# Question 8159: Whole Candidate Inverse Type Check and Two Independent Identities

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0015-08159 · q-20261012-advance-20261017-r02-inverse-function-0034



## Original question and givens

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 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. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.


- G001 — Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 0, 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. Candidate g:Fin(3)→Fin(5) has g=[0, 2, 4]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.





## Definition and scope

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.


- [mko-inverse-function](https://ocme.evemisslab.com/en/mko/mko-inverse-function/)





## Step-by-step derivation

S001 · impliesBefore

n=5;m=3;images=[1, 0, 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.

Step identifiers and dependenciesRule: inverse_typed_map_scope · Uses: G001





S002 · impliesBefore

target=0;original images=[1, 0, 0, 0, 0]

After

fiber=[1, 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.

Step identifiers and dependenciesRule: inverse_complete_target_fiber · Uses: G001, S001





S003 · impliesBefore

target=1;original images=[1, 0, 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.

Step identifiers and dependenciesRule: inverse_complete_target_fiber · Uses: G001, S002





S004 · impliesBefore

target=2;original images=[1, 0, 0, 0, 0]

After

fiber=[];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.

Step identifiers and dependenciesRule: inverse_complete_target_fiber · Uses: G001, S003





S005 · impliesBefore

n=5;m=3;fibers=[[1, 2, 3, 4], [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.

Step identifiers and dependenciesRule: inverse_bijection_gate · Uses: G001, S004





S006 · impliesBefore

candidate g=[0, 2, 4];target Fin(5)

After

mask=[Yes, Yes, Yes];whole table eligible=Yes

Conditions: none

Every candidate cell must be typed,including cells not reached by f.If any cell is invalid,neither composition is evaluated;do not skip that cell.

Step identifiers and dependenciesRule: whole_inverse_candidate_type_guard · Uses: G001, S005





S007 · impliesBefore

gof:input=0;f(input)=1;g=[0, 2, 4]

After

g(f(input))=2;identity target=0;matches=No

Conditions: none

Follow f to a codomain label,then read the typed g table.This direction has a different domain from the other composition.

Step identifiers and dependenciesRule: inverse_composition_lookup · Uses: G001, S006





S008 · impliesBefore

gof:input=1;f(input)=0;g=[0, 2, 4]

After

g(f(input))=0;identity target=1;matches=No

Conditions: none

Follow f to a codomain label,then read the typed g table.This direction has a different domain from the other composition.

Step identifiers and dependenciesRule: inverse_composition_lookup · Uses: G001, S007





S009 · impliesBefore

gof:input=2;f(input)=0;g=[0, 2, 4]

After

g(f(input))=0;identity target=2;matches=No

Conditions: none

Follow f to a codomain label,then read the typed g table.This direction has a different domain from the other composition.

Step identifiers and dependenciesRule: inverse_composition_lookup · Uses: G001, S008





S010 · impliesBefore

gof:input=3;f(input)=0;g=[0, 2, 4]

After

g(f(input))=0;identity target=3;matches=No

Conditions: none

Follow f to a codomain label,then read the typed g table.This direction has a different domain from the other composition.

Step identifiers and dependenciesRule: inverse_composition_lookup · Uses: G001, S009





S011 · impliesBefore

gof:input=4;f(input)=0;g=[0, 2, 4]

After

g(f(input))=0;identity target=4;matches=No

Conditions: none

Follow f to a codomain label,then read the typed g table.This direction has a different domain from the other composition.

Step identifiers and dependenciesRule: inverse_composition_lookup · Uses: G001, S010





S012 · impliesBefore

fog:input=0;g(input)=0;f=[1, 0, 0, 0, 0]

After

f(g(input))=1;identity target=0;matches=No

Conditions: none

Start the reverse composition independently at every original codomain target;g∘f values cannot replace f∘g values.

Step identifiers and dependenciesRule: inverse_composition_lookup · Uses: G001, S011





S013 · impliesBefore

fog:input=1;g(input)=2;f=[1, 0, 0, 0, 0]

After

f(g(input))=0;identity target=1;matches=No

Conditions: none

Start the reverse composition independently at every original codomain target;g∘f values cannot replace f∘g values.

Step identifiers and dependenciesRule: inverse_composition_lookup · Uses: G001, S012





S014 · impliesBefore

fog:input=2;g(input)=4;f=[1, 0, 0, 0, 0]

After

f(g(input))=0;identity target=2;matches=No

Conditions: none

Start the reverse composition independently at every original codomain target;g∘f values cannot replace f∘g values.

Step identifiers and dependenciesRule: inverse_composition_lookup · Uses: G001, S013





S015 · impliesBefore

gof=[2, 0, 0, 0, 0];fog=[1, 0, 0]

After

left identity=No;right identity=No

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.

Step identifiers and dependenciesRule: inverse_two_identities_separate · Uses: G001, S014









## Reverse checks


- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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.





## Result

([1, 0, 0, 0, 0], [Yes, Yes, Yes], Yes, [2, 0, 0, 0, 0], [1, 0, 0], No, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: a4d832553a120e670a5ec69bc4f328fe4e90a78b4a571867cd6aa7d1a247e53c
- candidate record SHA-256: 393ee60359489a9e9f2632203575946a72b76e3a0b3303e3c77d4474398e6e85
- semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18
- question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r02-inverse-function-0034.json)