Question 32794: Inverse Table of a Finite Bijection

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0064-32794 · q-20261004-advance-20261005-r04-inverse-function-0019

Original question and givens

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

mko-inverse-function

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, [-20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

After

derived = [[-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]]

After

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

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

After

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

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, -20, 39, [-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]]

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, -19, 1, [-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]]

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, -18, 3, [-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]]

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, -17, 5, [-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]]

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, -16, 7, [-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]]

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, -15, 9, [-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]]

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, -14, 11, [-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]]

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, -13, 13, [-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]]

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, -12, 15, [-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]]

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, -11, 17, [-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]]

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, -10, 19, [-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]]

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, -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]]

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, -8, 23, [-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]]

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, -7, 25, [-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]]

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, -6, 27, [-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]]

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, -5, 29, [-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]]

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, -4, 31, [-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]]

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, -3, 33, [-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]]

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, -2, 35, [-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]]

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, -1, 37, [-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]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

After

derived = [[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], 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, S044

S046 · implies

Before

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

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, S045

S047 · implies

Before

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

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, S046

S048 · implies

Before

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

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, S047

S049 · implies

Before

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

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, S048

S050 · implies

Before

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

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, S049

S051 · implies

Before

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

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, S050

S052 · implies

Before

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

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, S051

S053 · implies

Before

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

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, S052

S054 · implies

Before

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

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, S053

S055 · implies

Before

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

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, S054

S056 · 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], [[-20, 39], [-19, 1], [-18, 3], [-17, 5], [-16, 7], [-15, 9], [-14, 11], [-13, 13], [-12, 15], [-11, 17], [-10, 19], [-9, 21], [-8, 23], [-7, 25], [-6, 27], [-5, 29], [-4, 31], [-3, 33], [-2, 35], [-1, 37]]]

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, S055

S057 · implies

Before

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

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, S056

S058 · implies

Before

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

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, S057

S059 · implies

Before

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

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, S058

S060 · implies

Before

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

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, S059

S061 · implies

Before

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

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, S060

S062 · implies

Before

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

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, S061

S063 · implies

Before

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

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, S062

S064 · implies

Before

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

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, S063

S065 · implies

Before

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

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, S064

S066 · implies

Before

source roles = [ORIGINAL f, [-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], [[-19], [-18], [-17], [-16], [-15], [-14], [-13], [-12], [-11], [-10], [-9], [-8], [-7], [-6], [-5], [-4], [-3], [-2], [-1], [-20]]]

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, S065

S067 · implies

Before

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

After

derived = [[[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, -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, -20)}]

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, S066

Reverse checks

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, -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, -20)}

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: 73186cc4fc1afb24c7a29b578897a519fe15e6abc41d7ae00da2e45d4678fc92
candidate record SHA-256: f5ea2c2c358635c31e4ef5ed5d3eb1fe26e96f3dc6ba3ad2a5dbbc4e8a5e6f5f
semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON