Step-by-step derivation
S001 · implies
Before
raw relation pairs=[[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]];carrier Fin(6)
After
lexicographically sorted distinct ordered pairs=[[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]
Conditions: none
List distinct directed pairs lexicographically.Duplicates are invalid;do not silently add reverses,diagonals or a transitive closure.
Step identifiers and dependencies
Rule: complete_relation_no_duplicate_or_implicit_repair · Uses: G001
S002 · implies
Before
carrier label=0;required diagonal=[0, 0]
After
present=Yes
Conditions: none
Every carrier element needs its diagonal pair,not only labels appearing in the relation.
Step identifiers and dependencies
Rule: relation_every_carrier_diagonal · Uses: G001, S001
S003 · implies
Before
carrier label=1;required diagonal=[1, 1]
After
present=Yes
Conditions: none
Every carrier element needs its diagonal pair,not only labels appearing in the relation.
Step identifiers and dependencies
Rule: relation_every_carrier_diagonal · Uses: G001, S002
S004 · implies
Before
carrier label=2;required diagonal=[2, 2]
After
present=Yes
Conditions: none
Every carrier element needs its diagonal pair,not only labels appearing in the relation.
Step identifiers and dependencies
Rule: relation_every_carrier_diagonal · Uses: G001, S003
S005 · implies
Before
carrier label=3;required diagonal=[3, 3]
After
present=Yes
Conditions: none
Every carrier element needs its diagonal pair,not only labels appearing in the relation.
Step identifiers and dependencies
Rule: relation_every_carrier_diagonal · Uses: G001, S004
S006 · implies
Before
carrier label=4;required diagonal=[4, 4]
After
present=Yes
Conditions: none
Every carrier element needs its diagonal pair,not only labels appearing in the relation.
Step identifiers and dependencies
Rule: relation_every_carrier_diagonal · Uses: G001, S005
S007 · implies
Before
carrier label=5;required diagonal=[5, 5]
After
present=Yes
Conditions: none
Every carrier element needs its diagonal pair,not only labels appearing in the relation.
Step identifiers and dependencies
Rule: relation_every_carrier_diagonal · Uses: G001, S006
S008 · implies
Before
original pair=[0, 0];required reverse=[0, 0]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S007
S009 · implies
Before
original pair=[0, 1];required reverse=[1, 0]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S008
S010 · implies
Before
original pair=[0, 2];required reverse=[2, 0]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S009
S011 · implies
Before
original pair=[0, 3];required reverse=[3, 0]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S010
S012 · implies
Before
original pair=[0, 4];required reverse=[4, 0]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S011
S013 · implies
Before
original pair=[0, 5];required reverse=[5, 0]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S012
S014 · implies
Before
original pair=[1, 0];required reverse=[0, 1]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S013
S015 · implies
Before
original pair=[1, 1];required reverse=[1, 1]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S014
S016 · implies
Before
original pair=[1, 2];required reverse=[2, 1]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S015
S017 · implies
Before
original pair=[1, 3];required reverse=[3, 1]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S016
S018 · implies
Before
original pair=[1, 4];required reverse=[4, 1]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S017
S019 · implies
Before
original pair=[1, 5];required reverse=[5, 1]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S018
S020 · implies
Before
original pair=[2, 0];required reverse=[0, 2]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S019
S021 · implies
Before
original pair=[2, 1];required reverse=[1, 2]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S020
S022 · implies
Before
original pair=[2, 2];required reverse=[2, 2]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S021
S023 · implies
Before
original pair=[2, 3];required reverse=[3, 2]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S022
S024 · implies
Before
original pair=[2, 4];required reverse=[4, 2]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S023
S025 · implies
Before
original pair=[2, 5];required reverse=[5, 2]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S024
S026 · implies
Before
original pair=[3, 0];required reverse=[0, 3]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S025
S027 · implies
Before
original pair=[3, 1];required reverse=[1, 3]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S026
S028 · implies
Before
original pair=[3, 2];required reverse=[2, 3]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S027
S029 · implies
Before
original pair=[3, 3];required reverse=[3, 3]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S028
S030 · implies
Before
original pair=[3, 4];required reverse=[4, 3]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S029
S031 · implies
Before
original pair=[3, 5];required reverse=[5, 3]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S030
S032 · implies
Before
original pair=[4, 0];required reverse=[0, 4]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S031
S033 · implies
Before
original pair=[4, 1];required reverse=[1, 4]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S032
S034 · implies
Before
original pair=[4, 2];required reverse=[2, 4]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S033
S035 · implies
Before
original pair=[4, 3];required reverse=[3, 4]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S034
S036 · implies
Before
original pair=[4, 4];required reverse=[4, 4]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S035
S037 · implies
Before
original pair=[4, 5];required reverse=[5, 4]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S036
S038 · implies
Before
original pair=[5, 0];required reverse=[0, 5]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S037
S039 · implies
Before
original pair=[5, 1];required reverse=[1, 5]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S038
S040 · implies
Before
original pair=[5, 2];required reverse=[2, 5]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S039
S041 · implies
Before
original pair=[5, 3];required reverse=[3, 5]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S040
S042 · implies
Before
original pair=[5, 4];required reverse=[4, 5]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S041
S043 · implies
Before
original pair=[5, 5];required reverse=[5, 5]
After
reverse present=Yes
Conditions: none
Check for the reverse,but record the original[a,b] on failure,not its missing[b,a] reverse.An empty relation is vacuously symmetric.
Step identifiers and dependencies
Rule: relation_original_pair_missing_reverse_record · Uses: G001, S042
S044 · implies
Before
first original pair=[0, 0];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[0, 0, 0, Yes], [0, 0, 1, Yes], [0, 0, 2, Yes], [0, 0, 3, Yes], [0, 0, 4, Yes], [0, 0, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S043
S045 · implies
Before
first original pair=[0, 1];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[0, 1, 0, Yes], [0, 1, 1, Yes], [0, 1, 2, Yes], [0, 1, 3, Yes], [0, 1, 4, Yes], [0, 1, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S044
S046 · implies
Before
first original pair=[0, 2];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[0, 2, 0, Yes], [0, 2, 1, Yes], [0, 2, 2, Yes], [0, 2, 3, Yes], [0, 2, 4, Yes], [0, 2, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S045
S047 · implies
Before
first original pair=[0, 3];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[0, 3, 0, Yes], [0, 3, 1, Yes], [0, 3, 2, Yes], [0, 3, 3, Yes], [0, 3, 4, Yes], [0, 3, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S046
S048 · implies
Before
first original pair=[0, 4];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[0, 4, 0, Yes], [0, 4, 1, Yes], [0, 4, 2, Yes], [0, 4, 3, Yes], [0, 4, 4, Yes], [0, 4, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S047
S049 · implies
Before
first original pair=[0, 5];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[0, 5, 0, Yes], [0, 5, 1, Yes], [0, 5, 2, Yes], [0, 5, 3, Yes], [0, 5, 4, Yes], [0, 5, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S048
S050 · implies
Before
first original pair=[1, 0];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[1, 0, 0, Yes], [1, 0, 1, Yes], [1, 0, 2, Yes], [1, 0, 3, Yes], [1, 0, 4, Yes], [1, 0, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S049
S051 · implies
Before
first original pair=[1, 1];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[1, 1, 0, Yes], [1, 1, 1, Yes], [1, 1, 2, Yes], [1, 1, 3, Yes], [1, 1, 4, Yes], [1, 1, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S050
S052 · implies
Before
first original pair=[1, 2];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[1, 2, 0, Yes], [1, 2, 1, Yes], [1, 2, 2, Yes], [1, 2, 3, Yes], [1, 2, 4, Yes], [1, 2, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S051
S053 · implies
Before
first original pair=[1, 3];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[1, 3, 0, Yes], [1, 3, 1, Yes], [1, 3, 2, Yes], [1, 3, 3, Yes], [1, 3, 4, Yes], [1, 3, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S052
S054 · implies
Before
first original pair=[1, 4];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[1, 4, 0, Yes], [1, 4, 1, Yes], [1, 4, 2, Yes], [1, 4, 3, Yes], [1, 4, 4, Yes], [1, 4, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S053
S055 · implies
Before
first original pair=[1, 5];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[1, 5, 0, Yes], [1, 5, 1, Yes], [1, 5, 2, Yes], [1, 5, 3, Yes], [1, 5, 4, Yes], [1, 5, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S054
S056 · implies
Before
first original pair=[2, 0];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[2, 0, 0, Yes], [2, 0, 1, Yes], [2, 0, 2, Yes], [2, 0, 3, Yes], [2, 0, 4, Yes], [2, 0, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S055
S057 · implies
Before
first original pair=[2, 1];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[2, 1, 0, Yes], [2, 1, 1, Yes], [2, 1, 2, Yes], [2, 1, 3, Yes], [2, 1, 4, Yes], [2, 1, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S056
S058 · implies
Before
first original pair=[2, 2];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[2, 2, 0, Yes], [2, 2, 1, Yes], [2, 2, 2, Yes], [2, 2, 3, Yes], [2, 2, 4, Yes], [2, 2, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S057
S059 · implies
Before
first original pair=[2, 3];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[2, 3, 0, Yes], [2, 3, 1, Yes], [2, 3, 2, Yes], [2, 3, 3, Yes], [2, 3, 4, Yes], [2, 3, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S058
S060 · implies
Before
first original pair=[2, 4];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[2, 4, 0, Yes], [2, 4, 1, Yes], [2, 4, 2, Yes], [2, 4, 3, Yes], [2, 4, 4, Yes], [2, 4, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S059
S061 · implies
Before
first original pair=[2, 5];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[2, 5, 0, Yes], [2, 5, 1, Yes], [2, 5, 2, Yes], [2, 5, 3, Yes], [2, 5, 4, Yes], [2, 5, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S060
S062 · implies
Before
first original pair=[3, 0];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[3, 0, 0, Yes], [3, 0, 1, Yes], [3, 0, 2, Yes], [3, 0, 3, Yes], [3, 0, 4, Yes], [3, 0, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S061
S063 · implies
Before
first original pair=[3, 1];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[3, 1, 0, Yes], [3, 1, 1, Yes], [3, 1, 2, Yes], [3, 1, 3, Yes], [3, 1, 4, Yes], [3, 1, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S062
S064 · implies
Before
first original pair=[3, 2];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[3, 2, 0, Yes], [3, 2, 1, Yes], [3, 2, 2, Yes], [3, 2, 3, Yes], [3, 2, 4, Yes], [3, 2, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S063
S065 · implies
Before
first original pair=[3, 3];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[3, 3, 0, Yes], [3, 3, 1, Yes], [3, 3, 2, Yes], [3, 3, 3, Yes], [3, 3, 4, Yes], [3, 3, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S064
S066 · implies
Before
first original pair=[3, 4];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[3, 4, 0, Yes], [3, 4, 1, Yes], [3, 4, 2, Yes], [3, 4, 3, Yes], [3, 4, 4, Yes], [3, 4, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S065
S067 · implies
Before
first original pair=[3, 5];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[3, 5, 0, Yes], [3, 5, 1, Yes], [3, 5, 2, Yes], [3, 5, 3, Yes], [3, 5, 4, Yes], [3, 5, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S066
S068 · implies
Before
first original pair=[4, 0];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[4, 0, 0, Yes], [4, 0, 1, Yes], [4, 0, 2, Yes], [4, 0, 3, Yes], [4, 0, 4, Yes], [4, 0, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S067
S069 · implies
Before
first original pair=[4, 1];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[4, 1, 0, Yes], [4, 1, 1, Yes], [4, 1, 2, Yes], [4, 1, 3, Yes], [4, 1, 4, Yes], [4, 1, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S068
S070 · implies
Before
first original pair=[4, 2];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[4, 2, 0, Yes], [4, 2, 1, Yes], [4, 2, 2, Yes], [4, 2, 3, Yes], [4, 2, 4, Yes], [4, 2, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S069
S071 · implies
Before
first original pair=[4, 3];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[4, 3, 0, Yes], [4, 3, 1, Yes], [4, 3, 2, Yes], [4, 3, 3, Yes], [4, 3, 4, Yes], [4, 3, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S070
S072 · implies
Before
first original pair=[4, 4];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[4, 4, 0, Yes], [4, 4, 1, Yes], [4, 4, 2, Yes], [4, 4, 3, Yes], [4, 4, 4, Yes], [4, 4, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S071
S073 · implies
Before
first original pair=[4, 5];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[4, 5, 0, Yes], [4, 5, 1, Yes], [4, 5, 2, Yes], [4, 5, 3, Yes], [4, 5, 4, Yes], [4, 5, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S072
S074 · implies
Before
first original pair=[5, 0];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[5, 0, 0, Yes], [5, 0, 1, Yes], [5, 0, 2, Yes], [5, 0, 3, Yes], [5, 0, 4, Yes], [5, 0, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S073
S075 · implies
Before
first original pair=[5, 1];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[5, 1, 0, Yes], [5, 1, 1, Yes], [5, 1, 2, Yes], [5, 1, 3, Yes], [5, 1, 4, Yes], [5, 1, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S074
S076 · implies
Before
first original pair=[5, 2];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[5, 2, 0, Yes], [5, 2, 1, Yes], [5, 2, 2, Yes], [5, 2, 3, Yes], [5, 2, 4, Yes], [5, 2, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S075
S077 · implies
Before
first original pair=[5, 3];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[5, 3, 0, Yes], [5, 3, 1, Yes], [5, 3, 2, Yes], [5, 3, 3, Yes], [5, 3, 4, Yes], [5, 3, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S076
S078 · implies
Before
first original pair=[5, 4];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[5, 4, 0, Yes], [5, 4, 1, Yes], [5, 4, 2, Yes], [5, 4, 3, Yes], [5, 4, 4, Yes], [5, 4, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S077
S079 · implies
Before
first original pair=[5, 5];all composable second endpoints=[0, 1, 2, 3, 4, 5]
After
required transitivity rows [a,b,d,aRd]=[[5, 5, 0, Yes], [5, 5, 1, Yes], [5, 5, 2, Yes], [5, 5, 3, Yes], [5, 5, 4, Yes], [5, 5, 5, Yes]]
Conditions: none
List every supplied bRd continuation and check aRd.A failure is the triple[a,b,d].No continuations make this premise branch vacuous,not the whole relation an equivalence.
Step identifiers and dependencies
Rule: relation_every_composable_original_pair · Uses: G001, S078
S080 · implies
Before
least unseen representative=0;complete relation=[[0, 0], [0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 0], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 0], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [4, 0], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [5, 0], [5, 1], [5, 2], [5, 3], [5, 4], [5, 5]]
After
class=[0, 1, 2, 3, 4, 5];seen=[0, 1, 2, 3, 4, 5]
Conditions: none
Only after every axiom passes,derive classes from the original relation.Choose the least unseen representative and increasing class labels.The axioms ensure disjoint-or-equal classes covering the carrier.
Step identifiers and dependencies
Rule: equivalence_class_after_all_axioms · Uses: G001, S079
S081 · implies
Before
missing reflexive=[];missing reverses=[];missing transitive triples=[]
After
equivalence=Yes;classes=[[0, 1, 2, 3, 4, 5]]
Conditions: none
Emit classes only when all three obligations pass.On any failure leave classes empty,without repairing the relation by connected grouping.
Step identifiers and dependencies
Rule: equivalence_three_obligations_no_silent_closure · Uses: G001, S080