OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 32729: Checking Both Inverse Identities

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

Return to catalogue page 164

← Question derivation index · Original question batch

R0064-32729 · q-20261004-advance-20261005-r04-inverse-function-0054

Original question and givens

The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -3), (3, -2), (5, -1), (7, -5), (9, -4)}. Check both identities to decide whether g is the inverse of f.

  • G001 — The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. The candidate total g:B→A has table {(1, -3), (3, -2), (5, -1), (7, -5), (9, -4)}. 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.

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, [-5, -4, -3, -2, -1]]

After

derived = [[-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]]

After

derived = [[1, 3, 5, 7, 9]]

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, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, -5, 9, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

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, -4, 1, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

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, -3, 3, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

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, -2, 5, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

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, -1, 7, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

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, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S009

S011 · implies

Before

source roles = [ORIGINAL f, -4, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S010

S012 · implies

Before

source roles = [ORIGINAL f, -3, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S011

S013 · implies

Before

source roles = [ORIGINAL f, -2, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S012

S014 · implies

Before

source roles = [ORIGINAL f, -1, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S013

S015 · implies

Before

source roles = [ORIGINAL f, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[Yes, Yes, Yes, Yes, Yes], [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, S014

S016 · implies

Before

source roles = [ORIGINAL f, 1, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S015

S017 · implies

Before

source roles = [ORIGINAL f, 3, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S016

S018 · implies

Before

source roles = [ORIGINAL f, 5, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S017

S019 · implies

Before

source roles = [ORIGINAL f, 7, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S018

S020 · implies

Before

source roles = [ORIGINAL f, 9, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

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, S019

S021 · implies

Before

source roles = [ORIGINAL f, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9], [[-4], [-3], [-2], [-1], [-5]]]

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, S020

S022 · implies

Before

source roles = [CANDIDATE g domain, [1, 3, 5, 7, 9]]

After

derived = [[1, 3, 5, 7, 9]]

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, S021

S023 · implies

Before

source roles = [CANDIDATE g codomain, [-5, -4, -3, -2, -1]]

After

derived = [[-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, S022

S024 · implies

Before

source roles = [CANDIDATE g, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

After

derived = [[[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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, S023

S025 · implies

Before

source roles = [CANDIDATE g, 0, 1, -3, [1, 3, 5, 7, 9], [-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, S024

S026 · implies

Before

source roles = [CANDIDATE g, 1, 3, -2, [1, 3, 5, 7, 9], [-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, S025

S027 · implies

Before

source roles = [CANDIDATE g, 2, 5, -1, [1, 3, 5, 7, 9], [-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, S026

S028 · implies

Before

source roles = [CANDIDATE g, 3, 7, -5, [1, 3, 5, 7, 9], [-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, S027

S029 · implies

Before

source roles = [CANDIDATE g, 4, 9, -4, [1, 3, 5, 7, 9], [-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, S028

S030 · implies

Before

source roles = [CANDIDATE g, 1, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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 = [CANDIDATE g, 3, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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 = [CANDIDATE g, 5, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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 = [CANDIDATE g, 7, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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 = [CANDIDATE g, 9, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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 = [CANDIDATE g, [1, 3, 5, 7, 9], [-5, -4, -3, -2, -1], [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

After

derived = [[Yes, Yes, Yes, Yes, Yes], [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, S034

S036 · implies

Before

source roles = [CANDIDATE g, -5, [1, 3, 5, 7, 9], [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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, S035

S037 · implies

Before

source roles = [CANDIDATE g, -4, [1, 3, 5, 7, 9], [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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, S036

S038 · implies

Before

source roles = [CANDIDATE g, -3, [1, 3, 5, 7, 9], [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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, S037

S039 · implies

Before

source roles = [CANDIDATE g, -2, [1, 3, 5, 7, 9], [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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, S038

S040 · implies

Before

source roles = [CANDIDATE g, -1, [1, 3, 5, 7, 9], [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

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, S039

S041 · implies

Before

source roles = [CANDIDATE g, [1, 3, 5, 7, 9], [-5, -4, -3, -2, -1], [[7], [9], [1], [3], [5]]]

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, S040

S042 · implies

Before

source roles = [-5, 9, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

After

derived = [-4, No]

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, S041

S043 · implies

Before

source roles = [-4, 1, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

After

derived = [-3, No]

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, S042

S044 · implies

Before

source roles = [-3, 3, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

After

derived = [-2, No]

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, S043

S045 · implies

Before

source roles = [-2, 5, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

After

derived = [-1, No]

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, S044

S046 · implies

Before

source roles = [-1, 7, [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

After

derived = [-5, No]

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, S045

S047 · implies

Before

source roles = [1, -3, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [3, No]

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, S046

S048 · implies

Before

source roles = [3, -2, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [5, No]

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, S047

S049 · implies

Before

source roles = [5, -1, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [7, No]

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, S048

S050 · implies

Before

source roles = [7, -5, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [9, No]

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, S049

S051 · implies

Before

source roles = [9, -4, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [1, No]

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, S050

S052 · implies

Before

source roles = [[-5, -4, -3, -2, -1], [1, 3, 5, 7, 9], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]], [[1, -3], [3, -2], [5, -1], [7, -5], [9, -4]]]

After

derived = [[No, No, No, No, No], [No, No, No, No, No], No]

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, S051

Reverse checks

  • 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

No

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 8a1d6ea41bb810f4350bebbe5178c38d9dcd3adadf74c7f9d83225f7e4228ad0
  • candidate record SHA-256: 9f093f570695348205a1c91adfae9b951626d245aea7f35784ed35dcfd7c11f2
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON