/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 110-rule system { 0 0 0 1 0 ⟶ 2 2 1 1 1 0 , 0 1 0 1 2 2 ⟶ 1 1 2 0 0 2 , 2 2 0 0 2 2 ⟶ 2 2 1 1 1 2 0 , 0 1 0 0 2 2 2 1 ⟶ 2 1 1 1 2 0 0 1 1 , 0 1 2 0 1 1 2 1 ⟶ 1 0 2 2 1 1 0 1 , 0 2 2 2 2 0 1 2 2 ⟶ 1 1 1 1 0 1 0 0 2 2 , 2 0 0 0 1 0 2 0 0 ⟶ 1 0 0 1 2 0 1 1 1 1 2 , 2 2 1 2 0 0 1 1 2 ⟶ 0 1 1 1 1 1 2 2 2 1 , 0 0 0 0 0 1 1 0 1 0 1 ⟶ 0 1 2 1 0 1 0 0 1 1 1 1 , 0 0 1 0 2 0 0 0 0 0 2 ⟶ 1 0 0 1 2 1 2 1 0 1 1 1 , 0 2 0 1 0 2 0 1 0 1 2 ⟶ 1 0 0 2 1 1 1 1 2 1 1 2 0 , 1 0 0 0 2 0 2 1 2 2 0 ⟶ 1 0 1 0 0 2 2 2 1 1 1 1 2 , 2 2 1 2 0 2 0 1 0 1 2 ⟶ 1 1 1 1 0 2 1 2 2 2 1 0 , 1 0 1 0 2 2 0 1 0 0 0 0 ⟶ 1 0 0 2 1 2 0 0 0 1 1 1 1 2 , 1 2 1 2 0 1 2 2 2 0 0 1 ⟶ 1 1 1 1 2 1 1 1 1 1 1 1 1 0 1 1 2 , 0 1 2 2 2 2 2 2 0 0 2 2 1 ⟶ 2 2 0 2 2 0 1 0 1 1 1 1 1 2 2 1 , 0 2 0 1 1 0 2 2 0 0 1 2 0 ⟶ 2 1 2 1 0 2 1 1 1 1 1 1 0 1 , 0 2 0 2 0 1 0 2 2 1 1 0 0 ⟶ 1 1 1 1 0 2 2 2 2 2 1 1 1 2 0 1 , 2 0 2 0 0 2 2 2 2 1 2 0 2 ⟶ 0 0 2 0 1 0 0 2 1 1 1 1 1 2 1 2 , 2 1 0 1 1 0 1 0 2 1 2 0 2 ⟶ 1 1 1 1 1 1 0 0 0 1 0 2 2 2 0 2 , 2 1 1 0 2 2 1 2 0 2 2 1 2 ⟶ 1 1 1 0 2 1 1 1 1 1 0 1 2 2 2 0 , 2 1 2 0 0 0 1 0 1 0 0 0 0 ⟶ 2 2 0 1 2 0 0 1 1 1 1 1 1 0 , 0 0 0 2 0 1 1 2 1 1 2 0 1 1 ⟶ 1 1 2 0 2 0 2 2 1 2 0 1 1 1 1 , 2 1 2 0 1 2 1 2 1 2 1 1 0 0 ⟶ 1 1 2 0 2 1 2 2 1 2 1 1 0 0 , 1 2 0 1 0 0 1 2 2 0 0 0 0 0 2 ⟶ 1 1 1 2 1 0 1 1 1 1 2 2 1 1 0 0 0 , 2 1 1 0 0 0 0 1 0 1 2 2 1 2 1 ⟶ 2 0 1 1 1 1 2 2 1 1 2 2 1 1 1 2 , 0 1 0 1 0 1 0 1 0 2 1 1 0 0 0 2 ⟶ 0 0 1 1 1 1 2 0 1 0 1 0 0 0 0 2 , 1 0 2 2 1 0 1 0 1 1 2 2 1 2 0 0 ⟶ 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 2 2 0 2 , 2 0 0 0 2 2 0 2 1 0 1 0 2 2 1 1 ⟶ 2 1 1 1 1 0 0 0 1 1 2 1 2 0 2 1 1 1 , 2 1 2 1 2 0 1 2 0 2 0 0 1 0 2 0 ⟶ 1 1 1 0 1 2 2 1 0 1 2 2 2 1 1 1 2 , 2 2 0 2 2 0 2 2 1 2 0 2 1 0 2 2 ⟶ 1 2 1 1 1 2 2 0 1 2 2 1 1 1 2 2 1 1 1 , 0 2 2 0 0 2 1 2 2 0 2 1 0 0 0 0 2 ⟶ 0 1 0 2 1 2 2 1 2 1 2 2 0 1 1 1 1 2 2 , 1 1 2 0 0 0 2 0 0 1 1 0 0 2 1 2 2 ⟶ 1 1 2 1 1 1 1 2 2 1 1 2 1 0 0 0 1 1 0 , 2 2 2 2 0 1 0 2 2 0 2 0 0 0 0 0 0 ⟶ 0 1 2 1 2 2 0 1 0 0 0 1 1 1 1 2 2 1 , 0 2 2 2 2 0 2 0 0 1 1 1 0 2 2 1 1 2 ⟶ 0 1 0 2 1 1 1 1 1 1 0 2 2 1 1 1 2 1 2 2 , 1 2 2 0 2 2 0 0 1 2 1 2 0 1 0 2 1 2 ⟶ 1 0 0 2 2 1 0 2 1 1 2 0 1 1 1 1 0 2 0 0 , 2 1 2 0 2 0 0 2 2 1 2 2 0 2 0 2 2 1 ⟶ 0 2 0 0 1 1 1 2 0 2 0 1 0 2 0 2 2 1 1 , 0 0 2 2 0 0 1 0 0 1 1 2 1 0 1 0 2 1 2 ⟶ 0 0 2 2 0 0 1 1 2 0 0 1 0 2 1 0 1 1 1 1 1 1 1 , 0 1 0 0 1 1 0 0 0 1 0 0 2 1 2 1 2 0 1 ⟶ 2 0 0 2 2 1 1 1 2 0 0 2 0 1 1 0 2 1 0 1 , 0 1 2 0 2 1 0 2 0 2 2 1 2 1 2 2 0 2 2 ⟶ 0 2 0 2 0 1 1 1 1 1 2 2 2 0 2 2 0 2 2 2 , 0 1 2 1 2 2 0 1 0 1 0 2 0 1 0 1 0 1 0 ⟶ 1 1 1 0 1 2 0 0 1 1 0 0 0 0 1 2 0 2 2 , 0 0 0 0 0 2 1 2 2 2 1 0 2 0 1 2 2 0 1 2 ⟶ 0 2 2 1 1 1 1 1 1 2 2 2 2 1 0 2 0 0 0 0 0 2 , 0 1 0 1 2 1 0 1 0 0 2 1 2 0 1 0 2 0 1 2 ⟶ 2 1 2 1 1 2 0 1 0 1 2 2 0 0 0 1 1 1 2 2 2 , 1 0 2 1 1 2 2 0 1 1 1 2 0 2 0 2 2 2 0 2 ⟶ 1 1 2 1 2 1 2 2 1 1 1 1 2 2 0 2 1 0 0 0 1 , 1 2 1 1 2 1 1 2 1 0 1 1 1 2 0 1 2 2 0 0 ⟶ 1 1 1 1 0 1 0 0 0 1 1 1 1 0 0 1 0 1 2 1 2 , 2 0 0 1 2 0 2 2 2 2 0 0 1 1 2 1 2 2 2 2 ⟶ 1 0 1 0 0 0 1 1 1 2 2 2 1 1 1 2 2 2 2 0 2 1 , 2 0 0 2 0 0 1 0 0 0 1 1 2 0 0 1 1 0 2 1 ⟶ 0 1 1 0 2 2 1 1 1 2 2 1 0 2 0 1 1 1 1 2 1 , 2 2 0 0 1 1 2 1 1 0 1 2 0 1 2 0 2 2 0 0 ⟶ 1 2 0 2 2 2 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 2 2 , 0 1 0 1 1 0 1 2 1 2 2 2 1 2 1 0 2 2 0 0 0 ⟶ 1 0 2 0 0 0 1 0 1 1 1 0 0 2 1 1 1 1 0 1 0 0 1 , 2 0 1 1 1 0 1 0 2 0 1 1 2 1 0 1 1 2 2 0 1 ⟶ 2 1 1 1 2 0 0 0 2 0 1 2 0 1 0 2 1 1 1 1 1 , 2 0 2 2 0 1 0 1 0 2 0 1 0 2 2 2 2 0 1 2 0 ⟶ 1 1 1 2 1 0 0 0 1 2 1 2 2 1 1 1 1 2 1 0 0 1 1 1 1 1 1 0 , 0 0 2 2 1 2 0 1 0 0 2 1 1 2 0 2 1 0 1 0 0 0 ⟶ 2 1 0 2 2 2 1 1 1 1 1 1 1 1 2 0 1 0 0 0 1 1 0 0 , 0 1 0 2 1 1 2 1 2 2 0 1 1 2 0 1 2 0 2 1 2 1 ⟶ 1 2 1 1 0 0 0 1 2 1 0 2 1 1 2 0 1 2 0 1 1 1 1 , 0 2 2 0 2 0 0 1 2 1 2 0 2 0 0 0 1 0 2 2 1 0 ⟶ 0 0 1 1 1 2 2 0 1 2 0 1 2 1 1 1 1 0 2 0 1 0 1 , 1 1 2 1 0 1 1 2 2 2 2 2 0 1 1 0 2 2 0 2 1 2 ⟶ 1 1 1 1 0 2 0 1 0 1 0 2 0 0 1 2 2 0 0 1 1 2 1 , 2 1 1 1 2 2 0 1 0 1 0 2 0 0 0 0 1 2 0 2 2 2 ⟶ 1 2 1 2 0 1 1 1 1 0 0 1 1 0 2 1 2 1 1 0 0 0 1 , 2 1 2 0 1 1 2 2 0 1 1 0 1 2 1 0 2 2 2 2 1 2 ⟶ 1 1 1 1 1 2 0 1 1 1 2 2 2 1 0 1 1 2 0 2 1 0 0 , 2 1 2 2 0 2 1 2 0 2 0 2 0 2 2 0 0 1 0 0 2 1 ⟶ 0 2 1 0 0 1 0 2 1 2 1 1 0 0 0 2 2 2 1 2 1 1 1 , 2 1 2 2 1 0 2 0 1 0 2 1 1 0 0 2 0 0 1 1 0 2 ⟶ 2 2 2 0 2 0 0 0 1 2 0 0 2 0 2 0 2 0 2 1 1 1 1 , 0 0 1 2 2 1 2 1 2 0 1 1 0 2 2 0 0 2 1 0 0 1 2 ⟶ 2 0 1 1 0 1 2 2 0 2 1 1 2 0 1 1 1 1 0 2 2 1 1 1 , 0 2 0 0 1 1 0 2 1 1 0 1 2 2 1 0 0 0 0 0 2 0 2 ⟶ 0 1 1 1 0 2 2 2 0 0 1 0 2 1 2 0 1 0 0 0 0 0 2 , 0 2 2 1 2 1 0 2 1 2 0 1 0 1 1 1 2 2 0 2 0 2 1 ⟶ 2 0 1 1 1 1 1 1 1 2 0 2 2 1 1 1 0 2 0 2 2 2 1 1 0 , 1 0 2 2 2 1 2 1 1 0 0 1 0 0 0 2 0 2 2 2 2 2 1 ⟶ 1 1 1 2 0 0 1 0 2 1 1 2 2 1 1 1 2 0 1 0 0 1 1 0 1 , 0 0 2 1 0 0 2 1 2 2 0 1 2 0 0 0 2 1 0 2 1 0 2 1 ⟶ 0 1 2 1 2 2 1 1 2 2 1 0 2 0 1 2 1 1 1 0 0 2 0 1 1 , 0 1 0 2 0 0 1 2 1 2 2 2 0 0 0 0 0 0 1 2 0 0 0 0 ⟶ 2 1 0 0 2 0 1 0 1 2 2 1 1 1 2 2 1 1 1 1 2 2 2 2 0 0 , 0 1 0 2 1 0 2 0 1 1 2 2 1 2 1 0 0 0 1 1 1 0 2 2 ⟶ 1 1 1 1 1 2 2 0 0 2 2 0 1 2 0 0 2 2 2 0 1 0 2 1 2 , 0 1 2 1 1 0 1 1 2 1 2 1 0 0 0 1 1 2 0 1 0 0 0 1 ⟶ 0 1 2 2 1 1 0 2 2 1 0 2 1 2 0 1 1 2 1 1 1 1 2 1 1 , 0 2 1 0 1 0 0 1 0 0 0 2 0 0 2 0 2 2 1 0 2 0 2 0 ⟶ 1 0 2 2 0 1 1 2 2 1 2 2 2 2 1 1 1 1 0 0 2 2 1 0 0 2 , 1 2 0 1 1 0 0 0 0 1 2 0 0 2 2 2 1 0 0 0 2 2 0 0 ⟶ 1 0 1 2 2 2 1 2 1 2 1 1 1 1 1 1 0 0 0 1 1 0 0 2 1 0 1 , 1 2 2 0 0 1 2 0 1 2 0 2 0 1 0 1 2 2 1 1 2 1 2 2 ⟶ 1 1 1 1 1 2 0 0 2 2 0 2 1 1 1 0 0 0 1 2 2 0 0 1 1 , 2 1 1 2 1 2 1 2 1 2 0 0 1 0 1 0 1 2 0 2 2 0 2 0 ⟶ 2 0 0 0 0 2 1 0 2 1 1 2 0 2 0 1 0 2 0 1 0 1 1 1 1 , 0 0 0 0 0 0 0 0 2 0 0 1 0 2 1 0 0 2 1 2 2 1 0 1 1 ⟶ 1 1 1 1 0 0 1 0 1 2 2 1 1 1 1 0 2 0 2 0 1 1 0 0 0 0 , 0 1 0 0 2 1 2 0 1 1 0 2 2 1 2 0 1 0 0 0 2 0 2 2 0 ⟶ 0 2 1 1 0 2 2 2 1 1 2 1 2 0 0 1 1 1 1 1 0 0 1 2 0 2 2 , 0 2 2 1 0 1 0 2 2 0 2 0 0 0 1 0 2 2 2 0 2 2 1 2 1 ⟶ 1 0 0 0 2 2 1 1 1 1 2 1 0 2 0 0 2 0 2 1 1 0 0 0 2 1 , 0 2 2 2 2 0 1 0 2 0 0 1 2 2 1 2 2 0 0 0 0 2 1 0 1 ⟶ 2 1 1 1 2 1 2 0 2 1 1 0 0 2 1 1 0 2 1 2 2 0 2 2 1 1 , 1 0 0 0 2 0 1 0 1 0 2 0 1 0 0 1 0 1 2 2 0 1 2 0 2 ⟶ 1 0 2 2 1 0 1 2 1 2 2 0 0 0 1 1 1 1 0 0 2 1 1 1 2 1 , 1 1 2 0 0 1 0 1 2 0 0 2 2 0 0 0 2 2 2 0 2 0 2 0 2 ⟶ 1 1 1 1 2 2 1 1 0 2 2 1 2 0 0 0 1 0 2 1 0 1 2 1 0 0 0 , 1 2 1 1 0 0 1 1 2 0 0 0 1 0 1 0 1 0 0 2 1 2 2 0 2 ⟶ 1 1 1 2 2 2 2 0 0 1 0 1 1 2 1 1 1 0 0 1 1 1 2 0 0 1 , 1 2 2 0 0 2 2 2 1 1 0 0 1 2 0 1 2 1 2 1 1 2 1 2 0 ⟶ 1 2 1 0 2 0 2 1 1 2 2 2 1 0 2 1 0 2 0 0 2 2 1 1 1 , 2 0 0 0 0 2 0 1 0 0 1 0 0 1 0 0 0 0 2 1 0 2 0 2 1 ⟶ 1 0 2 0 1 2 1 1 1 0 1 0 2 2 2 0 1 0 2 2 2 2 1 1 1 0 1 , 2 1 2 2 1 1 2 2 0 1 2 2 0 2 0 1 0 2 1 1 0 0 2 0 2 ⟶ 1 0 1 2 1 0 1 1 1 1 0 1 2 2 2 1 1 1 1 1 0 0 2 1 0 2 2 , 0 0 1 1 2 1 0 1 2 2 2 0 0 1 1 2 0 0 2 1 2 1 0 0 0 2 ⟶ 0 1 2 1 1 0 2 1 1 1 0 0 1 1 0 1 1 2 2 2 1 1 1 0 0 0 0 0 , 0 2 1 0 0 2 1 0 2 0 0 1 1 0 1 2 0 0 2 0 0 2 2 2 2 0 ⟶ 2 1 1 1 0 2 2 0 0 2 1 0 1 0 1 2 0 0 2 1 1 1 2 0 1 2 2 2 , 0 2 1 2 0 0 0 2 0 2 0 1 1 2 1 2 0 1 2 2 0 2 2 0 1 2 ⟶ 1 0 0 1 1 2 2 2 2 2 2 1 1 2 1 1 0 0 1 1 1 2 0 0 2 0 2 , 0 0 2 0 2 0 2 2 1 0 2 1 2 1 0 2 1 1 0 2 1 0 2 0 0 1 0 ⟶ 1 1 1 1 1 0 1 0 0 2 2 2 0 1 2 0 2 1 2 0 0 0 1 1 1 2 0 0 0 , 0 1 0 2 2 2 2 0 2 1 2 2 0 0 0 1 0 1 1 2 0 1 0 1 0 0 2 ⟶ 1 1 1 1 2 0 2 2 2 0 0 0 0 1 0 2 1 2 1 2 1 0 2 2 1 2 0 2 1 , 0 1 1 2 1 0 2 0 1 0 0 0 1 2 1 1 0 1 2 2 1 2 2 2 2 1 2 ⟶ 0 0 0 2 1 1 1 1 0 2 2 1 1 2 2 0 1 2 1 2 1 1 1 1 2 0 1 1 , 0 2 1 2 1 2 2 2 0 2 2 0 2 0 2 1 0 1 1 1 1 2 1 0 1 2 0 ⟶ 0 2 2 1 0 1 0 2 2 1 2 2 0 0 1 1 2 1 1 1 1 1 1 2 1 1 1 1 , 1 0 2 2 2 1 0 1 1 0 2 0 1 2 0 0 0 2 1 1 2 1 2 1 1 2 1 ⟶ 1 1 1 0 1 2 1 1 2 2 2 0 1 0 0 1 2 2 2 2 1 1 1 0 0 0 2 2 , 1 2 1 0 1 0 0 0 0 1 2 2 2 1 2 1 0 1 1 2 2 2 0 2 0 1 1 ⟶ 1 2 2 2 0 1 1 1 2 0 1 1 2 1 2 0 0 0 2 1 2 1 0 2 1 2 1 1 , 1 2 1 2 2 2 0 0 2 1 2 1 1 0 0 0 2 2 1 0 2 2 1 0 0 0 1 ⟶ 1 1 1 1 1 2 2 0 1 2 0 2 0 2 1 2 1 1 1 1 0 0 0 2 1 0 0 0 1 , 2 1 1 0 2 2 2 1 0 2 0 1 0 0 2 1 0 0 0 2 1 1 0 0 1 1 1 ⟶ 1 2 1 1 1 2 0 0 0 0 2 0 2 2 1 1 1 2 2 1 1 1 1 1 1 0 1 2 , 2 2 0 2 0 0 1 1 0 1 0 2 0 2 0 1 0 2 1 2 0 2 2 1 0 0 2 ⟶ 2 2 2 0 1 1 0 1 1 2 2 2 2 2 1 1 0 0 1 1 1 0 0 1 1 1 2 0 0 , 0 2 2 1 1 2 2 0 1 2 1 2 2 0 2 2 1 2 2 2 2 2 2 0 0 2 1 0 ⟶ 2 2 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 2 1 1 0 1 2 0 2 2 , 2 0 0 2 2 0 0 0 0 0 0 2 0 1 0 2 2 0 2 2 0 0 0 2 2 0 2 2 ⟶ 1 2 0 1 0 2 2 1 2 0 1 1 1 1 2 2 0 1 1 2 0 1 0 1 0 1 1 2 2 2 , 2 0 1 2 2 2 0 1 2 0 0 2 0 1 1 2 1 0 0 0 1 2 1 1 2 0 2 0 ⟶ 0 2 1 2 0 1 1 0 1 1 1 1 0 0 0 1 2 2 2 1 0 0 1 0 0 1 1 0 0 , 2 1 0 2 1 0 2 0 0 0 1 2 2 2 1 2 0 0 1 1 2 1 0 0 0 0 0 0 ⟶ 1 0 2 1 1 1 1 1 2 0 2 1 0 1 1 1 2 2 1 1 0 2 2 1 2 1 0 2 0 , 2 1 1 0 1 1 0 1 2 1 2 1 2 2 0 1 2 0 1 0 0 0 2 0 2 2 2 0 ⟶ 1 1 1 0 2 2 0 1 1 0 1 1 2 2 2 1 0 2 0 2 0 0 1 0 2 2 0 2 , 2 1 2 1 2 2 1 2 1 2 1 1 0 2 2 0 2 1 2 1 2 0 1 0 0 2 1 0 ⟶ 1 1 1 2 2 1 1 2 1 2 1 2 1 2 0 1 1 2 1 0 0 2 2 0 2 1 1 1 1 2 , 2 1 2 2 0 1 2 0 2 2 2 2 1 2 2 1 0 0 0 0 2 0 1 0 2 1 2 2 ⟶ 2 2 1 1 2 1 0 0 0 1 1 1 1 1 0 2 2 2 2 0 1 1 1 0 1 0 2 0 1 , 2 2 0 0 0 2 0 1 0 1 2 1 0 2 0 2 2 1 0 0 0 1 0 2 0 1 0 2 ⟶ 1 0 1 2 0 0 1 2 0 0 2 0 0 2 0 2 0 0 0 2 2 2 1 1 1 1 0 1 2 1 , 2 2 1 0 2 1 2 1 2 0 2 0 0 1 0 2 0 1 2 1 0 0 0 0 1 1 2 0 ⟶ 2 2 2 1 2 2 0 2 2 0 2 0 2 2 1 0 2 1 1 1 2 1 0 0 1 1 1 1 1 1 , 0 0 1 0 0 0 2 0 2 0 1 2 2 0 1 2 1 0 1 0 0 0 0 2 2 0 2 0 2 ⟶ 0 1 0 2 0 0 1 2 2 0 1 2 2 0 1 0 0 0 0 0 0 0 2 0 0 2 1 2 2 , 0 1 0 1 1 0 1 0 0 2 0 2 2 2 0 0 1 2 2 0 2 1 2 1 1 0 0 1 1 ⟶ 0 1 2 2 2 2 1 1 2 1 2 2 1 1 1 0 2 2 2 1 1 1 1 0 1 0 2 0 2 2 , 0 1 1 0 1 0 1 2 0 2 0 2 0 0 0 2 1 1 2 1 1 1 0 0 2 0 1 0 0 ⟶ 2 1 1 0 2 0 0 0 0 2 0 2 0 1 0 1 1 1 1 1 2 1 0 0 1 2 0 0 0 , 1 2 0 1 1 1 0 0 0 2 0 0 0 1 2 2 2 1 0 1 1 2 1 2 0 1 2 0 2 ⟶ 1 2 2 1 2 2 0 0 0 2 0 0 1 0 0 1 1 1 1 1 2 2 0 0 0 2 1 2 1 , 1 2 1 1 0 1 2 2 1 0 1 2 2 2 1 0 2 0 1 0 1 0 2 0 0 0 1 0 0 ⟶ 1 1 0 0 0 1 1 1 1 0 0 0 2 0 0 2 2 0 2 0 1 0 1 2 0 2 0 1 2 2 , 2 0 1 2 2 1 0 0 2 2 2 1 0 1 0 0 2 0 2 1 0 0 1 0 0 1 2 2 0 ⟶ 2 0 0 1 2 2 0 2 2 1 0 1 0 0 2 1 1 2 2 0 0 1 1 1 2 1 0 2 2 0 , 2 2 1 0 1 0 1 2 1 2 0 0 1 0 0 2 2 1 2 2 2 0 0 0 2 1 0 0 2 ⟶ 2 0 2 1 1 1 2 2 1 1 0 2 0 2 1 0 0 2 2 0 1 1 2 2 2 0 1 1 2 0 , 1 1 0 1 1 2 0 2 2 0 0 1 2 0 0 0 2 1 2 1 2 2 0 0 2 2 2 0 1 1 ⟶ 1 1 2 1 1 2 1 2 2 1 1 0 0 2 0 0 2 0 2 0 0 0 2 0 2 2 2 0 1 1 } The system was reversed. After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 110-rule system { 0 1 0 0 0 ⟶ 0 1 1 1 2 2 , 2 2 1 0 1 0 ⟶ 2 0 0 2 1 1 , 2 2 0 0 2 2 ⟶ 0 2 1 1 1 2 2 , 1 2 2 2 0 0 1 0 ⟶ 1 1 0 0 2 1 1 1 2 , 1 2 1 1 0 2 1 0 ⟶ 1 0 1 1 2 2 0 1 , 2 2 1 0 2 2 2 2 0 ⟶ 2 2 0 0 1 0 1 1 1 1 , 0 0 2 0 1 0 0 0 2 ⟶ 2 1 1 1 1 0 2 1 0 0 1 , 2 1 1 0 0 2 1 2 2 ⟶ 1 2 2 2 1 1 1 1 1 0 , 1 0 1 0 1 1 0 0 0 0 0 ⟶ 1 1 1 1 0 0 1 0 1 2 1 0 , 2 0 0 0 0 0 2 0 1 0 0 ⟶ 1 1 1 0 1 2 1 2 1 0 0 1 , 2 1 0 1 0 2 0 1 0 2 0 ⟶ 0 2 1 1 2 1 1 1 1 2 0 0 1 , 0 2 2 1 2 0 2 0 0 0 1 ⟶ 2 1 1 1 1 2 2 2 0 0 1 0 1 , 2 1 0 1 0 2 0 2 1 2 2 ⟶ 0 1 2 2 2 1 2 0 1 1 1 1 , 0 0 0 0 1 0 2 2 0 1 0 1 ⟶ 2 1 1 1 1 0 0 0 2 1 2 0 0 1 , 1 0 0 2 2 2 1 0 2 1 2 1 ⟶ 2 1 1 0 1 1 1 1 1 1 1 1 2 1 1 1 1 , 1 2 2 0 0 2 2 2 2 2 2 1 0 ⟶ 1 2 2 1 1 1 1 1 0 1 0 2 2 0 2 2 , 0 2 1 0 0 2 2 0 1 1 0 2 0 ⟶ 1 0 1 1 1 1 1 1 2 0 1 2 1 2 , 0 0 1 1 2 2 0 1 0 2 0 2 0 ⟶ 1 0 2 1 1 1 2 2 2 2 2 0 1 1 1 1 , 2 0 2 1 2 2 2 2 0 0 2 0 2 ⟶ 2 1 2 1 1 1 1 1 2 0 0 1 0 2 0 0 , 2 0 2 1 2 0 1 0 1 1 0 1 2 ⟶ 2 0 2 2 2 0 1 0 0 0 1 1 1 1 1 1 , 2 1 2 2 0 2 1 2 2 0 1 1 2 ⟶ 0 2 2 2 1 0 1 1 1 1 1 2 0 1 1 1 , 0 0 0 0 1 0 1 0 0 0 2 1 2 ⟶ 0 1 1 1 1 1 1 0 0 2 1 0 2 2 , 1 1 0 2 1 1 2 1 1 0 2 0 0 0 ⟶ 1 1 1 1 0 2 1 2 2 0 2 0 2 1 1 , 0 0 1 1 2 1 2 1 2 1 0 2 1 2 ⟶ 0 0 1 1 2 1 2 2 1 2 0 2 1 1 , 2 0 0 0 0 0 2 2 1 0 0 1 0 2 1 ⟶ 0 0 0 1 1 2 2 1 1 1 1 0 1 2 1 1 1 , 1 2 1 2 2 1 0 1 0 0 0 0 1 1 2 ⟶ 2 1 1 1 2 2 1 1 2 2 1 1 1 1 0 2 , 2 0 0 0 1 1 2 0 1 0 1 0 1 0 1 0 ⟶ 2 0 0 0 0 1 0 1 0 2 1 1 1 1 0 0 , 0 0 2 1 2 2 1 1 0 1 0 1 2 2 0 1 ⟶ 2 0 2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 1 1 , 1 1 2 2 0 1 0 1 2 0 2 2 0 0 0 2 ⟶ 1 1 1 2 0 2 1 2 1 1 0 0 0 1 1 1 1 2 , 0 2 0 1 0 0 2 0 2 1 0 2 1 2 1 2 ⟶ 2 1 1 1 2 2 2 1 0 1 2 2 1 0 1 1 1 , 2 2 0 1 2 0 2 1 2 2 0 2 2 0 2 2 ⟶ 1 1 1 2 2 1 1 1 2 2 1 0 2 2 1 1 1 2 1 , 2 0 0 0 0 1 2 0 2 2 1 2 0 0 2 2 0 ⟶ 2 2 1 1 1 1 0 2 2 1 2 1 2 2 1 2 0 1 0 , 2 2 1 2 0 0 1 1 0 0 2 0 0 0 2 1 1 ⟶ 0 1 1 0 0 0 1 2 1 1 2 2 1 1 1 1 2 1 1 , 0 0 0 0 0 0 2 0 2 2 0 1 0 2 2 2 2 ⟶ 1 2 2 1 1 1 1 0 0 0 1 0 2 2 1 2 1 0 , 2 1 1 2 2 0 1 1 1 0 0 2 0 2 2 2 2 0 ⟶ 2 2 1 2 1 1 1 2 2 0 1 1 1 1 1 1 2 0 1 0 , 2 1 2 0 1 0 2 1 2 1 0 0 2 2 0 2 2 1 ⟶ 0 0 2 0 1 1 1 1 0 2 1 1 2 0 1 2 2 0 0 1 , 1 2 2 0 2 0 2 2 1 2 2 0 0 2 0 2 1 2 ⟶ 1 1 2 2 0 2 0 1 0 2 0 2 1 1 1 0 0 2 0 , 2 1 2 0 1 0 1 2 1 1 0 0 1 0 0 2 2 0 0 ⟶ 1 1 1 1 1 1 1 0 1 2 0 1 0 0 2 1 1 0 0 2 2 0 0 , 1 0 2 1 2 1 2 0 0 1 0 0 0 1 1 0 0 1 0 ⟶ 1 0 1 2 0 1 1 0 2 0 0 2 1 1 1 2 2 0 0 2 , 2 2 0 2 2 1 2 1 2 2 0 2 0 1 2 0 2 1 0 ⟶ 2 2 2 0 2 2 0 2 2 2 1 1 1 1 1 0 2 0 2 0 , 0 1 0 1 0 1 0 2 0 1 0 1 0 2 2 1 2 1 0 ⟶ 2 2 0 2 1 0 0 0 0 1 1 0 0 2 1 0 1 1 1 , 2 1 0 2 2 1 0 2 0 1 2 2 2 1 2 0 0 0 0 0 ⟶ 2 0 0 0 0 0 2 0 1 2 2 2 2 1 1 1 1 1 1 2 2 0 , 2 1 0 2 0 1 0 2 1 2 0 0 1 0 1 2 1 0 1 0 ⟶ 2 2 2 1 1 1 0 0 0 2 2 1 0 1 0 2 1 1 2 1 2 , 2 0 2 2 2 0 2 0 2 1 1 1 0 2 2 1 1 2 0 1 ⟶ 1 0 0 0 1 2 0 2 2 1 1 1 1 2 2 1 2 1 2 1 1 , 0 0 2 2 1 0 2 1 1 1 0 1 2 1 1 2 1 1 2 1 ⟶ 2 1 2 1 0 1 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 , 2 2 2 2 1 2 1 1 0 0 2 2 2 2 0 2 1 0 0 2 ⟶ 1 2 0 2 2 2 2 1 1 1 2 2 2 1 1 1 0 0 0 1 0 1 , 1 2 0 1 1 0 0 2 1 1 0 0 0 1 0 0 2 0 0 2 ⟶ 1 2 1 1 1 1 0 2 0 1 2 2 1 1 1 2 2 0 1 1 0 , 0 0 2 2 0 2 1 0 2 1 0 1 1 2 1 1 0 0 2 2 ⟶ 2 2 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 2 2 2 0 2 1 , 0 0 0 2 2 0 1 2 1 2 2 2 1 2 1 0 1 1 0 1 0 ⟶ 1 0 0 1 0 1 1 1 1 2 0 0 1 1 1 0 1 0 0 0 2 0 1 , 1 0 2 2 1 1 0 1 2 1 1 0 2 0 1 0 1 1 1 0 2 ⟶ 1 1 1 1 1 2 0 1 0 2 1 0 2 0 0 0 2 1 1 1 2 , 0 2 1 0 2 2 2 2 0 1 0 2 0 1 0 1 0 2 2 0 2 ⟶ 0 1 1 1 1 1 1 0 0 1 2 1 1 1 1 2 2 1 2 1 0 0 0 1 2 1 1 1 , 0 0 0 1 0 1 2 0 2 1 1 2 0 0 1 0 2 1 2 2 0 0 ⟶ 0 0 1 1 0 0 0 1 0 2 1 1 1 1 1 1 1 1 2 2 2 0 1 2 , 1 2 1 2 0 2 1 0 2 1 1 0 2 2 1 2 1 1 2 0 1 0 ⟶ 1 1 1 1 0 2 1 0 2 1 1 2 0 1 2 1 0 0 0 1 1 2 1 , 0 1 2 2 0 1 0 0 0 2 0 2 1 2 1 0 0 2 0 2 2 0 ⟶ 1 0 1 0 2 0 1 1 1 1 2 1 0 2 1 0 2 2 1 1 1 0 0 , 2 1 2 0 2 2 0 1 1 0 2 2 2 2 2 1 1 0 1 2 1 1 ⟶ 1 2 1 1 0 0 2 2 1 0 0 2 0 1 0 1 0 2 0 1 1 1 1 , 2 2 2 0 2 1 0 0 0 0 2 0 1 0 1 0 2 2 1 1 1 2 ⟶ 1 0 0 0 1 1 2 1 2 0 1 1 0 0 1 1 1 1 0 2 1 2 1 , 2 1 2 2 2 2 0 1 2 1 0 1 1 0 2 2 1 1 0 2 1 2 ⟶ 0 0 1 2 0 2 1 1 0 1 2 2 2 1 1 1 0 2 1 1 1 1 1 , 1 2 0 0 1 0 0 2 2 0 2 0 2 0 2 1 2 0 2 2 1 2 ⟶ 1 1 1 2 1 2 2 2 0 0 0 1 1 2 1 2 0 1 0 0 1 2 0 , 2 0 1 1 0 0 2 0 0 1 1 2 0 1 0 2 0 1 2 2 1 2 ⟶ 1 1 1 1 2 0 2 0 2 0 2 0 0 2 1 0 0 0 2 0 2 2 2 , 2 1 0 0 1 2 0 0 2 2 0 1 1 0 2 1 2 1 2 2 1 0 0 ⟶ 1 1 1 2 2 0 1 1 1 1 0 2 1 1 2 0 2 2 1 0 1 1 0 2 , 2 0 2 0 0 0 0 0 1 2 2 1 0 1 1 2 0 1 1 0 0 2 0 ⟶ 2 0 0 0 0 0 1 0 2 1 2 0 1 0 0 2 2 2 0 1 1 1 0 , 1 2 0 2 0 2 2 1 1 1 0 1 0 2 1 2 0 1 2 1 2 2 0 ⟶ 0 1 1 2 2 2 0 2 0 1 1 1 2 2 0 2 1 1 1 1 1 1 1 0 2 , 1 2 2 2 2 2 0 2 0 0 0 1 0 0 1 1 2 1 2 2 2 0 1 ⟶ 1 0 1 1 0 0 1 0 2 1 1 1 2 2 1 1 2 0 1 0 0 2 1 1 1 , 1 2 0 1 2 0 1 2 0 0 0 2 1 0 2 2 1 2 0 0 1 2 0 0 ⟶ 1 1 0 2 0 0 1 1 1 2 1 0 2 0 1 2 2 1 1 2 2 1 2 1 0 , 0 0 0 0 2 1 0 0 0 0 0 0 2 2 2 1 2 1 0 0 2 0 1 0 ⟶ 0 0 2 2 2 2 1 1 1 1 2 2 1 1 1 2 2 1 0 1 0 2 0 0 1 2 , 2 2 0 1 1 1 0 0 0 1 2 1 2 2 1 1 0 2 0 1 2 0 1 0 ⟶ 2 1 2 0 1 0 2 2 2 0 0 2 1 0 2 2 0 0 2 2 1 1 1 1 1 , 1 0 0 0 1 0 2 1 1 0 0 0 1 2 1 2 1 1 0 1 1 2 1 0 ⟶ 1 1 2 1 1 1 1 2 1 1 0 2 1 2 0 1 2 2 0 1 1 2 2 1 0 , 0 2 0 2 0 1 2 2 0 2 0 0 2 0 0 0 1 0 0 1 0 1 2 0 ⟶ 2 0 0 1 2 2 0 0 1 1 1 1 2 2 2 2 1 2 2 1 1 0 2 2 0 1 , 0 0 2 2 0 0 0 1 2 2 2 0 0 2 1 0 0 0 0 1 1 0 2 1 ⟶ 1 0 1 2 0 0 1 1 0 0 0 1 1 1 1 1 1 2 1 2 1 2 2 2 1 0 1 , 2 2 1 2 1 1 2 2 1 0 1 0 2 0 2 1 0 2 1 0 0 2 2 1 ⟶ 1 1 0 0 2 2 1 0 0 0 1 1 1 2 0 2 2 0 0 2 1 1 1 1 1 , 0 2 0 2 2 0 2 1 0 1 0 1 0 0 2 1 2 1 2 1 2 1 1 2 ⟶ 1 1 1 1 0 1 0 2 0 1 0 2 0 2 1 1 2 0 1 2 0 0 0 0 2 , 1 1 0 1 2 2 1 2 0 0 1 2 0 1 0 0 2 0 0 0 0 0 0 0 0 ⟶ 0 0 0 0 1 1 0 2 0 2 0 1 1 1 1 2 2 1 0 1 0 0 1 1 1 1 , 0 2 2 0 2 0 0 0 1 0 2 1 2 2 0 1 1 0 2 1 2 0 0 1 0 ⟶ 2 2 0 2 1 0 0 1 1 1 1 1 0 0 2 1 2 1 1 2 2 2 0 1 1 2 0 , 1 2 1 2 2 0 2 2 2 0 1 0 0 0 2 0 2 2 0 1 0 1 2 2 0 ⟶ 1 2 0 0 0 1 1 2 0 2 0 0 2 0 1 2 1 1 1 1 2 2 0 0 0 1 , 1 0 1 2 0 0 0 0 2 2 1 2 2 1 0 0 2 0 1 0 2 2 2 2 0 ⟶ 1 1 2 2 0 2 2 1 2 0 1 1 2 0 0 1 1 2 0 2 1 2 1 1 1 2 , 2 0 2 1 0 2 2 1 0 1 0 0 1 0 2 0 1 0 1 0 2 0 0 0 1 ⟶ 1 2 1 1 1 2 0 0 1 1 1 1 0 0 0 2 2 1 2 1 0 1 2 2 0 1 , 2 0 2 0 2 0 2 2 2 0 0 0 2 2 0 0 2 1 0 1 0 0 2 1 1 ⟶ 0 0 0 1 2 1 0 1 2 0 1 0 0 0 2 1 2 2 0 1 1 2 2 1 1 1 1 , 2 0 2 2 1 2 0 0 1 0 1 0 1 0 0 0 2 1 1 0 0 1 1 2 1 ⟶ 1 0 0 2 1 1 1 0 0 1 1 1 2 1 1 0 1 0 0 2 2 2 2 1 1 1 , 0 2 1 2 1 1 2 1 2 1 0 2 1 0 0 1 1 2 2 2 0 0 2 2 1 ⟶ 1 1 1 2 2 0 0 2 0 1 2 0 1 2 2 2 1 1 2 0 2 0 1 2 1 , 1 2 0 2 0 1 2 0 0 0 0 1 0 0 1 0 0 1 0 2 0 0 0 0 2 ⟶ 1 0 1 1 1 2 2 2 2 0 1 0 2 2 2 0 1 0 1 1 1 2 1 0 2 0 1 , 2 0 2 0 0 1 1 2 0 1 0 2 0 2 2 1 0 2 2 1 1 2 2 1 2 ⟶ 2 2 0 1 2 0 0 1 1 1 1 1 2 2 2 1 0 1 1 1 1 0 1 2 1 0 1 , 2 0 0 0 1 2 1 2 0 0 2 1 1 0 0 2 2 2 1 0 1 2 1 1 0 0 ⟶ 0 0 0 0 0 1 1 1 2 2 2 1 1 0 1 1 0 0 1 1 1 2 0 1 1 2 1 0 , 0 2 2 2 2 0 0 2 0 0 2 1 0 1 1 0 0 2 0 1 2 0 0 1 2 0 ⟶ 2 2 2 1 0 2 1 1 1 2 0 0 2 1 0 1 0 1 2 0 0 2 2 0 1 1 1 2 , 2 1 0 2 2 0 2 2 1 0 2 1 2 1 1 0 2 0 2 0 0 0 2 1 2 0 ⟶ 2 0 2 0 0 2 1 1 1 0 0 1 1 2 1 1 2 2 2 2 2 2 1 1 0 0 1 , 0 1 0 0 2 0 1 2 0 1 1 2 0 1 2 1 2 0 1 2 2 0 2 0 2 0 0 ⟶ 0 0 0 2 1 1 1 0 0 0 2 1 2 0 2 1 0 2 2 2 0 0 1 0 1 1 1 1 1 , 2 0 0 1 0 1 0 2 1 1 0 1 0 0 0 2 2 1 2 0 2 2 2 2 0 1 0 ⟶ 1 2 0 2 1 2 2 0 1 2 1 2 1 2 0 1 0 0 0 0 2 2 2 0 2 1 1 1 1 , 2 1 2 2 2 2 1 2 2 1 0 1 1 2 1 0 0 0 1 0 2 0 1 2 1 1 0 ⟶ 1 1 0 2 1 1 1 1 2 1 2 1 0 2 2 1 1 2 2 0 1 1 1 1 2 0 0 0 , 0 2 1 0 1 2 1 1 1 1 0 1 2 0 2 0 2 2 0 2 2 2 1 2 1 2 0 ⟶ 1 1 1 1 2 1 1 1 1 1 1 2 1 1 0 0 2 2 1 2 2 0 1 0 1 2 2 0 , 1 2 1 1 2 1 2 1 1 2 0 0 0 2 1 0 2 0 1 1 0 1 2 2 2 0 1 ⟶ 2 2 0 0 0 1 1 1 2 2 2 2 1 0 0 1 0 2 2 2 1 1 2 1 0 1 1 1 , 1 1 0 2 0 2 2 2 1 1 0 1 2 1 2 2 2 1 0 0 0 0 1 0 1 2 1 ⟶ 1 1 2 1 2 0 1 2 1 2 0 0 0 2 1 2 1 1 0 2 1 1 1 0 2 2 2 1 , 1 0 0 0 1 2 2 0 1 2 2 0 0 0 1 1 2 1 2 0 0 2 2 2 1 2 1 ⟶ 1 0 0 0 1 2 0 0 0 1 1 1 1 2 1 2 0 2 0 2 1 0 2 2 1 1 1 1 1 , 1 1 1 0 0 1 1 2 0 0 0 1 2 0 0 1 0 2 0 1 2 2 2 0 1 1 2 ⟶ 2 1 0 1 1 1 1 1 1 2 2 1 1 1 2 2 0 2 0 0 0 0 2 1 1 1 2 1 , 2 0 0 1 2 2 0 2 1 2 0 1 0 2 0 2 0 1 0 1 1 0 0 2 0 2 2 ⟶ 0 0 2 1 1 1 0 0 1 1 1 0 0 1 1 2 2 2 2 2 1 1 0 1 1 0 2 2 2 , 0 1 2 0 0 2 2 2 2 2 2 1 2 2 0 2 2 1 2 1 0 2 2 1 1 2 2 0 ⟶ 2 2 0 2 1 0 1 1 2 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 2 2 , 2 2 0 2 2 0 0 0 2 2 0 2 2 0 1 0 2 0 0 0 0 0 0 2 2 0 0 2 ⟶ 2 2 2 1 1 0 1 0 1 0 2 1 1 0 2 2 1 1 1 1 0 2 1 2 2 0 1 0 2 1 , 0 2 0 2 1 1 2 1 0 0 0 1 2 1 1 0 2 0 0 2 1 0 2 2 2 1 0 2 ⟶ 0 0 1 1 0 0 1 0 0 1 2 2 2 1 0 0 0 1 1 1 1 0 1 1 0 2 1 2 0 , 0 0 0 0 0 0 1 2 1 1 0 0 2 1 2 2 2 1 0 0 0 2 0 1 2 0 1 2 ⟶ 0 2 0 1 2 1 2 2 0 1 1 2 2 1 1 1 0 1 2 0 2 1 1 1 1 1 2 0 1 , 0 2 2 2 0 2 0 0 0 1 0 2 1 0 2 2 1 2 1 2 1 0 1 1 0 1 1 2 ⟶ 2 0 2 2 0 1 0 0 2 0 2 0 1 2 2 2 1 1 0 1 1 0 2 2 0 1 1 1 , 0 1 2 0 0 1 0 2 1 2 1 2 0 2 2 0 1 1 2 1 2 1 2 2 1 2 1 2 ⟶ 2 1 1 1 1 2 0 2 2 0 0 1 2 1 1 0 2 1 2 1 2 1 2 1 1 2 2 1 1 1 , 2 2 1 2 0 1 0 2 0 0 0 0 1 2 2 1 2 2 2 2 0 2 1 0 2 2 1 2 ⟶ 1 0 2 0 1 0 1 1 1 0 2 2 2 2 0 1 1 1 1 1 0 0 0 1 2 1 1 2 2 , 2 0 1 0 2 0 1 0 0 0 1 2 2 0 2 0 1 2 1 0 1 0 2 0 0 0 2 2 ⟶ 1 2 1 0 1 1 1 1 2 2 2 0 0 0 2 0 2 0 0 2 0 0 2 1 0 0 2 1 0 1 , 0 2 1 1 0 0 0 0 1 2 1 0 2 0 1 0 0 2 0 2 1 2 1 2 0 1 2 2 ⟶ 1 1 1 1 1 1 0 0 1 2 1 1 1 2 0 1 2 2 0 2 0 2 2 0 2 2 1 2 2 2 , 2 0 2 0 2 2 0 0 0 0 1 0 1 2 1 0 2 2 1 0 2 0 2 0 0 0 1 0 0 ⟶ 2 2 1 2 0 0 2 0 0 0 0 0 0 0 1 0 2 2 1 0 2 2 1 0 0 2 0 1 0 , 1 1 0 0 1 1 2 1 2 0 2 2 1 0 0 2 2 2 0 2 0 0 1 0 1 1 0 1 0 ⟶ 2 2 0 2 0 1 0 1 1 1 1 2 2 2 0 1 1 1 2 2 1 2 1 1 2 2 2 2 1 0 , 0 0 1 0 2 0 0 1 1 1 2 1 1 2 0 0 0 2 0 2 0 2 1 0 1 0 1 1 0 ⟶ 0 0 0 2 1 0 0 1 2 1 1 1 1 1 0 1 0 2 0 2 0 0 0 0 2 0 1 1 2 , 2 0 2 1 0 2 1 2 1 1 0 1 2 2 2 1 0 0 0 2 0 0 0 1 1 1 0 2 1 ⟶ 1 2 1 2 0 0 0 2 2 1 1 1 1 1 0 0 1 0 0 2 0 0 0 2 2 1 2 2 1 , 0 0 1 0 0 0 2 0 1 0 1 0 2 0 1 2 2 2 1 0 1 2 2 1 0 1 1 2 1 ⟶ 2 2 1 0 2 0 2 1 0 1 0 2 0 2 2 0 0 2 0 0 0 1 1 1 1 0 0 0 1 1 , 0 2 2 1 0 0 1 0 0 1 2 0 2 0 0 1 0 1 2 2 2 0 0 1 2 2 1 0 2 ⟶ 0 2 2 0 1 2 1 1 1 0 0 2 2 1 1 2 0 0 1 0 1 2 2 0 2 2 1 0 0 2 , 2 0 0 1 2 0 0 0 2 2 2 1 2 2 0 0 1 0 0 2 1 2 1 0 1 0 1 2 2 ⟶ 0 2 1 1 0 2 2 2 1 1 0 2 2 0 0 1 2 0 2 0 1 1 2 2 1 1 1 2 0 2 , 1 1 0 2 2 2 0 0 2 2 1 2 1 2 0 0 0 2 1 0 0 2 2 0 2 1 1 0 1 1 ⟶ 1 1 0 2 2 2 0 2 0 0 0 2 0 2 0 0 2 0 0 1 1 2 2 1 2 1 1 2 1 1 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,0) ↦ 2, (1,1) ↦ 3, (1,2) ↦ 4, (2,2) ↦ 5, (2,0) ↦ 6, (2,1) ↦ 7, (0,2) ↦ 8, (0,4) ↦ 9, (2,4) ↦ 10, (3,0) ↦ 11, (1,4) ↦ 12, (3,2) ↦ 13, (3,1) ↦ 14 }, it remains to prove termination of the 1760-rule system { 0 1 2 0 0 0 ⟶ 0 1 3 3 4 5 6 , 0 1 2 0 0 1 ⟶ 0 1 3 3 4 5 7 , 0 1 2 0 0 8 ⟶ 0 1 3 3 4 5 5 , 0 1 2 0 0 9 ⟶ 0 1 3 3 4 5 10 , 2 1 2 0 0 0 ⟶ 2 1 3 3 4 5 6 , 2 1 2 0 0 1 ⟶ 2 1 3 3 4 5 7 , 2 1 2 0 0 8 ⟶ 2 1 3 3 4 5 5 , 2 1 2 0 0 9 ⟶ 2 1 3 3 4 5 10 , 6 1 2 0 0 0 ⟶ 6 1 3 3 4 5 6 , 6 1 2 0 0 1 ⟶ 6 1 3 3 4 5 7 , 6 1 2 0 0 8 ⟶ 6 1 3 3 4 5 5 , 6 1 2 0 0 9 ⟶ 6 1 3 3 4 5 10 , 11 1 2 0 0 0 ⟶ 11 1 3 3 4 5 6 , 11 1 2 0 0 1 ⟶ 11 1 3 3 4 5 7 , 11 1 2 0 0 8 ⟶ 11 1 3 3 4 5 5 , 11 1 2 0 0 9 ⟶ 11 1 3 3 4 5 10 , 8 5 7 2 1 2 0 ⟶ 8 6 0 8 7 3 2 , 8 5 7 2 1 2 1 ⟶ 8 6 0 8 7 3 3 , 8 5 7 2 1 2 8 ⟶ 8 6 0 8 7 3 4 , 8 5 7 2 1 2 9 ⟶ 8 6 0 8 7 3 12 , 4 5 7 2 1 2 0 ⟶ 4 6 0 8 7 3 2 , 4 5 7 2 1 2 1 ⟶ 4 6 0 8 7 3 3 , 4 5 7 2 1 2 8 ⟶ 4 6 0 8 7 3 4 , 4 5 7 2 1 2 9 ⟶ 4 6 0 8 7 3 12 , 5 5 7 2 1 2 0 ⟶ 5 6 0 8 7 3 2 , 5 5 7 2 1 2 1 ⟶ 5 6 0 8 7 3 3 , 5 5 7 2 1 2 8 ⟶ 5 6 0 8 7 3 4 , 5 5 7 2 1 2 9 ⟶ 5 6 0 8 7 3 12 , 13 5 7 2 1 2 0 ⟶ 13 6 0 8 7 3 2 , 13 5 7 2 1 2 1 ⟶ 13 6 0 8 7 3 3 , 13 5 7 2 1 2 8 ⟶ 13 6 0 8 7 3 4 , 13 5 7 2 1 2 9 ⟶ 13 6 0 8 7 3 12 , 8 5 6 0 8 5 6 ⟶ 0 8 7 3 3 4 5 6 , 8 5 6 0 8 5 7 ⟶ 0 8 7 3 3 4 5 7 , 8 5 6 0 8 5 5 ⟶ 0 8 7 3 3 4 5 5 , 8 5 6 0 8 5 10 ⟶ 0 8 7 3 3 4 5 10 , 4 5 6 0 8 5 6 ⟶ 2 8 7 3 3 4 5 6 , 4 5 6 0 8 5 7 ⟶ 2 8 7 3 3 4 5 7 , 4 5 6 0 8 5 5 ⟶ 2 8 7 3 3 4 5 5 , 4 5 6 0 8 5 10 ⟶ 2 8 7 3 3 4 5 10 , 5 5 6 0 8 5 6 ⟶ 6 8 7 3 3 4 5 6 , 5 5 6 0 8 5 7 ⟶ 6 8 7 3 3 4 5 7 , 5 5 6 0 8 5 5 ⟶ 6 8 7 3 3 4 5 5 , 5 5 6 0 8 5 10 ⟶ 6 8 7 3 3 4 5 10 , 13 5 6 0 8 5 6 ⟶ 11 8 7 3 3 4 5 6 , 13 5 6 0 8 5 7 ⟶ 11 8 7 3 3 4 5 7 , 13 5 6 0 8 5 5 ⟶ 11 8 7 3 3 4 5 5 , 13 5 6 0 8 5 10 ⟶ 11 8 7 3 3 4 5 10 , 1 4 5 5 6 0 1 2 0 ⟶ 1 3 2 0 8 7 3 3 4 6 , 1 4 5 5 6 0 1 2 1 ⟶ 1 3 2 0 8 7 3 3 4 7 , 1 4 5 5 6 0 1 2 8 ⟶ 1 3 2 0 8 7 3 3 4 5 , 1 4 5 5 6 0 1 2 9 ⟶ 1 3 2 0 8 7 3 3 4 10 , 3 4 5 5 6 0 1 2 0 ⟶ 3 3 2 0 8 7 3 3 4 6 , 3 4 5 5 6 0 1 2 1 ⟶ 3 3 2 0 8 7 3 3 4 7 , 3 4 5 5 6 0 1 2 8 ⟶ 3 3 2 0 8 7 3 3 4 5 , 3 4 5 5 6 0 1 2 9 ⟶ 3 3 2 0 8 7 3 3 4 10 , 7 4 5 5 6 0 1 2 0 ⟶ 7 3 2 0 8 7 3 3 4 6 , 7 4 5 5 6 0 1 2 1 ⟶ 7 3 2 0 8 7 3 3 4 7 , 7 4 5 5 6 0 1 2 8 ⟶ 7 3 2 0 8 7 3 3 4 5 , 7 4 5 5 6 0 1 2 9 ⟶ 7 3 2 0 8 7 3 3 4 10 , 14 4 5 5 6 0 1 2 0 ⟶ 14 3 2 0 8 7 3 3 4 6 , 14 4 5 5 6 0 1 2 1 ⟶ 14 3 2 0 8 7 3 3 4 7 , 14 4 5 5 6 0 1 2 8 ⟶ 14 3 2 0 8 7 3 3 4 5 , 14 4 5 5 6 0 1 2 9 ⟶ 14 3 2 0 8 7 3 3 4 10 , 1 4 7 3 2 8 7 2 0 ⟶ 1 2 1 3 4 5 6 1 2 , 1 4 7 3 2 8 7 2 1 ⟶ 1 2 1 3 4 5 6 1 3 , 1 4 7 3 2 8 7 2 8 ⟶ 1 2 1 3 4 5 6 1 4 , 1 4 7 3 2 8 7 2 9 ⟶ 1 2 1 3 4 5 6 1 12 , 3 4 7 3 2 8 7 2 0 ⟶ 3 2 1 3 4 5 6 1 2 , 3 4 7 3 2 8 7 2 1 ⟶ 3 2 1 3 4 5 6 1 3 , 3 4 7 3 2 8 7 2 8 ⟶ 3 2 1 3 4 5 6 1 4 , 3 4 7 3 2 8 7 2 9 ⟶ 3 2 1 3 4 5 6 1 12 , 7 4 7 3 2 8 7 2 0 ⟶ 7 2 1 3 4 5 6 1 2 , 7 4 7 3 2 8 7 2 1 ⟶ 7 2 1 3 4 5 6 1 3 , 7 4 7 3 2 8 7 2 8 ⟶ 7 2 1 3 4 5 6 1 4 , 7 4 7 3 2 8 7 2 9 ⟶ 7 2 1 3 4 5 6 1 12 , 14 4 7 3 2 8 7 2 0 ⟶ 14 2 1 3 4 5 6 1 2 , 14 4 7 3 2 8 7 2 1 ⟶ 14 2 1 3 4 5 6 1 3 , 14 4 7 3 2 8 7 2 8 ⟶ 14 2 1 3 4 5 6 1 4 , 14 4 7 3 2 8 7 2 9 ⟶ 14 2 1 3 4 5 6 1 12 , 8 5 7 2 8 5 5 5 6 0 ⟶ 8 5 6 0 1 2 1 3 3 3 2 , 8 5 7 2 8 5 5 5 6 1 ⟶ 8 5 6 0 1 2 1 3 3 3 3 , 8 5 7 2 8 5 5 5 6 8 ⟶ 8 5 6 0 1 2 1 3 3 3 4 , 8 5 7 2 8 5 5 5 6 9 ⟶ 8 5 6 0 1 2 1 3 3 3 12 , 4 5 7 2 8 5 5 5 6 0 ⟶ 4 5 6 0 1 2 1 3 3 3 2 , 4 5 7 2 8 5 5 5 6 1 ⟶ 4 5 6 0 1 2 1 3 3 3 3 , 4 5 7 2 8 5 5 5 6 8 ⟶ 4 5 6 0 1 2 1 3 3 3 4 , 4 5 7 2 8 5 5 5 6 9 ⟶ 4 5 6 0 1 2 1 3 3 3 12 , 5 5 7 2 8 5 5 5 6 0 ⟶ 5 5 6 0 1 2 1 3 3 3 2 , 5 5 7 2 8 5 5 5 6 1 ⟶ 5 5 6 0 1 2 1 3 3 3 3 , 5 5 7 2 8 5 5 5 6 8 ⟶ 5 5 6 0 1 2 1 3 3 3 4 , 5 5 7 2 8 5 5 5 6 9 ⟶ 5 5 6 0 1 2 1 3 3 3 12 , 13 5 7 2 8 5 5 5 6 0 ⟶ 13 5 6 0 1 2 1 3 3 3 2 , 13 5 7 2 8 5 5 5 6 1 ⟶ 13 5 6 0 1 2 1 3 3 3 3 , 13 5 7 2 8 5 5 5 6 8 ⟶ 13 5 6 0 1 2 1 3 3 3 4 , 13 5 7 2 8 5 5 5 6 9 ⟶ 13 5 6 0 1 2 1 3 3 3 12 , 0 0 8 6 1 2 0 0 8 6 ⟶ 8 7 3 3 3 2 8 7 2 0 1 2 , 0 0 8 6 1 2 0 0 8 7 ⟶ 8 7 3 3 3 2 8 7 2 0 1 3 , 0 0 8 6 1 2 0 0 8 5 ⟶ 8 7 3 3 3 2 8 7 2 0 1 4 , 0 0 8 6 1 2 0 0 8 10 ⟶ 8 7 3 3 3 2 8 7 2 0 1 12 , 2 0 8 6 1 2 0 0 8 6 ⟶ 4 7 3 3 3 2 8 7 2 0 1 2 , 2 0 8 6 1 2 0 0 8 7 ⟶ 4 7 3 3 3 2 8 7 2 0 1 3 , 2 0 8 6 1 2 0 0 8 5 ⟶ 4 7 3 3 3 2 8 7 2 0 1 4 , 2 0 8 6 1 2 0 0 8 10 ⟶ 4 7 3 3 3 2 8 7 2 0 1 12 , 6 0 8 6 1 2 0 0 8 6 ⟶ 5 7 3 3 3 2 8 7 2 0 1 2 , 6 0 8 6 1 2 0 0 8 7 ⟶ 5 7 3 3 3 2 8 7 2 0 1 3 , 6 0 8 6 1 2 0 0 8 5 ⟶ 5 7 3 3 3 2 8 7 2 0 1 4 , 6 0 8 6 1 2 0 0 8 10 ⟶ 5 7 3 3 3 2 8 7 2 0 1 12 , 11 0 8 6 1 2 0 0 8 6 ⟶ 13 7 3 3 3 2 8 7 2 0 1 2 , 11 0 8 6 1 2 0 0 8 7 ⟶ 13 7 3 3 3 2 8 7 2 0 1 3 , 11 0 8 6 1 2 0 0 8 5 ⟶ 13 7 3 3 3 2 8 7 2 0 1 4 , 11 0 8 6 1 2 0 0 8 10 ⟶ 13 7 3 3 3 2 8 7 2 0 1 12 , 8 7 3 2 0 8 7 4 5 6 ⟶ 1 4 5 5 7 3 3 3 3 2 0 , 8 7 3 2 0 8 7 4 5 7 ⟶ 1 4 5 5 7 3 3 3 3 2 1 , 8 7 3 2 0 8 7 4 5 5 ⟶ 1 4 5 5 7 3 3 3 3 2 8 , 8 7 3 2 0 8 7 4 5 10 ⟶ 1 4 5 5 7 3 3 3 3 2 9 , 4 7 3 2 0 8 7 4 5 6 ⟶ 3 4 5 5 7 3 3 3 3 2 0 , 4 7 3 2 0 8 7 4 5 7 ⟶ 3 4 5 5 7 3 3 3 3 2 1 , 4 7 3 2 0 8 7 4 5 5 ⟶ 3 4 5 5 7 3 3 3 3 2 8 , 4 7 3 2 0 8 7 4 5 10 ⟶ 3 4 5 5 7 3 3 3 3 2 9 , 5 7 3 2 0 8 7 4 5 6 ⟶ 7 4 5 5 7 3 3 3 3 2 0 , 5 7 3 2 0 8 7 4 5 7 ⟶ 7 4 5 5 7 3 3 3 3 2 1 , 5 7 3 2 0 8 7 4 5 5 ⟶ 7 4 5 5 7 3 3 3 3 2 8 , 5 7 3 2 0 8 7 4 5 10 ⟶ 7 4 5 5 7 3 3 3 3 2 9 , 13 7 3 2 0 8 7 4 5 6 ⟶ 14 4 5 5 7 3 3 3 3 2 0 , 13 7 3 2 0 8 7 4 5 7 ⟶ 14 4 5 5 7 3 3 3 3 2 1 , 13 7 3 2 0 8 7 4 5 5 ⟶ 14 4 5 5 7 3 3 3 3 2 8 , 13 7 3 2 0 8 7 4 5 10 ⟶ 14 4 5 5 7 3 3 3 3 2 9 , 1 2 1 2 1 3 2 0 0 0 0 0 ⟶ 1 3 3 3 2 0 1 2 1 4 7 2 0 , 1 2 1 2 1 3 2 0 0 0 0 1 ⟶ 1 3 3 3 2 0 1 2 1 4 7 2 1 , 1 2 1 2 1 3 2 0 0 0 0 8 ⟶ 1 3 3 3 2 0 1 2 1 4 7 2 8 , 1 2 1 2 1 3 2 0 0 0 0 9 ⟶ 1 3 3 3 2 0 1 2 1 4 7 2 9 , 3 2 1 2 1 3 2 0 0 0 0 0 ⟶ 3 3 3 3 2 0 1 2 1 4 7 2 0 , 3 2 1 2 1 3 2 0 0 0 0 1 ⟶ 3 3 3 3 2 0 1 2 1 4 7 2 1 , 3 2 1 2 1 3 2 0 0 0 0 8 ⟶ 3 3 3 3 2 0 1 2 1 4 7 2 8 , 3 2 1 2 1 3 2 0 0 0 0 9 ⟶ 3 3 3 3 2 0 1 2 1 4 7 2 9 , 7 2 1 2 1 3 2 0 0 0 0 0 ⟶ 7 3 3 3 2 0 1 2 1 4 7 2 0 , 7 2 1 2 1 3 2 0 0 0 0 1 ⟶ 7 3 3 3 2 0 1 2 1 4 7 2 1 , 7 2 1 2 1 3 2 0 0 0 0 8 ⟶ 7 3 3 3 2 0 1 2 1 4 7 2 8 , 7 2 1 2 1 3 2 0 0 0 0 9 ⟶ 7 3 3 3 2 0 1 2 1 4 7 2 9 , 14 2 1 2 1 3 2 0 0 0 0 0 ⟶ 14 3 3 3 2 0 1 2 1 4 7 2 0 , 14 2 1 2 1 3 2 0 0 0 0 1 ⟶ 14 3 3 3 2 0 1 2 1 4 7 2 1 , 14 2 1 2 1 3 2 0 0 0 0 8 ⟶ 14 3 3 3 2 0 1 2 1 4 7 2 8 , 14 2 1 2 1 3 2 0 0 0 0 9 ⟶ 14 3 3 3 2 0 1 2 1 4 7 2 9 , 8 6 0 0 0 0 8 6 1 2 0 0 ⟶ 1 3 3 2 1 4 7 4 7 2 0 1 2 , 8 6 0 0 0 0 8 6 1 2 0 1 ⟶ 1 3 3 2 1 4 7 4 7 2 0 1 3 , 8 6 0 0 0 0 8 6 1 2 0 8 ⟶ 1 3 3 2 1 4 7 4 7 2 0 1 4 , 8 6 0 0 0 0 8 6 1 2 0 9 ⟶ 1 3 3 2 1 4 7 4 7 2 0 1 12 , 4 6 0 0 0 0 8 6 1 2 0 0 ⟶ 3 3 3 2 1 4 7 4 7 2 0 1 2 , 4 6 0 0 0 0 8 6 1 2 0 1 ⟶ 3 3 3 2 1 4 7 4 7 2 0 1 3 , 4 6 0 0 0 0 8 6 1 2 0 8 ⟶ 3 3 3 2 1 4 7 4 7 2 0 1 4 , 4 6 0 0 0 0 8 6 1 2 0 9 ⟶ 3 3 3 2 1 4 7 4 7 2 0 1 12 , 5 6 0 0 0 0 8 6 1 2 0 0 ⟶ 7 3 3 2 1 4 7 4 7 2 0 1 2 , 5 6 0 0 0 0 8 6 1 2 0 1 ⟶ 7 3 3 2 1 4 7 4 7 2 0 1 3 , 5 6 0 0 0 0 8 6 1 2 0 8 ⟶ 7 3 3 2 1 4 7 4 7 2 0 1 4 , 5 6 0 0 0 0 8 6 1 2 0 9 ⟶ 7 3 3 2 1 4 7 4 7 2 0 1 12 , 13 6 0 0 0 0 8 6 1 2 0 0 ⟶ 14 3 3 2 1 4 7 4 7 2 0 1 2 , 13 6 0 0 0 0 8 6 1 2 0 1 ⟶ 14 3 3 2 1 4 7 4 7 2 0 1 3 , 13 6 0 0 0 0 8 6 1 2 0 8 ⟶ 14 3 3 2 1 4 7 4 7 2 0 1 4 , 13 6 0 0 0 0 8 6 1 2 0 9 ⟶ 14 3 3 2 1 4 7 4 7 2 0 1 12 , 8 7 2 1 2 8 6 1 2 8 6 0 ⟶ 0 8 7 3 4 7 3 3 3 4 6 0 1 2 , 8 7 2 1 2 8 6 1 2 8 6 1 ⟶ 0 8 7 3 4 7 3 3 3 4 6 0 1 3 , 8 7 2 1 2 8 6 1 2 8 6 8 ⟶ 0 8 7 3 4 7 3 3 3 4 6 0 1 4 , 8 7 2 1 2 8 6 1 2 8 6 9 ⟶ 0 8 7 3 4 7 3 3 3 4 6 0 1 12 , 4 7 2 1 2 8 6 1 2 8 6 0 ⟶ 2 8 7 3 4 7 3 3 3 4 6 0 1 2 , 4 7 2 1 2 8 6 1 2 8 6 1 ⟶ 2 8 7 3 4 7 3 3 3 4 6 0 1 3 , 4 7 2 1 2 8 6 1 2 8 6 8 ⟶ 2 8 7 3 4 7 3 3 3 4 6 0 1 4 , 4 7 2 1 2 8 6 1 2 8 6 9 ⟶ 2 8 7 3 4 7 3 3 3 4 6 0 1 12 , 5 7 2 1 2 8 6 1 2 8 6 0 ⟶ 6 8 7 3 4 7 3 3 3 4 6 0 1 2 , 5 7 2 1 2 8 6 1 2 8 6 1 ⟶ 6 8 7 3 4 7 3 3 3 4 6 0 1 3 , 5 7 2 1 2 8 6 1 2 8 6 8 ⟶ 6 8 7 3 4 7 3 3 3 4 6 0 1 4 , 5 7 2 1 2 8 6 1 2 8 6 9 ⟶ 6 8 7 3 4 7 3 3 3 4 6 0 1 12 , 13 7 2 1 2 8 6 1 2 8 6 0 ⟶ 11 8 7 3 4 7 3 3 3 4 6 0 1 2 , 13 7 2 1 2 8 6 1 2 8 6 1 ⟶ 11 8 7 3 4 7 3 3 3 4 6 0 1 3 , 13 7 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7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 6 ⟶ 0 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 6 , 8 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 7 ⟶ 0 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 7 , 8 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 5 ⟶ 0 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 5 , 8 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 10 ⟶ 0 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 10 , 4 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 6 ⟶ 2 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 6 , 4 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 7 ⟶ 2 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 7 , 4 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 5 ⟶ 2 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 5 , 4 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 10 ⟶ 2 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 10 , 5 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 6 ⟶ 6 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 6 , 5 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 7 ⟶ 6 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 7 , 5 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 5 ⟶ 6 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 5 , 5 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 10 ⟶ 6 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 10 , 13 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 6 ⟶ 11 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 6 , 13 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 7 ⟶ 11 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 7 , 13 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 5 ⟶ 11 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 5 , 13 6 0 1 4 6 0 0 8 5 5 7 4 5 6 0 1 2 0 8 7 4 7 2 1 2 1 4 5 10 ⟶ 11 8 7 3 2 8 5 5 7 3 2 8 5 6 0 1 4 6 8 6 1 3 4 5 7 3 3 4 6 8 10 , 1 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 2 ⟶ 1 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 2 , 1 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 3 ⟶ 1 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 3 , 1 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 4 ⟶ 1 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 4 , 1 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 12 ⟶ 1 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 12 , 3 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 2 ⟶ 3 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 2 , 3 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 3 ⟶ 3 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 3 , 3 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 4 ⟶ 3 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 4 , 3 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 12 ⟶ 3 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 12 , 7 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 2 ⟶ 7 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 2 , 7 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 3 ⟶ 7 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 3 , 7 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 4 ⟶ 7 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 4 , 7 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 12 ⟶ 7 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 12 , 14 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 2 ⟶ 14 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 2 , 14 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 3 ⟶ 14 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 3 , 14 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 4 ⟶ 14 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 4 , 14 3 2 8 5 5 6 0 8 5 7 4 7 4 6 0 0 8 7 2 0 8 5 6 8 7 3 2 1 3 12 ⟶ 14 3 2 8 5 5 6 8 6 0 0 8 6 8 6 0 8 6 0 1 3 4 5 7 4 7 3 4 7 3 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 14 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 4 ↦ 1, 7 ↦ 2, 3 ↦ 3, 2 ↦ 4, 8 ↦ 5, 0 ↦ 6, 5 ↦ 7, 6 ↦ 8, 14 ↦ 9, 12 ↦ 10, 13 ↦ 11, 10 ↦ 12, 9 ↦ 13 }, it remains to prove termination of the 102-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 9 1 2 3 4 5 2 4 5 ⟶ 9 4 0 3 1 7 8 0 1 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 11 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 9 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 11 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 9 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 11 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 9 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 11 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 9 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 11 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 9 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 11 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 9 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 11 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 9 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 11 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 9 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 12 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 12 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 12 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 12 , 11 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 9 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 11 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 9 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 11 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 9 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 11 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 12 ⟶ 9 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 12 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 13 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 13 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 13 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 13 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 13 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 13 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 13 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 13 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 11 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 11 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 11 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 11 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 11 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 9 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 11 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 9 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 13 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 13 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 13 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 13 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 13 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 13 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 13 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 13 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 12 ↦ 11, 13 ↦ 12, 11 ↦ 13 }, it remains to prove termination of the 88-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 9 1 2 3 4 5 2 4 5 ⟶ 9 4 0 3 1 7 8 0 1 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 87-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 16: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 86-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 16: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 85-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 24: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 84-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 24: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 83-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 82-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 81-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 80-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 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⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 79-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 26: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 78-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 26: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 77-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 29: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 76-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 31: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 75-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 31: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 74-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 31: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 73-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 31: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 72-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 0 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 0 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 16: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 71-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 44: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 70-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 10 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 10 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 44: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 13 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12, 13 ↦ 13 }, it remains to prove termination of the 69-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 11 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 11 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 12 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 12 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 13 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 13 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 13 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 13 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 12 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 12 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 13 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 13 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 46: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 12 ↦ 11, 13 ↦ 12 }, it remains to prove termination of the 68-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 12 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 12 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 67-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 9 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 9 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 12 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 12 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 16: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 66-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 12 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 12 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 65-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 12 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 12 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 47: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 64-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 12 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 12 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 47: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 63-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 12 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 12 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 5 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 51: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 62-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 10 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 10 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 11 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 11 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 12 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 12 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 12 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 12 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 11 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 11 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 12 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 12 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 24: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 11 ↦ 10, 12 ↦ 11, 10 ↦ 12 }, it remains to prove termination of the 61-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 11 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 11 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 53: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 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0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 60-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 11 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 11 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 53: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 59-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 5 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 5 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 11 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 11 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 26: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 58-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 11 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 11 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 26: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 57-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 5 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 0 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 29: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 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0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 56-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 31: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 55-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 47: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 54-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 31: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11, 12 ↦ 12 }, it remains to prove termination of the 53-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 9 1 2 3 4 5 2 4 6 ⟶ 9 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 10 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 10 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 11 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 11 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 12 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 12 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 10 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 10 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 11 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 11 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 12 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 10 ↦ 9, 11 ↦ 10, 12 ↦ 11 }, it remains to prove termination of the 52-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 9 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 9 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 24: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ 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0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 51-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 9 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 9 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 24: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 50-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 9 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 9 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 16: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 49-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 2 1 2 3 4 5 2 4 5 ⟶ 2 4 0 3 1 7 8 0 1 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 9 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 9 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 30: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 48-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 0 1 2 3 4 5 2 4 5 ⟶ 0 4 0 3 1 7 8 0 1 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 9 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 9 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 18: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 47-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 9 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 9 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 31: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 46-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 5 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 5 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 31: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 45-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 26: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 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⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 44-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 5 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 5 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 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0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 43-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 9 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 9 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 10 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 10 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 11 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 11 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 10 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 10 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 10 ↦ 9, 11 ↦ 10 }, it remains to prove termination of the 42-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 6 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 6 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 25: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 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0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 41-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 5 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 1 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 48: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10 }, it remains to prove termination of the 40-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 10 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 10 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 29: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 39-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 24: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 38-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 44: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 37-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 3 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 3 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 44: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 36-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 53: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 35-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 8 6 0 4 0 4 5 2 3 4 0 4 6 6 5 7 2 1 8 5 7 7 7 8 0 4 6 ⟶ 2 1 8 5 2 1 7 8 0 1 2 1 2 1 8 0 4 6 6 6 5 7 7 8 5 2 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 51: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 34-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 8 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 8 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 24: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 33-rule system { 0 1 2 3 4 5 2 4 6 ⟶ 0 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 6 ⟶ 3 4 0 3 1 7 8 0 4 , 3 1 2 3 4 5 2 4 5 ⟶ 3 4 0 3 1 7 8 0 1 , 2 1 2 3 4 5 2 4 6 ⟶ 2 4 0 3 1 7 8 0 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 3 3 4 5 2 3 1 2 3 4 5 8 6 6 5 ⟶ 3 3 3 3 4 5 2 1 7 8 5 8 5 2 3 1 , 2 3 4 5 2 3 1 2 3 4 5 8 6 6 6 ⟶ 2 3 3 3 4 5 2 1 7 8 5 8 5 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 5 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 0 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 4 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 4 , 7 8 5 7 7 8 5 8 5 2 3 3 4 5 7 2 3 1 8 0 1 ⟶ 2 4 6 6 0 1 8 5 7 2 3 3 3 1 7 2 1 2 1 2 3 1 , 5 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 0 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 4 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 4 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 3 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 3 , 7 2 1 8 5 7 8 0 3 4 5 7 7 7 7 2 3 4 0 1 2 3 1 ⟶ 2 1 2 3 4 6 5 7 2 4 6 5 8 0 4 0 4 5 8 0 3 3 3 1 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 5 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 0 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 2 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 2 , 7 8 0 3 4 6 5 8 6 0 3 1 8 0 4 5 8 0 1 7 2 1 7 ⟶ 2 3 3 3 1 8 5 8 5 8 5 8 6 5 2 4 6 6 5 8 5 7 7 7 , 5 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 5 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 1 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 7 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 7 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 9 8 5 8 6 6 6 6 0 1 7 2 4 0 3 1 8 0 3 4 6 5 8 0 ⟶ 9 8 6 6 6 6 0 4 5 2 1 8 0 4 6 5 7 7 8 0 3 3 4 0 , 1 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 1 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 7 7 8 0 3 3 4 6 6 0 1 2 1 7 2 3 4 5 8 0 1 8 0 4 6 ⟶ 7 2 1 8 0 4 5 7 7 8 6 5 2 4 5 7 8 6 5 7 2 3 3 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 4 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 4 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 3 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 3 , 3 1 2 3 1 2 1 2 3 1 8 6 6 5 2 4 5 8 0 3 4 0 1 7 7 8 0 1 ⟶ 1 7 8 6 6 0 3 3 1 7 7 7 2 4 6 0 4 5 7 7 2 3 1 2 4 0 3 3 1 , 5 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 5 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 6 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 6 , 1 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 1 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 7 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 7 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 , 9 8 5 8 5 7 8 6 6 6 0 4 0 1 2 4 5 7 2 4 5 8 5 8 6 6 0 4 6 0 ⟶ 9 7 2 1 8 6 5 8 6 6 6 6 6 6 0 4 5 7 2 4 5 7 2 4 6 5 8 0 4 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 22: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 3 ↦ 0, 1 ↦ 1, 2 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 0 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 15-rule system { 0 1 2 0 3 4 2 3 5 ⟶ 0 3 6 0 1 7 8 6 3 , 0 1 2 0 3 4 2 3 4 ⟶ 0 3 6 0 1 7 8 6 1 , 0 0 3 4 2 0 1 2 0 3 4 8 5 5 5 ⟶ 0 0 0 0 3 4 2 1 7 8 4 8 4 2 0 3 , 2 0 3 4 2 0 1 2 0 3 4 8 5 5 5 ⟶ 2 0 0 0 3 4 2 1 7 8 4 8 4 2 0 3 , 7 8 4 7 7 8 4 8 4 2 0 0 3 4 7 2 0 1 8 6 3 ⟶ 2 3 5 5 6 1 8 4 7 2 0 0 0 1 7 2 1 2 1 2 0 3 , 4 8 4 8 5 5 5 5 6 1 7 2 3 6 0 1 8 6 0 3 5 4 8 6 ⟶ 4 8 5 5 5 5 6 3 4 2 1 8 6 3 5 4 7 7 8 6 0 0 3 6 , 1 8 4 8 5 5 5 5 6 1 7 2 3 6 0 1 8 6 0 3 5 4 8 6 ⟶ 1 8 5 5 5 5 6 3 4 2 1 8 6 3 5 4 7 7 8 6 0 0 3 6 , 7 8 4 8 5 5 5 5 6 1 7 2 3 6 0 1 8 6 0 3 5 4 8 6 ⟶ 7 8 5 5 5 5 6 3 4 2 1 8 6 3 5 4 7 7 8 6 0 0 3 6 , 9 8 4 8 5 5 5 5 6 1 7 2 3 6 0 1 8 6 0 3 5 4 8 6 ⟶ 9 8 5 5 5 5 6 3 4 2 1 8 6 3 5 4 7 7 8 6 0 0 3 6 , 0 1 2 0 1 2 1 2 0 1 8 5 5 4 2 3 4 8 6 0 3 6 1 7 7 8 6 3 ⟶ 1 7 8 5 5 6 0 0 1 7 7 7 2 3 5 6 3 4 7 7 2 0 1 2 3 6 0 0 3 , 4 8 4 8 4 7 8 5 5 5 6 3 6 1 2 3 4 7 2 3 4 8 4 8 5 5 6 3 5 6 ⟶ 4 7 2 1 8 5 4 8 5 5 5 5 5 5 6 3 4 7 2 3 4 7 2 3 5 4 8 6 3 6 , 1 8 4 8 4 7 8 5 5 5 6 3 6 1 2 3 4 7 2 3 4 8 4 8 5 5 6 3 5 5 ⟶ 1 7 2 1 8 5 4 8 5 5 5 5 5 5 6 3 4 7 2 3 4 7 2 3 5 4 8 6 3 5 , 1 8 4 8 4 7 8 5 5 5 6 3 6 1 2 3 4 7 2 3 4 8 4 8 5 5 6 3 5 6 ⟶ 1 7 2 1 8 5 4 8 5 5 5 5 5 5 6 3 4 7 2 3 4 7 2 3 5 4 8 6 3 6 , 7 8 4 8 4 7 8 5 5 5 6 3 6 1 2 3 4 7 2 3 4 8 4 8 5 5 6 3 5 6 ⟶ 7 7 2 1 8 5 4 8 5 5 5 5 5 5 6 3 4 7 2 3 4 7 2 3 5 4 8 6 3 6 , 9 8 4 8 4 7 8 5 5 5 6 3 6 1 2 3 4 7 2 3 4 8 4 8 5 5 6 3 5 6 ⟶ 9 7 2 1 8 5 4 8 5 5 5 5 5 5 6 3 4 7 2 3 4 7 2 3 5 4 8 6 3 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.