Question 25333: Explicit Representative Dependence Blocks This Quotient Multiplication

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

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R0049-25333 · q-20261015-advance-20261114-r02-finite-quotient-group-0083

Original question and givens

Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0, explicit subset=[0, 10] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary).

G001 — Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0, explicit subset=[0, 10] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-finite-quotient-group

Step-by-step derivation

S001 · implies

Before

source roles = [quotient-nonnormal-boundary]

After

derived = [[subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary]]

Conditions: none

List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17, 18, 19, 10, 11], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18, 19, 10, 11, 12], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19, 10, 11, 12, 13], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 15, 16, 17, 18, 19, 10, 11, 12, 13, 14], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 16, 17, 18, 19, 10, 11, 12, 13, 14, 15], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 17, 18, 19, 10, 11, 12, 13, 14, 15, 16], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 18, 19, 10, 11, 12, 13, 14, 15, 16, 17], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 19, 10, 11, 12, 13, 14, 15, 16, 17, 18], [10, 19, 18, 17, 16, 15, 14, 13, 12, 11, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [11, 10, 19, 18, 17, 16, 15, 14, 13, 12, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2], [12, 11, 10, 19, 18, 17, 16, 15, 14, 13, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3], [13, 12, 11, 10, 19, 18, 17, 16, 15, 14, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4], [14, 13, 12, 11, 10, 19, 18, 17, 16, 15, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5], [15, 14, 13, 12, 11, 10, 19, 18, 17, 16, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6], [16, 15, 14, 13, 12, 11, 10, 19, 18, 17, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7], [17, 16, 15, 14, 13, 12, 11, 10, 19, 18, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8], [18, 17, 16, 15, 14, 13, 12, 11, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9], [19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0]

After

derived = [20, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19]], Yes]

Conditions: none

Validate the complete closed Cayley table and actual identity label,without assuming0 or the old32-element cap.

Step identifiers and dependencies

Rule: Quotient_complete_group_table_actual_identity · Uses: G001, S001

S003 · implies

Before

source roles = [20, 0]

After

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 8000, [0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

Conditions: none

Each pair-matrix entry checks associativity for every z,covering n^3 obligations;each inverse returns the same identity from both sides.This is exhaustive,not sampling.

Step identifiers and dependencies

Rule: Quotient_ALL_group_associativity_triples_and_two_sided_inverses · Uses: G001, S002

S004 · implies

Before

source roles = [0, [0, 10]]

After

derived = [[[0, 0, 0, Yes], [0, 10, 10, Yes], [10, 0, 10, Yes], [10, 10, 0, Yes]], [[0, 0, Yes], [10, 10, Yes]], Yes, [[0, [0, 10]], [1, [0, 12]], [2, [0, 14]], [3, [0, 16]], [4, [0, 18]], [5, [0, 10]], [6, [0, 12]], [7, [0, 14]], [8, [0, 16]], [9, [0, 18]], [10, [0, 10]], [11, [0, 12]], [12, [0, 14]], [13, [0, 16]], [14, [0, 18]], [15, [0, 10]], [16, [0, 12]], [17, [0, 14]], [18, [0, 16]], [19, [0, 18]]], No]

Conditions: none

First qualify identity,all products and inverse closure as a genuine subgroup,then test conjugation for every g.Positive branches require normality and the boundary requires actual nonnormality,not failed subgroup qualification.

Step identifiers and dependencies

Rule: Quotient_genuine_subgroup_before_actual_normality · Uses: G001, S003

S005 · implies

Before

source roles = [[0, 10], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]

After

derived = [[[0, 10], [1, 11], [2, 12], [3, 13], [4, 14], [5, 15], [6, 16], [7, 17], [8, 18], [9, 19]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes]

Conditions: none

Every g belongs to gH.If aH meets bH,ah1=bh2 implies b^-1a in H and the complete cosets coincide.Enumerate every g,deduplicate and retain source order.Quotient elements are coset SETS,not least representatives.

Step identifiers and dependencies

Rule: Quotient_ALL_LEFT_coset_SETS_complete_disjoint_partition · Uses: G001, S004

S006 · implies

Before

source roles = [0, 0, [0, 10], [0, 10]]

After

derived = [[[0, 0, 0, 0], [0, 10, 10, 0], [10, 0, 10, 0], [10, 10, 0, 0]], [0], Yes]

Conditions: none

For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S005

S007 · implies

Before

source roles = [0, 1, [0, 10], [1, 11]]

After

derived = [[[0, 1, 1, 1], [0, 11, 11, 1], [10, 1, 19, 9], [10, 11, 9, 9]], [1, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S006

S008 · implies

Before

source roles = [0, 2, [0, 10], [2, 12]]

After

derived = [[[0, 2, 2, 2], [0, 12, 12, 2], [10, 2, 18, 8], [10, 12, 8, 8]], [2, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S007

S009 · implies

Before

source roles = [0, 3, [0, 10], [3, 13]]

After

derived = [[[0, 3, 3, 3], [0, 13, 13, 3], [10, 3, 17, 7], [10, 13, 7, 7]], [3, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S008

S010 · implies

Before

source roles = [0, 4, [0, 10], [4, 14]]

After

derived = [[[0, 4, 4, 4], [0, 14, 14, 4], [10, 4, 16, 6], [10, 14, 6, 6]], [4, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S009

S011 · implies

Before

source roles = [0, 5, [0, 10], [5, 15]]

After

derived = [[[0, 5, 5, 5], [0, 15, 15, 5], [10, 5, 15, 5], [10, 15, 5, 5]], [5], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S010

S012 · implies

Before

source roles = [0, 6, [0, 10], [6, 16]]

After

derived = [[[0, 6, 6, 6], [0, 16, 16, 6], [10, 6, 14, 4], [10, 16, 4, 4]], [4, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S011

S013 · implies

Before

source roles = [0, 7, [0, 10], [7, 17]]

After

derived = [[[0, 7, 7, 7], [0, 17, 17, 7], [10, 7, 13, 3], [10, 17, 3, 3]], [3, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S012

S014 · implies

Before

source roles = [0, 8, [0, 10], [8, 18]]

After

derived = [[[0, 8, 8, 8], [0, 18, 18, 8], [10, 8, 12, 2], [10, 18, 2, 2]], [2, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S013

S015 · implies

Before

source roles = [0, 9, [0, 10], [9, 19]]

After

derived = [[[0, 9, 9, 9], [0, 19, 19, 9], [10, 9, 11, 1], [10, 19, 1, 1]], [1, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S014

S016 · implies

Before

source roles = [1, 0, [1, 11], [0, 10]]

After

derived = [[[1, 0, 1, 1], [1, 10, 11, 1], [11, 0, 11, 1], [11, 10, 1, 1]], [1], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S015

S017 · implies

Before

source roles = [1, 1, [1, 11], [1, 11]]

After

derived = [[[1, 1, 2, 2], [1, 11, 12, 2], [11, 1, 10, 0], [11, 11, 0, 0]], [0, 2], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S016

S018 · implies

Before

source roles = [1, 2, [1, 11], [2, 12]]

After

derived = [[[1, 2, 3, 3], [1, 12, 13, 3], [11, 2, 19, 9], [11, 12, 9, 9]], [3, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S017

S019 · implies

Before

source roles = [1, 3, [1, 11], [3, 13]]

After

derived = [[[1, 3, 4, 4], [1, 13, 14, 4], [11, 3, 18, 8], [11, 13, 8, 8]], [4, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S018

S020 · implies

Before

source roles = [1, 4, [1, 11], [4, 14]]

After

derived = [[[1, 4, 5, 5], [1, 14, 15, 5], [11, 4, 17, 7], [11, 14, 7, 7]], [5, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S019

S021 · implies

Before

source roles = [1, 5, [1, 11], [5, 15]]

After

derived = [[[1, 5, 6, 6], [1, 15, 16, 6], [11, 5, 16, 6], [11, 15, 6, 6]], [6], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S020

S022 · implies

Before

source roles = [1, 6, [1, 11], [6, 16]]

After

derived = [[[1, 6, 7, 7], [1, 16, 17, 7], [11, 6, 15, 5], [11, 16, 5, 5]], [5, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S021

S023 · implies

Before

source roles = [1, 7, [1, 11], [7, 17]]

After

derived = [[[1, 7, 8, 8], [1, 17, 18, 8], [11, 7, 14, 4], [11, 17, 4, 4]], [4, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S022

S024 · implies

Before

source roles = [1, 8, [1, 11], [8, 18]]

After

derived = [[[1, 8, 9, 9], [1, 18, 19, 9], [11, 8, 13, 3], [11, 18, 3, 3]], [3, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S023

S025 · implies

Before

source roles = [1, 9, [1, 11], [9, 19]]

After

derived = [[[1, 9, 0, 0], [1, 19, 10, 0], [11, 9, 12, 2], [11, 19, 2, 2]], [0, 2], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S024

S026 · implies

Before

source roles = [2, 0, [2, 12], [0, 10]]

After

derived = [[[2, 0, 2, 2], [2, 10, 12, 2], [12, 0, 12, 2], [12, 10, 2, 2]], [2], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S025

S027 · implies

Before

source roles = [2, 1, [2, 12], [1, 11]]

After

derived = [[[2, 1, 3, 3], [2, 11, 13, 3], [12, 1, 11, 1], [12, 11, 1, 1]], [1, 3], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S026

S028 · implies

Before

source roles = [2, 2, [2, 12], [2, 12]]

After

derived = [[[2, 2, 4, 4], [2, 12, 14, 4], [12, 2, 10, 0], [12, 12, 0, 0]], [0, 4], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S027

S029 · implies

Before

source roles = [2, 3, [2, 12], [3, 13]]

After

derived = [[[2, 3, 5, 5], [2, 13, 15, 5], [12, 3, 19, 9], [12, 13, 9, 9]], [5, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S028

S030 · implies

Before

source roles = [2, 4, [2, 12], [4, 14]]

After

derived = [[[2, 4, 6, 6], [2, 14, 16, 6], [12, 4, 18, 8], [12, 14, 8, 8]], [6, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S029

S031 · implies

Before

source roles = [2, 5, [2, 12], [5, 15]]

After

derived = [[[2, 5, 7, 7], [2, 15, 17, 7], [12, 5, 17, 7], [12, 15, 7, 7]], [7], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S030

S032 · implies

Before

source roles = [2, 6, [2, 12], [6, 16]]

After

derived = [[[2, 6, 8, 8], [2, 16, 18, 8], [12, 6, 16, 6], [12, 16, 6, 6]], [6, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S031

S033 · implies

Before

source roles = [2, 7, [2, 12], [7, 17]]

After

derived = [[[2, 7, 9, 9], [2, 17, 19, 9], [12, 7, 15, 5], [12, 17, 5, 5]], [5, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S032

S034 · implies

Before

source roles = [2, 8, [2, 12], [8, 18]]

After

derived = [[[2, 8, 0, 0], [2, 18, 10, 0], [12, 8, 14, 4], [12, 18, 4, 4]], [0, 4], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S033

S035 · implies

Before

source roles = [2, 9, [2, 12], [9, 19]]

After

derived = [[[2, 9, 1, 1], [2, 19, 11, 1], [12, 9, 13, 3], [12, 19, 3, 3]], [1, 3], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S034

S036 · implies

Before

source roles = [3, 0, [3, 13], [0, 10]]

After

derived = [[[3, 0, 3, 3], [3, 10, 13, 3], [13, 0, 13, 3], [13, 10, 3, 3]], [3], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S035

S037 · implies

Before

source roles = [3, 1, [3, 13], [1, 11]]

After

derived = [[[3, 1, 4, 4], [3, 11, 14, 4], [13, 1, 12, 2], [13, 11, 2, 2]], [2, 4], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S036

S038 · implies

Before

source roles = [3, 2, [3, 13], [2, 12]]

After

derived = [[[3, 2, 5, 5], [3, 12, 15, 5], [13, 2, 11, 1], [13, 12, 1, 1]], [1, 5], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S037

S039 · implies

Before

source roles = [3, 3, [3, 13], [3, 13]]

After

derived = [[[3, 3, 6, 6], [3, 13, 16, 6], [13, 3, 10, 0], [13, 13, 0, 0]], [0, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S038

S040 · implies

Before

source roles = [3, 4, [3, 13], [4, 14]]

After

derived = [[[3, 4, 7, 7], [3, 14, 17, 7], [13, 4, 19, 9], [13, 14, 9, 9]], [7, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S039

S041 · implies

Before

source roles = [3, 5, [3, 13], [5, 15]]

After

derived = [[[3, 5, 8, 8], [3, 15, 18, 8], [13, 5, 18, 8], [13, 15, 8, 8]], [8], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S040

S042 · implies

Before

source roles = [3, 6, [3, 13], [6, 16]]

After

derived = [[[3, 6, 9, 9], [3, 16, 19, 9], [13, 6, 17, 7], [13, 16, 7, 7]], [7, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S041

S043 · implies

Before

source roles = [3, 7, [3, 13], [7, 17]]

After

derived = [[[3, 7, 0, 0], [3, 17, 10, 0], [13, 7, 16, 6], [13, 17, 6, 6]], [0, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S042

S044 · implies

Before

source roles = [3, 8, [3, 13], [8, 18]]

After

derived = [[[3, 8, 1, 1], [3, 18, 11, 1], [13, 8, 15, 5], [13, 18, 5, 5]], [1, 5], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S043

S045 · implies

Before

source roles = [3, 9, [3, 13], [9, 19]]

After

derived = [[[3, 9, 2, 2], [3, 19, 12, 2], [13, 9, 14, 4], [13, 19, 4, 4]], [2, 4], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S044

S046 · implies

Before

source roles = [4, 0, [4, 14], [0, 10]]

After

derived = [[[4, 0, 4, 4], [4, 10, 14, 4], [14, 0, 14, 4], [14, 10, 4, 4]], [4], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S045

S047 · implies

Before

source roles = [4, 1, [4, 14], [1, 11]]

After

derived = [[[4, 1, 5, 5], [4, 11, 15, 5], [14, 1, 13, 3], [14, 11, 3, 3]], [3, 5], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S046

S048 · implies

Before

source roles = [4, 2, [4, 14], [2, 12]]

After

derived = [[[4, 2, 6, 6], [4, 12, 16, 6], [14, 2, 12, 2], [14, 12, 2, 2]], [2, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S047

S049 · implies

Before

source roles = [4, 3, [4, 14], [3, 13]]

After

derived = [[[4, 3, 7, 7], [4, 13, 17, 7], [14, 3, 11, 1], [14, 13, 1, 1]], [1, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S048

S050 · implies

Before

source roles = [4, 4, [4, 14], [4, 14]]

After

derived = [[[4, 4, 8, 8], [4, 14, 18, 8], [14, 4, 10, 0], [14, 14, 0, 0]], [0, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S049

S051 · implies

Before

source roles = [4, 5, [4, 14], [5, 15]]

After

derived = [[[4, 5, 9, 9], [4, 15, 19, 9], [14, 5, 19, 9], [14, 15, 9, 9]], [9], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S050

S052 · implies

Before

source roles = [4, 6, [4, 14], [6, 16]]

After

derived = [[[4, 6, 0, 0], [4, 16, 10, 0], [14, 6, 18, 8], [14, 16, 8, 8]], [0, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S051

S053 · implies

Before

source roles = [4, 7, [4, 14], [7, 17]]

After

derived = [[[4, 7, 1, 1], [4, 17, 11, 1], [14, 7, 17, 7], [14, 17, 7, 7]], [1, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S052

S054 · implies

Before

source roles = [4, 8, [4, 14], [8, 18]]

After

derived = [[[4, 8, 2, 2], [4, 18, 12, 2], [14, 8, 16, 6], [14, 18, 6, 6]], [2, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S053

S055 · implies

Before

source roles = [4, 9, [4, 14], [9, 19]]

After

derived = [[[4, 9, 3, 3], [4, 19, 13, 3], [14, 9, 15, 5], [14, 19, 5, 5]], [3, 5], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S054

S056 · implies

Before

source roles = [5, 0, [5, 15], [0, 10]]

After

derived = [[[5, 0, 5, 5], [5, 10, 15, 5], [15, 0, 15, 5], [15, 10, 5, 5]], [5], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S055

S057 · implies

Before

source roles = [5, 1, [5, 15], [1, 11]]

After

derived = [[[5, 1, 6, 6], [5, 11, 16, 6], [15, 1, 14, 4], [15, 11, 4, 4]], [4, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S056

S058 · implies

Before

source roles = [5, 2, [5, 15], [2, 12]]

After

derived = [[[5, 2, 7, 7], [5, 12, 17, 7], [15, 2, 13, 3], [15, 12, 3, 3]], [3, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S057

S059 · implies

Before

source roles = [5, 3, [5, 15], [3, 13]]

After

derived = [[[5, 3, 8, 8], [5, 13, 18, 8], [15, 3, 12, 2], [15, 13, 2, 2]], [2, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S058

S060 · implies

Before

source roles = [5, 4, [5, 15], [4, 14]]

After

derived = [[[5, 4, 9, 9], [5, 14, 19, 9], [15, 4, 11, 1], [15, 14, 1, 1]], [1, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S059

S061 · implies

Before

source roles = [5, 5, [5, 15], [5, 15]]

After

derived = [[[5, 5, 0, 0], [5, 15, 10, 0], [15, 5, 10, 0], [15, 15, 0, 0]], [0], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S060

S062 · implies

Before

source roles = [5, 6, [5, 15], [6, 16]]

After

derived = [[[5, 6, 1, 1], [5, 16, 11, 1], [15, 6, 19, 9], [15, 16, 9, 9]], [1, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S061

S063 · implies

Before

source roles = [5, 7, [5, 15], [7, 17]]

After

derived = [[[5, 7, 2, 2], [5, 17, 12, 2], [15, 7, 18, 8], [15, 17, 8, 8]], [2, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S062

S064 · implies

Before

source roles = [5, 8, [5, 15], [8, 18]]

After

derived = [[[5, 8, 3, 3], [5, 18, 13, 3], [15, 8, 17, 7], [15, 18, 7, 7]], [3, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S063

S065 · implies

Before

source roles = [5, 9, [5, 15], [9, 19]]

After

derived = [[[5, 9, 4, 4], [5, 19, 14, 4], [15, 9, 16, 6], [15, 19, 6, 6]], [4, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S064

S066 · implies

Before

source roles = [6, 0, [6, 16], [0, 10]]

After

derived = [[[6, 0, 6, 6], [6, 10, 16, 6], [16, 0, 16, 6], [16, 10, 6, 6]], [6], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S065

S067 · implies

Before

source roles = [6, 1, [6, 16], [1, 11]]

After

derived = [[[6, 1, 7, 7], [6, 11, 17, 7], [16, 1, 15, 5], [16, 11, 5, 5]], [5, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S066

S068 · implies

Before

source roles = [6, 2, [6, 16], [2, 12]]

After

derived = [[[6, 2, 8, 8], [6, 12, 18, 8], [16, 2, 14, 4], [16, 12, 4, 4]], [4, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S067

S069 · implies

Before

source roles = [6, 3, [6, 16], [3, 13]]

After

derived = [[[6, 3, 9, 9], [6, 13, 19, 9], [16, 3, 13, 3], [16, 13, 3, 3]], [3, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S068

S070 · implies

Before

source roles = [6, 4, [6, 16], [4, 14]]

After

derived = [[[6, 4, 0, 0], [6, 14, 10, 0], [16, 4, 12, 2], [16, 14, 2, 2]], [0, 2], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S069

S071 · implies

Before

source roles = [6, 5, [6, 16], [5, 15]]

After

derived = [[[6, 5, 1, 1], [6, 15, 11, 1], [16, 5, 11, 1], [16, 15, 1, 1]], [1], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S070

S072 · implies

Before

source roles = [6, 6, [6, 16], [6, 16]]

After

derived = [[[6, 6, 2, 2], [6, 16, 12, 2], [16, 6, 10, 0], [16, 16, 0, 0]], [0, 2], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S071

S073 · implies

Before

source roles = [6, 7, [6, 16], [7, 17]]

After

derived = [[[6, 7, 3, 3], [6, 17, 13, 3], [16, 7, 19, 9], [16, 17, 9, 9]], [3, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S072

S074 · implies

Before

source roles = [6, 8, [6, 16], [8, 18]]

After

derived = [[[6, 8, 4, 4], [6, 18, 14, 4], [16, 8, 18, 8], [16, 18, 8, 8]], [4, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S073

S075 · implies

Before

source roles = [6, 9, [6, 16], [9, 19]]

After

derived = [[[6, 9, 5, 5], [6, 19, 15, 5], [16, 9, 17, 7], [16, 19, 7, 7]], [5, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S074

S076 · implies

Before

source roles = [7, 0, [7, 17], [0, 10]]

After

derived = [[[7, 0, 7, 7], [7, 10, 17, 7], [17, 0, 17, 7], [17, 10, 7, 7]], [7], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S075

S077 · implies

Before

source roles = [7, 1, [7, 17], [1, 11]]

After

derived = [[[7, 1, 8, 8], [7, 11, 18, 8], [17, 1, 16, 6], [17, 11, 6, 6]], [6, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S076

S078 · implies

Before

source roles = [7, 2, [7, 17], [2, 12]]

After

derived = [[[7, 2, 9, 9], [7, 12, 19, 9], [17, 2, 15, 5], [17, 12, 5, 5]], [5, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S077

S079 · implies

Before

source roles = [7, 3, [7, 17], [3, 13]]

After

derived = [[[7, 3, 0, 0], [7, 13, 10, 0], [17, 3, 14, 4], [17, 13, 4, 4]], [0, 4], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S078

S080 · implies

Before

source roles = [7, 4, [7, 17], [4, 14]]

After

derived = [[[7, 4, 1, 1], [7, 14, 11, 1], [17, 4, 13, 3], [17, 14, 3, 3]], [1, 3], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S079

S081 · implies

Before

source roles = [7, 5, [7, 17], [5, 15]]

After

derived = [[[7, 5, 2, 2], [7, 15, 12, 2], [17, 5, 12, 2], [17, 15, 2, 2]], [2], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S080

S082 · implies

Before

source roles = [7, 6, [7, 17], [6, 16]]

After

derived = [[[7, 6, 3, 3], [7, 16, 13, 3], [17, 6, 11, 1], [17, 16, 1, 1]], [1, 3], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S081

S083 · implies

Before

source roles = [7, 7, [7, 17], [7, 17]]

After

derived = [[[7, 7, 4, 4], [7, 17, 14, 4], [17, 7, 10, 0], [17, 17, 0, 0]], [0, 4], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S082

S084 · implies

Before

source roles = [7, 8, [7, 17], [8, 18]]

After

derived = [[[7, 8, 5, 5], [7, 18, 15, 5], [17, 8, 19, 9], [17, 18, 9, 9]], [5, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S083

S085 · implies

Before

source roles = [7, 9, [7, 17], [9, 19]]

After

derived = [[[7, 9, 6, 6], [7, 19, 16, 6], [17, 9, 18, 8], [17, 19, 8, 8]], [6, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S084

S086 · implies

Before

source roles = [8, 0, [8, 18], [0, 10]]

After

derived = [[[8, 0, 8, 8], [8, 10, 18, 8], [18, 0, 18, 8], [18, 10, 8, 8]], [8], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S085

S087 · implies

Before

source roles = [8, 1, [8, 18], [1, 11]]

After

derived = [[[8, 1, 9, 9], [8, 11, 19, 9], [18, 1, 17, 7], [18, 11, 7, 7]], [7, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S086

S088 · implies

Before

source roles = [8, 2, [8, 18], [2, 12]]

After

derived = [[[8, 2, 0, 0], [8, 12, 10, 0], [18, 2, 16, 6], [18, 12, 6, 6]], [0, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S087

S089 · implies

Before

source roles = [8, 3, [8, 18], [3, 13]]

After

derived = [[[8, 3, 1, 1], [8, 13, 11, 1], [18, 3, 15, 5], [18, 13, 5, 5]], [1, 5], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S088

S090 · implies

Before

source roles = [8, 4, [8, 18], [4, 14]]

After

derived = [[[8, 4, 2, 2], [8, 14, 12, 2], [18, 4, 14, 4], [18, 14, 4, 4]], [2, 4], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S089

S091 · implies

Before

source roles = [8, 5, [8, 18], [5, 15]]

After

derived = [[[8, 5, 3, 3], [8, 15, 13, 3], [18, 5, 13, 3], [18, 15, 3, 3]], [3], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S090

S092 · implies

Before

source roles = [8, 6, [8, 18], [6, 16]]

After

derived = [[[8, 6, 4, 4], [8, 16, 14, 4], [18, 6, 12, 2], [18, 16, 2, 2]], [2, 4], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S091

S093 · implies

Before

source roles = [8, 7, [8, 18], [7, 17]]

After

derived = [[[8, 7, 5, 5], [8, 17, 15, 5], [18, 7, 11, 1], [18, 17, 1, 1]], [1, 5], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S092

S094 · implies

Before

source roles = [8, 8, [8, 18], [8, 18]]

After

derived = [[[8, 8, 6, 6], [8, 18, 16, 6], [18, 8, 10, 0], [18, 18, 0, 0]], [0, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S093

S095 · implies

Before

source roles = [8, 9, [8, 18], [9, 19]]

After

derived = [[[8, 9, 7, 7], [8, 19, 17, 7], [18, 9, 19, 9], [18, 19, 9, 9]], [7, 9], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S094

S096 · implies

Before

source roles = [9, 0, [9, 19], [0, 10]]

After

derived = [[[9, 0, 9, 9], [9, 10, 19, 9], [19, 0, 19, 9], [19, 10, 9, 9]], [9], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S095

S097 · implies

Before

source roles = [9, 1, [9, 19], [1, 11]]

After

derived = [[[9, 1, 0, 0], [9, 11, 10, 0], [19, 1, 18, 8], [19, 11, 8, 8]], [0, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S096

S098 · implies

Before

source roles = [9, 2, [9, 19], [2, 12]]

After

derived = [[[9, 2, 1, 1], [9, 12, 11, 1], [19, 2, 17, 7], [19, 12, 7, 7]], [1, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S097

S099 · implies

Before

source roles = [9, 3, [9, 19], [3, 13]]

After

derived = [[[9, 3, 2, 2], [9, 13, 12, 2], [19, 3, 16, 6], [19, 13, 6, 6]], [2, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S098

S100 · implies

Before

source roles = [9, 4, [9, 19], [4, 14]]

After

derived = [[[9, 4, 3, 3], [9, 14, 13, 3], [19, 4, 15, 5], [19, 14, 5, 5]], [3, 5], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S099

S101 · implies

Before

source roles = [9, 5, [9, 19], [5, 15]]

After

derived = [[[9, 5, 4, 4], [9, 15, 14, 4], [19, 5, 14, 4], [19, 15, 4, 4]], [4], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S100

S102 · implies

Before

source roles = [9, 6, [9, 19], [6, 16]]

After

derived = [[[9, 6, 5, 5], [9, 16, 15, 5], [19, 6, 13, 3], [19, 16, 3, 3]], [3, 5], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S101

S103 · implies

Before

source roles = [9, 7, [9, 19], [7, 17]]

After

derived = [[[9, 7, 6, 6], [9, 17, 16, 6], [19, 7, 12, 2], [19, 17, 2, 2]], [2, 6], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S102

S104 · implies

Before

source roles = [9, 8, [9, 19], [8, 18]]

After

derived = [[[9, 8, 7, 7], [9, 18, 17, 7], [19, 8, 11, 1], [19, 18, 1, 1]], [1, 7], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S103

S105 · implies

Before

source roles = [9, 9, [9, 19], [9, 19]]

After

derived = [[[9, 9, 8, 8], [9, 19, 18, 8], [19, 9, 10, 0], [19, 19, 0, 0]], [0, 8], No]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S104

S106 · implies

Before

source roles = [[0, 1, [1, 9], No], [0, 10], [1, 11]]

After

derived = [[0, 1, 0, 1, 1, 10, 1, 9], No, No]

Conditions: none

Keep the source first bad ordered pair and left,left2,right,right2 search order.Different representatives of the same two cosets give different product cosets,so multiplication fails to be well-defined.The subgroup and coset partition remain valid;do not invent a quotient table.

Step identifiers and dependencies

Rule: Quotient_FIRST_bad_pair_exact_source_nested_representative_witness · Uses: G001, S105

Reverse checks

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

([0, 10], No, [[0, 10], [1, 11], [2, 12], [3, 13], [4, 14], [5, 15], [6, 16], [7, 17], [8, 18], [9, 19]], [0, 1, 0, 1, 1, 10, 1, 9], No, No, explicit_representative_dependence_blocks_quotient_group_multiplication)

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: 760f17d9376e7fc30462a290048b555adb375642dbcc7db0eff10dc528545ea8
candidate record SHA-256: bf3135b1d9703c266a77b787887d88a43aec196d125bbc88145d0cf6ecaa8ac4
semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON