Question 25983: The Minimum Cut Is Selected from the Complete Cut Family

Original question and givens

Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

• G001 — Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).

Definition and scope

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

• mko-max-flow-min-cut-theorem

Step-by-step derivation

source roles = [mfmc-minimum-cut-audit]

derived = [[cut_rows, minimum_capacity, minimum_cuts, certificate]]

source roles = [[s, a0, a1, a2, a3, a4, t], [[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], [9, 9, 10, 10, 11, 11, 12, 12, 13, 13], null, s, t]

derived = [[a0, a1, a2, a3, a4, s, t], [[a0, t], [a1, t], [a2, t], [a3, t], [a4, t], [s, a0], [s, a1], [s, a2], [s, a3], [s, a4]], [9, 10, 11, 12, 13, 9, 10, 11, 12, 13], null]

MFMC cut mask0 OUTWARD capacity := sum(9, 10, 11, 12, 13)

MFMC cut mask0 OUTWARD capacity = 55

source roles = [0, [a0, a1, a2, a3, a4], [s]]

derived = [[5, 6, 7, 8, 9], [], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55]

MFMC cut mask1 OUTWARD capacity := sum(9, 10, 11, 12, 13)

MFMC cut mask1 OUTWARD capacity = 55

source roles = [1, [a0, a1, a2, a3, a4], [a0, s]]

derived = [[0, 6, 7, 8, 9], [], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55]

MFMC cut mask2 OUTWARD capacity := sum(10, 9, 11, 12, 13)

MFMC cut mask2 OUTWARD capacity = 55

source roles = [2, [a0, a1, a2, a3, a4], [a1, s]]

derived = [[1, 5, 7, 8, 9], [], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55]

MFMC cut mask3 OUTWARD capacity := sum(9, 10, 11, 12, 13)

MFMC cut mask3 OUTWARD capacity = 55

source roles = [3, [a0, a1, a2, a3, a4], [a0, a1, s]]

derived = [[0, 1, 7, 8, 9], [], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55]

MFMC cut mask4 OUTWARD capacity := sum(11, 9, 10, 12, 13)

MFMC cut mask4 OUTWARD capacity = 55

source roles = [4, [a0, a1, a2, a3, a4], [a2, s]]

derived = [[2, 5, 6, 8, 9], [], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55]

MFMC cut mask5 OUTWARD capacity := sum(9, 11, 10, 12, 13)

MFMC cut mask5 OUTWARD capacity = 55

source roles = [5, [a0, a1, a2, a3, a4], [a0, a2, s]]

derived = [[0, 2, 6, 8, 9], [], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55]

MFMC cut mask6 OUTWARD capacity := sum(10, 11, 9, 12, 13)

MFMC cut mask6 OUTWARD capacity = 55

source roles = [6, [a0, a1, a2, a3, a4], [a1, a2, s]]

derived = [[1, 2, 5, 8, 9], [], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55]

MFMC cut mask7 OUTWARD capacity := sum(9, 10, 11, 12, 13)

MFMC cut mask7 OUTWARD capacity = 55

source roles = [7, [a0, a1, a2, a3, a4], [a0, a1, a2, s]]

derived = [[0, 1, 2, 8, 9], [], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55]

MFMC cut mask8 OUTWARD capacity := sum(12, 9, 10, 11, 13)

MFMC cut mask8 OUTWARD capacity = 55

source roles = [8, [a0, a1, a2, a3, a4], [a3, s]]

derived = [[3, 5, 6, 7, 9], [], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55]

MFMC cut mask9 OUTWARD capacity := sum(9, 12, 10, 11, 13)

MFMC cut mask9 OUTWARD capacity = 55

source roles = [9, [a0, a1, a2, a3, a4], [a0, a3, s]]

derived = [[0, 3, 6, 7, 9], [], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55]

MFMC cut mask10 OUTWARD capacity := sum(10, 12, 9, 11, 13)

MFMC cut mask10 OUTWARD capacity = 55

source roles = [10, [a0, a1, a2, a3, a4], [a1, a3, s]]

derived = [[1, 3, 5, 7, 9], [], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55]

MFMC cut mask11 OUTWARD capacity := sum(9, 10, 12, 11, 13)

MFMC cut mask11 OUTWARD capacity = 55

source roles = [11, [a0, a1, a2, a3, a4], [a0, a1, a3, s]]

derived = [[0, 1, 3, 7, 9], [], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55]

MFMC cut mask12 OUTWARD capacity := sum(11, 12, 9, 10, 13)

MFMC cut mask12 OUTWARD capacity = 55

source roles = [12, [a0, a1, a2, a3, a4], [a2, a3, s]]

derived = [[2, 3, 5, 6, 9], [], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55]

MFMC cut mask13 OUTWARD capacity := sum(9, 11, 12, 10, 13)

MFMC cut mask13 OUTWARD capacity = 55

source roles = [13, [a0, a1, a2, a3, a4], [a0, a2, a3, s]]

derived = [[0, 2, 3, 6, 9], [], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55]

MFMC cut mask14 OUTWARD capacity := sum(10, 11, 12, 9, 13)

MFMC cut mask14 OUTWARD capacity = 55

source roles = [14, [a0, a1, a2, a3, a4], [a1, a2, a3, s]]

derived = [[1, 2, 3, 5, 9], [], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55]

MFMC cut mask15 OUTWARD capacity := sum(9, 10, 11, 12, 13)

MFMC cut mask15 OUTWARD capacity = 55

source roles = [15, [a0, a1, a2, a3, a4], [a0, a1, a2, a3, s]]

derived = [[0, 1, 2, 3, 9], [], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55]

MFMC cut mask16 OUTWARD capacity := sum(13, 9, 10, 11, 12)

MFMC cut mask16 OUTWARD capacity = 55

source roles = [16, [a0, a1, a2, a3, a4], [a4, s]]

derived = [[4, 5, 6, 7, 8], [], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55]

MFMC cut mask17 OUTWARD capacity := sum(9, 13, 10, 11, 12)

MFMC cut mask17 OUTWARD capacity = 55

source roles = [17, [a0, a1, a2, a3, a4], [a0, a4, s]]

derived = [[0, 4, 6, 7, 8], [], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55]

MFMC cut mask18 OUTWARD capacity := sum(10, 13, 9, 11, 12)

MFMC cut mask18 OUTWARD capacity = 55

source roles = [18, [a0, a1, a2, a3, a4], [a1, a4, s]]

derived = [[1, 4, 5, 7, 8], [], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55]

MFMC cut mask19 OUTWARD capacity := sum(9, 10, 13, 11, 12)

MFMC cut mask19 OUTWARD capacity = 55

source roles = [19, [a0, a1, a2, a3, a4], [a0, a1, a4, s]]

derived = [[0, 1, 4, 7, 8], [], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55]

MFMC cut mask20 OUTWARD capacity := sum(11, 13, 9, 10, 12)

MFMC cut mask20 OUTWARD capacity = 55

source roles = [20, [a0, a1, a2, a3, a4], [a2, a4, s]]

derived = [[2, 4, 5, 6, 8], [], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55]

MFMC cut mask21 OUTWARD capacity := sum(9, 11, 13, 10, 12)

MFMC cut mask21 OUTWARD capacity = 55

source roles = [21, [a0, a1, a2, a3, a4], [a0, a2, a4, s]]

derived = [[0, 2, 4, 6, 8], [], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55]

MFMC cut mask22 OUTWARD capacity := sum(10, 11, 13, 9, 12)

MFMC cut mask22 OUTWARD capacity = 55

source roles = [22, [a0, a1, a2, a3, a4], [a1, a2, a4, s]]

derived = [[1, 2, 4, 5, 8], [], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55]

MFMC cut mask23 OUTWARD capacity := sum(9, 10, 11, 13, 12)

MFMC cut mask23 OUTWARD capacity = 55

source roles = [23, [a0, a1, a2, a3, a4], [a0, a1, a2, a4, s]]

derived = [[0, 1, 2, 4, 8], [], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55]

MFMC cut mask24 OUTWARD capacity := sum(12, 13, 9, 10, 11)

MFMC cut mask24 OUTWARD capacity = 55

source roles = [24, [a0, a1, a2, a3, a4], [a3, a4, s]]

derived = [[3, 4, 5, 6, 7], [], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55]

MFMC cut mask25 OUTWARD capacity := sum(9, 12, 13, 10, 11)

MFMC cut mask25 OUTWARD capacity = 55

source roles = [25, [a0, a1, a2, a3, a4], [a0, a3, a4, s]]

derived = [[0, 3, 4, 6, 7], [], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55]

MFMC cut mask26 OUTWARD capacity := sum(10, 12, 13, 9, 11)

MFMC cut mask26 OUTWARD capacity = 55

source roles = [26, [a0, a1, a2, a3, a4], [a1, a3, a4, s]]

derived = [[1, 3, 4, 5, 7], [], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55]

MFMC cut mask27 OUTWARD capacity := sum(9, 10, 12, 13, 11)

MFMC cut mask27 OUTWARD capacity = 55

source roles = [27, [a0, a1, a2, a3, a4], [a0, a1, a3, a4, s]]

derived = [[0, 1, 3, 4, 7], [], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55]

MFMC cut mask28 OUTWARD capacity := sum(11, 12, 13, 9, 10)

MFMC cut mask28 OUTWARD capacity = 55

source roles = [28, [a0, a1, a2, a3, a4], [a2, a3, a4, s]]

derived = [[2, 3, 4, 5, 6], [], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55]

MFMC cut mask29 OUTWARD capacity := sum(9, 11, 12, 13, 10)

MFMC cut mask29 OUTWARD capacity = 55

source roles = [29, [a0, a1, a2, a3, a4], [a0, a2, a3, a4, s]]

derived = [[0, 2, 3, 4, 6], [], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55]

MFMC cut mask30 OUTWARD capacity := sum(10, 11, 12, 13, 9)

MFMC cut mask30 OUTWARD capacity = 55

source roles = [30, [a0, a1, a2, a3, a4], [a1, a2, a3, a4, s]]

derived = [[1, 2, 3, 4, 5], [], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55]

MFMC cut mask31 OUTWARD capacity := sum(9, 10, 11, 12, 13)

MFMC cut mask31 OUTWARD capacity = 55

source roles = [31, [a0, a1, a2, a3, a4], [a0, a1, a2, a3, a4, s]]

derived = [[0, 1, 2, 3, 4], [], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]

source roles = [[a0, a1, a2, a3, a4, s, t], [a0, a1, a2, a3, a4]]

derived = [32, 32, 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]]]

Reverse checks

• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Recompute MFMC cut mask31 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Recompute MFMC cut mask30 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Recompute MFMC cut mask29 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Recompute MFMC cut mask28 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Recompute MFMC cut mask27 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Recompute MFMC cut mask26 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Recompute MFMC cut mask25 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Recompute MFMC cut mask24 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Recompute MFMC cut mask23 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Recompute MFMC cut mask22 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Recompute MFMC cut mask21 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Recompute MFMC cut mask20 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Recompute MFMC cut mask19 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Recompute MFMC cut mask18 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Recompute MFMC cut mask17 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute MFMC cut mask16 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Recompute MFMC cut mask15 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Recompute MFMC cut mask14 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Recompute MFMC cut mask13 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Recompute MFMC cut mask12 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute MFMC cut mask11 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute MFMC cut mask10 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Recompute MFMC cut mask9 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute MFMC cut mask8 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute MFMC cut mask7 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute MFMC cut mask6 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute MFMC cut mask5 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute MFMC cut mask4 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute MFMC cut mask3 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• 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 — Recompute MFMC cut mask2 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• 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 — Recompute MFMC cut mask1 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• 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 — Recompute MFMC cut mask0 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• 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

([[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)

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.