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Question 26118: Compose A-to-B then B-to-C without Reversing the Reduction

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

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R0051-26118 · q-20261015-advance-20261115-r07-many-one-reduction-0068

Original question and givens

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).

  • G001 — 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).

Definition and scope

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

Step-by-step derivation

S001 · implies

Before

source roles = [mreduction-composition-direction]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · Uses: G001, S001

S003 · implies

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · Uses: G001, S002

S004 · implies

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · Uses: G001, S003

S005 · implies

Before

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

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · Uses: G001, S004

S006 · implies

Before

source roles = [Reduction ACTUAL B-after-A composition, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 11], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11]]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · Uses: G001, S005

S007 · implies

Before

source roles = [Reduction ACTUAL B-after-A composition, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [11], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [10, 11], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11]]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · Uses: G001, S006

S008 · implies

Before

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

After

derived = [[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]

Conditions: none

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

Step identifiers and dependencies

Rule: Reduction_GENERAL_composition_direction_second_after_first · Uses: G001, S007

Reverse checks

  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

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

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

This question review and source binding

Review date: 2026-10-26

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

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: d7f82e3d40b6d531a2e66cb0c54090c1178aff766c90cd6deb06858c9166b123
  • candidate record SHA-256: 3be105bb059f3412d5e13f1bdf201d48bf7d703e3ba57e0501acea70bcad8b62
  • semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3
  • question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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