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Question 32787: Inverse Table of a Finite Bijection

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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R0064-32787 · q-20261004-advance-20261005-r04-inverse-function-0012

Original question and givens

The declared total function f:A→B has A={-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}, and table {(-13, 25), (-12, 1), (-11, 3), (-10, 5), (-9, 7), (-8, 9), (-7, 11), (-6, 13), (-5, 15), (-4, 17), (-3, 19), (-2, 21), (-1, 23)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

  • G001 — The declared total function f:A→B has A={-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25}, and table {(-13, 25), (-12, 1), (-11, 3), (-10, 5), (-9, 7), (-8, 9), (-7, 11), (-6, 13), (-5, 15), (-4, 17), (-3, 19), (-2, 21), (-1, 23)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

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

source roles = [qf-inverse-function-table, inverse_table, no-required-domain, []]

After

derived = [[complete_inverse_pairset_STRING]]

Conditions: none

Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status.

Step identifiers and dependencies

Rule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001

S002 · implies

Before

source roles = [ORIGINAL f domain, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

After

derived = [[-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

Conditions: none

RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes.

Step identifiers and dependencies

Rule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S001

S003 · implies

Before

source roles = [ORIGINAL f codomain, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

Conditions: none

RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes.

Step identifiers and dependencies

Rule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S002

S004 · implies

Before

source roles = [ORIGINAL f, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

Conditions: none

FIRST type EVERY raw pair and enforce raw length≤200,THEN normalize identical rows as one ordered pair.Do not merge different outputs of one input.Empty domains do not waive RAW checks.

Step identifiers and dependencies

Rule: FiniteMap_RAW200_bound_BEFORE_identical_row_ASSET_normalization · Uses: G001, S003

S005 · implies

Before

source roles = [ORIGINAL f, 0, -13, 25, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S004

S006 · implies

Before

source roles = [ORIGINAL f, 1, -12, 1, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S005

S007 · implies

Before

source roles = [ORIGINAL f, 2, -11, 3, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S006

S008 · implies

Before

source roles = [ORIGINAL f, 3, -10, 5, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S007

S009 · implies

Before

source roles = [ORIGINAL f, 4, -9, 7, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S008

S010 · implies

Before

source roles = [ORIGINAL f, 5, -8, 9, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S009

S011 · implies

Before

source roles = [ORIGINAL f, 6, -7, 11, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S010

S012 · implies

Before

source roles = [ORIGINAL f, 7, -6, 13, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S011

S013 · implies

Before

source roles = [ORIGINAL f, 8, -5, 15, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S012

S014 · implies

Before

source roles = [ORIGINAL f, 9, -4, 17, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S013

S015 · implies

Before

source roles = [ORIGINAL f, 10, -3, 19, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S014

S016 · implies

Before

source roles = [ORIGINAL f, 11, -2, 21, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S015

S017 · implies

Before

source roles = [ORIGINAL f, 12, -1, 23, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S016

S018 · implies

Before

source roles = [ORIGINAL f, -13, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S017

S019 · implies

Before

source roles = [ORIGINAL f, -12, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S018

S020 · implies

Before

source roles = [ORIGINAL f, -11, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S019

S021 · implies

Before

source roles = [ORIGINAL f, -10, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S020

S022 · implies

Before

source roles = [ORIGINAL f, -9, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S021

S023 · implies

Before

source roles = [ORIGINAL f, -8, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S022

S024 · implies

Before

source roles = [ORIGINAL f, -7, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S023

S025 · implies

Before

source roles = [ORIGINAL f, -6, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S024

S026 · implies

Before

source roles = [ORIGINAL f, -5, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S025

S027 · implies

Before

source roles = [ORIGINAL f, -4, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S026

S028 · implies

Before

source roles = [ORIGINAL f, -3, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S027

S029 · implies

Before

source roles = [ORIGINAL f, -2, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S028

S030 · implies

Before

source roles = [ORIGINAL f, -1, [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S029

S031 · implies

Before

source roles = [ORIGINAL f, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], Yes]

Conditions: none

Permit lookups ONLY after BOTH all-row typing and all-input totality.Empty-set vacuity cannot conceal an invalid row.

Step identifiers and dependencies

Rule: FiniteMap_TOTAL_qualification_BEFORE_lookups_even_empty_domain · Uses: G001, S030

S032 · implies

Before

source roles = [ORIGINAL f, 1, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-12], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S031

S033 · implies

Before

source roles = [ORIGINAL f, 3, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-11], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S032

S034 · implies

Before

source roles = [ORIGINAL f, 5, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-10], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S033

S035 · implies

Before

source roles = [ORIGINAL f, 7, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-9], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S034

S036 · implies

Before

source roles = [ORIGINAL f, 9, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-8], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S035

S037 · implies

Before

source roles = [ORIGINAL f, 11, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-7], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S036

S038 · implies

Before

source roles = [ORIGINAL f, 13, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-6], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S037

S039 · implies

Before

source roles = [ORIGINAL f, 15, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-5], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S038

S040 · implies

Before

source roles = [ORIGINAL f, 17, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-4], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S039

S041 · implies

Before

source roles = [ORIGINAL f, 19, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-3], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S040

S042 · implies

Before

source roles = [ORIGINAL f, 21, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-2], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S041

S043 · implies

Before

source roles = [ORIGINAL f, 23, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-1], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S042

S044 · implies

Before

source roles = [ORIGINAL f, 25, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[-13], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S043

S045 · implies

Before

source roles = [ORIGINAL f, [-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [[-12], [-11], [-10], [-9], [-8], [-7], [-6], [-5], [-4], [-3], [-2], [-1], [-13]]]

After

derived = [Yes, Yes, Yes]

Conditions: none

Injective means every fiber≤1,surjective every fiber≥1,and BOTH mean bijective.Unattained B does not defeat injectivity.Empty→empty is bijective;empty→nonempty is only injective.

Step identifiers and dependencies

Rule: FiniteMap_injective_surjective_BOTH_needed_for_inverse · Uses: G001, S044

S046 · implies

Before

source roles = [[-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [[-13, 25], [-12, 1], [-11, 3], [-10, 5], [-9, 7], [-8, 9], [-7, 11], [-6, 13], [-5, 15], [-4, 17], [-3, 19], [-2, 21], [-1, 23]]]

After

derived = [[[1, -12], [3, -11], [5, -10], [7, -9], [9, -8], [11, -7], [13, -6], [15, -5], [17, -4], [19, -3], [21, -2], [23, -1], [25, -13]], {(1, -12), (3, -11), (5, -10), (7, -9), (9, -8), (11, -7), (13, -6), (15, -5), (17, -4), (19, -3), (21, -2), (23, -1), (25, -13)}]

Conditions: none

AFTER actual bijectivity,swap EVERY(x,y)to(y,x)and sort numericLEX for the COMPLETE B→A inverse table.The empty bijection has inverse∅;an arbitrary converse is not automatically an inverse function.

Step identifiers and dependencies

Rule: Inverse_TABLE_actual_bijection_complete_swapped_numeric_LEX_pairs · Uses: G001, S045

Reverse checks

  • S046 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S045 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S044 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S043 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S042 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S041 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S040 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S039 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S038 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S037 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S036 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S035 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S034 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S033 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S032 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S031 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S030 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S029 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S028 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S027 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S026 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S025 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S024 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S023 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S022 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S021 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S020 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S019 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S018 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S016 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S014 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S012 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S008 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S004 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S003 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

{(1, -12), (3, -11), (5, -10), (7, -9), (9, -8), (11, -7), (13, -6), (15, -5), (17, -4), (19, -3), (21, -2), (23, -1), (25, -13)}

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: 96feb00c15d15585a434bffc20549cb8e5e1dd6449f6cf3f9d96cbdfcf822c88
  • candidate record SHA-256: a1e26c4242d6a438e2778975d70ac42dd0a4677fd82bc62b36b6aedd2d878c90
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON