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R0033-17254 · q-20261014-advance-20261102-r07-regular-language-0004
Use alphabet=Fin(2)={0,…,1} and the language whose number of occurrences of target_symbol=1 is congruent to residue=0 modulo modulus=5. 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=5. 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.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
mod-count state=0;original symbol=0;target symbol=1;indicator=0;modulus=5
After
next state=(0+0) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001
Before
mod-count state=0;original symbol=1;target symbol=1;indicator=1;modulus=5
After
next state=(0+1) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S001
Before
mod-count state=1;original symbol=0;target symbol=1;indicator=0;modulus=5
After
next state=(1+0) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S002
Before
mod-count state=1;original symbol=1;target symbol=1;indicator=1;modulus=5
After
next state=(1+1) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S003
Before
mod-count state=2;original symbol=0;target symbol=1;indicator=0;modulus=5
After
next state=(2+0) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S004
Before
mod-count state=2;original symbol=1;target symbol=1;indicator=1;modulus=5
After
next state=(2+1) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S005
Before
mod-count state=3;original symbol=0;target symbol=1;indicator=0;modulus=5
After
next state=(3+0) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S006
Before
mod-count state=3;original symbol=1;target symbol=1;indicator=1;modulus=5
After
next state=(3+1) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S007
Before
mod-count state=4;original symbol=0;target symbol=1;indicator=0;modulus=5
After
next state=(4+0) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S008
Before
mod-count state=4;original symbol=1;target symbol=1;indicator=1;modulus=5
After
next state=(4+1) mod5=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.
Rule: regular_mod_count_every_state_symbol_cell · Uses: G001, S009
Before
states Fin(5);complete typed table=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 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.
Rule: regular_mod_count_all_prefix_induction_base_step · Uses: G001, S010
Before
for EVERY finite word,state equals target-symbol count mod5;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.
Rule: regular_mod_count_language_characterized_not_test_suite · Uses: G001, S011
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.([0, 1, 2, 3, 4], [[0, 1], [1, 2], [2, 3], [3, 4], [4, 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.
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: 068fbc713d3ca26f359d2e22531cd8815c15534e2a233f6486360463884f51aacandidate record SHA-256: ca60e1f0194d3728866e2c1287f2c453878b244d2b126454ccc79ef60eccb211semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93Lean: NOT_RUN · human review: NOT_PERFORMED