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R0015-08176 · q-20261012-advance-20261017-r02-inverse-function-0051
Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, -1]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.
G001 — Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Ordered queries=[0, -1]. Give (images, bijectivity, two codomain type guards, two inverse values). If f is not bijective use not-an-inverse-function for each value; otherwise an untyped query yields outside-codomain.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
n=0;m=0;images=[]
After
complete typed total map;inverse is not reciprocal
Conditions: none
Validate the whole original total table first.Equal finite cardinalities do not ensure this particular table is bijective,and a zero function value does not prohibit an inverse.
Rule: inverse_typed_map_scope · Uses: G001
Before
n=0;m=0;fibers=[]
After
bijective=Yes
Conditions: none
For the already total map,exactly one preimage per codomain target is equivalent to bijectivity.The empty-to-empty map qualifies;checking n=m alone is insufficient.
Rule: inverse_bijection_gate · Uses: G001, S001
Before
query0=0;bijective=Yes;in Fin(0)=No
After
inverse value/status=outside the codomain
Conditions: none
Retain the query type mask separately.A nonbijective original map takes precedence and yields not-an-inverse-function.Read the unique preimage only if the inverse exists and the target is typed.
Rule: inverse_query_guard_precedence · Uses: G001, S002
Before
query1=-1;bijective=Yes;in Fin(0)=No
After
inverse value/status=outside the codomain
Conditions: none
Retain the query type mask separately.A nonbijective original map takes precedence and yields not-an-inverse-function.Read the unique preimage only if the inverse exists and the target is typed.
Rule: inverse_query_guard_precedence · 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.([], Yes, [No, No], [outside-codomain, outside-codomain])
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: a155203dd60e3f0dc54f4d511273aa9e5a9989171c0742bdc276c5238b70fc9ecandidate record SHA-256: ce311af57b7bc3fc09304b73a44cea2ed7aed71420c48b33d88e04d46324a622semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED