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R0050-25974 · q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0049
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=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29], candidate flows=[24, 24, 25, 25, 26, 26, 27, 27, 28, 28]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, 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=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29], candidate flows=[24, 24, 25, 25, 26, 26, 27, 27, 28, 28]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).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.
Before
source roles = [mfmc-flow-cut-upper-bound]
After
derived = [[source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate]]
Conditions: none
List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
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]], [25, 25, 26, 26, 27, 27, 28, 28, 29, 29], [24, 24, 25, 25, 26, 26, 27, 27, 28, 28], s, t]
After
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]], [25, 26, 27, 28, 29, 25, 26, 27, 28, 29], [24, 25, 26, 27, 28, 24, 25, 26, 27, 28]]
Conditions: none
Use2..12 vertices.Zip directed edges with capacity/flow by original index before sorting;reverse arcs remain distinct.Source and sink differ;reject loops/duplicates but allow zero capacities,no edges and isolated vertices.
Rule: MFMC_complete_directed_network_ZIP_before_source_order · Uses: G001, S001
Before
MFMC cut mask0 OUTWARD capacity := sum(25, 26, 27, 28, 29)
After
MFMC cut mask0 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [0, [a0, a1, a2, a3, a4], [s]]
After
derived = [[5, 6, 7, 8, 9], [], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S003
Before
MFMC cut mask1 OUTWARD capacity := sum(25, 26, 27, 28, 29)
After
MFMC cut mask1 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [1, [a0, a1, a2, a3, a4], [a0, s]]
After
derived = [[0, 6, 7, 8, 9], [], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S005
Before
MFMC cut mask2 OUTWARD capacity := sum(26, 25, 27, 28, 29)
After
MFMC cut mask2 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
source roles = [2, [a0, a1, a2, a3, a4], [a1, s]]
After
derived = [[1, 5, 7, 8, 9], [], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S007
Before
MFMC cut mask3 OUTWARD capacity := sum(25, 26, 27, 28, 29)
After
MFMC cut mask3 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
source roles = [3, [a0, a1, a2, a3, a4], [a0, a1, s]]
After
derived = [[0, 1, 7, 8, 9], [], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S009
Before
MFMC cut mask4 OUTWARD capacity := sum(27, 25, 26, 28, 29)
After
MFMC cut mask4 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
source roles = [4, [a0, a1, a2, a3, a4], [a2, s]]
After
derived = [[2, 5, 6, 8, 9], [], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S011
Before
MFMC cut mask5 OUTWARD capacity := sum(25, 27, 26, 28, 29)
After
MFMC cut mask5 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
source roles = [5, [a0, a1, a2, a3, a4], [a0, a2, s]]
After
derived = [[0, 2, 6, 8, 9], [], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S013
Before
MFMC cut mask6 OUTWARD capacity := sum(26, 27, 25, 28, 29)
After
MFMC cut mask6 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
source roles = [6, [a0, a1, a2, a3, a4], [a1, a2, s]]
After
derived = [[1, 2, 5, 8, 9], [], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S015
Before
MFMC cut mask7 OUTWARD capacity := sum(25, 26, 27, 28, 29)
After
MFMC cut mask7 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
source roles = [7, [a0, a1, a2, a3, a4], [a0, a1, a2, s]]
After
derived = [[0, 1, 2, 8, 9], [], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S017
Before
MFMC cut mask8 OUTWARD capacity := sum(28, 25, 26, 27, 29)
After
MFMC cut mask8 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
source roles = [8, [a0, a1, a2, a3, a4], [a3, s]]
After
derived = [[3, 5, 6, 7, 9], [], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S019
Before
MFMC cut mask9 OUTWARD capacity := sum(25, 28, 26, 27, 29)
After
MFMC cut mask9 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
source roles = [9, [a0, a1, a2, a3, a4], [a0, a3, s]]
After
derived = [[0, 3, 6, 7, 9], [], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S021
Before
MFMC cut mask10 OUTWARD capacity := sum(26, 28, 25, 27, 29)
After
MFMC cut mask10 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
source roles = [10, [a0, a1, a2, a3, a4], [a1, a3, s]]
After
derived = [[1, 3, 5, 7, 9], [], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S023
Before
MFMC cut mask11 OUTWARD capacity := sum(25, 26, 28, 27, 29)
After
MFMC cut mask11 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
source roles = [11, [a0, a1, a2, a3, a4], [a0, a1, a3, s]]
After
derived = [[0, 1, 3, 7, 9], [], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S025
Before
MFMC cut mask12 OUTWARD capacity := sum(27, 28, 25, 26, 29)
After
MFMC cut mask12 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S026
Before
source roles = [12, [a0, a1, a2, a3, a4], [a2, a3, s]]
After
derived = [[2, 3, 5, 6, 9], [], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S027
Before
MFMC cut mask13 OUTWARD capacity := sum(25, 27, 28, 26, 29)
After
MFMC cut mask13 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
source roles = [13, [a0, a1, a2, a3, a4], [a0, a2, a3, s]]
After
derived = [[0, 2, 3, 6, 9], [], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S029
Before
MFMC cut mask14 OUTWARD capacity := sum(26, 27, 28, 25, 29)
After
MFMC cut mask14 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
source roles = [14, [a0, a1, a2, a3, a4], [a1, a2, a3, s]]
After
derived = [[1, 2, 3, 5, 9], [], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S031
Before
MFMC cut mask15 OUTWARD capacity := sum(25, 26, 27, 28, 29)
After
MFMC cut mask15 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S032
Before
source roles = [15, [a0, a1, a2, a3, a4], [a0, a1, a2, a3, s]]
After
derived = [[0, 1, 2, 3, 9], [], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S033
Before
MFMC cut mask16 OUTWARD capacity := sum(29, 25, 26, 27, 28)
After
MFMC cut mask16 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
source roles = [16, [a0, a1, a2, a3, a4], [a4, s]]
After
derived = [[4, 5, 6, 7, 8], [], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S035
Before
MFMC cut mask17 OUTWARD capacity := sum(25, 29, 26, 27, 28)
After
MFMC cut mask17 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
source roles = [17, [a0, a1, a2, a3, a4], [a0, a4, s]]
After
derived = [[0, 4, 6, 7, 8], [], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S037
Before
MFMC cut mask18 OUTWARD capacity := sum(26, 29, 25, 27, 28)
After
MFMC cut mask18 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
source roles = [18, [a0, a1, a2, a3, a4], [a1, a4, s]]
After
derived = [[1, 4, 5, 7, 8], [], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S039
Before
MFMC cut mask19 OUTWARD capacity := sum(25, 26, 29, 27, 28)
After
MFMC cut mask19 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
source roles = [19, [a0, a1, a2, a3, a4], [a0, a1, a4, s]]
After
derived = [[0, 1, 4, 7, 8], [], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S041
Before
MFMC cut mask20 OUTWARD capacity := sum(27, 29, 25, 26, 28)
After
MFMC cut mask20 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
source roles = [20, [a0, a1, a2, a3, a4], [a2, a4, s]]
After
derived = [[2, 4, 5, 6, 8], [], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S043
Before
MFMC cut mask21 OUTWARD capacity := sum(25, 27, 29, 26, 28)
After
MFMC cut mask21 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S044
Before
source roles = [21, [a0, a1, a2, a3, a4], [a0, a2, a4, s]]
After
derived = [[0, 2, 4, 6, 8], [], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S045
Before
MFMC cut mask22 OUTWARD capacity := sum(26, 27, 29, 25, 28)
After
MFMC cut mask22 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
source roles = [22, [a0, a1, a2, a3, a4], [a1, a2, a4, s]]
After
derived = [[1, 2, 4, 5, 8], [], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S047
Before
MFMC cut mask23 OUTWARD capacity := sum(25, 26, 27, 29, 28)
After
MFMC cut mask23 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
source roles = [23, [a0, a1, a2, a3, a4], [a0, a1, a2, a4, s]]
After
derived = [[0, 1, 2, 4, 8], [], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S049
Before
MFMC cut mask24 OUTWARD capacity := sum(28, 29, 25, 26, 27)
After
MFMC cut mask24 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S050
Before
source roles = [24, [a0, a1, a2, a3, a4], [a3, a4, s]]
After
derived = [[3, 4, 5, 6, 7], [], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S051
Before
MFMC cut mask25 OUTWARD capacity := sum(25, 28, 29, 26, 27)
After
MFMC cut mask25 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
source roles = [25, [a0, a1, a2, a3, a4], [a0, a3, a4, s]]
After
derived = [[0, 3, 4, 6, 7], [], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S053
Before
MFMC cut mask26 OUTWARD capacity := sum(26, 28, 29, 25, 27)
After
MFMC cut mask26 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
source roles = [26, [a0, a1, a2, a3, a4], [a1, a3, a4, s]]
After
derived = [[1, 3, 4, 5, 7], [], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S055
Before
MFMC cut mask27 OUTWARD capacity := sum(25, 26, 28, 29, 27)
After
MFMC cut mask27 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S056
Before
source roles = [27, [a0, a1, a2, a3, a4], [a0, a1, a3, a4, s]]
After
derived = [[0, 1, 3, 4, 7], [], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S057
Before
MFMC cut mask28 OUTWARD capacity := sum(27, 28, 29, 25, 26)
After
MFMC cut mask28 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
source roles = [28, [a0, a1, a2, a3, a4], [a2, a3, a4, s]]
After
derived = [[2, 3, 4, 5, 6], [], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S059
Before
MFMC cut mask29 OUTWARD capacity := sum(25, 27, 28, 29, 26)
After
MFMC cut mask29 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
source roles = [29, [a0, a1, a2, a3, a4], [a0, a2, a3, a4, s]]
After
derived = [[0, 2, 3, 4, 6], [], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S061
Before
MFMC cut mask30 OUTWARD capacity := sum(26, 27, 28, 29, 25)
After
MFMC cut mask30 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S062
Before
source roles = [30, [a0, a1, a2, a3, a4], [a1, a2, a3, a4, s]]
After
derived = [[1, 2, 3, 4, 5], [], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S063
Before
MFMC cut mask31 OUTWARD capacity := sum(25, 26, 27, 28, 29)
After
MFMC cut mask31 OUTWARD capacity = 135
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
source roles = [31, [a0, a1, a2, a3, a4], [a0, a1, a2, a3, a4, s]]
After
derived = [[0, 1, 2, 3, 4], [], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]
Conditions: none
S contains source and excludes sink.Enumerate every other-vertex subset;sort inside S by UTF16 but retain numeric-mask outer order.Capacity sums OUTWARD edges only,not incoming edges;ties are not re-sorted in English.
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S065
Before
source roles = [[a0, a1, a2, a3, a4, s, t], [a0, a1, a2, a3, a4]]
After
derived = [32, 32, 135, [[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]]]
Conditions: none
Every s–t cut uniquely corresponds to a subset of other vertices,so all2^(n−2) cuts are compared.Select the minimum and all minimizers from this same complete family,retaining mask tie order.
Rule: MFMC_ALL_cut_family_complete_minimum_and_ALL_ties · Uses: G001, S066
Before
MFMC vertex a0 ALL incoming := sum(24)
After
MFMC vertex a0 ALL incoming = 24
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
MFMC vertex a0 ALL outgoing := sum(24)
After
MFMC vertex a0 ALL outgoing = 24
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S068
Before
MFMC vertex a1 ALL incoming := sum(25)
After
MFMC vertex a1 ALL incoming = 25
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
MFMC vertex a1 ALL outgoing := sum(25)
After
MFMC vertex a1 ALL outgoing = 25
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
MFMC vertex a2 ALL incoming := sum(26)
After
MFMC vertex a2 ALL incoming = 26
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
MFMC vertex a2 ALL outgoing := sum(26)
After
MFMC vertex a2 ALL outgoing = 26
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
MFMC vertex a3 ALL incoming := sum(27)
After
MFMC vertex a3 ALL incoming = 27
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
MFMC vertex a3 ALL outgoing := sum(27)
After
MFMC vertex a3 ALL outgoing = 27
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S074
Before
MFMC vertex a4 ALL incoming := sum(28)
After
MFMC vertex a4 ALL incoming = 28
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
MFMC vertex a4 ALL outgoing := sum(28)
After
MFMC vertex a4 ALL outgoing = 28
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
MFMC vertex s ALL incoming := sum()
After
MFMC vertex s ALL incoming = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
MFMC vertex s ALL outgoing := sum(24, 25, 26, 27, 28)
After
MFMC vertex s ALL outgoing = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
MFMC vertex t ALL incoming := sum(24, 25, 26, 27, 28)
After
MFMC vertex t ALL incoming = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
MFMC vertex t ALL outgoing := sum()
After
MFMC vertex t ALL outgoing = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S080
Before
MFMC SOURCE net out not gross := sub(130, 0)
After
MFMC SOURCE net out not gross = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
MFMC SINK net in not gross := sub(130, 0)
After
MFMC SINK net in not gross = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
source roles = [[[a0, t], [a1, t], [a2, t], [a3, t], [a4, t], [s, a0], [s, a1], [s, a2], [s, a3], [s, a4]], [24, 25, 26, 27, 28, 24, 25, 26, 27, 28], [25, 26, 27, 28, 29, 25, 26, 27, 28, 29], s, t]
After
derived = [[[[a0, t], 24, 25, Yes], [[a1, t], 25, 26, Yes], [[a2, t], 26, 27, Yes], [[a3, t], 27, 28, Yes], [[a4, t], 28, 29, Yes], [[s, a0], 24, 25, Yes], [[s, a1], 25, 26, Yes], [[s, a2], 26, 27, Yes], [[s, a3], 27, 28, Yes], [[s, a4], 28, 29, Yes]], [[a0, 24, 24, Yes], [a1, 25, 25, Yes], [a2, 26, 26, Yes], [a3, 27, 27, Yes], [a4, 28, 28, Yes]], Yes, Yes, 130, 130, Yes, Yes]
Conditions: none
Feasibility requires every edge capacity bound,every internal conservation equation and equal source/sink NET values.Do not discard incoming-source or outgoing-sink arcs.A feasible flow may have negative net value;do not add value>=0.
Rule: MFMC_EVERY_capacity_EVERY_internal_balance_source_sink_NET_feasibility · Uses: G001, S083
Before
MFMC cut mask0 OUTWARD flow := sum(24, 25, 26, 27, 28)
After
MFMC cut mask0 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S084
Before
MFMC cut mask0 INWARD flow := sum()
After
MFMC cut mask0 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
MFMC cut mask0 NET flow := sub(130, 0)
After
MFMC cut mask0 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S086
Before
source roles = [[s], [5, 6, 7, 8, 9], [], 130]
After
derived = [[[s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S087
Before
MFMC cut mask1 OUTWARD flow := sum(24, 25, 26, 27, 28)
After
MFMC cut mask1 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
MFMC cut mask1 INWARD flow := sum()
After
MFMC cut mask1 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
MFMC cut mask1 NET flow := sub(130, 0)
After
MFMC cut mask1 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S090
Before
source roles = [[a0, s], [0, 6, 7, 8, 9], [], 130]
After
derived = [[[a0, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S091
Before
MFMC cut mask2 OUTWARD flow := sum(25, 24, 26, 27, 28)
After
MFMC cut mask2 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S092
Before
MFMC cut mask2 INWARD flow := sum()
After
MFMC cut mask2 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
MFMC cut mask2 NET flow := sub(130, 0)
After
MFMC cut mask2 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S094
Before
source roles = [[a1, s], [1, 5, 7, 8, 9], [], 130]
After
derived = [[[a1, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S095
Before
MFMC cut mask3 OUTWARD flow := sum(24, 25, 26, 27, 28)
After
MFMC cut mask3 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
MFMC cut mask3 INWARD flow := sum()
After
MFMC cut mask3 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
MFMC cut mask3 NET flow := sub(130, 0)
After
MFMC cut mask3 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S098
Before
source roles = [[a0, a1, s], [0, 1, 7, 8, 9], [], 130]
After
derived = [[[a0, a1, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S099
Before
MFMC cut mask4 OUTWARD flow := sum(26, 24, 25, 27, 28)
After
MFMC cut mask4 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S100
Before
MFMC cut mask4 INWARD flow := sum()
After
MFMC cut mask4 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
MFMC cut mask4 NET flow := sub(130, 0)
After
MFMC cut mask4 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S102
Before
source roles = [[a2, s], [2, 5, 6, 8, 9], [], 130]
After
derived = [[[a2, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S103
Before
MFMC cut mask5 OUTWARD flow := sum(24, 26, 25, 27, 28)
After
MFMC cut mask5 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S104
Before
MFMC cut mask5 INWARD flow := sum()
After
MFMC cut mask5 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S105
Before
MFMC cut mask5 NET flow := sub(130, 0)
After
MFMC cut mask5 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S106
Before
source roles = [[a0, a2, s], [0, 2, 6, 8, 9], [], 130]
After
derived = [[[a0, a2, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S107
Before
MFMC cut mask6 OUTWARD flow := sum(25, 26, 24, 27, 28)
After
MFMC cut mask6 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S108
Before
MFMC cut mask6 INWARD flow := sum()
After
MFMC cut mask6 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
MFMC cut mask6 NET flow := sub(130, 0)
After
MFMC cut mask6 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S110
Before
source roles = [[a1, a2, s], [1, 2, 5, 8, 9], [], 130]
After
derived = [[[a1, a2, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S111
Before
MFMC cut mask7 OUTWARD flow := sum(24, 25, 26, 27, 28)
After
MFMC cut mask7 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
MFMC cut mask7 INWARD flow := sum()
After
MFMC cut mask7 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
MFMC cut mask7 NET flow := sub(130, 0)
After
MFMC cut mask7 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S114
Before
source roles = [[a0, a1, a2, s], [0, 1, 2, 8, 9], [], 130]
After
derived = [[[a0, a1, a2, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S115
Before
MFMC cut mask8 OUTWARD flow := sum(27, 24, 25, 26, 28)
After
MFMC cut mask8 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S116
Before
MFMC cut mask8 INWARD flow := sum()
After
MFMC cut mask8 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S117
Before
MFMC cut mask8 NET flow := sub(130, 0)
After
MFMC cut mask8 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S118
Before
source roles = [[a3, s], [3, 5, 6, 7, 9], [], 130]
After
derived = [[[a3, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S119
Before
MFMC cut mask9 OUTWARD flow := sum(24, 27, 25, 26, 28)
After
MFMC cut mask9 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S120
Before
MFMC cut mask9 INWARD flow := sum()
After
MFMC cut mask9 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
MFMC cut mask9 NET flow := sub(130, 0)
After
MFMC cut mask9 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S122
Before
source roles = [[a0, a3, s], [0, 3, 6, 7, 9], [], 130]
After
derived = [[[a0, a3, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S123
Before
MFMC cut mask10 OUTWARD flow := sum(25, 27, 24, 26, 28)
After
MFMC cut mask10 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S124
Before
MFMC cut mask10 INWARD flow := sum()
After
MFMC cut mask10 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S125
Before
MFMC cut mask10 NET flow := sub(130, 0)
After
MFMC cut mask10 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S126
Before
source roles = [[a1, a3, s], [1, 3, 5, 7, 9], [], 130]
After
derived = [[[a1, a3, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S127
Before
MFMC cut mask11 OUTWARD flow := sum(24, 25, 27, 26, 28)
After
MFMC cut mask11 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S128
Before
MFMC cut mask11 INWARD flow := sum()
After
MFMC cut mask11 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S129
Before
MFMC cut mask11 NET flow := sub(130, 0)
After
MFMC cut mask11 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S130
Before
source roles = [[a0, a1, a3, s], [0, 1, 3, 7, 9], [], 130]
After
derived = [[[a0, a1, a3, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S131
Before
MFMC cut mask12 OUTWARD flow := sum(26, 27, 24, 25, 28)
After
MFMC cut mask12 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S132
Before
MFMC cut mask12 INWARD flow := sum()
After
MFMC cut mask12 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S133
Before
MFMC cut mask12 NET flow := sub(130, 0)
After
MFMC cut mask12 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S134
Before
source roles = [[a2, a3, s], [2, 3, 5, 6, 9], [], 130]
After
derived = [[[a2, a3, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S135
Before
MFMC cut mask13 OUTWARD flow := sum(24, 26, 27, 25, 28)
After
MFMC cut mask13 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S136
Before
MFMC cut mask13 INWARD flow := sum()
After
MFMC cut mask13 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S137
Before
MFMC cut mask13 NET flow := sub(130, 0)
After
MFMC cut mask13 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S138
Before
source roles = [[a0, a2, a3, s], [0, 2, 3, 6, 9], [], 130]
After
derived = [[[a0, a2, a3, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S139
Before
MFMC cut mask14 OUTWARD flow := sum(25, 26, 27, 24, 28)
After
MFMC cut mask14 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S140
Before
MFMC cut mask14 INWARD flow := sum()
After
MFMC cut mask14 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S141
Before
MFMC cut mask14 NET flow := sub(130, 0)
After
MFMC cut mask14 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S142
Before
source roles = [[a1, a2, a3, s], [1, 2, 3, 5, 9], [], 130]
After
derived = [[[a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S143
Before
MFMC cut mask15 OUTWARD flow := sum(24, 25, 26, 27, 28)
After
MFMC cut mask15 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S144
Before
MFMC cut mask15 INWARD flow := sum()
After
MFMC cut mask15 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S145
Before
MFMC cut mask15 NET flow := sub(130, 0)
After
MFMC cut mask15 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S146
Before
source roles = [[a0, a1, a2, a3, s], [0, 1, 2, 3, 9], [], 130]
After
derived = [[[a0, a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S147
Before
MFMC cut mask16 OUTWARD flow := sum(28, 24, 25, 26, 27)
After
MFMC cut mask16 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S148
Before
MFMC cut mask16 INWARD flow := sum()
After
MFMC cut mask16 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S149
Before
MFMC cut mask16 NET flow := sub(130, 0)
After
MFMC cut mask16 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S150
Before
source roles = [[a4, s], [4, 5, 6, 7, 8], [], 130]
After
derived = [[[a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S151
Before
MFMC cut mask17 OUTWARD flow := sum(24, 28, 25, 26, 27)
After
MFMC cut mask17 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S152
Before
MFMC cut mask17 INWARD flow := sum()
After
MFMC cut mask17 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S153
Before
MFMC cut mask17 NET flow := sub(130, 0)
After
MFMC cut mask17 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S154
Before
source roles = [[a0, a4, s], [0, 4, 6, 7, 8], [], 130]
After
derived = [[[a0, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S155
Before
MFMC cut mask18 OUTWARD flow := sum(25, 28, 24, 26, 27)
After
MFMC cut mask18 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S156
Before
MFMC cut mask18 INWARD flow := sum()
After
MFMC cut mask18 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S157
Before
MFMC cut mask18 NET flow := sub(130, 0)
After
MFMC cut mask18 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S158
Before
source roles = [[a1, a4, s], [1, 4, 5, 7, 8], [], 130]
After
derived = [[[a1, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S159
Before
MFMC cut mask19 OUTWARD flow := sum(24, 25, 28, 26, 27)
After
MFMC cut mask19 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S160
Before
MFMC cut mask19 INWARD flow := sum()
After
MFMC cut mask19 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S161
Before
MFMC cut mask19 NET flow := sub(130, 0)
After
MFMC cut mask19 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S162
Before
source roles = [[a0, a1, a4, s], [0, 1, 4, 7, 8], [], 130]
After
derived = [[[a0, a1, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S163
Before
MFMC cut mask20 OUTWARD flow := sum(26, 28, 24, 25, 27)
After
MFMC cut mask20 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S164
Before
MFMC cut mask20 INWARD flow := sum()
After
MFMC cut mask20 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S165
Before
MFMC cut mask20 NET flow := sub(130, 0)
After
MFMC cut mask20 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S166
Before
source roles = [[a2, a4, s], [2, 4, 5, 6, 8], [], 130]
After
derived = [[[a2, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S167
Before
MFMC cut mask21 OUTWARD flow := sum(24, 26, 28, 25, 27)
After
MFMC cut mask21 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S168
Before
MFMC cut mask21 INWARD flow := sum()
After
MFMC cut mask21 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S169
Before
MFMC cut mask21 NET flow := sub(130, 0)
After
MFMC cut mask21 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S170
Before
source roles = [[a0, a2, a4, s], [0, 2, 4, 6, 8], [], 130]
After
derived = [[[a0, a2, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S171
Before
MFMC cut mask22 OUTWARD flow := sum(25, 26, 28, 24, 27)
After
MFMC cut mask22 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S172
Before
MFMC cut mask22 INWARD flow := sum()
After
MFMC cut mask22 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S173
Before
MFMC cut mask22 NET flow := sub(130, 0)
After
MFMC cut mask22 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S174
Before
source roles = [[a1, a2, a4, s], [1, 2, 4, 5, 8], [], 130]
After
derived = [[[a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S175
Before
MFMC cut mask23 OUTWARD flow := sum(24, 25, 26, 28, 27)
After
MFMC cut mask23 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S176
Before
MFMC cut mask23 INWARD flow := sum()
After
MFMC cut mask23 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S177
Before
MFMC cut mask23 NET flow := sub(130, 0)
After
MFMC cut mask23 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S178
Before
source roles = [[a0, a1, a2, a4, s], [0, 1, 2, 4, 8], [], 130]
After
derived = [[[a0, a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S179
Before
MFMC cut mask24 OUTWARD flow := sum(27, 28, 24, 25, 26)
After
MFMC cut mask24 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S180
Before
MFMC cut mask24 INWARD flow := sum()
After
MFMC cut mask24 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S181
Before
MFMC cut mask24 NET flow := sub(130, 0)
After
MFMC cut mask24 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S182
Before
source roles = [[a3, a4, s], [3, 4, 5, 6, 7], [], 130]
After
derived = [[[a3, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S183
Before
MFMC cut mask25 OUTWARD flow := sum(24, 27, 28, 25, 26)
After
MFMC cut mask25 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S184
Before
MFMC cut mask25 INWARD flow := sum()
After
MFMC cut mask25 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S185
Before
MFMC cut mask25 NET flow := sub(130, 0)
After
MFMC cut mask25 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S186
Before
source roles = [[a0, a3, a4, s], [0, 3, 4, 6, 7], [], 130]
After
derived = [[[a0, a3, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S187
Before
MFMC cut mask26 OUTWARD flow := sum(25, 27, 28, 24, 26)
After
MFMC cut mask26 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S188
Before
MFMC cut mask26 INWARD flow := sum()
After
MFMC cut mask26 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S189
Before
MFMC cut mask26 NET flow := sub(130, 0)
After
MFMC cut mask26 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S190
Before
source roles = [[a1, a3, a4, s], [1, 3, 4, 5, 7], [], 130]
After
derived = [[[a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S191
Before
MFMC cut mask27 OUTWARD flow := sum(24, 25, 27, 28, 26)
After
MFMC cut mask27 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S192
Before
MFMC cut mask27 INWARD flow := sum()
After
MFMC cut mask27 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S193
Before
MFMC cut mask27 NET flow := sub(130, 0)
After
MFMC cut mask27 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S194
Before
source roles = [[a0, a1, a3, a4, s], [0, 1, 3, 4, 7], [], 130]
After
derived = [[[a0, a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S195
Before
MFMC cut mask28 OUTWARD flow := sum(26, 27, 28, 24, 25)
After
MFMC cut mask28 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S196
Before
MFMC cut mask28 INWARD flow := sum()
After
MFMC cut mask28 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S197
Before
MFMC cut mask28 NET flow := sub(130, 0)
After
MFMC cut mask28 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S198
Before
source roles = [[a2, a3, a4, s], [2, 3, 4, 5, 6], [], 130]
After
derived = [[[a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S199
Before
MFMC cut mask29 OUTWARD flow := sum(24, 26, 27, 28, 25)
After
MFMC cut mask29 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S200
Before
MFMC cut mask29 INWARD flow := sum()
After
MFMC cut mask29 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S201
Before
MFMC cut mask29 NET flow := sub(130, 0)
After
MFMC cut mask29 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S202
Before
source roles = [[a0, a2, a3, a4, s], [0, 2, 3, 4, 6], [], 130]
After
derived = [[[a0, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S203
Before
MFMC cut mask30 OUTWARD flow := sum(25, 26, 27, 28, 24)
After
MFMC cut mask30 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S204
Before
MFMC cut mask30 INWARD flow := sum()
After
MFMC cut mask30 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S205
Before
MFMC cut mask30 NET flow := sub(130, 0)
After
MFMC cut mask30 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S206
Before
source roles = [[a1, a2, a3, a4, s], [1, 2, 3, 4, 5], [], 130]
After
derived = [[[a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S207
Before
MFMC cut mask31 OUTWARD flow := sum(24, 25, 26, 27, 28)
After
MFMC cut mask31 OUTWARD flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S208
Before
MFMC cut mask31 INWARD flow := sum()
After
MFMC cut mask31 INWARD flow = 0
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S209
Before
MFMC cut mask31 NET flow := sub(130, 0)
After
MFMC cut mask31 NET flow = 130
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S210
Before
source roles = [[a0, a1, a2, a3, a4, s], [0, 1, 2, 3, 4], [], 130]
After
derived = [[[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]]
Conditions: none
Summing net outflows over S cancels internal arcs;internal conservation leaves the source net.Because inward flow is nonnegative and each outward flow is capacity-bounded,value=out−in<=out<=cut capacity,for every cut.
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S211
S212 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S211 · premise_and_result_audit — Recompute MFMC cut mask31 NET flow with exact rational arithmetic, not rounded decimals.S210 · premise_and_result_audit — Recompute MFMC cut mask31 INWARD flow with exact rational arithmetic, not rounded decimals.S209 · premise_and_result_audit — Recompute MFMC cut mask31 OUTWARD flow with exact rational arithmetic, not rounded decimals.S208 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S207 · premise_and_result_audit — Recompute MFMC cut mask30 NET flow with exact rational arithmetic, not rounded decimals.S206 · premise_and_result_audit — Recompute MFMC cut mask30 INWARD flow with exact rational arithmetic, not rounded decimals.S205 · premise_and_result_audit — Recompute MFMC cut mask30 OUTWARD flow with exact rational arithmetic, not rounded decimals.S204 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S203 · premise_and_result_audit — Recompute MFMC cut mask29 NET flow with exact rational arithmetic, not rounded decimals.S202 · premise_and_result_audit — Recompute MFMC cut mask29 INWARD flow with exact rational arithmetic, not rounded decimals.S201 · premise_and_result_audit — Recompute MFMC cut mask29 OUTWARD flow with exact rational arithmetic, not rounded decimals.S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S199 · premise_and_result_audit — Recompute MFMC cut mask28 NET flow with exact rational arithmetic, not rounded decimals.S198 · premise_and_result_audit — Recompute MFMC cut mask28 INWARD flow with exact rational arithmetic, not rounded decimals.S197 · premise_and_result_audit — Recompute MFMC cut mask28 OUTWARD flow with exact rational arithmetic, not rounded decimals.S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S195 · premise_and_result_audit — Recompute MFMC cut mask27 NET flow with exact rational arithmetic, not rounded decimals.S194 · premise_and_result_audit — Recompute MFMC cut mask27 INWARD flow with exact rational arithmetic, not rounded decimals.S193 · premise_and_result_audit — Recompute MFMC cut mask27 OUTWARD flow with exact rational arithmetic, not rounded decimals.S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S191 · premise_and_result_audit — Recompute MFMC cut mask26 NET flow with exact rational arithmetic, not rounded decimals.S190 · premise_and_result_audit — Recompute MFMC cut mask26 INWARD flow with exact rational arithmetic, not rounded decimals.S189 · premise_and_result_audit — Recompute MFMC cut mask26 OUTWARD flow with exact rational arithmetic, not rounded decimals.S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S187 · premise_and_result_audit — Recompute MFMC cut mask25 NET flow with exact rational arithmetic, not rounded decimals.S186 · premise_and_result_audit — Recompute MFMC cut mask25 INWARD flow with exact rational arithmetic, not rounded decimals.S185 · premise_and_result_audit — Recompute MFMC cut mask25 OUTWARD flow with exact rational arithmetic, not rounded decimals.S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S183 · premise_and_result_audit — Recompute MFMC cut mask24 NET flow with exact rational arithmetic, not rounded decimals.S182 · premise_and_result_audit — Recompute MFMC cut mask24 INWARD flow with exact rational arithmetic, not rounded decimals.S181 · premise_and_result_audit — Recompute MFMC cut mask24 OUTWARD flow with exact rational arithmetic, not rounded decimals.S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S179 · premise_and_result_audit — Recompute MFMC cut mask23 NET flow with exact rational arithmetic, not rounded decimals.S178 · premise_and_result_audit — Recompute MFMC cut mask23 INWARD flow with exact rational arithmetic, not rounded decimals.S177 · premise_and_result_audit — Recompute MFMC cut mask23 OUTWARD flow with exact rational arithmetic, not rounded decimals.S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S175 · premise_and_result_audit — Recompute MFMC cut mask22 NET flow with exact rational arithmetic, not rounded decimals.S174 · premise_and_result_audit — Recompute MFMC cut mask22 INWARD flow with exact rational arithmetic, not rounded decimals.S173 · premise_and_result_audit — Recompute MFMC cut mask22 OUTWARD flow with exact rational arithmetic, not rounded decimals.S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S171 · premise_and_result_audit — Recompute MFMC cut mask21 NET flow with exact rational arithmetic, not rounded decimals.S170 · premise_and_result_audit — Recompute MFMC cut mask21 INWARD flow with exact rational arithmetic, not rounded decimals.S169 · premise_and_result_audit — Recompute MFMC cut mask21 OUTWARD flow with exact rational arithmetic, not rounded decimals.S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S167 · premise_and_result_audit — Recompute MFMC cut mask20 NET flow with exact rational arithmetic, not rounded decimals.S166 · premise_and_result_audit — Recompute MFMC cut mask20 INWARD flow with exact rational arithmetic, not rounded decimals.S165 · premise_and_result_audit — Recompute MFMC cut mask20 OUTWARD flow with exact rational arithmetic, not rounded decimals.S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S163 · premise_and_result_audit — Recompute MFMC cut mask19 NET flow with exact rational arithmetic, not rounded decimals.S162 · premise_and_result_audit — Recompute MFMC cut mask19 INWARD flow with exact rational arithmetic, not rounded decimals.S161 · premise_and_result_audit — Recompute MFMC cut mask19 OUTWARD flow with exact rational arithmetic, not rounded decimals.S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S159 · premise_and_result_audit — Recompute MFMC cut mask18 NET flow with exact rational arithmetic, not rounded decimals.S158 · premise_and_result_audit — Recompute MFMC cut mask18 INWARD flow with exact rational arithmetic, not rounded decimals.S157 · premise_and_result_audit — Recompute MFMC cut mask18 OUTWARD flow with exact rational arithmetic, not rounded decimals.S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S155 · premise_and_result_audit — Recompute MFMC cut mask17 NET flow with exact rational arithmetic, not rounded decimals.S154 · premise_and_result_audit — Recompute MFMC cut mask17 INWARD flow with exact rational arithmetic, not rounded decimals.S153 · premise_and_result_audit — Recompute MFMC cut mask17 OUTWARD flow with exact rational arithmetic, not rounded decimals.S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S151 · premise_and_result_audit — Recompute MFMC cut mask16 NET flow with exact rational arithmetic, not rounded decimals.S150 · premise_and_result_audit — Recompute MFMC cut mask16 INWARD flow with exact rational arithmetic, not rounded decimals.S149 · premise_and_result_audit — Recompute MFMC cut mask16 OUTWARD flow with exact rational arithmetic, not rounded decimals.S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S147 · premise_and_result_audit — Recompute MFMC cut mask15 NET flow with exact rational arithmetic, not rounded decimals.S146 · premise_and_result_audit — Recompute MFMC cut mask15 INWARD flow with exact rational arithmetic, not rounded decimals.S145 · premise_and_result_audit — Recompute MFMC cut mask15 OUTWARD flow with exact rational arithmetic, not rounded decimals.S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S143 · premise_and_result_audit — Recompute MFMC cut mask14 NET flow with exact rational arithmetic, not rounded decimals.S142 · premise_and_result_audit — Recompute MFMC cut mask14 INWARD flow with exact rational arithmetic, not rounded decimals.S141 · premise_and_result_audit — Recompute MFMC cut mask14 OUTWARD flow with exact rational arithmetic, not rounded decimals.S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S139 · premise_and_result_audit — Recompute MFMC cut mask13 NET flow with exact rational arithmetic, not rounded decimals.S138 · premise_and_result_audit — Recompute MFMC cut mask13 INWARD flow with exact rational arithmetic, not rounded decimals.S137 · premise_and_result_audit — Recompute MFMC cut mask13 OUTWARD flow with exact rational arithmetic, not rounded decimals.S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S135 · premise_and_result_audit — Recompute MFMC cut mask12 NET flow with exact rational arithmetic, not rounded decimals.S134 · premise_and_result_audit — Recompute MFMC cut mask12 INWARD flow with exact rational arithmetic, not rounded decimals.S133 · premise_and_result_audit — Recompute MFMC cut mask12 OUTWARD flow with exact rational arithmetic, not rounded decimals.S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S131 · premise_and_result_audit — Recompute MFMC cut mask11 NET flow with exact rational arithmetic, not rounded decimals.S130 · premise_and_result_audit — Recompute MFMC cut mask11 INWARD flow with exact rational arithmetic, not rounded decimals.S129 · premise_and_result_audit — Recompute MFMC cut mask11 OUTWARD flow with exact rational arithmetic, not rounded decimals.S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S127 · premise_and_result_audit — Recompute MFMC cut mask10 NET flow with exact rational arithmetic, not rounded decimals.S126 · premise_and_result_audit — Recompute MFMC cut mask10 INWARD flow with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Recompute MFMC cut mask10 OUTWARD flow with exact rational arithmetic, not rounded decimals.S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S123 · premise_and_result_audit — Recompute MFMC cut mask9 NET flow with exact rational arithmetic, not rounded decimals.S122 · premise_and_result_audit — Recompute MFMC cut mask9 INWARD flow with exact rational arithmetic, not rounded decimals.S121 · premise_and_result_audit — Recompute MFMC cut mask9 OUTWARD flow with exact rational arithmetic, not rounded decimals.S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S119 · premise_and_result_audit — Recompute MFMC cut mask8 NET flow with exact rational arithmetic, not rounded decimals.S118 · premise_and_result_audit — Recompute MFMC cut mask8 INWARD flow with exact rational arithmetic, not rounded decimals.S117 · premise_and_result_audit — Recompute MFMC cut mask8 OUTWARD flow with exact rational arithmetic, not rounded decimals.S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S115 · premise_and_result_audit — Recompute MFMC cut mask7 NET flow with exact rational arithmetic, not rounded decimals.S114 · premise_and_result_audit — Recompute MFMC cut mask7 INWARD flow with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Recompute MFMC cut mask7 OUTWARD flow with exact rational arithmetic, not rounded decimals.S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S111 · premise_and_result_audit — Recompute MFMC cut mask6 NET flow with exact rational arithmetic, not rounded decimals.S110 · premise_and_result_audit — Recompute MFMC cut mask6 INWARD flow with exact rational arithmetic, not rounded decimals.S109 · premise_and_result_audit — Recompute MFMC cut mask6 OUTWARD flow with exact rational arithmetic, not rounded decimals.S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S107 · premise_and_result_audit — Recompute MFMC cut mask5 NET flow with exact rational arithmetic, not rounded decimals.S106 · premise_and_result_audit — Recompute MFMC cut mask5 INWARD flow with exact rational arithmetic, not rounded decimals.S105 · premise_and_result_audit — Recompute MFMC cut mask5 OUTWARD flow with exact rational arithmetic, not rounded decimals.S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S103 · premise_and_result_audit — Recompute MFMC cut mask4 NET flow with exact rational arithmetic, not rounded decimals.S102 · premise_and_result_audit — Recompute MFMC cut mask4 INWARD flow with exact rational arithmetic, not rounded decimals.S101 · premise_and_result_audit — Recompute MFMC cut mask4 OUTWARD flow with exact rational arithmetic, not rounded decimals.S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S099 · premise_and_result_audit — Recompute MFMC cut mask3 NET flow with exact rational arithmetic, not rounded decimals.S098 · premise_and_result_audit — Recompute MFMC cut mask3 INWARD flow with exact rational arithmetic, not rounded decimals.S097 · premise_and_result_audit — Recompute MFMC cut mask3 OUTWARD flow with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S095 · premise_and_result_audit — Recompute MFMC cut mask2 NET flow with exact rational arithmetic, not rounded decimals.S094 · premise_and_result_audit — Recompute MFMC cut mask2 INWARD flow with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Recompute MFMC cut mask2 OUTWARD flow with exact rational arithmetic, not rounded decimals.S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S091 · premise_and_result_audit — Recompute MFMC cut mask1 NET flow with exact rational arithmetic, not rounded decimals.S090 · premise_and_result_audit — Recompute MFMC cut mask1 INWARD flow with exact rational arithmetic, not rounded decimals.S089 · premise_and_result_audit — Recompute MFMC cut mask1 OUTWARD flow with exact rational arithmetic, not rounded decimals.S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S087 · premise_and_result_audit — Recompute MFMC cut mask0 NET flow with exact rational arithmetic, not rounded decimals.S086 · premise_and_result_audit — Recompute MFMC cut mask0 INWARD flow with exact rational arithmetic, not rounded decimals.S085 · premise_and_result_audit — Recompute MFMC cut mask0 OUTWARD flow with exact rational arithmetic, not rounded decimals.S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S083 · premise_and_result_audit — Recompute MFMC SINK net in not gross with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Recompute MFMC SOURCE net out not gross with exact rational arithmetic, not rounded decimals.S081 · premise_and_result_audit — Recompute MFMC vertex t ALL outgoing with exact rational arithmetic, not rounded decimals.S080 · premise_and_result_audit — Recompute MFMC vertex t ALL incoming with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Recompute MFMC vertex s ALL outgoing with exact rational arithmetic, not rounded decimals.S078 · premise_and_result_audit — Recompute MFMC vertex s ALL incoming with exact rational arithmetic, not rounded decimals.S077 · premise_and_result_audit — Recompute MFMC vertex a4 ALL outgoing with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Recompute MFMC vertex a4 ALL incoming with exact rational arithmetic, not rounded decimals.S075 · premise_and_result_audit — Recompute MFMC vertex a3 ALL outgoing with exact rational arithmetic, not rounded decimals.S074 · premise_and_result_audit — Recompute MFMC vertex a3 ALL incoming with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Recompute MFMC vertex a2 ALL outgoing with exact rational arithmetic, not rounded decimals.S072 · premise_and_result_audit — Recompute MFMC vertex a2 ALL incoming with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Recompute MFMC vertex a1 ALL outgoing with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Recompute MFMC vertex a1 ALL incoming with exact rational arithmetic, not rounded decimals.S069 · premise_and_result_audit — Recompute MFMC vertex a0 ALL outgoing with exact rational arithmetic, not rounded decimals.S068 · premise_and_result_audit — Recompute MFMC vertex a0 ALL incoming with exact rational arithmetic, not rounded decimals.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.(130, 130, [[[s], 130, 0, 130, 135, Yes, Yes], [[a0, s], 130, 0, 130, 135, Yes, Yes], [[a1, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, s], 130, 0, 130, 135, Yes, Yes], [[a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f81f6bf2d48b4ba47a44d7678de695fb4a4cbb97c0df42e3168eeb2bbfe8d5a1candidate record SHA-256: 6013173dc4254a0b7cdc68547645c7447ee927c2dbddd13d77162f43c837b035semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95fLean: NOT_RUN · human review: NOT_PERFORMED