Question 32740: Checking Both Inverse Identities

On this page

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

← Question derivation index · Original question batch

R0064-32740 · q-20261004-advance-20261005-r04-inverse-function-0065

Original question and givens

The declared total function f:A→B has A={-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}, and table {(-16, 31), (-15, 1), (-14, 3), (-13, 5), (-12, 7), (-11, 9), (-10, 11), (-9, 13), (-8, 15), (-7, 17), (-6, 19), (-5, 21), (-4, 23), (-3, 25), (-2, 27), (-1, 29)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -16)}. Check both identities to decide whether g is the inverse of f.

G001 — The declared total function f:A→B has A={-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}, and table {(-16, 31), (-15, 1), (-14, 3), (-13, 5), (-12, 7), (-11, 9), (-10, 11), (-9, 13), (-8, 15), (-7, 17), (-6, 19), (-5, 21), (-4, 23), (-3, 25), (-2, 27), (-1, 29)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -15), (3, -14), (5, -13), (7, -12), (9, -11), (11, -10), (13, -9), (15, -8), (17, -7), (19, -6), (21, -5), (23, -4), (25, -3), (27, -2), (29, -1), (31, -16)}. Check both identities to decide whether g is the inverse of f.

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-candidate, inverse_candidate, no-required-domain, []]

After

derived = [[BOTH_restorations_YesNo_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, [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

After

derived = [[-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]]

After

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

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

After

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

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, -16, 31, [-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]]

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, -15, 1, [-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]]

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, -14, 3, [-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]]

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, -13, 5, [-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]]

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, -12, 7, [-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]]

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, -11, 9, [-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]]

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, -10, 11, [-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]]

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, -9, 13, [-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]]

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, -8, 15, [-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]]

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, -7, 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]]

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, -6, 19, [-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]]

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, -5, 21, [-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]]

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, -4, 23, [-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]]

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, -3, 25, [-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]]

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, -2, 27, [-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]]

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, -1, 29, [-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]]

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

After

derived = [1, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S020

S022 · implies

Before

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

After

derived = [1, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S021

S023 · implies

Before

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

After

derived = [1, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S022

S024 · implies

Before

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

After

derived = [1, Yes]

Conditions: none

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

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S023

S025 · implies

Before

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

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

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

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

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

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

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

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

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

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

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

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

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

After

derived = [[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], 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, S036

S038 · implies

Before

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

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, S037

S039 · implies

Before

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

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, S038

S040 · implies

Before

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

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, S039

S041 · implies

Before

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

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, S040

S042 · implies

Before

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

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, S041

S043 · implies

Before

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

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, S042

S044 · implies

Before

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

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, S043

S045 · implies

Before

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

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, S044

S046 · implies

Before

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

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, S045

S047 · implies

Before

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

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, S046

S048 · implies

Before

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

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, S047

S049 · implies

Before

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

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, S048

S050 · implies

Before

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

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, S049

S051 · implies

Before

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

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, S050

S052 · implies

Before

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

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, S051

S053 · implies

Before

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

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, S052

S054 · implies

Before

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

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, S053

S055 · implies

Before

source roles = [CANDIDATE g domain, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31]]

After

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

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, S054

S056 · implies

Before

source roles = [CANDIDATE g codomain, [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

After

derived = [[-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, S055

S057 · implies

Before

source roles = [CANDIDATE g, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

After

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

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, S056

S058 · implies

Before

source roles = [CANDIDATE g, 0, 1, -15, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S057

S059 · implies

Before

source roles = [CANDIDATE g, 1, 3, -14, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S058

S060 · implies

Before

source roles = [CANDIDATE g, 2, 5, -13, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S059

S061 · implies

Before

source roles = [CANDIDATE g, 3, 7, -12, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S060

S062 · implies

Before

source roles = [CANDIDATE g, 4, 9, -11, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S061

S063 · implies

Before

source roles = [CANDIDATE g, 5, 11, -10, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S062

S064 · implies

Before

source roles = [CANDIDATE g, 6, 13, -9, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S063

S065 · implies

Before

source roles = [CANDIDATE g, 7, 15, -8, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S064

S066 · implies

Before

source roles = [CANDIDATE g, 8, 17, -7, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S065

S067 · implies

Before

source roles = [CANDIDATE g, 9, 19, -6, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S066

S068 · implies

Before

source roles = [CANDIDATE g, 10, 21, -5, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S067

S069 · implies

Before

source roles = [CANDIDATE g, 11, 23, -4, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S068

S070 · implies

Before

source roles = [CANDIDATE g, 12, 25, -3, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S069

S071 · implies

Before

source roles = [CANDIDATE g, 13, 27, -2, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S070

S072 · implies

Before

source roles = [CANDIDATE g, 14, 29, -1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S071

S073 · implies

Before

source roles = [CANDIDATE g, 15, 31, -16, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1]]

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, S072

S074 · implies

Before

source roles = [CANDIDATE g, 1, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S073

S075 · implies

Before

source roles = [CANDIDATE g, 3, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S074

S076 · implies

Before

source roles = [CANDIDATE g, 5, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S075

S077 · implies

Before

source roles = [CANDIDATE g, 7, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S076

S078 · implies

Before

source roles = [CANDIDATE g, 9, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S077

S079 · implies

Before

source roles = [CANDIDATE g, 11, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S078

S080 · implies

Before

source roles = [CANDIDATE g, 13, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S079

S081 · implies

Before

source roles = [CANDIDATE g, 15, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S080

S082 · implies

Before

source roles = [CANDIDATE g, 17, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S081

S083 · implies

Before

source roles = [CANDIDATE g, 19, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S082

S084 · implies

Before

source roles = [CANDIDATE g, 21, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S083

S085 · implies

Before

source roles = [CANDIDATE g, 23, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S084

S086 · implies

Before

source roles = [CANDIDATE g, 25, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S085

S087 · implies

Before

source roles = [CANDIDATE g, 27, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S086

S088 · implies

Before

source roles = [CANDIDATE g, 29, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S087

S089 · implies

Before

source roles = [CANDIDATE g, 31, [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S088

S090 · implies

Before

source roles = [CANDIDATE g, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

After

derived = [[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], 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, S089

S091 · implies

Before

source roles = [CANDIDATE g, -16, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

After

derived = [[31], 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, S090

S092 · implies

Before

source roles = [CANDIDATE g, -15, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S091

S093 · implies

Before

source roles = [CANDIDATE g, -14, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S092

S094 · implies

Before

source roles = [CANDIDATE g, -13, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S093

S095 · implies

Before

source roles = [CANDIDATE g, -12, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S094

S096 · implies

Before

source roles = [CANDIDATE g, -11, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S095

S097 · implies

Before

source roles = [CANDIDATE g, -10, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S096

S098 · implies

Before

source roles = [CANDIDATE g, -9, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S097

S099 · implies

Before

source roles = [CANDIDATE g, -8, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S098

S100 · implies

Before

source roles = [CANDIDATE g, -7, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S099

S101 · implies

Before

source roles = [CANDIDATE g, -6, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S100

S102 · implies

Before

source roles = [CANDIDATE g, -5, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

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, S101

S103 · implies

Before

source roles = [CANDIDATE g, -4, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

After

derived = [[23], 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, S102

S104 · implies

Before

source roles = [CANDIDATE g, -3, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

After

derived = [[25], 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, S103

S105 · implies

Before

source roles = [CANDIDATE g, -2, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

After

derived = [[27], 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, S104

S106 · implies

Before

source roles = [CANDIDATE g, -1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [[1, -15], [3, -14], [5, -13], [7, -12], [9, -11], [11, -10], [13, -9], [15, -8], [17, -7], [19, -6], [21, -5], [23, -4], [25, -3], [27, -2], [29, -1], [31, -16]]]

After

derived = [[29], 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, S105

S107 · implies

Before

source roles = [CANDIDATE g, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31], [-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1], [[31], [1], [3], [5], [7], [9], [11], [13], [15], [17], [19], [21], [23], [25], [27], [29]]]

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, S106

S108 · implies

Before

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

After

derived = [-16, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S107

S109 · implies

Before

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

After

derived = [-15, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S108

S110 · implies

Before

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

After

derived = [-14, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S109

S111 · implies

Before

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

After

derived = [-13, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S110

S112 · implies

Before

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

After

derived = [-12, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S111

S113 · implies

Before

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

After

derived = [-11, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S112

S114 · implies

Before

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

After

derived = [-10, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S113

S115 · implies

Before

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

After

derived = [-9, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S114

S116 · implies

Before

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

After

derived = [-8, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S115

S117 · implies

Before

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

After

derived = [-7, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S116

S118 · implies

Before

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

After

derived = [-6, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S117

S119 · implies

Before

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

After

derived = [-5, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S118

S120 · implies

Before

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

After

derived = [-4, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S119

S121 · implies

Before

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

After

derived = [-3, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S120

S122 · implies

Before

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

After

derived = [-2, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S121

S123 · implies

Before

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

After

derived = [-1, Yes]

Conditions: none

Apply f into B,THEN the typed total g back into A.Check restoration for EVERY original input without omitting failures.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_g_after_f_restores_ORIGINAL_A · Uses: G001, S122

S124 · implies

Before

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

After

derived = [1, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S123

S125 · implies

Before

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

After

derived = [3, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S124

S126 · implies

Before

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

After

derived = [5, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S125

S127 · implies

Before

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

After

derived = [7, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S126

S128 · implies

Before

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

After

derived = [9, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S127

S129 · implies

Before

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

After

derived = [11, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S128

S130 · implies

Before

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

After

derived = [13, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S129

S131 · implies

Before

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

After

derived = [15, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S130

S132 · implies

Before

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

After

derived = [17, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S131

S133 · implies

Before

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

After

derived = [19, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S132

S134 · implies

Before

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

After

derived = [21, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S133

S135 · implies

Before

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

After

derived = [23, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S134

S136 · implies

Before

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

After

derived = [25, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S135

S137 · implies

Before

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

After

derived = [27, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S136

S138 · implies

Before

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

After

derived = [29, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S137

S139 · implies

Before

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

After

derived = [31, Yes]

Conditions: none

In the OTHER direction,start at EVERY original B label,apply g then f,and restore itself.Retain this check separately.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_EACH_f_after_g_restores_ORIGINAL_B · Uses: G001, S138

S140 · implies

Before

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

After

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

Conditions: none

AFTER f is bijective and g is fully typed/total,BOTH complete restoration inventories must pass.A qualified wrong inverse is a legal No;missing/outside rows fail qualification instead.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_BOTH_typed_total_restoration_inventories · Uses: G001, S139

Reverse checks

S140 · 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.
S139 · 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.
S138 · 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.
S137 · 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.
S136 · 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.
S135 · 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.
S134 · 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.
S133 · 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.
S132 · 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.
S131 · 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.
S130 · 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.
S129 · 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.
S128 · 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.
S127 · 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.
S126 · 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.
S125 · 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.
S124 · 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.
S123 · 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.
S122 · 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.
S121 · 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.
S120 · 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.
S119 · 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.
S118 · 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.
S117 · 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.
S116 · 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.
S115 · 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.
S114 · 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.
S113 · 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.
S112 · 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.
S111 · 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.
S110 · 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.
S109 · 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.
S108 · 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.
S107 · 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.
S106 · 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.
S105 · 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.
S104 · 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.
S103 · 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.
S102 · 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.
S101 · 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.
S100 · 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.
S099 · 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.
S098 · 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.
S097 · 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.
S096 · 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.
S095 · 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.
S094 · 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.
S093 · 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.
S092 · 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.
S091 · 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.
S090 · 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.
S089 · 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.
S088 · 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.
S087 · 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.
S086 · 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.
S085 · 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.
S084 · 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.
S083 · 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.
S082 · 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.
S081 · 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.
S080 · 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.
S079 · 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.
S078 · 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.
S077 · 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.
S076 · 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.
S075 · 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.
S074 · 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.
S073 · 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.
S072 · 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.
S071 · 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.
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

Yes

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: 2bb9641046c006645b572f9d0737d49c5a140af2846ab13b1854e386b2018a3f
candidate record SHA-256: 05b7d822b9f2e1e5dc99712e03b44ed690f720451b36126e650933f229181f48
semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON