, 0, 3]

After

derived = [0, 3, [0, 3, 0, a^0b^3, 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.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S007





S009 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,1 := sub(3, 1)

After

Pumping ORIGINAL remaining a after i0 split1,1 = 2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

source roles = [3, 1, 1]

After

derived = [1, 4, [1, 1, 1, a^2b^3, 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.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S009





S011 · calculationBefore

Pumping ORIGINAL remaining a after i0 split1,2 := sub(3, 2)

After

Pumping ORIGINAL remaining a after i0 split1,2 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · impliesBefore

source roles = [3, 1, 2]

After

derived = [0, 3, [1, 2, 0, a^1b^3, 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.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S011





S013 · calculationBefore

Pumping ORIGINAL remaining a after i0 split2,1 := sub(3, 1)

After

Pumping ORIGINAL remaining a after i0 split2,1 = 2

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

source roles = [3, 2, 1]

After

derived = [0, 3, [2, 1, 0, a^2b^3, 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.

Step identifiers and dependenciesRule: Pumping_EVERY_legal_split_i0_refutation_remainingA_NOT_fullZ · Uses: G001, S013





S015 · calculationBefore

Pumping ORIGINAL triangular numerator := mul(3, 4)

After

Pumping ORIGINAL triangular numerator = 12

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

Pumping ORIGINAL triangular all-splits count := div(12, 2)

After

Pumping ORIGINAL triangular all-splits count = 6

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · impliesBefore

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

After

derived = [6, 6, 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.

Step identifiers and dependenciesRule: Pumping_ALL_splits_and_GENERAL_every_proposed_p_contradiction · Uses: G001, S016









## Reverse checks


- S017 · 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.
- S016 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular all-splits count with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular numerator 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 split2,1 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 split1,2 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 split1,1 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

(3, a^3b^3, [[0, 1, 2, a^2b^3, No], [0, 2, 1, a^1b^3, No], [0, 3, 0, a^0b^3, No], [1, 1, 1, a^2b^3, No], [1, 2, 0, a^1b^3, No], [2, 1, 0, a^2b^3, No]], 6, 6, 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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: c5502baf84ea6f8f6b7ece2e14d0589debc7b8aeb39a848cc53447e6ab697bbc
- candidate record SHA-256: 74ac7fd8a1927a45d8caa5e9a776858f25af1d28e4f5886d14bf37fd6541395b
- semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
- question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
- Lean: NOT_RUN · human review: NOT_PERFORMED

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