Original question and givens
The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {7, 12} an edge in E?
G001 — The finite simple undirected graph has V={7, 8, 9, 10, 11, 12} and E={{7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}}. There are no loops or multiple edges, and repeated representations of the same unordered edge count once. Is the candidate endpoint set {7, 12} 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, [7, 8, 9, 10, 11, 12]]
After
derived = [[7, 8, 9, 10, 11, 12]]
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, [7, 8]]
After
derived = [[7, 8]]
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, [8, 9]]
After
derived = [[8, 9]]
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, [9, 10]]
After
derived = [[9, 10]]
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, [10, 11]]
After
derived = [[10, 11]]
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, 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, S006
S008 · implies
Before
source roles = [[7, 8, 9, 10, 11, 12], [[7, 8], [8, 9], [9, 10], [10, 11], [11, 12]], No]
After
derived = [[[7, 8], [8, 9], [9, 10], [10, 11], [11, 12]], [[7, [8]], [8, [7, 9]], [9, [8, 10]], [10, [9, 11]], [11, [10, 12]], [12, [11]]]]
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, S007
S009 · implies
Before
source roles = [7, 12]
After
derived = [[7, 12], 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, S008