/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS (5) DependencyPairsProof [EQUIVALENT, 47 ms] (6) QDP (7) QDPOrderProof [EQUIVALENT, 461 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 5970 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 807 ms] (12) QDP (13) PisEmptyProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 0(0(0(1(x1)))) 1(0(0(1(x1)))) -> 0(1(0(0(x1)))) Q is empty. ---------------------------------------- (1) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(0(0(x1)))) -> 0(0(0(1(x1)))) 0(1(0(0(1(x1))))) -> 0(0(1(0(0(x1))))) 1(1(0(0(1(x1))))) -> 1(0(1(0(0(x1))))) Q is empty. ---------------------------------------- (3) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) Q is empty. ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}^1(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}^1(0_{1_1}(1_{0_1}(x1))) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{1_1}^1(1_{0_1}(x1)) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}^1(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}^1(0_{1_1}(1_{1_1}(x1))) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{1_1}^1(1_{1_1}(x1)) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 1_{1_1}^1(x1) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{0_1}(x1)))) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(0_{0_1}(x1)) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(x1) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}^1(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(x1)))) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}^1(0_{1_1}(x1)) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{1_1}^1(x1) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{0_1}(x1)))) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(0_{0_1}(x1)) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(x1) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(x1)))) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}^1(0_{1_1}(x1)) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{1_1}^1(x1) The TRS R consists of the following rules: 0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}^1(0_{1_1}(1_{0_1}(x1))) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{1_1}^1(1_{0_1}(x1)) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}^1(0_{1_1}(1_{1_1}(x1))) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{1_1}^1(1_{1_1}(x1)) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 1_{1_1}^1(x1) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{0_1}(x1)))) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(0_{0_1}(x1)) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(x1) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}^1(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(x1)))) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}^1(0_{1_1}(x1)) 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{1_1}^1(x1) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{0_1}(x1)))) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(0_{0_1}(x1)) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}^1(x1) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(x1)))) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}^1(0_{1_1}(x1)) 1_{1_1}^1(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{1_1}^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0_{0_1}(x_1)) = 1 + 2*x_1 POL(0_{0_1}^1(x_1)) = 4 + 4*x_1 POL(0_{1_1}(x_1)) = 3 + 2*x_1 POL(0_{1_1}^1(x_1)) = 1 + 5*x_1 POL(1_{0_1}(x_1)) = 2*x_1 POL(1_{1_1}(x_1)) = 2 + 2*x_1 POL(1_{1_1}^1(x_1)) = 4*x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}^1(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}^1(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) The TRS R consists of the following rules: 0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}^1(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0_{0_1}^1(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(0_{0_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [0A, -I, -I], [-I, 0A, 1A]] * x_1 >>> <<< POL(0_{1_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 1A], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(1_{0_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(1_{1_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, -I, 1A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}^1(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) The TRS R consists of the following rules: 0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0_{0_1}^1(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}^1(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0_{0_1}^1(x_1)) = [[1A]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(0_{0_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, -I, -I], [0A, 0A, 1A]] * x_1 >>> <<< POL(0_{1_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, 0A, 0A], [-I, -I, 0A]] * x_1 >>> <<< POL(1_{0_1}(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(1_{1_1}(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, 0A], [-I, 1A, 1A], [-I, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) ---------------------------------------- (12) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0_{0_1}(0_{0_1}(0_{0_1}(0_{0_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1)))) 0_{0_1}(0_{0_1}(0_{0_1}(0_{1_1}(x1)))) -> 0_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1)))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 0_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{0_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{0_1}(x1))))) 1_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(1_{1_1}(x1))))) -> 1_{0_1}(0_{1_1}(1_{0_1}(0_{0_1}(0_{1_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (14) YES