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R0054-27582 · q-20261015-advance-20261118-r06-regular-expression-0057
Use alphabet=[a, b, c] and typed regular-expression AST={"type":"union","children":[{"type":"concat","children":[{"type":"symbol","value":"a"},{"type":"symbol","value":"b"}]},{"type":"star","child":{"type":"symbol","value":"b"}}]}. Words are arrays of alphabet symbols; distinguish empty language from the language containing the empty word. Decide word=[a]. Give (alphabet, canonical_typed_AST, word, member, accepted_words_at_same_length, membership_scope).
G001 — Use alphabet=[a, b, c] and typed regular-expression AST={"type":"union","children":[{"type":"concat","children":[{"type":"symbol","value":"a"},{"type":"symbol","value":"b"}]},{"type":"star","child":{"type":"symbol","value":"b"}}]}. Words are arrays of alphabet symbols; distinguish empty language from the language containing the empty word. Decide word=[a]. Give (alphabet, canonical_typed_AST, word, member, accepted_words_at_same_length, membership_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 = [regex-word-membership]
After
derived = [[alphabet, canonical_typed_AST, word, member, accepted_words_at_same_length, membership_scope]]
Conditions: none
List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [Regex ORIGINAL membership.ast.children[0].children[0], [a, b, c], {'type': 'symbol', 'value': 'a'}]After
derived = [{'type': 'symbol', 'value': 'a'}, {type:"symbol",value:"a"}]Conditions: none
For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.
Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S001
Before
source roles = [Regex ORIGINAL membership.ast.children[0].children[1], [a, b, c], {'type': 'symbol', 'value': 'b'}]After
derived = [{'type': 'symbol', 'value': 'b'}, {type:"symbol",value:"b"}]Conditions: none
For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.
Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S002
Before
source roles = [Regex ORIGINAL membership.ast.children[0], [a, b, c], {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}]}]After
derived = [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}]}, {children:[{type:"symbol",value:"a"},{type:"symbol",value:"b"}],type:"concat"}]Conditions: none
For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.
Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S003
Before
source roles = [Regex ORIGINAL membership.ast.children[1].child, [a, b, c], {'type': 'symbol', 'value': 'b'}]After
derived = [{'type': 'symbol', 'value': 'b'}, {type:"symbol",value:"b"}]Conditions: none
For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.
Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S004
Before
source roles = [Regex ORIGINAL membership.ast.children[1], [a, b, c], {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]After
derived = [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {child:{type:"symbol",value:"b"},type:"star"}]Conditions: none
For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.
Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S005
Before
source roles = [Regex ORIGINAL membership.ast, [a, b, c], {'type': 'union', 'children': [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]}]After
derived = [{'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}]}]}, {children:[{child:{type:"symbol",value:"b"},type:"star"},{children:[{type:"symbol",value:"a"},{type:"symbol",value:"b"}],type:"concat"}],type:"union"}]Conditions: none
For EACH node qualify type,exact keys and symbol membership.A union checks raw arity>=2 BEFORE extensional deduplication,so its canonical form may have one child.Concatenation retains order and star retains its child.Do not revalidate canonical output as raw input with a new arity gate.
Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S006
Before
source roles = [[a, b, c], 1]
After
derived = [4]
Conditions: none
The bounded universe has sum |alphabet|^k words,including epsilon,and source budget<=4096.This defines complete bounded enumeration,not unbounded equivalence.
Rule: Regex_COMPLETE_bounded_word_universe_not_unbounded_equivalence · Uses: G001, S007
Before
source roles = [Regex ORIGINAL membership.language.union[0].star_base, symbol, {'type': 'symbol', 'value': 'b'}, 1]After
derived = [[[b]]]
Conditions: none
The empty language has no words;epsilon has the single empty word;a symbol has its one token within the queried length bound.Words are token arrays,not reinterpreted character splits.
Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S008
Before
source roles = [Regex ORIGINAL membership.language.union[0], 1, 0, [[]], [[b]]]
After
derived = [[[b]], [[b], []]]
Conditions: none
Start with epsilon and add ALL base concatenations within the length bound.Only genuinely new words enter the next frontier.Epsilon cycles add none.The finite alphabet/length bound and4096 budget give a finite universe;completion requires an empty frontier,not an arbitrary iteration cutoff.
Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S009
Before
source roles = [Regex ORIGINAL membership.language.union[0], 1, 1, [[b]], [[b]]]
After
derived = [[], [[b], []]]
Conditions: none
Start with epsilon and add ALL base concatenations within the length bound.Only genuinely new words enter the next frontier.Epsilon cycles add none.The finite alphabet/length bound and4096 budget give a finite universe;completion requires an empty frontier,not an arbitrary iteration cutoff.
Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S010
Before
source roles = [Regex ORIGINAL membership.language.union[1].concat[0], symbol, {'type': 'symbol', 'value': 'a'}, 1]After
derived = [[[a]]]
Conditions: none
The empty language has no words;epsilon has the single empty word;a symbol has its one token within the queried length bound.Words are token arrays,not reinterpreted character splits.
Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S011
Before
source roles = [Regex ORIGINAL membership.language.union[1], 0, 1, [[]], [[a]]]
After
derived = [[[a]]]
Conditions: none
In ORIGINAL concatenation order take EVERY prefix/next-child word concatenation and discard only excessive length.If total length<=m,every factor length<=m,so truncation loses no allowed combination.
Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S012
Before
source roles = [Regex ORIGINAL membership.language.union[1].concat[1], symbol, {'type': 'symbol', 'value': 'b'}, 1]After
derived = [[[b]]]
Conditions: none
The empty language has no words;epsilon has the single empty word;a symbol has its one token within the queried length bound.Words are token arrays,not reinterpreted character splits.
Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S013
Before
source roles = [Regex ORIGINAL membership.language.union[1], 1, 1, [[a]], [[b]]]
After
derived = [[]]
Conditions: none
In ORIGINAL concatenation order take EVERY prefix/next-child word concatenation and discard only excessive length.If total length<=m,every factor length<=m,so truncation loses no allowed combination.
Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S014
Before
source roles = [Regex ORIGINAL membership.language, 1, [[[b], []], []]]
After
derived = [[[b], []]]
Conditions: none
Union includes ALL child languages,retaining each word once without reordering its symbols.Structural induction accounts for inclusion/exclusion of every word within the bound.
Rule: Regex_union_ALL_child_languages_extensional_bound · Uses: G001, S015
Before
source roles = [[a, b, c], {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}]}]}, [a], 1]After
derived = [No, [[b]]]
Conditions: none
Recompute through the FULL queried length and retain every accepted word at that length.The empty word differs from a symbol;absence from an insufficient shorter bound is not nonmembership.
Rule: Regex_WHOLE_queried_word_exact_membership_complete_same_length_slice · Uses: G001, S016
S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S016 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S014 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S012 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S010 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S008 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S006 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S005 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.([a, b, c], {children:[{child:{type:"symbol",value:"b"},type:"star"},{children:[{type:"symbol",value:"a"},{type:"symbol",value:"b"}],type:"concat"}],type:"union"}, [a], No, [[b]], complete_membership_not_absence_from_an_insufficient_bound)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: 45f699e3812c817bb67cefb27c6300de88e08277149824c8cb6733d1fd2e9b05candidate record SHA-256: d2dc5e50bb9c51d219d8b58d3e77ec1b590ef4b215612facbec33d2d24dca630semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437Lean: NOT_RUN · human review: NOT_PERFORMED