Step-by-step derivation
S001 · implies
Before
source roles = [mfmc-equal-value-optimality]
After
derived = [[flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_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.
Step identifiers and dependencies
Rule: source_ordered_fields_without_JS · Uses: G001
S002 · implies
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]], [13, 13, 14, 14, 15, 15, 16, 16, 17, 17], [13, 13, 14, 14, 15, 15, 16, 16, 17, 17], 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]], [13, 14, 15, 16, 17, 13, 14, 15, 16, 17], [13, 14, 15, 16, 17, 13, 14, 15, 16, 17]]
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.
Step identifiers and dependencies
Rule: MFMC_complete_directed_network_ZIP_before_source_order · Uses: G001, S001
S003 · calculation
Before
MFMC cut mask0 OUTWARD capacity := sum(13, 14, 15, 16, 17)
After
MFMC cut mask0 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
source roles = [0, [a0, a1, a2, a3, a4], [s]]
After
derived = [[5, 6, 7, 8, 9], [], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S003
S005 · calculation
Before
MFMC cut mask1 OUTWARD capacity := sum(13, 14, 15, 16, 17)
After
MFMC cut mask1 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [1, [a0, a1, a2, a3, a4], [a0, s]]
After
derived = [[0, 6, 7, 8, 9], [], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S005
S007 · calculation
Before
MFMC cut mask2 OUTWARD capacity := sum(14, 13, 15, 16, 17)
After
MFMC cut mask2 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
source roles = [2, [a0, a1, a2, a3, a4], [a1, s]]
After
derived = [[1, 5, 7, 8, 9], [], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S007
S009 · calculation
Before
MFMC cut mask3 OUTWARD capacity := sum(13, 14, 15, 16, 17)
After
MFMC cut mask3 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
source roles = [3, [a0, a1, a2, a3, a4], [a0, a1, s]]
After
derived = [[0, 1, 7, 8, 9], [], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S009
S011 · calculation
Before
MFMC cut mask4 OUTWARD capacity := sum(15, 13, 14, 16, 17)
After
MFMC cut mask4 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S010
S012 · implies
Before
source roles = [4, [a0, a1, a2, a3, a4], [a2, s]]
After
derived = [[2, 5, 6, 8, 9], [], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S011
S013 · calculation
Before
MFMC cut mask5 OUTWARD capacity := sum(13, 15, 14, 16, 17)
After
MFMC cut mask5 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · implies
Before
source roles = [5, [a0, a1, a2, a3, a4], [a0, a2, s]]
After
derived = [[0, 2, 6, 8, 9], [], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S013
S015 · calculation
Before
MFMC cut mask6 OUTWARD capacity := sum(14, 15, 13, 16, 17)
After
MFMC cut mask6 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · implies
Before
source roles = [6, [a0, a1, a2, a3, a4], [a1, a2, s]]
After
derived = [[1, 2, 5, 8, 9], [], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S015
S017 · calculation
Before
MFMC cut mask7 OUTWARD capacity := sum(13, 14, 15, 16, 17)
After
MFMC cut mask7 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · implies
Before
source roles = [7, [a0, a1, a2, a3, a4], [a0, a1, a2, s]]
After
derived = [[0, 1, 2, 8, 9], [], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S017
S019 · calculation
Before
MFMC cut mask8 OUTWARD capacity := sum(16, 13, 14, 15, 17)
After
MFMC cut mask8 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
source roles = [8, [a0, a1, a2, a3, a4], [a3, s]]
After
derived = [[3, 5, 6, 7, 9], [], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S019
S021 · calculation
Before
MFMC cut mask9 OUTWARD capacity := sum(13, 16, 14, 15, 17)
After
MFMC cut mask9 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · implies
Before
source roles = [9, [a0, a1, a2, a3, a4], [a0, a3, s]]
After
derived = [[0, 3, 6, 7, 9], [], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S021
S023 · calculation
Before
MFMC cut mask10 OUTWARD capacity := sum(14, 16, 13, 15, 17)
After
MFMC cut mask10 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · implies
Before
source roles = [10, [a0, a1, a2, a3, a4], [a1, a3, s]]
After
derived = [[1, 3, 5, 7, 9], [], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S023
S025 · calculation
Before
MFMC cut mask11 OUTWARD capacity := sum(13, 14, 16, 15, 17)
After
MFMC cut mask11 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · implies
Before
source roles = [11, [a0, a1, a2, a3, a4], [a0, a1, a3, s]]
After
derived = [[0, 1, 3, 7, 9], [], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S025
S027 · calculation
Before
MFMC cut mask12 OUTWARD capacity := sum(15, 16, 13, 14, 17)
After
MFMC cut mask12 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · implies
Before
source roles = [12, [a0, a1, a2, a3, a4], [a2, a3, s]]
After
derived = [[2, 3, 5, 6, 9], [], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S027
S029 · calculation
Before
MFMC cut mask13 OUTWARD capacity := sum(13, 15, 16, 14, 17)
After
MFMC cut mask13 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S028
S030 · implies
Before
source roles = [13, [a0, a1, a2, a3, a4], [a0, a2, a3, s]]
After
derived = [[0, 2, 3, 6, 9], [], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S029
S031 · calculation
Before
MFMC cut mask14 OUTWARD capacity := sum(14, 15, 16, 13, 17)
After
MFMC cut mask14 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S030
S032 · implies
Before
source roles = [14, [a0, a1, a2, a3, a4], [a1, a2, a3, s]]
After
derived = [[1, 2, 3, 5, 9], [], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S031
S033 · calculation
Before
MFMC cut mask15 OUTWARD capacity := sum(13, 14, 15, 16, 17)
After
MFMC cut mask15 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · implies
Before
source roles = [15, [a0, a1, a2, a3, a4], [a0, a1, a2, a3, s]]
After
derived = [[0, 1, 2, 3, 9], [], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S033
S035 · calculation
Before
MFMC cut mask16 OUTWARD capacity := sum(17, 13, 14, 15, 16)
After
MFMC cut mask16 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S034
S036 · implies
Before
source roles = [16, [a0, a1, a2, a3, a4], [a4, s]]
After
derived = [[4, 5, 6, 7, 8], [], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S035
S037 · calculation
Before
MFMC cut mask17 OUTWARD capacity := sum(13, 17, 14, 15, 16)
After
MFMC cut mask17 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S036
S038 · implies
Before
source roles = [17, [a0, a1, a2, a3, a4], [a0, a4, s]]
After
derived = [[0, 4, 6, 7, 8], [], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S037
S039 · calculation
Before
MFMC cut mask18 OUTWARD capacity := sum(14, 17, 13, 15, 16)
After
MFMC cut mask18 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · implies
Before
source roles = [18, [a0, a1, a2, a3, a4], [a1, a4, s]]
After
derived = [[1, 4, 5, 7, 8], [], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S039
S041 · calculation
Before
MFMC cut mask19 OUTWARD capacity := sum(13, 14, 17, 15, 16)
After
MFMC cut mask19 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S040
S042 · implies
Before
source roles = [19, [a0, a1, a2, a3, a4], [a0, a1, a4, s]]
After
derived = [[0, 1, 4, 7, 8], [], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S041
S043 · calculation
Before
MFMC cut mask20 OUTWARD capacity := sum(15, 17, 13, 14, 16)
After
MFMC cut mask20 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S042
S044 · implies
Before
source roles = [20, [a0, a1, a2, a3, a4], [a2, a4, s]]
After
derived = [[2, 4, 5, 6, 8], [], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S043
S045 · calculation
Before
MFMC cut mask21 OUTWARD capacity := sum(13, 15, 17, 14, 16)
After
MFMC cut mask21 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S044
S046 · implies
Before
source roles = [21, [a0, a1, a2, a3, a4], [a0, a2, a4, s]]
After
derived = [[0, 2, 4, 6, 8], [], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S045
S047 · calculation
Before
MFMC cut mask22 OUTWARD capacity := sum(14, 15, 17, 13, 16)
After
MFMC cut mask22 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S046
S048 · implies
Before
source roles = [22, [a0, a1, a2, a3, a4], [a1, a2, a4, s]]
After
derived = [[1, 2, 4, 5, 8], [], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S047
S049 · calculation
Before
MFMC cut mask23 OUTWARD capacity := sum(13, 14, 15, 17, 16)
After
MFMC cut mask23 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S048
S050 · implies
Before
source roles = [23, [a0, a1, a2, a3, a4], [a0, a1, a2, a4, s]]
After
derived = [[0, 1, 2, 4, 8], [], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S049
S051 · calculation
Before
MFMC cut mask24 OUTWARD capacity := sum(16, 17, 13, 14, 15)
After
MFMC cut mask24 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S050
S052 · implies
Before
source roles = [24, [a0, a1, a2, a3, a4], [a3, a4, s]]
After
derived = [[3, 4, 5, 6, 7], [], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S051
S053 · calculation
Before
MFMC cut mask25 OUTWARD capacity := sum(13, 16, 17, 14, 15)
After
MFMC cut mask25 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S052
S054 · implies
Before
source roles = [25, [a0, a1, a2, a3, a4], [a0, a3, a4, s]]
After
derived = [[0, 3, 4, 6, 7], [], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S053
S055 · calculation
Before
MFMC cut mask26 OUTWARD capacity := sum(14, 16, 17, 13, 15)
After
MFMC cut mask26 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S054
S056 · implies
Before
source roles = [26, [a0, a1, a2, a3, a4], [a1, a3, a4, s]]
After
derived = [[1, 3, 4, 5, 7], [], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S055
S057 · calculation
Before
MFMC cut mask27 OUTWARD capacity := sum(13, 14, 16, 17, 15)
After
MFMC cut mask27 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S056
S058 · implies
Before
source roles = [27, [a0, a1, a2, a3, a4], [a0, a1, a3, a4, s]]
After
derived = [[0, 1, 3, 4, 7], [], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S057
S059 · calculation
Before
MFMC cut mask28 OUTWARD capacity := sum(15, 16, 17, 13, 14)
After
MFMC cut mask28 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S058
S060 · implies
Before
source roles = [28, [a0, a1, a2, a3, a4], [a2, a3, a4, s]]
After
derived = [[2, 3, 4, 5, 6], [], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S059
S061 · calculation
Before
MFMC cut mask29 OUTWARD capacity := sum(13, 15, 16, 17, 14)
After
MFMC cut mask29 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S060
S062 · implies
Before
source roles = [29, [a0, a1, a2, a3, a4], [a0, a2, a3, a4, s]]
After
derived = [[0, 2, 3, 4, 6], [], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S061
S063 · calculation
Before
MFMC cut mask30 OUTWARD capacity := sum(14, 15, 16, 17, 13)
After
MFMC cut mask30 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S062
S064 · implies
Before
source roles = [30, [a0, a1, a2, a3, a4], [a1, a2, a3, a4, s]]
After
derived = [[1, 2, 3, 4, 5], [], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S063
S065 · calculation
Before
MFMC cut mask31 OUTWARD capacity := sum(13, 14, 15, 16, 17)
After
MFMC cut mask31 OUTWARD capacity = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S064
S066 · implies
Before
source roles = [31, [a0, a1, a2, a3, a4], [a0, a1, a2, a3, a4, s]]
After
derived = [[0, 1, 2, 3, 4], [], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]
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.
Step identifiers and dependencies
Rule: MFMC_EVERY_numeric_mask_cut_OUTWARD_edges_only · Uses: G001, S065
S067 · implies
Before
source roles = [[a0, a1, a2, a3, a4, s, t], [a0, a1, a2, a3, a4]]
After
derived = [32, 32, 75, [[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.
Step identifiers and dependencies
Rule: MFMC_ALL_cut_family_complete_minimum_and_ALL_ties · Uses: G001, S066
S068 · calculation
Before
MFMC vertex a0 ALL incoming := sum(13)
After
MFMC vertex a0 ALL incoming = 13
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S067
S069 · calculation
Before
MFMC vertex a0 ALL outgoing := sum(13)
After
MFMC vertex a0 ALL outgoing = 13
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S068
S070 · calculation
Before
MFMC vertex a1 ALL incoming := sum(14)
After
MFMC vertex a1 ALL incoming = 14
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S069
S071 · calculation
Before
MFMC vertex a1 ALL outgoing := sum(14)
After
MFMC vertex a1 ALL outgoing = 14
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S070
S072 · calculation
Before
MFMC vertex a2 ALL incoming := sum(15)
After
MFMC vertex a2 ALL incoming = 15
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S071
S073 · calculation
Before
MFMC vertex a2 ALL outgoing := sum(15)
After
MFMC vertex a2 ALL outgoing = 15
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S072
S074 · calculation
Before
MFMC vertex a3 ALL incoming := sum(16)
After
MFMC vertex a3 ALL incoming = 16
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S073
S075 · calculation
Before
MFMC vertex a3 ALL outgoing := sum(16)
After
MFMC vertex a3 ALL outgoing = 16
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S074
S076 · calculation
Before
MFMC vertex a4 ALL incoming := sum(17)
After
MFMC vertex a4 ALL incoming = 17
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S075
S077 · calculation
Before
MFMC vertex a4 ALL outgoing := sum(17)
After
MFMC vertex a4 ALL outgoing = 17
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S076
S078 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S077
S079 · calculation
Before
MFMC vertex s ALL outgoing := sum(13, 14, 15, 16, 17)
After
MFMC vertex s ALL outgoing = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S078
S080 · calculation
Before
MFMC vertex t ALL incoming := sum(13, 14, 15, 16, 17)
After
MFMC vertex t ALL incoming = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S079
S081 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S080
S082 · calculation
Before
MFMC SOURCE net out not gross := sub(75, 0)
After
MFMC SOURCE net out not gross = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S081
S083 · calculation
Before
MFMC SINK net in not gross := sub(75, 0)
After
MFMC SINK net in not gross = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S082
S084 · implies
Before
source roles = [[[a0, t], [a1, t], [a2, t], [a3, t], [a4, t], [s, a0], [s, a1], [s, a2], [s, a3], [s, a4]], [13, 14, 15, 16, 17, 13, 14, 15, 16, 17], [13, 14, 15, 16, 17, 13, 14, 15, 16, 17], s, t]
After
derived = [[[[a0, t], 13, 13, Yes], [[a1, t], 14, 14, Yes], [[a2, t], 15, 15, Yes], [[a3, t], 16, 16, Yes], [[a4, t], 17, 17, Yes], [[s, a0], 13, 13, Yes], [[s, a1], 14, 14, Yes], [[s, a2], 15, 15, Yes], [[s, a3], 16, 16, Yes], [[s, a4], 17, 17, Yes]], [[a0, 13, 13, Yes], [a1, 14, 14, Yes], [a2, 15, 15, Yes], [a3, 16, 16, Yes], [a4, 17, 17, Yes]], Yes, Yes, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_capacity_EVERY_internal_balance_source_sink_NET_feasibility · Uses: G001, S083
S085 · calculation
Before
MFMC cut mask0 OUTWARD flow := sum(13, 14, 15, 16, 17)
After
MFMC cut mask0 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S084
S086 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S085
S087 · calculation
Before
MFMC cut mask0 NET flow := sub(75, 0)
After
MFMC cut mask0 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S086
S088 · implies
Before
source roles = [[s], [5, 6, 7, 8, 9], [], 75]
After
derived = [[[s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S087
S089 · calculation
Before
MFMC cut mask1 OUTWARD flow := sum(13, 14, 15, 16, 17)
After
MFMC cut mask1 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S088
S090 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S089
S091 · calculation
Before
MFMC cut mask1 NET flow := sub(75, 0)
After
MFMC cut mask1 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S090
S092 · implies
Before
source roles = [[a0, s], [0, 6, 7, 8, 9], [], 75]
After
derived = [[[a0, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S091
S093 · calculation
Before
MFMC cut mask2 OUTWARD flow := sum(14, 13, 15, 16, 17)
After
MFMC cut mask2 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S092
S094 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S093
S095 · calculation
Before
MFMC cut mask2 NET flow := sub(75, 0)
After
MFMC cut mask2 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S094
S096 · implies
Before
source roles = [[a1, s], [1, 5, 7, 8, 9], [], 75]
After
derived = [[[a1, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S095
S097 · calculation
Before
MFMC cut mask3 OUTWARD flow := sum(13, 14, 15, 16, 17)
After
MFMC cut mask3 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S096
S098 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S097
S099 · calculation
Before
MFMC cut mask3 NET flow := sub(75, 0)
After
MFMC cut mask3 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S098
S100 · implies
Before
source roles = [[a0, a1, s], [0, 1, 7, 8, 9], [], 75]
After
derived = [[[a0, a1, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S099
S101 · calculation
Before
MFMC cut mask4 OUTWARD flow := sum(15, 13, 14, 16, 17)
After
MFMC cut mask4 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S100
S102 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S101
S103 · calculation
Before
MFMC cut mask4 NET flow := sub(75, 0)
After
MFMC cut mask4 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S102
S104 · implies
Before
source roles = [[a2, s], [2, 5, 6, 8, 9], [], 75]
After
derived = [[[a2, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S103
S105 · calculation
Before
MFMC cut mask5 OUTWARD flow := sum(13, 15, 14, 16, 17)
After
MFMC cut mask5 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S104
S106 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S105
S107 · calculation
Before
MFMC cut mask5 NET flow := sub(75, 0)
After
MFMC cut mask5 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S106
S108 · implies
Before
source roles = [[a0, a2, s], [0, 2, 6, 8, 9], [], 75]
After
derived = [[[a0, a2, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S107
S109 · calculation
Before
MFMC cut mask6 OUTWARD flow := sum(14, 15, 13, 16, 17)
After
MFMC cut mask6 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S108
S110 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S109
S111 · calculation
Before
MFMC cut mask6 NET flow := sub(75, 0)
After
MFMC cut mask6 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S110
S112 · implies
Before
source roles = [[a1, a2, s], [1, 2, 5, 8, 9], [], 75]
After
derived = [[[a1, a2, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S111
S113 · calculation
Before
MFMC cut mask7 OUTWARD flow := sum(13, 14, 15, 16, 17)
After
MFMC cut mask7 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S112
S114 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S113
S115 · calculation
Before
MFMC cut mask7 NET flow := sub(75, 0)
After
MFMC cut mask7 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S114
S116 · implies
Before
source roles = [[a0, a1, a2, s], [0, 1, 2, 8, 9], [], 75]
After
derived = [[[a0, a1, a2, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S115
S117 · calculation
Before
MFMC cut mask8 OUTWARD flow := sum(16, 13, 14, 15, 17)
After
MFMC cut mask8 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S116
S118 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S117
S119 · calculation
Before
MFMC cut mask8 NET flow := sub(75, 0)
After
MFMC cut mask8 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S118
S120 · implies
Before
source roles = [[a3, s], [3, 5, 6, 7, 9], [], 75]
After
derived = [[[a3, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S119
S121 · calculation
Before
MFMC cut mask9 OUTWARD flow := sum(13, 16, 14, 15, 17)
After
MFMC cut mask9 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S120
S122 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S121
S123 · calculation
Before
MFMC cut mask9 NET flow := sub(75, 0)
After
MFMC cut mask9 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S122
S124 · implies
Before
source roles = [[a0, a3, s], [0, 3, 6, 7, 9], [], 75]
After
derived = [[[a0, a3, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S123
S125 · calculation
Before
MFMC cut mask10 OUTWARD flow := sum(14, 16, 13, 15, 17)
After
MFMC cut mask10 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S124
S126 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S125
S127 · calculation
Before
MFMC cut mask10 NET flow := sub(75, 0)
After
MFMC cut mask10 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S126
S128 · implies
Before
source roles = [[a1, a3, s], [1, 3, 5, 7, 9], [], 75]
After
derived = [[[a1, a3, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S127
S129 · calculation
Before
MFMC cut mask11 OUTWARD flow := sum(13, 14, 16, 15, 17)
After
MFMC cut mask11 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S128
S130 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S129
S131 · calculation
Before
MFMC cut mask11 NET flow := sub(75, 0)
After
MFMC cut mask11 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S130
S132 · implies
Before
source roles = [[a0, a1, a3, s], [0, 1, 3, 7, 9], [], 75]
After
derived = [[[a0, a1, a3, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S131
S133 · calculation
Before
MFMC cut mask12 OUTWARD flow := sum(15, 16, 13, 14, 17)
After
MFMC cut mask12 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S132
S134 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S133
S135 · calculation
Before
MFMC cut mask12 NET flow := sub(75, 0)
After
MFMC cut mask12 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S134
S136 · implies
Before
source roles = [[a2, a3, s], [2, 3, 5, 6, 9], [], 75]
After
derived = [[[a2, a3, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S135
S137 · calculation
Before
MFMC cut mask13 OUTWARD flow := sum(13, 15, 16, 14, 17)
After
MFMC cut mask13 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S136
S138 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S137
S139 · calculation
Before
MFMC cut mask13 NET flow := sub(75, 0)
After
MFMC cut mask13 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S138
S140 · implies
Before
source roles = [[a0, a2, a3, s], [0, 2, 3, 6, 9], [], 75]
After
derived = [[[a0, a2, a3, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S139
S141 · calculation
Before
MFMC cut mask14 OUTWARD flow := sum(14, 15, 16, 13, 17)
After
MFMC cut mask14 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S140
S142 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S141
S143 · calculation
Before
MFMC cut mask14 NET flow := sub(75, 0)
After
MFMC cut mask14 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S142
S144 · implies
Before
source roles = [[a1, a2, a3, s], [1, 2, 3, 5, 9], [], 75]
After
derived = [[[a1, a2, a3, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S143
S145 · calculation
Before
MFMC cut mask15 OUTWARD flow := sum(13, 14, 15, 16, 17)
After
MFMC cut mask15 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S144
S146 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S145
S147 · calculation
Before
MFMC cut mask15 NET flow := sub(75, 0)
After
MFMC cut mask15 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S146
S148 · implies
Before
source roles = [[a0, a1, a2, a3, s], [0, 1, 2, 3, 9], [], 75]
After
derived = [[[a0, a1, a2, a3, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S147
S149 · calculation
Before
MFMC cut mask16 OUTWARD flow := sum(17, 13, 14, 15, 16)
After
MFMC cut mask16 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S148
S150 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S149
S151 · calculation
Before
MFMC cut mask16 NET flow := sub(75, 0)
After
MFMC cut mask16 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S150
S152 · implies
Before
source roles = [[a4, s], [4, 5, 6, 7, 8], [], 75]
After
derived = [[[a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S151
S153 · calculation
Before
MFMC cut mask17 OUTWARD flow := sum(13, 17, 14, 15, 16)
After
MFMC cut mask17 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S152
S154 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S153
S155 · calculation
Before
MFMC cut mask17 NET flow := sub(75, 0)
After
MFMC cut mask17 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S154
S156 · implies
Before
source roles = [[a0, a4, s], [0, 4, 6, 7, 8], [], 75]
After
derived = [[[a0, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S155
S157 · calculation
Before
MFMC cut mask18 OUTWARD flow := sum(14, 17, 13, 15, 16)
After
MFMC cut mask18 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S156
S158 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S157
S159 · calculation
Before
MFMC cut mask18 NET flow := sub(75, 0)
After
MFMC cut mask18 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S158
S160 · implies
Before
source roles = [[a1, a4, s], [1, 4, 5, 7, 8], [], 75]
After
derived = [[[a1, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S159
S161 · calculation
Before
MFMC cut mask19 OUTWARD flow := sum(13, 14, 17, 15, 16)
After
MFMC cut mask19 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S160
S162 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S161
S163 · calculation
Before
MFMC cut mask19 NET flow := sub(75, 0)
After
MFMC cut mask19 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S162
S164 · implies
Before
source roles = [[a0, a1, a4, s], [0, 1, 4, 7, 8], [], 75]
After
derived = [[[a0, a1, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S163
S165 · calculation
Before
MFMC cut mask20 OUTWARD flow := sum(15, 17, 13, 14, 16)
After
MFMC cut mask20 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S164
S166 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S165
S167 · calculation
Before
MFMC cut mask20 NET flow := sub(75, 0)
After
MFMC cut mask20 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S166
S168 · implies
Before
source roles = [[a2, a4, s], [2, 4, 5, 6, 8], [], 75]
After
derived = [[[a2, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S167
S169 · calculation
Before
MFMC cut mask21 OUTWARD flow := sum(13, 15, 17, 14, 16)
After
MFMC cut mask21 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S168
S170 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S169
S171 · calculation
Before
MFMC cut mask21 NET flow := sub(75, 0)
After
MFMC cut mask21 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S170
S172 · implies
Before
source roles = [[a0, a2, a4, s], [0, 2, 4, 6, 8], [], 75]
After
derived = [[[a0, a2, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S171
S173 · calculation
Before
MFMC cut mask22 OUTWARD flow := sum(14, 15, 17, 13, 16)
After
MFMC cut mask22 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S172
S174 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S173
S175 · calculation
Before
MFMC cut mask22 NET flow := sub(75, 0)
After
MFMC cut mask22 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S174
S176 · implies
Before
source roles = [[a1, a2, a4, s], [1, 2, 4, 5, 8], [], 75]
After
derived = [[[a1, a2, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S175
S177 · calculation
Before
MFMC cut mask23 OUTWARD flow := sum(13, 14, 15, 17, 16)
After
MFMC cut mask23 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S176
S178 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S177
S179 · calculation
Before
MFMC cut mask23 NET flow := sub(75, 0)
After
MFMC cut mask23 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S178
S180 · implies
Before
source roles = [[a0, a1, a2, a4, s], [0, 1, 2, 4, 8], [], 75]
After
derived = [[[a0, a1, a2, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S179
S181 · calculation
Before
MFMC cut mask24 OUTWARD flow := sum(16, 17, 13, 14, 15)
After
MFMC cut mask24 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S180
S182 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S181
S183 · calculation
Before
MFMC cut mask24 NET flow := sub(75, 0)
After
MFMC cut mask24 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S182
S184 · implies
Before
source roles = [[a3, a4, s], [3, 4, 5, 6, 7], [], 75]
After
derived = [[[a3, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S183
S185 · calculation
Before
MFMC cut mask25 OUTWARD flow := sum(13, 16, 17, 14, 15)
After
MFMC cut mask25 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S184
S186 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S185
S187 · calculation
Before
MFMC cut mask25 NET flow := sub(75, 0)
After
MFMC cut mask25 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S186
S188 · implies
Before
source roles = [[a0, a3, a4, s], [0, 3, 4, 6, 7], [], 75]
After
derived = [[[a0, a3, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S187
S189 · calculation
Before
MFMC cut mask26 OUTWARD flow := sum(14, 16, 17, 13, 15)
After
MFMC cut mask26 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S188
S190 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S189
S191 · calculation
Before
MFMC cut mask26 NET flow := sub(75, 0)
After
MFMC cut mask26 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S190
S192 · implies
Before
source roles = [[a1, a3, a4, s], [1, 3, 4, 5, 7], [], 75]
After
derived = [[[a1, a3, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S191
S193 · calculation
Before
MFMC cut mask27 OUTWARD flow := sum(13, 14, 16, 17, 15)
After
MFMC cut mask27 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S192
S194 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S193
S195 · calculation
Before
MFMC cut mask27 NET flow := sub(75, 0)
After
MFMC cut mask27 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S194
S196 · implies
Before
source roles = [[a0, a1, a3, a4, s], [0, 1, 3, 4, 7], [], 75]
After
derived = [[[a0, a1, a3, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S195
S197 · calculation
Before
MFMC cut mask28 OUTWARD flow := sum(15, 16, 17, 13, 14)
After
MFMC cut mask28 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S196
S198 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S197
S199 · calculation
Before
MFMC cut mask28 NET flow := sub(75, 0)
After
MFMC cut mask28 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S198
S200 · implies
Before
source roles = [[a2, a3, a4, s], [2, 3, 4, 5, 6], [], 75]
After
derived = [[[a2, a3, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S199
S201 · calculation
Before
MFMC cut mask29 OUTWARD flow := sum(13, 15, 16, 17, 14)
After
MFMC cut mask29 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S200
S202 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S201
S203 · calculation
Before
MFMC cut mask29 NET flow := sub(75, 0)
After
MFMC cut mask29 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S202
S204 · implies
Before
source roles = [[a0, a2, a3, a4, s], [0, 2, 3, 4, 6], [], 75]
After
derived = [[[a0, a2, a3, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S203
S205 · calculation
Before
MFMC cut mask30 OUTWARD flow := sum(14, 15, 16, 17, 13)
After
MFMC cut mask30 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S204
S206 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S205
S207 · calculation
Before
MFMC cut mask30 NET flow := sub(75, 0)
After
MFMC cut mask30 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S206
S208 · implies
Before
source roles = [[a1, a2, a3, a4, s], [1, 2, 3, 4, 5], [], 75]
After
derived = [[[a1, a2, a3, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S207
S209 · calculation
Before
MFMC cut mask31 OUTWARD flow := sum(13, 14, 15, 16, 17)
After
MFMC cut mask31 OUTWARD flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S208
S210 · calculation
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.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S209
S211 · calculation
Before
MFMC cut mask31 NET flow := sub(75, 0)
After
MFMC cut mask31 NET flow = 75
Conditions: none
Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S210
S212 · implies
Before
source roles = [[a0, a1, a2, a3, a4, s], [0, 1, 2, 3, 4], [], 75]
After
derived = [[[a0, a1, a2, a3, a4, s], 75, 0, 75, 75, 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.
Step identifiers and dependencies
Rule: MFMC_EVERY_cut_conservation_cancellation_and_weak_upper_bound · Uses: G001, S211
S213 · implies
Before
source roles = [Yes, 75, 75, [[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]]]
After
derived = [[[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]], Yes, Yes, Yes, No]
Conditions: none
Any feasible flow is bounded by this equal-valued cut,and any cut is bounded below by this feasible flow,so the current flow is maximum and equal-valued cuts minimum.This uses universal weak bounds plus feasibility,not bare equality.The final No only withholds integrality/algorithm proof.
Step identifiers and dependencies
Rule: MFMC_feasible_equal_value_GENERAL_bounds_certify_BOTH_optima · Uses: G001, S212