# Question 17296: Trace One Word under the Already Declared Mod-Count Language [Return to catalogue page 87](https://ocme.evemisslab.com/en/questions/page/87/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261102-r07-regular-language-foundations/#q-20261014-advance-20261102-r07-regular-language-0046) 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. R0033-17296 · q-20261014-advance-20261102-r07-regular-language-0046 ## Original question and givens Use alphabet=Fin(3)={0,…,2} and the language whose number of occurrences of target_symbol=1 is congruent to residue=1 modulo modulus=3. Construct the complete DFA on states 0 through modulus−1, start 0 and accepting subset [residue]. A target symbol increments the state modulo the modulus; every other symbol leaves it unchanged. The invariant is state equals target-symbol count modulo the modulus for every prefix of every finite word, not just selected tests. No regex, NFA, epsilon move, minimization or pumping-lemma result is added. Word=[0, 0, 1, 0, 1]. Give (word,state_trace,final_state,accepted?), retaining the initial state and every transition. - G001 — Use alphabet=Fin(3)={0,…,2} and the language whose number of occurrences of target_symbol=1 is congruent to residue=1 modulo modulus=3. Construct the complete DFA on states 0 through modulus−1, start 0 and accepting subset [residue]. A target symbol increments the state modulo the modulus; every other symbol leaves it unchanged. The invariant is state equals target-symbol count modulo the modulus for every prefix of every finite word, not just selected tests. No regex, NFA, epsilon move, minimization or pumping-lemma result is added. Word=[0, 0, 1, 0, 1]. Give (word,state_trace,final_state,accepted?), retaining the initial state and every transition. ## 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-regular-language](https://ocme.evemisslab.com/en/mko/mko-regular-language/) ## Step-by-step derivation S001 · impliesBefore mod-count state=0;original symbol=0;target symbol=1;indicator=0;modulus=3 After next state=(0+0) mod3=0 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001 S002 · impliesBefore mod-count state=0;original symbol=1;target symbol=1;indicator=1;modulus=3 After next state=(0+1) mod3=1 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001, S001 S003 · impliesBefore mod-count state=0;original symbol=2;target symbol=1;indicator=0;modulus=3 After next state=(0+0) mod3=0 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001, S002 S004 · impliesBefore mod-count state=1;original symbol=0;target symbol=1;indicator=0;modulus=3 After next state=(1+0) mod3=1 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001, S003 S005 · impliesBefore mod-count state=1;original symbol=1;target symbol=1;indicator=1;modulus=3 After next state=(1+1) mod3=2 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001, S004 S006 · impliesBefore mod-count state=1;original symbol=2;target symbol=1;indicator=0;modulus=3 After next state=(1+0) mod3=1 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001, S005 S007 · impliesBefore mod-count state=2;original symbol=0;target symbol=1;indicator=0;modulus=3 After next state=(2+0) mod3=2 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001, S006 S008 · impliesBefore mod-count state=2;original symbol=1;target symbol=1;indicator=1;modulus=3 After next state=(2+1) mod3=0 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001, S007 S009 · impliesBefore mod-count state=2;original symbol=2;target symbol=1;indicator=0;modulus=3 After next state=(2+0) mod3=2 Conditions: none Increment only for the target symbol,otherwise by0,and reduce modulo the same modulus.Check every state-symbol cell,including wraparound from the last residue to0. Step identifiers and dependenciesRule: regular_mod_count_every_state_symbol_cell · Uses: G001, S008 S010 · impliesBefore states Fin(3);complete typed table=[[0, 1, 0], [1, 2, 1], [2, 0, 2]];start0;accepting residue=1 After empty prefix count=0 equals start;append any symbol adds exactly its target indicator to BOTH count and state modulo the modulus Conditions: none Both count and state start at0.Appending any symbol updates both by the same indicator,so induction preserves state equal to target-symbol count modulo m for every prefix of every word. Step identifiers and dependenciesRule: regular_mod_count_all_prefix_induction_base_step · Uses: G001, S009 S011 · impliesBefore for EVERY finite word,state equals target-symbol count mod3;accepting set=[1] After accepted iff count congruent to1;whole declared language has this complete DFA certificate Conditions: none Final-state membership in the specified residue characterizes the entire declared language.This uses the complete DFA and universal invariant,not finite tests,and adds no regex,epsilon moves,minimization or pumping lemma. Step identifiers and dependenciesRule: regular_mod_count_language_characterized_not_test_suite · Uses: G001, S010 S012 · impliesBefore selected mod-count word:original ordered word=[0, 0, 1, 0, 1];start=0 After trace before consuming symbols=[0] Conditions: none Include the initial state.An empty word performs no transition;do not reorder symbols merely because a final answer might coincide. Step identifiers and dependenciesRule: regular_word_initial_state_empty_word_retained · Uses: G001, S011 S013 · impliesBefore selected mod-count word:word position=0;symbol=0;current state=0;complete original row=[0, 1, 0] After next state=0;ordered prefix trace=[0, 0] Conditions: none Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state. Step identifiers and dependenciesRule: regular_every_symbol_source_bound_transition · Uses: G001, S012 S014 · impliesBefore selected mod-count word:word position=1;symbol=0;current state=0;complete original row=[0, 1, 0] After next state=0;ordered prefix trace=[0, 0, 0] Conditions: none Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state. Step identifiers and dependenciesRule: regular_every_symbol_source_bound_transition · Uses: G001, S013 S015 · impliesBefore selected mod-count word:word position=2;symbol=1;current state=0;complete original row=[0, 1, 0] After next state=1;ordered prefix trace=[0, 0, 0, 1] Conditions: none Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state. Step identifiers and dependenciesRule: regular_every_symbol_source_bound_transition · Uses: G001, S014 S016 · impliesBefore selected mod-count word:word position=3;symbol=0;current state=1;complete original row=[1, 2, 1] After next state=1;ordered prefix trace=[0, 0, 0, 1, 1] Conditions: none Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state. Step identifiers and dependenciesRule: regular_every_symbol_source_bound_transition · Uses: G001, S015 S017 · impliesBefore selected mod-count word:word position=4;symbol=1;current state=1;complete original row=[1, 2, 1] After next state=2;ordered prefix trace=[0, 0, 0, 1, 1, 2] Conditions: none Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state. Step identifiers and dependenciesRule: regular_every_symbol_source_bound_transition · Uses: G001, S016 S018 · calculationBefore selected mod-count word exact target-symbol indicator count := sum(0, 0, 1, 0, 1) After selected mod-count word exact target-symbol indicator count = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore selected mod-count word:original word=[0, 0, 1, 0, 1];exact target count=2;final state=2;accepting residue=1 After accepted=No;final state equals count modulo3 Conditions: none Independently count original-symbol indicators and cross-check the final trace state.An empty word has empty sum0 and no invented transition. Step identifiers and dependenciesRule: regular_selected_word_count_reverse_check · Uses: G001, S018 ## Reverse checks - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute selected mod-count word exact target-symbol indicator count with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([0, 0, 1, 0, 1], [0, 0, 0, 1, 1, 2], 2, No) 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: 9172e149de0999771a52339feed62144257d2ee69677fff6f1123f2569682408 - candidate record SHA-256: 6130531c8a15f6e547a7aa7777b3ddd750f7b5b7c348bb7912185edd0ddc4d63 - semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3 - question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261102-r07-regular-language-0046.json)