Original question and givens
The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {11, 17} an edge in E?
G001 — The finite simple undirected graph has V={11, 12, 13, 14, 15, 16} and E={{11, 12}, {11, 13}, {11, 14}, {11, 15}, {11, 16}, {12, 13}, {12, 14}, {12, 15}, {12, 16}, {13, 14}, {13, 15}, {13, 16}, {14, 15}, {14, 16}, {15, 16}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {11, 17} an edge in E?
Step-by-step derivation
S001 · implies
Before
source roles = [qf-simple-graph-edge-membership, [E, V, operation, u, v]]
After
derived = [edge_membership, no-invented-domain]
Conditions: none
Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.
Step identifiers and dependencies
Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001
S002 · implies
Before
source roles = [LDL graph vertices ORIGINAL, 100, [11, 12, 13, 14, 15, 16]]
After
derived = [[11, 12, 13, 14, 15, 16]]
Conditions: none
Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits.
Step identifiers and dependencies
Rule: Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort · Uses: G001, S001
S003 · implies
Before
source roles = [LDL raw edges ORIGINAL, 0, [11, 12]]
After
derived = [[11, 12]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S002
S004 · implies
Before
source roles = [LDL raw edges ORIGINAL, 1, [11, 13]]
After
derived = [[11, 13]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S003
S005 · implies
Before
source roles = [LDL raw edges ORIGINAL, 2, [11, 14]]
After
derived = [[11, 14]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S004
S006 · implies
Before
source roles = [LDL raw edges ORIGINAL, 3, [11, 15]]
After
derived = [[11, 15]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S005
S007 · implies
Before
source roles = [LDL raw edges ORIGINAL, 4, [11, 16]]
After
derived = [[11, 16]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S006
S008 · implies
Before
source roles = [LDL raw edges ORIGINAL, 5, [12, 13]]
After
derived = [[12, 13]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S007
S009 · implies
Before
source roles = [LDL raw edges ORIGINAL, 6, [12, 14]]
After
derived = [[12, 14]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S008
S010 · implies
Before
source roles = [LDL raw edges ORIGINAL, 7, [12, 15]]
After
derived = [[12, 15]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S009
S011 · implies
Before
source roles = [LDL raw edges ORIGINAL, 8, [12, 16]]
After
derived = [[12, 16]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S010
S012 · implies
Before
source roles = [LDL raw edges ORIGINAL, 9, [13, 14]]
After
derived = [[13, 14]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S011
S013 · implies
Before
source roles = [LDL raw edges ORIGINAL, 10, [13, 15]]
After
derived = [[13, 15]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S012
S014 · implies
Before
source roles = [LDL raw edges ORIGINAL, 11, [13, 16]]
After
derived = [[13, 16]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S013
S015 · implies
Before
source roles = [LDL raw edges ORIGINAL, 12, [14, 15]]
After
derived = [[14, 15]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S014
S016 · implies
Before
source roles = [LDL raw edges ORIGINAL, 13, [14, 16]]
After
derived = [[14, 16]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S015
S017 · implies
Before
source roles = [LDL raw edges ORIGINAL, 14, [15, 16]]
After
derived = [[15, 16]]
Conditions: none
BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded.
Step identifiers and dependencies
Rule: Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup · Uses: G001, S016
S018 · implies
Before
source roles = [[11, 12, 13, 14, 15, 16], [[11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [12, 13], [12, 14], [12, 15], [12, 16], [13, 14], [13, 15], [13, 16], [14, 15], [14, 16], [15, 16]], No]
After
derived = [[[11, 12], [11, 13], [11, 14], [11, 15], [11, 16], [12, 13], [12, 14], [12, 15], [12, 16], [13, 14], [13, 15], [13, 16], [14, 15], [14, 16], [15, 16]], [[11, [12, 13, 14, 15, 16]], [12, [11, 13, 14, 15, 16]], [13, [11, 12, 14, 15, 16]], [14, [11, 12, 13, 15, 16]], [15, [11, 12, 13, 14, 16]], [16, [11, 12, 13, 14, 15]]]]
Conditions: none
Type the WHOLE raw edge list,then normalize undirected endpoints/deduplicate;reversed duplicates are allowed.INPUT edges have no loops/outside endpoints,while isolated vertices stay in V.TREE raw E does NOT acquire LDL’s200-entry limit.
Step identifiers and dependencies
Rule: Graph_WHOLE_simple_undirected_edges_normalized_NO_loops_or_outside · Uses: G001, S017
S019 · implies
Before
source roles = [11, 17]
After
derived = [[11, 17], No]
Conditions: none
An undirected QUERY ignores endpoint order.A same-endpoint or outside-vertex query is a LEGAL No;do not confuse it with INPUT edge qualification.
Step identifiers and dependencies
Rule: Graph_EDGE_query_outside_or_same_endpoint_LEGAL_No · Uses: G001, S018