# Question 26113: Compose A-to-B then B-to-C without Reversing the Reduction

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0051-26113 · q-20261015-advance-20261115-r07-many-one-reduction-0063



## Original question and givens

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).


- G001 — Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).





## 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-many-one-reduction](https://ocme.evemisslab.com/en/mko/mko-many-one-reduction/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [mreduction-composition-direction]

After

derived = [[first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate]]

Conditions: none

List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference.

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [Reduction ORIGINAL A-to-B, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]]

After

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [[0, 0, Yes], [1, 1, Yes], [2, 0, Yes], [3, 1, Yes], [4, 0, Yes], [5, 1, Yes], [6, 0, Yes], [7, 1, Yes], [8, 0, Yes], [9, 1, Yes], [10, 0, Yes], [11, 1, Yes], [12, 0, Yes], [13, 1, Yes], [14, 0, Yes]], Yes, null]

Conditions: none

Sort source values numerically with their aligned images,and target values numerically.Null,outside and Boolean images are not valid integer target values;False is not0.Qualification retains actual raw images and the first invalid row,not an invented total map.

Step identifiers and dependenciesRule: Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · Uses: G001, S001





S003 · impliesBefore

source roles = [Reduction ORIGINAL A-to-B, [1, 3, 5, 7, 9, 11, 13], [1], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]]

After

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

Conditions: none

For every x audit x in A iff f(x) in B,in both yes and no directions.Invalid images have null target predicates,notFalse.Totality and full membership equivalence jointly qualify this finite reduction.

Step identifiers and dependenciesRule: Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · Uses: G001, S002





S004 · impliesBefore

source roles = [Reduction ORIGINAL B-to-C, [0, 1], [10, 11], [10, 11]]

After

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

Conditions: none

Sort source values numerically with their aligned images,and target values numerically.Null,outside and Boolean images are not valid integer target values;False is not0.Qualification retains actual raw images and the first invalid row,not an invented total map.

Step identifiers and dependenciesRule: Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · Uses: G001, S003





S005 · impliesBefore

source roles = [Reduction ORIGINAL B-to-C, [1], [11], [0, 1], [10, 11], [10, 11]]

After

derived = [[[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], Yes, Yes]

Conditions: none

For every x audit x in A iff f(x) in B,in both yes and no directions.Invalid images have null target predicates,notFalse.Totality and full membership equivalence jointly qualify this finite reduction.

Step identifiers and dependenciesRule: Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · Uses: G001, S004





S006 · impliesBefore

source roles = [Reduction ACTUAL B-after-A composition, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [10, 11], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10]]

After

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [10, 11], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, 10, Yes], [1, 11, Yes], [2, 10, Yes], [3, 11, Yes], [4, 10, Yes], [5, 11, Yes], [6, 10, Yes], [7, 11, Yes], [8, 10, Yes], [9, 11, Yes], [10, 10, Yes], [11, 11, Yes], [12, 10, Yes], [13, 11, Yes], [14, 10, Yes]], Yes, null]

Conditions: none

Sort source values numerically with their aligned images,and target values numerically.Null,outside and Boolean images are not valid integer target values;False is not0.Qualification retains actual raw images and the first invalid row,not an invented total map.

Step identifiers and dependenciesRule: Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · Uses: G001, S005





S007 · impliesBefore

source roles = [Reduction ACTUAL B-after-A composition, [1, 3, 5, 7, 9, 11, 13], [11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [10, 11], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10]]

After

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

Conditions: none

For every x audit x in A iff f(x) in B,in both yes and no directions.Invalid images have null target predicates,notFalse.Totality and full membership equivalence jointly qualify this finite reduction.

Step identifiers and dependenciesRule: Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · Uses: G001, S006





S008 · impliesBefore

source roles = [[[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]]]

After

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

Conditions: none

For any x,apply f_AB then f_BC:x in A iff f_AB(x) in B iff f_BC(f_AB(x)) in C.Totality makes each application defined.The direction is A<=mB<=mC to A<=mC,not reverse/Turing reduction or infinite computability.

Step identifiers and dependenciesRule: Reduction_GENERAL_composition_direction_second_after_first · Uses: G001, S007









## Reverse checks


- 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, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

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: 5afe1cea2a3096c227d8ec603dc4b201712012de1ce1d51dfd7f7c08606cf5be
- candidate record SHA-256: 60592f1a7d2d99952675639b623b281befb4160a49ae46c95693dc8737bd1e76
- semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
- question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r07-many-one-reduction-0063.json)