YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 7 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) QDPOrderProof [EQUIVALENT, 619 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 0 ms] (12) QDP (13) PisEmptyProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(b(d(b(d(a(x1))))))) -> a(a(c(a(a(b(d(x1))))))) a(a(c(x1))) -> c(c(a(a(x1)))) c(c(c(x1))) -> b(d(c(b(d(x1))))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1))))))) c(a(a(x1))) -> a(a(c(c(x1)))) c(c(c(x1))) -> d(b(c(d(b(x1))))) Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: A(d(b(d(b(a(a(x1))))))) -> A(a(c(a(a(x1))))) A(d(b(d(b(a(a(x1))))))) -> A(c(a(a(x1)))) A(d(b(d(b(a(a(x1))))))) -> C(a(a(x1))) C(a(a(x1))) -> A(a(c(c(x1)))) C(a(a(x1))) -> A(c(c(x1))) C(a(a(x1))) -> C(c(x1)) C(a(a(x1))) -> C(x1) C(c(c(x1))) -> C(d(b(x1))) The TRS R consists of the following rules: a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1))))))) c(a(a(x1))) -> a(a(c(c(x1)))) c(c(c(x1))) -> d(b(c(d(b(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: A(d(b(d(b(a(a(x1))))))) -> A(c(a(a(x1)))) A(d(b(d(b(a(a(x1))))))) -> A(a(c(a(a(x1))))) A(d(b(d(b(a(a(x1))))))) -> C(a(a(x1))) C(a(a(x1))) -> A(a(c(c(x1)))) C(a(a(x1))) -> A(c(c(x1))) C(a(a(x1))) -> C(c(x1)) C(a(a(x1))) -> C(x1) The TRS R consists of the following rules: a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1))))))) c(a(a(x1))) -> a(a(c(c(x1)))) c(c(c(x1))) -> d(b(c(d(b(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. A(d(b(d(b(a(a(x1))))))) -> A(c(a(a(x1)))) A(d(b(d(b(a(a(x1))))))) -> A(a(c(a(a(x1))))) A(d(b(d(b(a(a(x1))))))) -> C(a(a(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(d(x_1)) = [[0A], [0A], [0A]] + [[0A, 1A, -I], [0A, 0A, 0A], [0A, -I, -I]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, 0A, -I], [-I, 0A, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [0A], [1A]] + [[0A, -I, -I], [0A, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(c(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(C(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1))))))) c(a(a(x1))) -> a(a(c(c(x1)))) c(c(c(x1))) -> d(b(c(d(b(x1))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: C(a(a(x1))) -> A(a(c(c(x1)))) C(a(a(x1))) -> A(c(c(x1))) C(a(a(x1))) -> C(c(x1)) C(a(a(x1))) -> C(x1) The TRS R consists of the following rules: a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1))))))) c(a(a(x1))) -> a(a(c(c(x1)))) c(c(c(x1))) -> d(b(c(d(b(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: C(a(a(x1))) -> C(x1) C(a(a(x1))) -> C(c(x1)) The TRS R consists of the following rules: a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1))))))) c(a(a(x1))) -> a(a(c(c(x1)))) c(c(c(x1))) -> d(b(c(d(b(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. C(a(a(x1))) -> C(x1) C(a(a(x1))) -> C(c(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(C(x_1)) = 4*x_1 POL(a(x_1)) = 4 + 4*x_1 POL(b(x_1)) = 0 POL(c(x_1)) = x_1 POL(d(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: c(a(a(x1))) -> a(a(c(c(x1)))) c(c(c(x1))) -> d(b(c(d(b(x1))))) a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1))))))) ---------------------------------------- (12) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: a(d(b(d(b(a(a(x1))))))) -> d(b(a(a(c(a(a(x1))))))) c(a(a(x1))) -> a(a(c(c(x1)))) c(c(c(x1))) -> d(b(c(d(b(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