/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 75-rule system { 0 1 2 1 ⟶ 0 2 2 2 1 , 0 2 1 1 2 0 2 ⟶ 2 2 2 0 2 1 2 2 , 2 0 0 0 0 1 0 ⟶ 1 2 2 2 1 2 1 0 1 , 0 0 0 1 1 0 0 1 0 1 1 0 0 ⟶ 2 2 1 2 0 2 2 1 1 1 2 2 1 0 , 0 1 1 0 0 1 1 2 1 2 0 0 2 ⟶ 1 2 2 2 1 2 1 0 2 2 0 0 1 2 2 2 2 , 1 0 0 0 1 0 1 1 0 1 1 0 1 2 0 1 ⟶ 2 2 1 1 2 1 2 0 0 1 2 0 0 1 1 0 0 0 , 1 0 0 2 0 2 1 0 0 2 0 2 2 0 2 2 ⟶ 0 0 2 1 2 0 2 0 0 2 0 1 2 0 2 2 , 1 1 0 1 1 2 0 0 0 0 0 0 1 2 0 0 ⟶ 2 0 1 1 0 0 2 0 0 2 2 2 0 2 1 2 2 , 2 0 0 2 2 2 0 1 2 1 2 0 1 1 0 0 ⟶ 2 2 2 2 2 2 2 1 2 0 0 2 1 0 1 0 1 2 1 2 , 1 1 1 1 1 1 2 1 0 1 2 1 0 2 2 0 1 ⟶ 1 1 2 2 0 2 2 2 2 1 0 1 2 0 2 1 0 0 0 , 0 2 0 0 2 1 0 0 0 0 1 0 2 1 1 1 0 1 ⟶ 0 1 2 2 1 0 0 2 2 1 1 1 1 2 1 1 2 1 0 , 1 1 2 1 2 1 1 1 1 1 1 2 2 1 0 1 2 1 ⟶ 2 0 2 2 2 2 2 1 1 1 2 0 1 0 2 1 2 2 1 2 2 2 1 2 2 2 , 0 0 1 0 2 0 1 0 1 1 2 0 0 1 0 0 0 1 1 ⟶ 0 2 2 2 0 2 0 1 2 1 1 0 1 2 2 0 1 0 0 2 2 2 2 , 0 2 0 2 1 0 2 2 1 0 0 2 0 0 0 2 0 0 1 ⟶ 2 2 0 0 1 2 1 2 1 2 2 2 2 1 0 2 1 1 2 0 , 1 0 0 2 2 0 2 0 2 0 0 2 1 1 0 0 0 1 2 ⟶ 2 2 1 0 2 1 0 2 1 2 1 2 1 0 2 2 2 1 2 2 1 2 , 0 1 0 0 1 2 2 0 1 0 1 1 1 2 1 1 1 1 0 2 0 ⟶ 1 0 1 1 1 1 2 2 1 1 2 1 0 0 0 2 2 0 2 0 0 2 , 1 1 0 1 0 1 1 2 1 0 0 0 1 0 2 0 1 0 0 1 0 1 ⟶ 0 1 1 1 1 1 0 0 2 1 0 0 2 1 2 0 2 2 1 2 2 1 1 , 0 2 0 2 2 1 0 1 2 1 0 0 1 0 2 0 0 0 0 0 2 2 0 ⟶ 2 1 1 2 2 1 2 2 2 1 0 0 2 2 0 1 0 1 2 0 0 2 1 1 , 1 0 0 1 2 1 2 0 0 0 2 1 0 2 0 2 0 0 0 0 0 1 1 2 2 ⟶ 1 1 1 1 2 0 1 2 2 2 1 2 1 0 1 2 2 0 2 2 2 0 0 1 0 2 , 2 1 2 2 1 2 2 2 0 0 2 1 1 1 0 1 1 2 0 1 1 2 0 2 2 ⟶ 2 2 2 2 0 0 1 2 1 2 0 0 2 1 2 0 2 2 1 2 0 0 2 2 0 1 , 1 0 0 2 1 1 1 1 1 2 2 1 1 1 0 2 1 0 1 2 1 2 0 0 1 2 ⟶ 2 1 2 2 1 2 1 0 2 2 2 1 0 2 2 1 0 1 0 2 2 2 2 1 2 1 2 1 1 0 2 , 0 1 2 1 2 2 0 0 0 1 2 1 0 2 0 0 0 1 2 2 2 0 0 0 1 2 1 ⟶ 2 2 1 2 2 2 1 2 2 2 2 1 2 2 0 2 1 1 1 0 1 0 2 2 1 1 0 1 , 1 2 2 0 2 0 1 2 1 2 2 0 2 1 1 1 1 2 0 1 0 0 1 2 0 1 2 ⟶ 1 2 0 0 2 2 0 2 2 0 0 2 0 0 1 2 0 0 2 2 2 0 1 2 1 0 1 2 , 0 2 0 2 0 1 0 0 1 0 2 0 0 0 0 1 0 0 2 2 2 1 1 2 0 2 1 2 2 2 ⟶ 2 2 1 2 0 1 1 1 2 2 1 2 2 2 0 1 2 2 2 1 2 1 1 1 0 0 1 2 2 0 0 , 1 0 2 0 2 0 2 0 0 1 1 1 2 0 1 0 0 0 1 2 0 1 1 2 0 0 2 1 0 0 ⟶ 1 0 1 2 2 1 2 1 2 0 1 1 0 0 2 2 2 2 1 2 0 1 1 1 1 1 2 2 2 2 1 1 , 1 0 2 2 0 2 1 1 0 2 0 1 2 0 2 1 0 1 0 2 1 0 0 1 0 0 2 0 2 2 ⟶ 1 1 0 1 2 2 1 1 1 2 1 2 2 0 0 2 2 1 1 2 0 1 2 1 2 0 2 0 2 0 0 2 , 1 2 0 2 0 0 1 2 2 1 1 0 1 2 0 1 0 1 0 0 2 0 1 0 1 0 1 0 2 0 ⟶ 2 1 0 2 2 2 0 2 1 2 0 0 2 1 2 2 2 0 1 2 2 2 1 1 0 1 1 0 1 1 0 2 , 2 2 2 1 2 0 0 1 1 1 2 0 0 2 0 1 1 0 2 1 0 2 0 2 1 1 0 0 0 2 ⟶ 2 2 2 1 2 1 2 2 2 1 0 0 2 2 1 0 2 2 0 0 2 0 0 0 2 1 1 2 1 1 1 2 2 2 , 0 1 1 0 0 2 2 1 0 1 1 1 2 2 1 2 0 0 1 1 0 0 2 0 2 2 0 1 1 1 1 ⟶ 0 2 2 0 1 0 1 1 2 1 2 0 0 0 0 0 1 0 1 0 1 2 2 2 2 0 0 0 1 2 1 2 , 0 1 1 1 2 1 1 0 1 2 2 2 0 1 2 1 0 0 1 2 2 1 1 2 0 1 1 1 2 1 1 ⟶ 2 2 1 1 2 1 2 1 2 2 1 0 0 2 1 2 2 2 0 0 0 2 2 0 2 1 1 1 0 2 2 2 , 1 2 1 1 1 1 2 1 0 1 0 2 2 0 0 1 0 1 0 1 2 1 1 1 1 0 2 2 2 2 1 ⟶ 2 0 1 2 2 2 1 2 0 0 2 2 1 1 2 2 2 0 1 2 1 2 1 2 2 1 2 1 1 2 0 1 0 , 2 0 1 0 1 2 1 2 2 2 0 1 0 0 0 1 2 0 1 1 1 1 2 2 0 0 0 1 0 0 0 ⟶ 2 2 0 2 2 0 2 1 1 2 1 2 2 1 0 0 1 0 2 0 2 2 2 1 0 2 2 0 1 0 2 2 1 2 , 0 2 2 2 0 1 2 1 0 2 0 1 2 2 0 1 2 1 1 1 1 1 2 0 1 1 2 2 0 1 0 1 ⟶ 2 2 0 1 0 1 2 0 0 2 0 2 2 2 1 2 2 2 1 0 1 1 1 0 1 1 2 0 1 1 1 1 , 1 0 0 2 2 1 0 2 1 2 0 2 0 2 0 2 0 1 1 2 2 1 0 0 0 1 0 1 2 1 2 0 ⟶ 2 2 0 1 0 1 1 1 2 2 1 2 1 0 2 2 2 2 1 2 1 1 2 0 2 2 2 0 0 0 1 1 1 , 1 2 1 1 1 0 1 2 1 0 1 2 1 1 2 2 0 2 2 1 1 1 1 0 1 2 0 1 1 1 1 1 ⟶ 2 0 1 2 0 2 0 2 1 0 0 0 0 2 2 2 2 2 1 1 0 2 2 2 0 2 2 2 0 1 0 2 2 1 1 , 0 1 0 0 1 2 1 2 0 1 1 1 0 1 0 0 2 1 2 0 2 1 1 0 1 0 0 1 1 0 1 0 1 ⟶ 2 0 0 0 1 0 1 2 2 2 0 1 0 2 0 2 2 2 1 0 1 0 0 2 2 2 0 1 1 1 1 1 1 1 , 0 2 0 0 2 1 1 0 0 0 2 2 1 0 2 1 0 1 0 0 1 1 2 0 1 1 2 2 0 1 1 1 0 ⟶ 0 1 2 1 2 2 1 0 2 1 1 2 2 1 2 1 1 0 2 1 0 0 0 2 2 1 2 2 2 2 1 0 0 2 , 1 2 2 2 0 1 0 0 1 2 1 0 1 1 0 1 1 1 2 1 1 1 2 1 0 0 0 1 0 0 0 1 2 ⟶ 2 2 2 2 2 2 2 1 0 1 0 2 2 0 0 2 2 2 0 0 0 2 0 0 1 1 2 2 1 2 2 2 1 0 1 0 1 0 2 0 , 1 1 0 2 0 2 0 0 0 1 1 0 1 0 1 0 1 1 2 1 0 2 0 0 2 2 1 0 0 2 0 2 0 2 ⟶ 0 2 1 0 0 0 2 2 2 1 2 2 2 2 2 1 0 0 2 2 2 0 0 0 0 1 1 0 1 1 2 1 2 2 2 , 1 1 2 1 0 2 0 0 0 2 0 0 2 0 1 0 0 0 0 2 1 0 0 0 0 1 2 2 0 1 1 2 0 1 ⟶ 2 2 1 2 2 2 2 1 0 2 1 2 2 2 1 1 2 2 1 2 0 1 0 1 1 2 2 1 1 1 0 2 1 2 2 0 0 2 , 1 0 1 1 1 1 1 2 2 1 0 2 0 0 1 0 0 0 0 2 1 1 1 1 0 0 0 0 2 1 0 1 1 2 1 ⟶ 0 2 0 1 1 2 1 1 1 2 2 0 2 0 2 2 1 2 1 1 2 0 1 2 1 1 2 1 2 1 1 0 1 0 2 1 , 1 1 2 0 0 0 1 1 0 1 0 2 2 1 2 1 0 2 2 2 0 2 1 1 2 0 0 0 2 0 1 2 0 0 1 ⟶ 2 2 2 1 1 2 2 0 0 2 0 1 1 2 2 1 1 2 0 1 2 1 0 2 1 2 0 2 2 1 0 0 2 0 0 0 , 0 0 0 0 0 0 2 1 0 1 0 1 0 1 0 1 2 0 0 1 1 0 2 1 0 2 0 0 0 0 2 2 1 2 2 0 ⟶ 1 2 0 0 0 0 1 1 1 2 0 1 2 1 2 2 2 2 1 2 0 1 1 2 1 1 0 1 2 0 1 0 0 0 2 0 0 1 , 0 0 1 2 2 1 1 2 0 0 1 0 2 2 0 0 2 0 1 0 1 0 1 2 2 1 0 2 1 1 0 1 0 0 0 1 ⟶ 1 1 2 2 1 2 0 2 2 2 1 2 1 2 0 0 1 2 2 0 0 1 0 1 2 0 0 1 0 1 1 0 1 1 1 1 1 0 , 0 1 1 0 2 0 0 2 1 1 2 0 2 0 1 0 2 0 2 2 2 1 0 0 0 1 1 0 0 2 2 0 0 0 0 2 ⟶ 2 2 2 2 1 0 2 0 0 0 0 0 2 2 1 0 1 0 1 2 0 0 0 0 1 2 2 1 2 2 2 2 2 0 1 2 1 1 0 2 , 0 2 0 0 2 1 1 1 0 2 2 1 1 1 0 0 0 1 0 0 0 2 0 1 1 0 1 0 0 2 0 0 0 2 0 0 ⟶ 2 1 2 0 2 1 0 2 1 0 1 1 0 2 2 2 2 1 1 0 2 2 1 2 1 2 0 1 0 1 2 1 2 2 2 0 0 0 , 1 1 2 2 0 1 2 1 2 1 2 2 0 2 0 0 2 2 1 1 0 1 0 0 0 2 1 1 1 1 2 1 2 0 2 2 ⟶ 1 1 1 2 0 0 2 1 2 0 1 2 2 2 2 0 2 2 2 1 0 1 0 0 1 2 1 1 1 1 2 0 2 1 0 2 , 0 2 0 0 2 0 0 1 0 2 0 0 0 0 2 1 0 0 0 2 2 0 1 0 2 2 0 0 2 0 2 0 0 1 0 1 2 ⟶ 2 2 1 2 1 0 1 1 2 2 2 2 1 0 2 0 0 1 1 2 2 1 0 1 1 1 0 1 1 0 2 1 0 2 2 0 1 2 2 2 , 0 2 0 2 0 2 1 0 0 0 1 1 1 2 1 0 1 2 1 2 0 1 1 2 0 2 0 0 1 0 2 2 2 2 0 0 2 ⟶ 2 2 1 0 1 2 0 1 2 1 2 2 2 2 2 1 0 2 1 2 0 2 1 1 2 1 1 1 0 0 0 2 0 2 0 2 2 1 , 1 1 0 2 0 2 1 1 1 2 2 2 1 1 2 1 0 2 0 2 0 2 1 2 1 1 1 2 0 2 1 1 2 2 0 1 0 ⟶ 1 2 2 2 1 2 2 0 1 2 2 1 1 2 1 0 0 2 2 0 2 0 1 2 1 2 2 0 1 2 2 2 1 1 0 0 2 0 , 2 1 0 1 0 0 1 0 0 0 1 1 2 2 1 0 0 1 1 1 0 1 0 1 2 1 2 1 0 1 2 1 0 2 0 0 1 ⟶ 1 1 2 2 0 0 2 0 2 1 2 0 2 0 1 2 2 2 1 1 1 0 0 1 0 0 1 1 1 1 1 1 2 1 0 2 2 2 2 , 2 1 0 0 1 1 1 1 0 1 1 0 1 2 0 0 2 2 0 2 1 2 1 2 2 2 0 0 1 0 2 1 2 1 0 2 1 2 ⟶ 2 0 1 0 0 1 2 0 2 1 2 1 2 1 0 2 0 2 2 2 2 2 2 1 0 0 0 2 2 2 0 2 0 0 2 2 1 0 1 1 0 , 1 1 0 0 0 2 1 2 1 1 0 0 1 1 1 2 2 0 0 0 1 0 1 0 1 0 2 1 0 0 1 0 1 1 1 2 2 1 0 ⟶ 2 0 0 0 2 2 1 1 2 1 1 2 2 1 2 1 0 1 1 0 0 1 2 0 0 0 0 0 0 0 0 1 0 1 0 1 1 2 2 1 2 , 2 0 2 0 2 1 2 1 1 0 1 1 2 1 2 2 0 1 0 0 2 2 0 1 1 0 1 1 1 2 2 0 1 0 0 1 2 0 2 ⟶ 2 0 1 0 0 2 2 0 0 2 2 2 2 0 2 0 0 2 1 1 1 0 1 2 0 2 2 1 2 0 0 2 1 1 1 2 0 1 2 1 2 2 , 0 1 1 0 2 1 0 2 1 2 0 0 1 0 2 0 0 2 0 2 1 0 1 0 1 2 2 0 1 0 0 1 2 2 0 1 0 0 0 0 ⟶ 0 2 2 2 2 2 2 1 1 2 2 1 1 2 0 1 1 2 1 2 2 1 2 0 1 1 2 0 1 1 2 1 1 0 1 2 0 1 1 2 2 , 0 1 0 1 0 0 2 0 2 0 1 0 2 1 0 1 0 1 2 0 0 0 1 0 2 2 0 2 0 0 0 1 1 1 1 1 0 1 0 0 0 ⟶ 1 1 0 0 1 2 2 0 0 2 1 1 2 2 0 0 1 1 1 0 2 2 2 0 1 0 2 2 0 2 0 2 1 2 1 1 2 1 0 1 2 2 1 , 1 1 0 2 0 1 2 0 0 1 1 0 0 2 1 0 1 0 1 2 1 2 2 0 1 1 2 0 1 2 1 0 1 1 0 1 0 1 0 1 2 ⟶ 1 0 0 0 2 1 0 2 0 0 0 0 0 1 0 2 2 0 0 2 2 2 2 2 0 2 0 0 0 1 0 0 1 0 1 0 2 2 2 2 0 0 0 , 1 1 0 2 2 1 0 2 2 2 1 0 0 0 2 0 0 2 2 0 1 2 1 2 2 1 1 0 0 1 2 0 0 2 2 2 0 0 0 1 1 ⟶ 1 1 1 2 2 2 2 2 2 1 1 2 0 2 0 1 0 2 1 0 0 0 1 2 2 2 1 2 2 2 0 2 2 0 1 1 2 2 2 2 2 1 2 2 , 2 2 1 2 0 0 0 0 0 1 2 2 0 1 0 0 2 1 1 0 1 2 1 0 0 1 0 1 0 2 0 0 1 0 2 2 2 1 1 0 2 ⟶ 2 1 1 2 1 1 2 1 1 0 2 0 0 0 1 2 2 2 0 2 0 0 2 0 1 0 0 2 1 1 1 1 2 2 2 0 0 2 0 2 2 2 2 , 2 0 1 2 1 1 1 2 1 2 1 2 0 0 0 0 1 2 2 1 1 1 0 2 1 1 0 0 0 0 1 0 0 0 1 1 0 2 0 1 1 1 ⟶ 2 1 2 1 2 1 1 2 2 1 0 1 2 0 2 1 0 0 2 1 0 2 2 2 2 1 0 2 2 0 1 2 1 0 2 2 2 1 1 0 2 2 1 0 , 2 1 2 1 1 0 0 0 0 1 0 1 2 2 0 1 0 2 1 1 2 1 1 2 2 1 0 1 1 1 1 1 1 0 2 2 2 1 0 0 0 0 ⟶ 2 1 2 1 2 2 2 2 2 0 0 1 0 0 1 1 2 1 2 2 0 0 2 0 1 1 2 2 1 0 0 1 1 2 2 1 0 2 2 2 1 1 1 2 , 0 1 2 1 0 0 2 0 0 2 1 1 1 2 2 1 0 2 0 2 1 2 2 0 2 0 1 0 1 2 1 1 0 0 0 1 0 0 1 2 1 0 1 ⟶ 0 1 2 0 2 1 0 2 2 1 2 1 1 0 1 1 0 1 2 2 2 0 1 0 1 2 2 2 2 0 1 1 2 2 0 1 1 2 1 2 0 2 1 2 2 2 , 0 2 2 2 1 2 2 0 2 0 2 0 0 0 2 0 0 1 1 2 1 1 1 0 1 2 2 2 1 0 1 2 2 0 0 0 2 1 1 1 1 0 0 ⟶ 0 2 2 2 0 0 2 1 2 0 2 1 0 1 2 0 2 1 1 2 0 1 2 1 1 2 2 2 1 0 0 2 2 0 0 0 0 1 0 1 1 1 0 , 0 0 2 1 0 0 2 1 2 2 0 1 2 0 0 0 2 0 0 0 0 0 0 0 1 0 0 2 0 2 0 0 2 2 0 2 1 1 1 1 2 1 0 2 0 ⟶ 2 2 1 2 1 1 2 0 2 2 2 1 1 2 2 0 1 2 0 0 0 0 0 1 0 1 1 0 2 0 0 0 0 2 2 1 2 2 2 0 2 0 2 0 1 2 1 0 0 , 0 2 1 2 2 1 1 1 1 0 1 1 0 2 0 1 0 2 2 0 2 0 0 1 1 2 0 0 0 0 2 1 2 2 1 1 2 2 0 1 1 2 0 1 0 ⟶ 2 2 0 2 2 1 1 0 2 2 0 0 2 1 0 0 1 1 0 0 2 1 2 1 1 1 2 1 0 1 2 2 2 2 2 1 1 1 2 2 0 2 2 0 1 2 , 1 0 1 2 1 1 0 2 2 0 1 2 2 0 0 2 1 0 1 0 0 1 2 1 0 2 1 1 0 0 0 1 0 0 0 2 1 1 0 1 2 2 0 1 0 ⟶ 1 0 2 1 0 0 0 2 2 0 1 1 0 1 2 2 1 0 2 1 2 2 2 1 1 1 1 2 2 2 1 1 1 1 1 0 0 0 0 0 1 0 2 2 0 0 0 2 , 1 0 2 0 0 1 0 1 1 1 1 1 2 1 0 1 0 0 1 2 0 2 0 0 0 1 0 0 2 1 2 0 1 1 0 0 0 1 0 2 1 2 1 1 1 ⟶ 2 0 1 0 1 0 1 1 2 0 2 2 2 0 2 0 2 0 1 1 2 0 1 1 2 2 0 1 1 0 1 0 0 2 2 2 1 0 2 0 1 0 0 2 1 0 , 1 1 1 0 0 2 1 0 2 2 0 2 0 0 1 1 0 1 2 2 1 0 1 1 2 0 0 2 2 1 1 2 0 0 0 1 1 0 0 0 0 0 1 2 0 ⟶ 1 0 0 2 1 0 2 0 1 1 1 1 2 0 1 2 0 2 0 0 2 2 2 1 2 1 2 2 1 2 0 0 2 0 1 1 1 2 0 1 1 1 1 0 0 1 2 , 0 0 2 0 2 0 2 0 2 0 2 0 0 2 2 2 1 1 1 0 0 1 1 1 1 1 1 2 1 2 1 2 0 1 0 1 1 1 1 2 0 1 2 1 1 1 ⟶ 1 1 2 2 0 2 0 2 1 2 2 1 2 0 2 1 2 1 1 2 2 2 2 1 0 1 1 1 0 1 0 0 1 0 2 0 2 1 2 2 0 2 1 2 1 2 1 2 , 0 2 0 2 2 0 2 0 1 2 1 2 0 0 0 1 0 1 1 2 1 1 2 0 0 0 0 1 1 1 1 2 1 0 2 0 1 1 0 1 0 1 0 0 2 1 ⟶ 2 1 1 2 0 1 2 0 2 0 1 2 0 1 2 2 2 2 1 1 1 2 2 2 0 2 1 2 2 0 0 1 0 0 0 0 2 0 1 2 1 1 2 1 2 0 1 0 , 0 2 2 0 0 2 0 0 1 0 1 0 1 0 1 2 0 2 0 0 0 0 2 2 0 0 1 0 0 0 0 1 2 0 1 0 2 0 0 0 2 0 2 0 2 1 ⟶ 2 2 2 0 1 2 0 0 1 0 1 0 2 1 0 2 0 1 0 0 1 2 0 2 0 0 1 0 0 0 0 1 0 0 0 0 2 2 0 0 2 0 0 2 0 0 , 1 1 0 2 2 2 1 1 0 1 0 0 0 2 1 1 1 1 1 1 0 2 0 0 0 0 2 2 0 1 1 0 1 1 1 0 1 2 1 1 0 0 1 2 0 1 ⟶ 1 2 2 2 2 2 1 0 2 0 0 0 1 1 2 2 2 1 2 1 0 0 0 0 0 2 2 0 0 0 0 0 0 1 2 2 2 0 2 1 0 1 1 0 0 0 0 , 2 0 2 2 2 1 2 2 2 2 2 2 2 0 0 1 0 0 2 2 2 1 1 0 0 2 2 0 0 1 0 2 2 0 2 1 2 2 0 1 0 1 2 0 0 1 ⟶ 2 2 2 2 2 1 0 2 0 0 1 2 2 2 1 2 0 1 2 2 1 1 2 0 0 1 0 2 2 1 1 2 0 1 0 1 0 2 2 2 1 1 2 0 2 2 2 , 0 0 0 2 1 1 2 0 1 1 0 2 0 1 1 2 2 1 2 2 1 2 2 1 0 1 0 0 2 1 1 1 0 1 1 0 2 2 2 1 0 1 2 1 0 0 0 ⟶ 0 0 1 1 1 1 1 0 2 2 2 1 2 1 0 2 2 1 2 2 1 2 0 1 2 0 0 0 1 1 2 1 0 0 1 0 2 2 2 1 0 1 1 1 0 0 0 , 2 1 0 2 0 0 0 2 0 2 2 1 1 2 2 1 1 2 1 0 1 1 0 0 2 0 0 1 0 2 1 1 0 0 2 2 1 1 1 1 0 2 2 0 1 0 2 0 ⟶ 2 1 2 2 0 1 0 1 0 0 0 0 0 1 2 1 1 1 2 1 2 0 2 1 0 1 0 2 1 1 2 0 1 1 0 1 0 2 1 2 0 0 2 2 2 2 1 1 2 } 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 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 0 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 9-rule system { 0 1 1 2 1 2 0 1 1 2 1 2 2 1 2 2 ⟶ 1 1 2 0 2 1 2 1 1 2 1 0 2 1 2 2 , 1 2 2 2 1 0 2 0 1 2 1 0 2 2 1 0 2 0 0 0 0 0 2 1 0 0 2 2 1 0 1 0 ⟶ 2 2 1 0 1 0 2 1 1 2 1 2 2 2 0 2 2 2 0 1 0 0 0 1 0 0 2 1 0 0 0 0 , 1 1 0 2 2 0 0 2 1 1 0 1 2 2 1 1 2 1 0 1 0 1 0 2 2 0 1 2 0 0 1 0 1 1 1 0 ⟶ 0 0 2 2 0 2 1 2 2 2 0 2 0 2 1 1 0 2 2 1 1 0 1 0 2 1 1 0 1 0 0 1 0 0 0 0 0 1 , 0 0 2 2 1 0 2 0 2 0 2 2 1 2 1 1 2 2 0 0 1 0 1 1 1 2 0 0 0 0 2 0 2 1 2 2 ⟶ 0 0 0 2 1 1 2 0 2 1 0 2 2 2 2 1 2 2 2 0 1 0 1 1 0 2 0 0 0 0 2 1 2 0 1 2 , 1 2 2 2 0 2 2 1 2 1 2 1 1 1 2 1 1 0 0 2 0 0 0 1 0 2 2 2 0 1 0 2 2 1 1 1 2 0 0 0 0 1 1 ⟶ 1 2 2 2 1 1 2 0 2 1 2 0 1 0 2 1 2 0 0 2 1 0 2 0 0 2 2 2 0 1 1 2 2 1 1 1 1 0 1 0 0 0 1 , 1 2 2 1 1 2 1 1 0 1 0 1 0 1 0 2 1 2 1 1 1 1 2 2 1 1 0 1 1 1 1 0 2 1 0 1 2 1 1 1 2 1 2 1 2 0 ⟶ 2 2 2 1 0 2 1 1 0 1 0 1 2 0 1 2 1 0 1 1 0 2 1 2 1 1 0 1 1 1 1 0 1 1 1 1 2 2 1 1 2 1 1 2 1 1 , 2 1 2 2 2 0 2 2 2 2 2 2 2 1 1 0 1 1 2 2 2 0 0 1 1 2 2 1 1 0 1 2 2 1 2 0 2 2 1 0 1 0 2 1 1 0 ⟶ 2 2 2 2 2 0 1 2 1 1 0 2 2 2 0 2 1 0 2 2 0 0 2 1 1 0 1 2 2 0 0 2 1 0 1 0 1 2 2 2 0 0 2 1 2 2 2 , 1 1 1 2 0 0 2 1 0 0 1 2 1 0 0 2 2 0 2 2 0 2 2 0 1 0 1 1 2 0 0 0 1 0 0 1 2 2 2 0 1 0 2 0 1 1 1 ⟶ 1 1 0 0 0 0 0 1 2 2 2 0 2 0 1 2 2 0 2 2 0 2 1 0 2 1 1 1 0 0 2 0 1 1 0 1 2 2 2 0 1 0 0 0 1 1 1 , 2 0 1 2 1 1 1 2 1 2 2 0 0 2 2 0 0 2 0 1 0 0 1 1 2 1 1 0 1 2 0 0 1 1 2 2 0 0 0 0 1 2 2 1 0 1 2 1 ⟶ 2 0 2 2 1 0 1 0 1 1 1 1 1 0 2 0 0 0 2 0 2 1 2 0 1 0 1 2 0 0 2 1 0 0 1 0 1 2 0 2 1 1 2 2 2 2 0 0 2 } 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 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 6-rule system { 0 1 1 2 1 2 0 1 1 2 1 2 2 1 2 2 ⟶ 1 1 2 0 2 1 2 1 1 2 1 0 2 1 2 2 , 1 2 2 2 1 0 2 0 1 2 1 0 2 2 1 0 2 0 0 0 0 0 2 1 0 0 2 2 1 0 1 0 ⟶ 2 2 1 0 1 0 2 1 1 2 1 2 2 2 0 2 2 2 0 1 0 0 0 1 0 0 2 1 0 0 0 0 , 0 0 2 2 1 0 2 0 2 0 2 2 1 2 1 1 2 2 0 0 1 0 1 1 1 2 0 0 0 0 2 0 2 1 2 2 ⟶ 0 0 0 2 1 1 2 0 2 1 0 2 2 2 2 1 2 2 2 0 1 0 1 1 0 2 0 0 0 0 2 1 2 0 1 2 , 1 2 2 2 0 2 2 1 2 1 2 1 1 1 2 1 1 0 0 2 0 0 0 1 0 2 2 2 0 1 0 2 2 1 1 1 2 0 0 0 0 1 1 ⟶ 1 2 2 2 1 1 2 0 2 1 2 0 1 0 2 1 2 0 0 2 1 0 2 0 0 2 2 2 0 1 1 2 2 1 1 1 1 0 1 0 0 0 1 , 1 2 2 1 1 2 1 1 0 1 0 1 0 1 0 2 1 2 1 1 1 1 2 2 1 1 0 1 1 1 1 0 2 1 0 1 2 1 1 1 2 1 2 1 2 0 ⟶ 2 2 2 1 0 2 1 1 0 1 0 1 2 0 1 2 1 0 1 1 0 2 1 2 1 1 0 1 1 1 1 0 1 1 1 1 2 2 1 1 2 1 1 2 1 1 , 1 1 1 2 0 0 2 1 0 0 1 2 1 0 0 2 2 0 2 2 0 2 2 0 1 0 1 1 2 0 0 0 1 0 0 1 2 2 2 0 1 0 2 0 1 1 1 ⟶ 1 1 0 0 0 0 0 1 2 2 2 0 2 0 1 2 2 0 2 2 0 2 1 0 2 1 1 1 0 0 2 0 1 1 0 1 2 2 2 0 1 0 0 0 1 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 47: 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 }, it remains to prove termination of the 5-rule system { 0 1 1 2 1 2 0 1 1 2 1 2 2 1 2 2 ⟶ 1 1 2 0 2 1 2 1 1 2 1 0 2 1 2 2 , 1 2 2 2 1 0 2 0 1 2 1 0 2 2 1 0 2 0 0 0 0 0 2 1 0 0 2 2 1 0 1 0 ⟶ 2 2 1 0 1 0 2 1 1 2 1 2 2 2 0 2 2 2 0 1 0 0 0 1 0 0 2 1 0 0 0 0 , 0 0 2 2 1 0 2 0 2 0 2 2 1 2 1 1 2 2 0 0 1 0 1 1 1 2 0 0 0 0 2 0 2 1 2 2 ⟶ 0 0 0 2 1 1 2 0 2 1 0 2 2 2 2 1 2 2 2 0 1 0 1 1 0 2 0 0 0 0 2 1 2 0 1 2 , 1 2 2 2 0 2 2 1 2 1 2 1 1 1 2 1 1 0 0 2 0 0 0 1 0 2 2 2 0 1 0 2 2 1 1 1 2 0 0 0 0 1 1 ⟶ 1 2 2 2 1 1 2 0 2 1 2 0 1 0 2 1 2 0 0 2 1 0 2 0 0 2 2 2 0 1 1 2 2 1 1 1 1 0 1 0 0 0 1 , 1 1 1 2 0 0 2 1 0 0 1 2 1 0 0 2 2 0 2 2 0 2 2 0 1 0 1 1 2 0 0 0 1 0 0 1 2 2 2 0 1 0 2 0 1 1 1 ⟶ 1 1 0 0 0 0 0 1 2 2 2 0 2 0 1 2 2 0 2 2 0 2 1 0 2 1 1 1 0 0 2 0 1 1 0 1 2 2 2 0 1 0 0 0 1 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 33: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 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 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 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 }, it remains to prove termination of the 4-rule system { 0 1 1 2 1 2 0 1 1 2 1 2 2 1 2 2 ⟶ 1 1 2 0 2 1 2 1 1 2 1 0 2 1 2 2 , 0 0 2 2 1 0 2 0 2 0 2 2 1 2 1 1 2 2 0 0 1 0 1 1 1 2 0 0 0 0 2 0 2 1 2 2 ⟶ 0 0 0 2 1 1 2 0 2 1 0 2 2 2 2 1 2 2 2 0 1 0 1 1 0 2 0 0 0 0 2 1 2 0 1 2 , 1 2 2 2 0 2 2 1 2 1 2 1 1 1 2 1 1 0 0 2 0 0 0 1 0 2 2 2 0 1 0 2 2 1 1 1 2 0 0 0 0 1 1 ⟶ 1 2 2 2 1 1 2 0 2 1 2 0 1 0 2 1 2 0 0 2 1 0 2 0 0 2 2 2 0 1 1 2 2 1 1 1 1 0 1 0 0 0 1 , 1 1 1 2 0 0 2 1 0 0 1 2 1 0 0 2 2 0 2 2 0 2 2 0 1 0 1 1 2 0 0 0 1 0 0 1 2 2 2 0 1 0 2 0 1 1 1 ⟶ 1 1 0 0 0 0 0 1 2 2 2 0 2 0 1 2 2 0 2 2 0 2 1 0 2 1 1 1 0 0 2 0 1 1 0 1 2 2 2 0 1 0 0 0 1 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 37: 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2 }, it remains to prove termination of the 3-rule system { 0 1 1 2 1 2 0 1 1 2 1 2 2 1 2 2 ⟶ 1 1 2 0 2 1 2 1 1 2 1 0 2 1 2 2 , 1 2 2 2 0 2 2 1 2 1 2 1 1 1 2 1 1 0 0 2 0 0 0 1 0 2 2 2 0 1 0 2 2 1 1 1 2 0 0 0 0 1 1 ⟶ 1 2 2 2 1 1 2 0 2 1 2 0 1 0 2 1 2 0 0 2 1 0 2 0 0 2 2 2 0 1 1 2 2 1 1 1 1 0 1 0 0 0 1 , 1 1 1 2 0 0 2 1 0 0 1 2 1 0 0 2 2 0 2 2 0 2 2 0 1 0 1 1 2 0 0 0 1 0 0 1 2 2 2 0 1 0 2 0 1 1 1 ⟶ 1 1 0 0 0 0 0 1 2 2 2 0 2 0 1 2 2 0 2 2 0 2 1 0 2 1 1 1 0 0 2 0 1 1 0 1 2 2 2 0 1 0 0 0 1 1 1 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 17: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 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 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 1 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 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 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 2 ↦ 1, 0 ↦ 2 }, it remains to prove termination of the 2-rule system { 0 1 1 1 2 1 1 0 1 0 1 0 0 0 1 0 0 2 2 1 2 2 2 0 2 1 1 1 2 0 2 1 1 0 0 0 1 2 2 2 2 0 0 ⟶ 0 1 1 1 0 0 1 2 1 0 1 2 0 2 1 0 1 2 2 1 0 2 1 2 2 1 1 1 2 0 0 1 1 0 0 0 0 2 0 2 2 2 0 , 0 0 0 1 2 2 1 0 2 2 0 1 0 2 2 1 1 2 1 1 2 1 1 2 0 2 0 0 1 2 2 2 0 2 2 0 1 1 1 2 0 2 1 2 0 0 0 ⟶ 0 0 2 2 2 2 2 0 1 1 1 2 1 2 0 1 1 2 1 1 2 1 0 2 1 0 0 0 2 2 1 2 0 0 2 0 1 1 1 2 0 2 2 2 0 0 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 44: 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 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 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 }, it remains to prove termination of the 1-rule system { 0 0 0 1 2 2 1 0 2 2 0 1 0 2 2 1 1 2 1 1 2 1 1 2 0 2 0 0 1 2 2 2 0 2 2 0 1 1 1 2 0 2 1 2 0 0 0 ⟶ 0 0 2 2 2 2 2 0 1 1 1 2 1 2 0 1 1 2 1 1 2 1 0 2 1 0 0 0 2 2 1 2 0 0 2 0 1 1 1 2 0 2 2 2 0 0 0 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 48: 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 ⎟ ⎝ ⎠ After renaming modulo the bijection { }, it remains to prove termination of the 0-rule system { } The system is trivially terminating.