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Question 27595: Whole-Word Membership Uses a Sufficient Complete Length Slice

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0054-27595 · q-20261015-advance-20261118-r06-regular-expression-0070

Original question and givens

Use alphabet=[a, b, c] and typed regular-expression AST={"type":"union","children":[{"type":"union","children":[{"type":"union","children":[{"type":"concat","children":[{"type":"symbol","value":"b"},{"type":"symbol","value":"c"},{"type":"symbol","value":"a"},{"type":"symbol","value":"b"},{"type":"symbol","value":"c"}]},{"type":"star","child":{"type":"symbol","value":"c"}}]},{"type":"star","child":{"type":"symbol","value":"a"}}]},{"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=[b, c, a, b]. 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":"union","children":[{"type":"union","children":[{"type":"concat","children":[{"type":"symbol","value":"b"},{"type":"symbol","value":"c"},{"type":"symbol","value":"a"},{"type":"symbol","value":"b"},{"type":"symbol","value":"c"}]},{"type":"star","child":{"type":"symbol","value":"c"}}]},{"type":"star","child":{"type":"symbol","value":"a"}}]},{"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=[b, c, a, b]. Give (alphabet, canonical_typed_AST, word, member, accepted_words_at_same_length, membership_scope).

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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[0].children[0], [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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S001

S003 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[0].children[1], [a, b, c], {'type': 'symbol', 'value': 'c'}]

After

derived = [{'type': 'symbol', 'value': 'c'}, {type:"symbol",value:"c"}]

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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S002

S004 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[0].children[2], [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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S003

S005 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[0].children[3], [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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S004

S006 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[0].children[4], [a, b, c], {'type': 'symbol', 'value': 'c'}]

After

derived = [{'type': 'symbol', 'value': 'c'}, {type:"symbol",value:"c"}]

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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S005

S007 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[0], [a, b, c], {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}]

After

derived = [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}, {children:[{type:"symbol",value:"b"},{type:"symbol",value:"c"},{type:"symbol",value:"a"},{type:"symbol",value:"b"},{type:"symbol",value:"c"}],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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S006

S008 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[1].child, [a, b, c], {'type': 'symbol', 'value': 'c'}]

After

derived = [{'type': 'symbol', 'value': 'c'}, {type:"symbol",value:"c"}]

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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S007

S009 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0].children[1], [a, b, c], {'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}]

After

derived = [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {child:{type:"symbol",value:"c"},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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S008

S010 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[0], [a, b, c], {'type': 'union', 'children': [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}]}]

After

derived = [{'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}]}, {children:[{child:{type:"symbol",value:"c"},type:"star"},{children:[{type:"symbol",value:"b"},{type:"symbol",value:"c"},{type:"symbol",value:"a"},{type:"symbol",value:"b"},{type:"symbol",value:"c"}],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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S009

S011 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[1].child, [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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S010

S012 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0].children[1], [a, b, c], {'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}]

After

derived = [{'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}, {child:{type:"symbol",value:"a"},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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S011

S013 · implies

Before

source roles = [Regex ORIGINAL membership.ast.children[0], [a, b, c], {'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}]}]

After

derived = [{'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}]}]}, {children:[{child:{type:"symbol",value:"a"},type:"star"},{children:[{child:{type:"symbol",value:"c"},type:"star"},{children:[{type:"symbol",value:"b"},{type:"symbol",value:"c"},{type:"symbol",value:"a"},{type:"symbol",value:"b"},{type:"symbol",value:"c"}],type:"concat"}],type:"union"}],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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S012

S014 · implies

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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S013

S015 · implies

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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S014

S016 · implies

Before

source roles = [Regex ORIGINAL membership.ast, [a, b, c], {'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]}]

After

derived = [{'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}]}]}]}, {children:[{child:{type:"symbol",value:"b"},type:"star"},{children:[{child:{type:"symbol",value:"a"},type:"star"},{children:[{child:{type:"symbol",value:"c"},type:"star"},{children:[{type:"symbol",value:"b"},{type:"symbol",value:"c"},{type:"symbol",value:"a"},{type:"symbol",value:"b"},{type:"symbol",value:"c"}],type:"concat"}],type:"union"}],type:"union"}],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.

Step identifiers and dependencies

Rule: Regex_EACH_typed_node_exact_keys_symbol_scope_union_set_concat_order · Uses: G001, S015

S017 · implies

Before

source roles = [[a, b, c], 4]

After

derived = [121]

Conditions: none

The bounded universe has sum |alphabet|^k words,including epsilon,and source budget<=4096.This defines complete bounded enumeration,not unbounded equivalence.

Step identifiers and dependencies

Rule: Regex_COMPLETE_bounded_word_universe_not_unbounded_equivalence · Uses: G001, S016

S018 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[0].star_base, symbol, {'type': 'symbol', 'value': 'b'}, 4]

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.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S017

S019 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[0], 4, 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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S018

S020 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[0], 4, 1, [[b]], [[b]]]

After

derived = [[[b, b]], [[b, 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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S019

S021 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[0], 4, 2, [[b, b]], [[b]]]

After

derived = [[[b, b, b]], [[b, b, b], [b, 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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S020

S022 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[0], 4, 3, [[b, b, b]], [[b]]]

After

derived = [[[b, b, b, b]], [[b, b, b, b], [b, b, b], [b, 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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S021

S023 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[0], 4, 4, [[b, b, b, b]], [[b]]]

After

derived = [[], [[b, b, b, b], [b, b, b], [b, 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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S022

S024 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[0].star_base, symbol, {'type': 'symbol', 'value': 'a'}, 4]

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.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S023

S025 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[0], 4, 0, [[]], [[a]]]

After

derived = [[[a]], [[a], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S024

S026 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[0], 4, 1, [[a]], [[a]]]

After

derived = [[[a, a]], [[a, a], [a], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S025

S027 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[0], 4, 2, [[a, a]], [[a]]]

After

derived = [[[a, a, a]], [[a, a, a], [a, a], [a], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S026

S028 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[0], 4, 3, [[a, a, a]], [[a]]]

After

derived = [[[a, a, a, a]], [[a, a, a, a], [a, a, a], [a, a], [a], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S027

S029 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[0], 4, 4, [[a, a, a, a]], [[a]]]

After

derived = [[], [[a, a, a, a], [a, a, a], [a, a], [a], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S028

S030 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[0].star_base, symbol, {'type': 'symbol', 'value': 'c'}, 4]

After

derived = [[[c]]]

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.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S029

S031 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[0], 4, 0, [[]], [[c]]]

After

derived = [[[c]], [[c], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S030

S032 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[0], 4, 1, [[c]], [[c]]]

After

derived = [[[c, c]], [[c, c], [c], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S031

S033 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[0], 4, 2, [[c, c]], [[c]]]

After

derived = [[[c, c, c]], [[c, c, c], [c, c], [c], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S032

S034 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[0], 4, 3, [[c, c, c]], [[c]]]

After

derived = [[[c, c, c, c]], [[c, c, c, c], [c, c, c], [c, c], [c], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S033

S035 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[0], 4, 4, [[c, c, c, c]], [[c]]]

After

derived = [[], [[c, c, c, c], [c, c, c], [c, c], [c], []]]

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.

Step identifiers and dependencies

Rule: Regex_STAR_each_extensional_frontier_closure_EPS_cycles_terminate · Uses: G001, S034

S036 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1].concat[0], symbol, {'type': 'symbol', 'value': 'b'}, 4]

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.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S035

S037 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1], 0, 4, [[]], [[b]]]

After

derived = [[[b]]]

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.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S036

S038 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1].concat[1], symbol, {'type': 'symbol', 'value': 'c'}, 4]

After

derived = [[[c]]]

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.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S037

S039 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1], 1, 4, [[b]], [[c]]]

After

derived = [[[b, c]]]

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.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S038

S040 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1].concat[2], symbol, {'type': 'symbol', 'value': 'a'}, 4]

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.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S039

S041 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1], 2, 4, [[b, c]], [[a]]]

After

derived = [[[b, c, 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.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S040

S042 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1].concat[3], symbol, {'type': 'symbol', 'value': 'b'}, 4]

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.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S041

S043 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1], 3, 4, [[b, c, a]], [[b]]]

After

derived = [[[b, c, a, b]]]

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.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S042

S044 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1].concat[4], symbol, {'type': 'symbol', 'value': 'c'}, 4]

After

derived = [[[c]]]

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.

Step identifiers and dependencies

Rule: Regex_atomic_language_EMPTY_versus_EPS_versus_literal_symbol · Uses: G001, S043

S045 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1].union[1], 4, 4, [[b, c, a, b]], [[c]]]

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.

Step identifiers and dependencies

Rule: Regex_EACH_ordered_concatenation_complete_cartesian_word_products · Uses: G001, S044

S046 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1].union[1], 4, [[[c, c, c, c], [c, c, c], [c, c], [c], []], []]]

After

derived = [[[c, c, c, c], [c, c, c], [c, c], [c], []]]

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.

Step identifiers and dependencies

Rule: Regex_union_ALL_child_languages_extensional_bound · Uses: G001, S045

S047 · implies

Before

source roles = [Regex ORIGINAL membership.language.union[1], 4, [[[a, a, a, a], [a, a, a], [a, a], [a], []], [[c, c, c, c], [c, c, c], [c, c], [c], []]]]

After

derived = [[[a, a, a, a], [a, a, a], [a, a], [a], [c, c, c, c], [c, c, c], [c, c], [c], []]]

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.

Step identifiers and dependencies

Rule: Regex_union_ALL_child_languages_extensional_bound · Uses: G001, S046

S048 · implies

Before

source roles = [Regex ORIGINAL membership.language, 4, [[[b, b, b, b], [b, b, b], [b, b], [b], []], [[a, a, a, a], [a, a, a], [a, a], [a], [c, c, c, c], [c, c, c], [c, c], [c], []]]]

After

derived = [[[a, a, a, a], [a, a, a], [a, a], [a], [b, b, b, b], [b, b, b], [b, b], [b], [c, c, c, c], [c, c, c], [c, c], [c], []]]

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.

Step identifiers and dependencies

Rule: Regex_union_ALL_child_languages_extensional_bound · Uses: G001, S047

S049 · implies

Before

source roles = [[a, b, c], {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'a'}}, {'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {'type': 'concat', 'children': [{'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}, {'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}, {'type': 'symbol', 'value': 'c'}]}]}]}]}, [b, c, a, b], 4]

After

derived = [No, [[a, a, a, a], [b, b, b, b], [c, c, c, c]]]

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.

Step identifiers and dependencies

Rule: Regex_WHOLE_queried_word_exact_membership_complete_same_length_slice · Uses: G001, S048

Reverse checks

  • S049 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S048 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S047 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S046 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S045 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S044 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S043 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S042 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S041 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S040 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S039 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S038 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S037 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S036 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S035 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S034 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S033 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S032 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S030 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S028 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S026 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S024 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S022 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S020 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S018 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • 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.

Result

([a, b, c], {children:[{child:{type:"symbol",value:"b"},type:"star"},{children:[{child:{type:"symbol",value:"a"},type:"star"},{children:[{child:{type:"symbol",value:"c"},type:"star"},{children:[{type:"symbol",value:"b"},{type:"symbol",value:"c"},{type:"symbol",value:"a"},{type:"symbol",value:"b"},{type:"symbol",value:"c"}],type:"concat"}],type:"union"}],type:"union"}],type:"union"}, [b, c, a, b], No, [[a, a, a, a], [b, b, b, b], [c, c, c, c]], 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.

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: b37eca84f192227d36aa6123b43e12d0ff1fef87a11e739a3099d7c36b436a78
  • candidate record SHA-256: a633e8019e7d59c29b8c0a0f1577b67075d358e6dcf1de4d65e289c74a3bed67
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON