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Question 27522: Regular Expressions Have Typed Nodes and an Explicit Alphabet

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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R0054-27522 · q-20261015-advance-20261118-r06-regular-expression-0022

Original question and givens

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

  • G001 — Use alphabet=[a, b, c] and typed regular-expression AST={"type":"union","children":[{"type":"union","children":[{"type":"union","children":[{"type":"union","children":[{"type":"concat","children":[{"type":"symbol","value":"a"},{"type":"symbol","value":"b"}]},{"type":"star","child":{"type":"symbol","value":"b"}}]},{"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. Give (alphabet, canonical_typed_AST, AST_qualified, qualification_statement).

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-ast-qualification]

After

derived = [[alphabet, canonical_typed_AST, AST_qualified, qualification_statement]]

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 = [root.children[0].children[0].children[0].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.

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 = [root.children[0].children[0].children[0].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.

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 = [root.children[0].children[0].children[0].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.

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 = [root.children[0].children[0].children[0].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, S004

S006 · implies

Before

source roles = [root.children[0].children[0].children[0].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, S005

S007 · implies

Before

source roles = [root.children[0].children[0].children[0], [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.

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 = [root.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 = [root.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 = [root.children[0].children[0], [a, b, c], {'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}]}]

After

derived = [{'type': 'union', 'children': [{'type': 'star', 'child': {'type': 'symbol', 'value': 'c'}}, {'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:"c"},type:"star"},{children:[{child:{type:"symbol",value:"b"},type:"star"},{children:[{type:"symbol",value:"a"},{type:"symbol",value:"b"}],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, S009

S011 · implies

Before

source roles = [root.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 = [root.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 = [root.children[0], [a, b, c], {'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'union', 'children': [{'type': 'concat', 'children': [{'type': 'symbol', 'value': 'a'}, {'type': 'symbol', 'value': 'b'}]}, {'type': 'star', 'child': {'type': 'symbol', 'value': 'b'}}]}, {'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': '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:"a"},type:"star"},{children:[{child:{type:"symbol",value:"c"},type:"star"},{children:[{child:{type:"symbol",value:"b"},type:"star"},{children:[{type:"symbol",value:"a"},{type:"symbol",value:"b"}],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, S012

S014 · implies

Before

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

Reverse checks

  • 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:[{child:{type:"symbol",value:"b"},type:"star"},{children:[{type:"symbol",value:"a"},{type:"symbol",value:"b"}],type:"concat"}],type:"union"}],type:"union"}],type:"union"}],type:"union"}, Yes, typed_AST_symbols_in_alphabet)

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: 9f823ea04add13a77b34909fd3cfaf3e393322ae49689dd9077680dd4071575e
  • candidate record SHA-256: a305d7714061c681f2a5d68e7e9b676b70b0ac777cfda6f513f36471a8d0c89c
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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