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R0015-08352 · q-20261012-advance-20261017-r03-relation-0077
The explicitly typed relation R⊆Fin(2)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).
G001 — The explicitly typed relation R⊆Fin(2)×Fin(0) has ordered pairs=[]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).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.
Before
types=(2,0);raw ordered pairs=[]
After
normalized relation=[]
Conditions: none
Validate both types of every ordered pair,then deduplicate identical pairs.Swapping coordinates forms the converse,not a sorting of one pair.A relation may be multivalued or partial.
Rule: typed_relation_set_normalization · Uses: G001
Before
source=0;pairs=[]
After
outgoing values=[];count=0
Conditions: none
Collect all distinct outgoing targets of every declared source;an absent source has an empty fiber.
Rule: relation_each_source_value_fiber · Uses: G001, S001
Before
source=1;pairs=[]
After
outgoing values=[];count=0
Conditions: none
Collect all distinct outgoing targets of every declared source;an absent source has an empty fiber.
Rule: relation_each_source_value_fiber · Uses: G001, S002
Before
source fiber sizes=[0, 0]
After
single-valued=Yes;exactly-one total=No;table/status=not a total function
Conditions: none
At most one value permits missing sources;exactly one makes a total function.On an empty source both universal conditions are true and the table is empty.Repeating a pair does not increase a fiber size.
Rule: partial_single_valued_not_total · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([2, 0], [], [0, 0], Yes, No, not-a-total-function)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 998fbf6604705d5425beb13b29ff2b7c306cee630e0c6127a1d0e3555708618dcandidate record SHA-256: d2063bc8b1f6aca7f820d72f37bb7bfc00c9320a163f257084d4b080df57308csemantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED