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R0015-08188 · q-20261012-advance-20261017-r02-inverse-function-0063
Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] 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, 2]. 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(5)={0,…,4}→Fin(3) with images=[2, 1, 0, 0, 0] 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, 2]. 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=5;m=3;images=[2, 1, 0, 0, 0]
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
target=0;original images=[2, 1, 0, 0, 0]
After
fiber=[2, 3, 4];unique=No
Conditions: none
Scan every original input and list all preimages of this declared target.An empty or multiple-preimage fiber cannot supply a cell of a total inverse.
Rule: inverse_complete_target_fiber · Uses: G001, S001
Before
target=1;original images=[2, 1, 0, 0, 0]
After
fiber=[1];unique=Yes
Conditions: none
Scan every original input and list all preimages of this declared target.An empty or multiple-preimage fiber cannot supply a cell of a total inverse.
Rule: inverse_complete_target_fiber · Uses: G001, S002
Before
target=2;original images=[2, 1, 0, 0, 0]
After
fiber=[0];unique=Yes
Conditions: none
Scan every original input and list all preimages of this declared target.An empty or multiple-preimage fiber cannot supply a cell of a total inverse.
Rule: inverse_complete_target_fiber · Uses: G001, S003
Before
n=5;m=3;fibers=[[2, 3, 4], [1], [0]]
After
bijective=No
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, S004
Before
query0=0;bijective=No;in Fin(3)=Yes
After
inverse value/status=does not define an inverse function
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, S005
Before
query1=2;bijective=No;in Fin(3)=Yes
After
inverse value/status=does not define an inverse function
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, S006
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([2, 1, 0, 0, 0], No, [Yes, Yes], [not-an-inverse-function, not-an-inverse-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: 9a44fdcd21d28f52634f9dae13834e773bccb66c0f87a66c386a33cea941ae5ecandidate record SHA-256: 9b93d003311ccee33934e973871294face9c0dd6323423f226a7f783b7d3b482semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED