Total Iff Maps, Directed Composition and Forward Decider Transfer
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261115-r07-many-one-reduction-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2], target carrier=[0, 1, 2], aligned mapping=[0, 1, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, null, No]], No, [2, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, null, No]], No, [2, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
2. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
3. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
4. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
5. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
6. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, null, No]], No, [7, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, null, No]], No, [7, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
7. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
8. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
9. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
10. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
11. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, null, No]], No, [12, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, null, No]], No, [12, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
12. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
13. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
14. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
15. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
16. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, null, No]], No, [17, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, null, No]], No, [17, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
17. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
18. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
19. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
20. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
21. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, null, No]], No, [22, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, null, No]], No, [22, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
22. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
23. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes], [22, 2, Yes], [23, 0, Yes], [24, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes], [22, 2, Yes], [23, 0, Yes], [24, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
24. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, 0, Yes], [23, 1, Yes], [24, 2, Yes], [25, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, 0, Yes], [23, 1, Yes], [24, 2, Yes], [25, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
25. Complete Total Mapping Qualification Precedes Membership Equivalence
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes], [24, 0, Yes], [25, 1, Yes], [26, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)
For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes], [24, 0, Yes], [25, 1, Yes], [26, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate
26. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2], target carrier=[0, 1], aligned mapping=[0, 1, 0], source yes-set=[1], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
27. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
28. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
29. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
30. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
31. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
32. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
33. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
34. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
35. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
36. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
37. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13], [1], [[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]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
38. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13], [1], [[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, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13], [1], [[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, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
39. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15], [1], [[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], [15, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1], [[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], [15, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
40. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
41. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
42. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
43. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
44. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
45. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
46. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
47. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
48. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
49. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
50. A Many-One Reduction Preserves Both Yes and No Membership
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).
View answer and explanation
([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)
Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-iff-audit · candidate
51. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1], [1], [11]], aligned maps A-to-B=[0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
52. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
53. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3], [1], [11]], aligned maps A-to-B=[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).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
54. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
55. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5], [1], [11]], aligned maps A-to-B=[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).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
56. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
57. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7], [1], [11]], aligned maps A-to-B=[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).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
58. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
59. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9], [1], [11]], aligned maps A-to-B=[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).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
60. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
61. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11], [1], [11]], aligned maps A-to-B=[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).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
62. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 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], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
63. Compose A-to-B then B-to-C without Reversing the Reduction
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).
View answer and explanation
([[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)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
64. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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], [15, Yes, 1, Yes, 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, 11], [[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], [15, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, 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, 11], [[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], [15, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
65. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15], [1], [11]], aligned maps A-to-B=[0, 1, 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).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
66. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, 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, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, 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, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
67. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 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).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
68. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, 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, 11, 10, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, 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, 11, 10, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
69. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 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).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
70. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, 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, 11, 10, 11, 10, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, 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, 11, 10, 11, 10, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
71. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 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).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
72. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, 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, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, 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, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
73. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
74. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, 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, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, 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, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
75. Compose A-to-B then B-to-C without Reversing the Reduction
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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).
View answer and explanation
([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes], [26, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)
Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, 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, 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], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes], [26, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-composition-direction · candidate
76. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2], target carrier=[0, 1], aligned mapping=[0, 1, 0], source yes-set=[1], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
77. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
78. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, null], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
79. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
80. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(valid_transfer, [[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]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[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]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
81. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(reverse_misuse, [[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]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[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]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
82. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(partial_map, [[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, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[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, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
83. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
84. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(valid_transfer, [[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]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[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]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
85. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(reverse_misuse, [[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]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[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]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
86. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(partial_map, [[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, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[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, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
87. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
88. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(valid_transfer, [[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, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[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, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
89. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(reverse_misuse, [[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], [15, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[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], [15, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
90. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(partial_map, [[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], [15, Yes, 1, Yes, Yes], [16, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[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], [15, Yes, 1, Yes, Yes], [16, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
91. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
92. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(valid_transfer, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
93. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(reverse_misuse, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
94. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(partial_map, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
95. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
96. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(valid_transfer, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
97. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(reverse_misuse, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
98. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(partial_map, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
99. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No], [22, No, 0, Yes, No], [23, No, 0, Yes, No], [24, No, 0, Yes, No], [25, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No], [22, No, 0, Yes, No], [23, No, 0, Yes, No], [24, No, 0, Yes, No], [25, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
100. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider
Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).
All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.
View answer and explanation
(valid_transfer, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)
Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[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], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).
Reviewed step-by-step derivation
Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate
Canonical question JSONL · English JSONL · Validation report