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Question 8156: Whole Candidate Inverse Type Check and Two Independent Identities

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0015-08156 · q-20261012-advance-20261017-r02-inverse-function-0031

Original question and givens

Use the complete total map Fin(2)={0,…,1}→Fin(2) with images=[1, 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(2)→Fin(2) has g=[0, 1]. 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(2)={0,…,1}→Fin(2) with images=[1, 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(2)→Fin(2) has g=[0, 1]. 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.

Step-by-step derivation

S001 · implies

Before

n=2;m=2;images=[1, 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 dependencies

Rule: inverse_typed_map_scope · Uses: G001

S002 · implies

Before

target=0;original images=[1, 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.

Step identifiers and dependencies

Rule: inverse_complete_target_fiber · Uses: G001, S001

S003 · implies

Before

target=1;original images=[1, 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 dependencies

Rule: inverse_complete_target_fiber · Uses: G001, S002

S004 · implies

Before

n=2;m=2;fibers=[[1], [0]]

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.

Step identifiers and dependencies

Rule: inverse_bijection_gate · Uses: G001, S003

S005 · implies

Before

candidate g=[0, 1];target Fin(2)

After

mask=[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 dependencies

Rule: whole_inverse_candidate_type_guard · Uses: G001, S004

S006 · implies

Before

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

After

g(f(input))=1;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 dependencies

Rule: inverse_composition_lookup · Uses: G001, S005

S007 · implies

Before

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

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 dependencies

Rule: inverse_composition_lookup · Uses: G001, S006

S008 · implies

Before

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

Rule: inverse_composition_lookup · Uses: G001, S007

S009 · implies

Before

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

Rule: inverse_composition_lookup · Uses: G001, S008

S010 · implies

Before

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

Rule: inverse_two_identities_separate · Uses: G001, S009

Reverse checks

  • 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], [Yes, Yes], Yes, [1, 0], [1, 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: 66d11fef88f096843be00f26c9b903ec51d958d6ad97838dd9bd17c77427d868
  • candidate record SHA-256: c78de8354122c139fbed740b6bf14d904c533dd86784f2b547bfcba95148241d
  • semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18
  • question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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