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R0053-27302 · q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0052
Use positive pumping length p=2 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=2 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 = [2]
After
derived = [a^2b^2, 4, 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(2, 1)
After
Pumping ORIGINAL remaining a after i0 split0,1 = 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, S002
Before
source roles = [2, 0, 1]
After
derived = [1, 3, [0, 1, 1, a^1b^2, 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 remaining a after i0 split0,2 := sub(2, 2)
After
Pumping ORIGINAL remaining a after i0 split0,2 = 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, S004
Before
source roles = [2, 0, 2]
After
derived = [0, 2, [0, 2, 0, a^0b^2, 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, S005
Before
Pumping ORIGINAL remaining a after i0 split1,1 := sub(2, 1)
After
Pumping ORIGINAL remaining a after i0 split1,1 = 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, S006
Before
source roles = [2, 1, 1]
After
derived = [0, 2, [1, 1, 0, a^1b^2, 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, S007
Before
Pumping ORIGINAL triangular numerator := mul(2, 3)
After
Pumping ORIGINAL triangular numerator = 6
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Pumping ORIGINAL triangular all-splits count := div(6, 2)
After
Pumping ORIGINAL triangular all-splits count = 3
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
source roles = [2, [[0, 1, 1, a^1b^2, No], [0, 2, 0, a^0b^2, No], [1, 1, 0, a^1b^2, No]]]
After
derived = [3, 3, 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, S010
S011 · 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.S010 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular all-splits count with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute Pumping ORIGINAL triangular numerator 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 split1,1 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.(2, a^2b^2, [[0, 1, 1, a^1b^2, No], [0, 2, 0, a^0b^2, No], [1, 1, 0, a^1b^2, No]], 3, 3, 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: ebf3105853eba6bc57641959548a0bb27705ca87ffdff151359276ddcf50247dcandidate record SHA-256: e708a2f098325f40a4f79ac339e1266ee57df2b03a0cb0fcb6c5cf7bb9b1cec7semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED