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R0015-08333 · q-20261012-advance-20261017-r03-relation-0058
The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1]]. 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], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).
G001 — The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1]]. 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], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).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=(3,3);raw ordered pairs=[[0, 0], [0, 1]]
After
normalized relation=[[0, 0], [0, 1]]
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
phase=converse;pair=[0, 0]
After
swapped=[0, 0]
Conditions: none
Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function.
Rule: relation_swap_ordered_coordinates · Uses: G001, S001
Before
phase=converse;pair=[0, 1]
After
swapped=[1, 0]
Conditions: none
Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function.
Rule: relation_swap_ordered_coordinates · Uses: G001, S002
Before
phase=double converse;pair=[0, 0]
After
swapped=[0, 0]
Conditions: none
Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function.
Rule: relation_swap_ordered_coordinates · Uses: G001, S003
Before
phase=double converse;pair=[1, 0]
After
swapped=[0, 1]
Conditions: none
Swap the ordered coordinates together with their source/target types.The converse relation is not automatically an inverse function.
Rule: relation_swap_ordered_coordinates · Uses: G001, S004
Before
old types=[3, 3];converse types=[3, 3];converse=[[0, 0], [1, 0]]
After
double converse=[[0, 0], [0, 1]];equal=Yes
Conditions: none
Swapping twice returns the original types and pair set even when the relation is not a function.
Rule: double_converse_typed_audit · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([3, 3], [3, 3], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes)
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: e37b794a8f80a01bbf445b568e99be3374441e746640ad15d0b8c6b0a2e09824candidate record SHA-256: 50cd76efbe16caf3fe2d25b61bd9c990704d5606a71bc3a5aff6a21612122f31semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED