# Question 17267: A State Invariant Covers Every Word, Not Just Test Cases [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-0017) 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. R0033-17267 · q-20261014-advance-20261102-r07-regular-language-0017 ## Original question and givens Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=18. 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. Give (states,transition,start,accepting,regular?,state_equals_target_symbol_count_mod_modulus_for_all_prefixes). Rows follow state order and columns symbol order. - G001 — Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=18. 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. Give (states,transition,start,accepting,regular?,state_equals_target_symbol_count_mod_modulus_for_all_prefixes). Rows follow state order and columns symbol order. ## 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=18 After next state=(0+0) mod18=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=18 After next state=(0+1) mod18=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=1;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(1+0) mod18=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, S002 S004 · impliesBefore mod-count state=1;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(1+1) mod18=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, S003 S005 · impliesBefore mod-count state=2;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(2+0) mod18=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=2;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(2+1) mod18=3 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=3;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(3+0) mod18=3 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=3;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(3+1) mod18=4 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=4;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(4+0) mod18=4 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 mod-count state=4;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(4+1) mod18=5 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, S009 S011 · impliesBefore mod-count state=5;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(5+0) mod18=5 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, S010 S012 · impliesBefore mod-count state=5;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(5+1) mod18=6 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, S011 S013 · impliesBefore mod-count state=6;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(6+0) mod18=6 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, S012 S014 · impliesBefore mod-count state=6;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(6+1) mod18=7 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, S013 S015 · impliesBefore mod-count state=7;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(7+0) mod18=7 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, S014 S016 · impliesBefore mod-count state=7;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(7+1) mod18=8 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, S015 S017 · impliesBefore mod-count state=8;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(8+0) mod18=8 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, S016 S018 · impliesBefore mod-count state=8;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(8+1) mod18=9 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, S017 S019 · impliesBefore mod-count state=9;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(9+0) mod18=9 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, S018 S020 · impliesBefore mod-count state=9;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(9+1) mod18=10 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, S019 S021 · impliesBefore mod-count state=10;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(10+0) mod18=10 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, S020 S022 · impliesBefore mod-count state=10;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(10+1) mod18=11 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, S021 S023 · impliesBefore mod-count state=11;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(11+0) mod18=11 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, S022 S024 · impliesBefore mod-count state=11;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(11+1) mod18=12 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, S023 S025 · impliesBefore mod-count state=12;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(12+0) mod18=12 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, S024 S026 · impliesBefore mod-count state=12;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(12+1) mod18=13 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, S025 S027 · impliesBefore mod-count state=13;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(13+0) mod18=13 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, S026 S028 · impliesBefore mod-count state=13;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(13+1) mod18=14 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, S027 S029 · impliesBefore mod-count state=14;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(14+0) mod18=14 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, S028 S030 · impliesBefore mod-count state=14;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(14+1) mod18=15 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, S029 S031 · impliesBefore mod-count state=15;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(15+0) mod18=15 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, S030 S032 · impliesBefore mod-count state=15;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(15+1) mod18=16 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, S031 S033 · impliesBefore mod-count state=16;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(16+0) mod18=16 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, S032 S034 · impliesBefore mod-count state=16;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(16+1) mod18=17 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, S033 S035 · impliesBefore mod-count state=17;original symbol=0;target symbol=1;indicator=0;modulus=18 After next state=(17+0) mod18=17 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, S034 S036 · impliesBefore mod-count state=17;original symbol=1;target symbol=1;indicator=1;modulus=18 After next state=(17+1) mod18=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, S035 S037 · impliesBefore states Fin(18);complete typed table=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 0]];start0;accepting residue=0 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, S036 S038 · impliesBefore for EVERY finite word,state equals target-symbol count mod18;accepting set=[0] After accepted iff count congruent to0;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, S037 ## Reverse checks - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 0]], 0, [0], Yes, state_equals_target_symbol_count_mod_modulus_for_all_prefixes) 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: 28d37362bae9f85a13bac7fcab148c65c9729397448f9101f9dd5a321a3bb029 - candidate record SHA-256: 59554a015a9b35bc5bfe0160831255fd41b3466b754821c7b2ae881092b1338d - 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-0017.json)