Original question and givens
Use the complete finite grammar: nonterminals=[S2, A2], terminals=[a2, b2], start=S2, productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a2, b2, b2], supplied forms=[[S2], [a2, A2], [a2, b2, A2], [a2, b2, b2, A2], [a2, b2, b2]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).
G001 — Use the complete finite grammar: nonterminals=[S2, A2], terminals=[a2, b2], start=S2, productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a2, b2, b2], supplied forms=[[S2], [a2, A2], [a2, b2, A2], [a2, b2, b2, A2], [a2, b2, b2]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=cfg-membership-witness;original snapshot and mathematics retained
After
explicit ordered source output fields=[word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate]
Conditions: none
Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL production index=0;lhs=S2,rhs=[a2, A2];N=[S2, A2],T=[a2, b2],allow_epsilon=Yes
After
lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained
Conditions: none
Each production has one nonterminal on the left.Check every right-hand symbol and the explicit epsilon convention.Duplicate production positions may exist;sorting them would change first-witness indices.
Step identifiers and dependencies
Rule: cfg_each_original_production_single_nonterminal_legal_rhs · Uses: G001, S001
S003 · implies
Before
ORIGINAL production index=1;lhs=A2,rhs=[b2, A2];N=[S2, A2],T=[a2, b2],allow_epsilon=Yes
After
lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained
Conditions: none
Each production has one nonterminal on the left.Check every right-hand symbol and the explicit epsilon convention.Duplicate production positions may exist;sorting them would change first-witness indices.
Step identifiers and dependencies
Rule: cfg_each_original_production_single_nonterminal_legal_rhs · Uses: G001, S002
S004 · implies
Before
ORIGINAL production index=2;lhs=A2,rhs=[];N=[S2, A2],T=[a2, b2],allow_epsilon=Yes
After
lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained
Conditions: none
Each production has one nonterminal on the left.Check every right-hand symbol and the explicit epsilon convention.Duplicate production positions may exist;sorting them would change first-witness indices.
Step identifiers and dependencies
Rule: cfg_each_original_production_single_nonterminal_legal_rhs · Uses: G001, S003
S005 · implies
Before
ORIGINAL unique disjoint N=[S2, A2],T=[a2, b2];start=S2;complete productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]];allow_epsilon=Yes
After
start belongs to nonterminals;finite grammar qualified;NO derivation,membership,regularity,unambiguity or PDA equivalence inferred
Conditions: none
Grammar qualification alone supplies no derivation for a word and no regularity or unambiguity claim.Retain both alphabet orders.
Step identifiers and dependencies
Rule: cfg_qualified_alphabets_start_not_language_property · Uses: G001, S004
S006 · implies
Before
ORIGINAL forms=[[S2], [a2, A2], [a2, b2, A2], [a2, b2, b2, A2], [a2, b2, b2]];actual grammar start=S2
After
first form exactly [start]=Yes;empty forms or wrong start fail even if the last form is terminal
Conditions: none
A derivation must actually start at[start];a terminal final form alone is insufficient.Empty forms are not an epsilon derivation:epsilon is an actual empty-string form[] requiring a legal chain.
Step identifiers and dependencies
Rule: cfg_derivation_actual_start_required · Uses: G001, S005
S007 · implies
Before
ORIGINAL transition index=0;from=[S2];to=[a2, A2];original productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]]
After
FIRST matching [position,production_index,lhs,rhs]=[0, 0, S2, [a2, A2]];replace EXACTLY ONE occurrence,retaining both surrounding substrings
Conditions: none
Scan original positions and then original production indices for the first actual replacement.Replace one occurrence and preserve both surrounding strings,not every matching symbol.On failure retain prior partial witnesses and stop.
Step identifiers and dependencies
Rule: cfg_each_transition_first_position_then_original_production · Uses: G001, S006
S008 · implies
Before
ORIGINAL transition index=1;from=[a2, A2];to=[a2, b2, A2];original productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]]
After
FIRST matching [position,production_index,lhs,rhs]=[1, 1, A2, [b2, A2]];replace EXACTLY ONE occurrence,retaining both surrounding substrings
Conditions: none
Scan original positions and then original production indices for the first actual replacement.Replace one occurrence and preserve both surrounding strings,not every matching symbol.On failure retain prior partial witnesses and stop.
Step identifiers and dependencies
Rule: cfg_each_transition_first_position_then_original_production · Uses: G001, S007
S009 · implies
Before
ORIGINAL transition index=2;from=[a2, b2, A2];to=[a2, b2, b2, A2];original productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]]
After
FIRST matching [position,production_index,lhs,rhs]=[2, 1, A2, [b2, A2]];replace EXACTLY ONE occurrence,retaining both surrounding substrings
Conditions: none
Scan original positions and then original production indices for the first actual replacement.Replace one occurrence and preserve both surrounding strings,not every matching symbol.On failure retain prior partial witnesses and stop.
Step identifiers and dependencies
Rule: cfg_each_transition_first_position_then_original_production · Uses: G001, S008
S010 · implies
Before
ORIGINAL transition index=3;from=[a2, b2, b2, A2];to=[a2, b2, b2];original productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]]
After
FIRST matching [position,production_index,lhs,rhs]=[3, 2, A2, []];replace EXACTLY ONE occurrence,retaining both surrounding substrings
Conditions: none
Scan original positions and then original production indices for the first actual replacement.Replace one occurrence and preserve both surrounding strings,not every matching symbol.On failure retain prior partial witnesses and stop.
Step identifiers and dependencies
Rule: cfg_each_transition_first_position_then_original_production · Uses: G001, S009
S011 · implies
Before
ORIGINAL actual forms=[[S2], [a2, A2], [a2, b2, A2], [a2, b2, b2, A2], [a2, b2, b2]];accepted prior witnesses=[[0, 0, S2, [a2, A2]], [1, 1, A2, [b2, A2]], [2, 1, A2, [b2, A2]], [3, 2, A2, []]]
After
valid_derivation=Yes;this is separate from whether the actual final form contains only terminals
Conditions: none
Validity concerns the complete start-to-form chain.A terminal string after a failed step cannot repair it.A zero-step[start] derivation can be valid without being a terminal word.
Step identifiers and dependencies
Rule: cfg_partial_trace_validity_not_final_terminal_flag · Uses: G001, S010
S012 · implies
Before
ORIGINAL word=[a2, b2, b2];actual last form=[a2, b2, b2];valid chain=Yes
After
terminal_word=Yes;last equals original word=Yes;membership certificate=Yes
Conditions: none
Membership requires a valid chain from the start,a terminal-only word and exact final equality.An epsilon witness must actually derive[];an empty language is not a language containing epsilon.
Step identifiers and dependencies
Rule: cfg_membership_three_obligations_actual_terminal_target · Uses: G001, S011
Result
([a2, b2, b2], [[0, 0, S2, [a2, A2]], [1, 1, A2, [b2, A2]], [2, 1, A2, [b2, A2]], [3, 2, A2, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.