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Question 32795: 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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R0064-32795 · q-20261004-advance-20261005-r04-inverse-function-0020

Original question and givens

The declared total function f:A→B has A={-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41}, and table {(-21, 41), (-20, 1), (-19, 3), (-18, 5), (-17, 7), (-16, 9), (-15, 11), (-14, 13), (-13, 15), (-12, 17), (-11, 19), (-10, 21), (-9, 23), (-8, 25), (-7, 27), (-6, 29), (-5, 31), (-4, 33), (-3, 35), (-2, 37), (-1, 39)}. 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={-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41}, and table {(-21, 41), (-20, 1), (-19, 3), (-18, 5), (-17, 7), (-16, 9), (-15, 11), (-14, 13), (-13, 15), (-12, 17), (-11, 19), (-10, 21), (-9, 23), (-8, 25), (-7, 27), (-6, 29), (-5, 31), (-4, 33), (-3, 35), (-2, 37), (-1, 39)}. 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, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

After

derived = [[-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

After

derived = [[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41]]

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, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, -21, 41, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -20, 1, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -19, 3, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -18, 5, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -17, 7, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -16, 9, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -15, 11, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -14, 13, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -13, 15, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -12, 17, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -11, 19, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -10, 21, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -9, 23, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, -8, 25, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, S017

S019 · implies

Before

source roles = [ORIGINAL f, 14, -7, 27, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, S018

S020 · implies

Before

source roles = [ORIGINAL f, 15, -6, 29, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, S019

S021 · implies

Before

source roles = [ORIGINAL f, 16, -5, 31, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, S020

S022 · implies

Before

source roles = [ORIGINAL f, 17, -4, 33, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, S021

S023 · implies

Before

source roles = [ORIGINAL f, 18, -3, 35, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, S022

S024 · implies

Before

source roles = [ORIGINAL f, 19, -2, 37, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, S023

S025 · implies

Before

source roles = [ORIGINAL f, 20, -1, 39, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41]]

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, S024

S026 · implies

Before

source roles = [ORIGINAL f, -21, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, -20, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, -19, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, -18, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, -17, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, -16, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S030

S032 · implies

Before

source roles = [ORIGINAL f, -15, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S031

S033 · implies

Before

source roles = [ORIGINAL f, -14, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S032

S034 · implies

Before

source roles = [ORIGINAL f, -13, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S033

S035 · implies

Before

source roles = [ORIGINAL f, -12, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S034

S036 · implies

Before

source roles = [ORIGINAL f, -11, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S035

S037 · implies

Before

source roles = [ORIGINAL f, -10, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S036

S038 · implies

Before

source roles = [ORIGINAL f, -9, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S037

S039 · implies

Before

source roles = [ORIGINAL f, -8, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S038

S040 · implies

Before

source roles = [ORIGINAL f, -7, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S039

S041 · implies

Before

source roles = [ORIGINAL f, -6, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S040

S042 · implies

Before

source roles = [ORIGINAL f, -5, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S041

S043 · implies

Before

source roles = [ORIGINAL f, -4, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S042

S044 · implies

Before

source roles = [ORIGINAL f, -3, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S043

S045 · implies

Before

source roles = [ORIGINAL f, -2, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S044

S046 · implies

Before

source roles = [ORIGINAL f, -1, [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S045

S047 · implies

Before

source roles = [ORIGINAL f, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, 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, 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, S046

S048 · implies

Before

source roles = [ORIGINAL f, 1, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[-20], 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, S047

S049 · implies

Before

source roles = [ORIGINAL f, 3, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[-19], 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, S048

S050 · implies

Before

source roles = [ORIGINAL f, 5, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[-18], 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, S049

S051 · implies

Before

source roles = [ORIGINAL f, 7, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[-17], 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, S050

S052 · implies

Before

source roles = [ORIGINAL f, 9, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[-16], 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, S051

S053 · implies

Before

source roles = [ORIGINAL f, 11, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[-15], 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, S052

S054 · implies

Before

source roles = [ORIGINAL f, 13, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[-14], 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, S053

S055 · implies

Before

source roles = [ORIGINAL f, 15, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S054

S056 · implies

Before

source roles = [ORIGINAL f, 17, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S055

S057 · implies

Before

source roles = [ORIGINAL f, 19, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S056

S058 · implies

Before

source roles = [ORIGINAL f, 21, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S057

S059 · implies

Before

source roles = [ORIGINAL f, 23, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S058

S060 · implies

Before

source roles = [ORIGINAL f, 25, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S059

S061 · implies

Before

source roles = [ORIGINAL f, 27, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S060

S062 · implies

Before

source roles = [ORIGINAL f, 29, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S061

S063 · implies

Before

source roles = [ORIGINAL f, 31, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S062

S064 · implies

Before

source roles = [ORIGINAL f, 33, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S063

S065 · implies

Before

source roles = [ORIGINAL f, 35, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S064

S066 · implies

Before

source roles = [ORIGINAL f, 37, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S065

S067 · implies

Before

source roles = [ORIGINAL f, 39, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

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, S066

S068 · implies

Before

source roles = [ORIGINAL f, 41, [-21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[-21], 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, S067

S069 · implies

Before

source roles = [ORIGINAL f, [-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41], [[-20], [-19], [-18], [-17], [-16], [-15], [-14], [-13], [-12], [-11], [-10], [-9], [-8], [-7], [-6], [-5], [-4], [-3], [-2], [-1], [-21]]]

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, S068

S070 · implies

Before

source roles = [[-21, -20, -19, -18, -17, -16, -15, -14, -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, 27, 29, 31, 33, 35, 37, 39, 41], [[-21, 41], [-20, 1], [-19, 3], [-18, 5], [-17, 7], [-16, 9], [-15, 11], [-14, 13], [-13, 15], [-12, 17], [-11, 19], [-10, 21], [-9, 23], [-8, 25], [-7, 27], [-6, 29], [-5, 31], [-4, 33], [-3, 35], [-2, 37], [-1, 39]]]

After

derived = [[[1, -20], [3, -19], [5, -18], [7, -17], [9, -16], [11, -15], [13, -14], [15, -13], [17, -12], [19, -11], [21, -10], [23, -9], [25, -8], [27, -7], [29, -6], [31, -5], [33, -4], [35, -3], [37, -2], [39, -1], [41, -21]], {(1, -20), (3, -19), (5, -18), (7, -17), (9, -16), (11, -15), (13, -14), (15, -13), (17, -12), (19, -11), (21, -10), (23, -9), (25, -8), (27, -7), (29, -6), (31, -5), (33, -4), (35, -3), (37, -2), (39, -1), (41, -21)}]

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, S069

Reverse checks

  • S070 · 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.
  • S069 · 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.
  • S068 · 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.
  • S067 · 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.
  • S066 · 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.
  • S065 · 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.
  • S064 · 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.
  • S063 · 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.
  • S062 · 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.
  • S061 · 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.
  • S060 · 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.
  • S059 · 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.
  • S058 · 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.
  • S057 · 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.
  • S056 · 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.
  • S055 · 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.
  • S054 · 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.
  • S053 · 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.
  • S052 · 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.
  • S051 · 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.
  • S050 · 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.
  • S049 · 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.
  • S048 · 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.
  • S047 · 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.
  • 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, -20), (3, -19), (5, -18), (7, -17), (9, -16), (11, -15), (13, -14), (15, -13), (17, -12), (19, -11), (21, -10), (23, -9), (25, -8), (27, -7), (29, -6), (31, -5), (33, -4), (35, -3), (37, -2), (39, -1), (41, -21)}

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: 5bdccc64baa9dcea6a14a9388398a7861b6472adb629ff58cdd3509debe7e2e5
  • candidate record SHA-256: f195562a19229ecc3144841ba0225abc00bffdc2c1aceed16fe7535408d287b4
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON