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Question 9226: Topology Qualification before Discrete or Indiscrete Classification

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

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← Question derivation index · Original question batch

R0017-09226 · q-20261013-advance-20261018-r01-topological-space-0051

Original question and givens

Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.

  • G001 — Complete finite data: X=Fin(4)={0,…,3}. Candidate open family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize repeated elements/subsets as sets and order subsets by their binary masks sum(2^x). A pair counterexample is the first ordered pair in that order. The open family and underlying set are different objects; sampling is not an axiom check. Give (n, normalized family, topology?, invalid-candidate/discrete/indiscrete/other-topology). Classify discrete first; on an empty X the discrete and indiscrete families coincide.

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

space=given;n=4;raw family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]

After

universe=[0, 1, 2, 3];canonical opens=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]

Conditions: none

Type every candidate as a subset of the same Fin(n),deduplicate elements and members,and order by binary masks.The open family and underlying set are distinct data layers.

Step identifiers and dependencies

Rule: finite_topology_normalize_relative_universe · Uses: G001

S002 · implies

Before

space=given;canonical family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]

After

self pairs are idempotent;unordered distinct pairs cover all ordered pairs

Conditions: none

Self pairs are covered by A∪A=A and A∩A=A.Commutativity means all distinct unordered pairs cover every ordered pair;this is not sampling.

Step identifiers and dependencies

Rule: binary_set_operation_symmetry_coverage · Uses: G001, S001

S003 · implies

Before

space=given;A=[];B=[0]

After

union=[0];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S002

S004 · implies

Before

space=given;A=[];B=[1]

After

union=[1];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S003

S005 · implies

Before

space=given;A=[];B=[0, 1]

After

union=[0, 1];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S004

S006 · implies

Before

space=given;A=[];B=[2]

After

union=[2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S005

S007 · implies

Before

space=given;A=[];B=[0, 2]

After

union=[0, 2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S006

S008 · implies

Before

space=given;A=[];B=[1, 2]

After

union=[1, 2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S007

S009 · implies

Before

space=given;A=[];B=[0, 1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S008

S010 · implies

Before

space=given;A=[];B=[3]

After

union=[3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S009

S011 · implies

Before

space=given;A=[];B=[0, 3]

After

union=[0, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S010

S012 · implies

Before

space=given;A=[];B=[1, 3]

After

union=[1, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S011

S013 · implies

Before

space=given;A=[];B=[0, 1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S012

S014 · implies

Before

space=given;A=[];B=[2, 3]

After

union=[2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S013

S015 · implies

Before

space=given;A=[];B=[0, 2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S014

S016 · implies

Before

space=given;A=[];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S015

S017 · implies

Before

space=given;A=[];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S016

S018 · implies

Before

space=given;A=[0];B=[1]

After

union=[0, 1];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S017

S019 · implies

Before

space=given;A=[0];B=[0, 1]

After

union=[0, 1];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S018

S020 · implies

Before

space=given;A=[0];B=[2]

After

union=[0, 2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S019

S021 · implies

Before

space=given;A=[0];B=[0, 2]

After

union=[0, 2];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S020

S022 · implies

Before

space=given;A=[0];B=[1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S021

S023 · implies

Before

space=given;A=[0];B=[0, 1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S022

S024 · implies

Before

space=given;A=[0];B=[3]

After

union=[0, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S023

S025 · implies

Before

space=given;A=[0];B=[0, 3]

After

union=[0, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S024

S026 · implies

Before

space=given;A=[0];B=[1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S025

S027 · implies

Before

space=given;A=[0];B=[0, 1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S026

S028 · implies

Before

space=given;A=[0];B=[2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S027

S029 · implies

Before

space=given;A=[0];B=[0, 2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S028

S030 · implies

Before

space=given;A=[0];B=[1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S029

S031 · implies

Before

space=given;A=[0];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S030

S032 · implies

Before

space=given;A=[1];B=[0, 1]

After

union=[0, 1];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S031

S033 · implies

Before

space=given;A=[1];B=[2]

After

union=[1, 2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S032

S034 · implies

Before

space=given;A=[1];B=[0, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S033

S035 · implies

Before

space=given;A=[1];B=[1, 2]

After

union=[1, 2];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S034

S036 · implies

Before

space=given;A=[1];B=[0, 1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S035

S037 · implies

Before

space=given;A=[1];B=[3]

After

union=[1, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S036

S038 · implies

Before

space=given;A=[1];B=[0, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S037

S039 · implies

Before

space=given;A=[1];B=[1, 3]

After

union=[1, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S038

S040 · implies

Before

space=given;A=[1];B=[0, 1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S039

S041 · implies

Before

space=given;A=[1];B=[2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S040

S042 · implies

Before

space=given;A=[1];B=[0, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S041

S043 · implies

Before

space=given;A=[1];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S042

S044 · implies

Before

space=given;A=[1];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S043

S045 · implies

Before

space=given;A=[0, 1];B=[2]

After

union=[0, 1, 2];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S044

S046 · implies

Before

space=given;A=[0, 1];B=[0, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S045

S047 · implies

Before

space=given;A=[0, 1];B=[1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S046

S048 · implies

Before

space=given;A=[0, 1];B=[0, 1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[0, 1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S047

S049 · implies

Before

space=given;A=[0, 1];B=[3]

After

union=[0, 1, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S048

S050 · implies

Before

space=given;A=[0, 1];B=[0, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S049

S051 · implies

Before

space=given;A=[0, 1];B=[1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S050

S052 · implies

Before

space=given;A=[0, 1];B=[0, 1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[0, 1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S051

S053 · implies

Before

space=given;A=[0, 1];B=[2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S052

S054 · implies

Before

space=given;A=[0, 1];B=[0, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S053

S055 · implies

Before

space=given;A=[0, 1];B=[1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S054

S056 · implies

Before

space=given;A=[0, 1];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S055

S057 · implies

Before

space=given;A=[2];B=[0, 2]

After

union=[0, 2];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S056

S058 · implies

Before

space=given;A=[2];B=[1, 2]

After

union=[1, 2];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S057

S059 · implies

Before

space=given;A=[2];B=[0, 1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S058

S060 · implies

Before

space=given;A=[2];B=[3]

After

union=[2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S059

S061 · implies

Before

space=given;A=[2];B=[0, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S060

S062 · implies

Before

space=given;A=[2];B=[1, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S061

S063 · implies

Before

space=given;A=[2];B=[0, 1, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S062

S064 · implies

Before

space=given;A=[2];B=[2, 3]

After

union=[2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S063

S065 · implies

Before

space=given;A=[2];B=[0, 2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S064

S066 · implies

Before

space=given;A=[2];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S065

S067 · implies

Before

space=given;A=[2];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S066

S068 · implies

Before

space=given;A=[0, 2];B=[1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S067

S069 · implies

Before

space=given;A=[0, 2];B=[0, 1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[0, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S068

S070 · implies

Before

space=given;A=[0, 2];B=[3]

After

union=[0, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S069

S071 · implies

Before

space=given;A=[0, 2];B=[0, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S070

S072 · implies

Before

space=given;A=[0, 2];B=[1, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S071

S073 · implies

Before

space=given;A=[0, 2];B=[0, 1, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S072

S074 · implies

Before

space=given;A=[0, 2];B=[2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S073

S075 · implies

Before

space=given;A=[0, 2];B=[0, 2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[0, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S074

S076 · implies

Before

space=given;A=[0, 2];B=[1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S075

S077 · implies

Before

space=given;A=[0, 2];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S076

S078 · implies

Before

space=given;A=[1, 2];B=[0, 1, 2]

After

union=[0, 1, 2];in family=Yes;intersection=[1, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S077

S079 · implies

Before

space=given;A=[1, 2];B=[3]

After

union=[1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S078

S080 · implies

Before

space=given;A=[1, 2];B=[0, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S079

S081 · implies

Before

space=given;A=[1, 2];B=[1, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S080

S082 · implies

Before

space=given;A=[1, 2];B=[0, 1, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S081

S083 · implies

Before

space=given;A=[1, 2];B=[2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S082

S084 · implies

Before

space=given;A=[1, 2];B=[0, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S083

S085 · implies

Before

space=given;A=[1, 2];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[1, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S084

S086 · implies

Before

space=given;A=[1, 2];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S085

S087 · implies

Before

space=given;A=[0, 1, 2];B=[3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S086

S088 · implies

Before

space=given;A=[0, 1, 2];B=[0, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S087

S089 · implies

Before

space=given;A=[0, 1, 2];B=[1, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S088

S090 · implies

Before

space=given;A=[0, 1, 2];B=[0, 1, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 1];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S089

S091 · implies

Before

space=given;A=[0, 1, 2];B=[2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S090

S092 · implies

Before

space=given;A=[0, 1, 2];B=[0, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S091

S093 · implies

Before

space=given;A=[0, 1, 2];B=[1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S092

S094 · implies

Before

space=given;A=[0, 1, 2];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 1, 2];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S093

S095 · implies

Before

space=given;A=[3];B=[0, 3]

After

union=[0, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S094

S096 · implies

Before

space=given;A=[3];B=[1, 3]

After

union=[1, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S095

S097 · implies

Before

space=given;A=[3];B=[0, 1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S096

S098 · implies

Before

space=given;A=[3];B=[2, 3]

After

union=[2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S097

S099 · implies

Before

space=given;A=[3];B=[0, 2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S098

S100 · implies

Before

space=given;A=[3];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S099

S101 · implies

Before

space=given;A=[3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S100

S102 · implies

Before

space=given;A=[0, 3];B=[1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S101

S103 · implies

Before

space=given;A=[0, 3];B=[0, 1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[0, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S102

S104 · implies

Before

space=given;A=[0, 3];B=[2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S103

S105 · implies

Before

space=given;A=[0, 3];B=[0, 2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[0, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S104

S106 · implies

Before

space=given;A=[0, 3];B=[1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S105

S107 · implies

Before

space=given;A=[0, 3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S106

S108 · implies

Before

space=given;A=[1, 3];B=[0, 1, 3]

After

union=[0, 1, 3];in family=Yes;intersection=[1, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S107

S109 · implies

Before

space=given;A=[1, 3];B=[2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S108

S110 · implies

Before

space=given;A=[1, 3];B=[0, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S109

S111 · implies

Before

space=given;A=[1, 3];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[1, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S110

S112 · implies

Before

space=given;A=[1, 3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S111

S113 · implies

Before

space=given;A=[0, 1, 3];B=[2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S112

S114 · implies

Before

space=given;A=[0, 1, 3];B=[0, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S113

S115 · implies

Before

space=given;A=[0, 1, 3];B=[1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S114

S116 · implies

Before

space=given;A=[0, 1, 3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 1, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S115

S117 · implies

Before

space=given;A=[2, 3];B=[0, 2, 3]

After

union=[0, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S116

S118 · implies

Before

space=given;A=[2, 3];B=[1, 2, 3]

After

union=[1, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S117

S119 · implies

Before

space=given;A=[2, 3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S118

S120 · implies

Before

space=given;A=[0, 2, 3];B=[1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S119

S121 · implies

Before

space=given;A=[0, 2, 3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[0, 2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S120

S122 · implies

Before

space=given;A=[1, 2, 3];B=[0, 1, 2, 3]

After

union=[0, 1, 2, 3];in family=Yes;intersection=[1, 2, 3];in family=Yes

Conditions: none

Form union and intersection from the set definitions and test each against the original family.Preserve the original pair and result for the first failure.

Step identifiers and dependencies

Rule: finite_topology_complete_binary_pair · Uses: G001, S121

S123 · implies

Before

space=given;empty=Yes;full=Yes;union closed=Yes;intersection closed=Yes

After

topology=Yes;first union failure=none;first intersection failure=none

Conditions: none

All four conditions are required.On a finite underlying set there are finitely many distinct opens,so an arbitrary union reduces to a finite distinct-member union,covered by binary closure and the empty union.The empty intersection needs the whole set.Vacuous binary closure does not waive empty/whole-set membership.

Step identifiers and dependencies

Rule: finite_topology_axioms_joint_gate · Uses: G001, S122

S124 · implies

Before

topology qualified=Yes;family size=16;powerset size=16

After

classification=discrete

Conditions: none

Qualify the topology first.All2^n subsets give the discrete topology;otherwise only empty and whole give the indiscrete topology.On the empty space both coincide,and the protocol classifies discrete first.

Step identifiers and dependencies

Rule: topology_classification_after_qualification · Uses: G001, S123

Reverse checks

  • S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(4, [[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], Yes, discrete)

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: 8c04e4134e118df32162c72dac7dc7d4ceb60324a6e8da6c257758d7d349af97
  • candidate record SHA-256: 793a93b802f6811ca1df498d6ae989182fc3d522f20da4d96e1986ecee31d399
  • semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
  • question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON