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R0015-08129 · q-20261012-advance-20261017-r02-inverse-function-0004
Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 1, 2] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.
G001 — Use the complete total map Fin(3)={0,…,2}→Fin(3) with images=[0, 1, 2] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, bijectivity, inverse table or not-an-inverse-function, g∘f values or not-evaluated, f∘g values or not-evaluated). Do not confuse inverse functions with reciprocal function values.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=3;m=3;images=[0, 1, 2]
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=[0, 1, 2]
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, S001
Before
target=1;original images=[0, 1, 2]
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=[0, 1, 2]
After
fiber=[2];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=3;m=3;fibers=[[0], [1], [2]]
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, S004
Before
unique fibers=[[0], [1], [2]]
After
inverse table=[0, 1, 2]
Conditions: none
Construct the inverse table in codomain order from each target's unique preimage.
Rule: construct_inverse_only_after_bijection · Uses: G001, S005
Before
gof:input=0;f(input)=0;g=[0, 1, 2]
After
g(f(input))=0;identity target=0;matches=Yes
Conditions: none
Follow f to a codomain label,then read the typed g table.This direction has a different domain from the other composition.
Rule: inverse_composition_lookup · Uses: G001, S006
Before
gof:input=1;f(input)=1;g=[0, 1, 2]
After
g(f(input))=1;identity target=1;matches=Yes
Conditions: none
Follow f to a codomain label,then read the typed g table.This direction has a different domain from the other composition.
Rule: inverse_composition_lookup · Uses: G001, S007
Before
gof:input=2;f(input)=2;g=[0, 1, 2]
After
g(f(input))=2;identity target=2;matches=Yes
Conditions: none
Follow f to a codomain label,then read the typed g table.This direction has a different domain from the other composition.
Rule: inverse_composition_lookup · Uses: G001, S008
Before
fog:input=0;g(input)=0;f=[0, 1, 2]
After
f(g(input))=0;identity target=0;matches=Yes
Conditions: none
Start the reverse composition independently at every original codomain target;g∘f values cannot replace f∘g values.
Rule: inverse_composition_lookup · Uses: G001, S009
Before
fog:input=1;g(input)=1;f=[0, 1, 2]
After
f(g(input))=1;identity target=1;matches=Yes
Conditions: none
Start the reverse composition independently at every original codomain target;g∘f values cannot replace f∘g values.
Rule: inverse_composition_lookup · Uses: G001, S010
Before
fog:input=2;g(input)=2;f=[0, 1, 2]
After
f(g(input))=2;identity target=2;matches=Yes
Conditions: none
Start the reverse composition independently at every original codomain target;g∘f values cannot replace f∘g values.
Rule: inverse_composition_lookup · Uses: G001, S011
Before
gof=[0, 1, 2];fog=[0, 1, 2]
After
left identity=Yes;right identity=Yes
Conditions: none
Check each identity over its own complete index set.A one-sided identity does not imply the other;conditions over an empty index set are true.
Rule: inverse_two_identities_separate · Uses: G001, S012
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([0, 1, 2], Yes, [0, 1, 2], [0, 1, 2], [0, 1, 2])
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: 030d45d5de383d1313e91237c594c60d961de2053ff1946b06ac9a43c54ca22bcandidate record SHA-256: 185b2ab6683be22e20c86659aec048703d0bbf8e85dc77f6c36b7acdc6f24cdbsemantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED