48.72/13.24 YES 49.03/13.28 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 49.03/13.28 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 49.03/13.28 49.03/13.28 49.03/13.28 Termination w.r.t. Q of the given QTRS could be proven: 49.03/13.28 49.03/13.28 (0) QTRS 49.03/13.28 (1) QTRS Reverse [EQUIVALENT, 0 ms] 49.03/13.28 (2) QTRS 49.03/13.28 (3) DependencyPairsProof [EQUIVALENT, 22 ms] 49.03/13.28 (4) QDP 49.03/13.28 (5) QDPOrderProof [EQUIVALENT, 137 ms] 49.03/13.28 (6) QDP 49.03/13.28 (7) QDPOrderProof [EQUIVALENT, 36 ms] 49.03/13.28 (8) QDP 49.03/13.28 (9) QDPOrderProof [EQUIVALENT, 53 ms] 49.03/13.28 (10) QDP 49.03/13.28 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 49.03/13.28 (12) QDP 49.03/13.28 (13) QDPOrderProof [EQUIVALENT, 17 ms] 49.03/13.28 (14) QDP 49.03/13.28 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 49.03/13.28 (16) TRUE 49.03/13.28 49.03/13.28 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (0) 49.03/13.28 Obligation: 49.03/13.28 Q restricted rewrite system: 49.03/13.28 The TRS R consists of the following rules: 49.03/13.28 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 a(b(x1)) -> c(a(x1)) 49.03/13.28 b(c(x1)) -> c(b(a(x1))) 49.03/13.28 49.03/13.28 Q is empty. 49.03/13.28 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (1) QTRS Reverse (EQUIVALENT) 49.03/13.28 We applied the QTRS Reverse Processor [REVERSE]. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (2) 49.03/13.28 Obligation: 49.03/13.28 Q restricted rewrite system: 49.03/13.28 The TRS R consists of the following rules: 49.03/13.28 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 49.03/13.28 Q is empty. 49.03/13.28 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (3) DependencyPairsProof (EQUIVALENT) 49.03/13.28 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (4) 49.03/13.28 Obligation: 49.03/13.28 Q DP problem: 49.03/13.28 The TRS P consists of the following rules: 49.03/13.28 49.03/13.28 A(a(x1)) -> B(x1) 49.03/13.28 B(a(x1)) -> A(c(x1)) 49.03/13.28 B(a(x1)) -> C(x1) 49.03/13.28 C(b(x1)) -> A(b(c(x1))) 49.03/13.28 C(b(x1)) -> B(c(x1)) 49.03/13.28 C(b(x1)) -> C(x1) 49.03/13.28 49.03/13.28 The TRS R consists of the following rules: 49.03/13.28 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 49.03/13.28 Q is empty. 49.03/13.28 We have to consider all minimal (P,Q,R)-chains. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (5) QDPOrderProof (EQUIVALENT) 49.03/13.28 We use the reduction pair processor [LPAR04,JAR06]. 49.03/13.28 49.03/13.28 49.03/13.28 The following pairs can be oriented strictly and are deleted. 49.03/13.28 49.03/13.28 C(b(x1)) -> A(b(c(x1))) 49.03/13.28 The remaining pairs can at least be oriented weakly. 49.03/13.28 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(a(x_1)) = [[0A], [1A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 1A], [-I, 0A, -I]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(B(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(c(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 0A], [-I, 0A, 1A], [-I, -I, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(C(x_1)) = [[1A]] + [[0A, 0A, 1A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(b(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 1A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 49.03/13.28 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.03/13.28 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 49.03/13.28 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (6) 49.03/13.28 Obligation: 49.03/13.28 Q DP problem: 49.03/13.28 The TRS P consists of the following rules: 49.03/13.28 49.03/13.28 A(a(x1)) -> B(x1) 49.03/13.28 B(a(x1)) -> A(c(x1)) 49.03/13.28 B(a(x1)) -> C(x1) 49.03/13.28 C(b(x1)) -> B(c(x1)) 49.03/13.28 C(b(x1)) -> C(x1) 49.03/13.28 49.03/13.28 The TRS R consists of the following rules: 49.03/13.28 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 49.03/13.28 Q is empty. 49.03/13.28 We have to consider all minimal (P,Q,R)-chains. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (7) QDPOrderProof (EQUIVALENT) 49.03/13.28 We use the reduction pair processor [LPAR04,JAR06]. 49.03/13.28 49.03/13.28 49.03/13.28 The following pairs can be oriented strictly and are deleted. 49.03/13.28 49.03/13.28 C(b(x1)) -> C(x1) 49.03/13.28 The remaining pairs can at least be oriented weakly. 49.03/13.28 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(a(x_1)) = [[-I], [0A], [0A]] + [[0A, 1A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(B(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(c(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(C(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(b(x_1)) = [[1A], [0A], [-I]] + [[1A, 0A, 1A], [0A, -I, -I], [-I, 1A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 49.03/13.28 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.03/13.28 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 49.03/13.28 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (8) 49.03/13.28 Obligation: 49.03/13.28 Q DP problem: 49.03/13.28 The TRS P consists of the following rules: 49.03/13.28 49.03/13.28 A(a(x1)) -> B(x1) 49.03/13.28 B(a(x1)) -> A(c(x1)) 49.03/13.28 B(a(x1)) -> C(x1) 49.03/13.28 C(b(x1)) -> B(c(x1)) 49.03/13.28 49.03/13.28 The TRS R consists of the following rules: 49.03/13.28 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 49.03/13.28 Q is empty. 49.03/13.28 We have to consider all minimal (P,Q,R)-chains. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (9) QDPOrderProof (EQUIVALENT) 49.03/13.28 We use the reduction pair processor [LPAR04,JAR06]. 49.03/13.28 49.03/13.28 49.03/13.28 The following pairs can be oriented strictly and are deleted. 49.03/13.28 49.03/13.28 A(a(x1)) -> B(x1) 49.03/13.28 The remaining pairs can at least be oriented weakly. 49.03/13.28 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(A(x_1)) = [[-I]] + [[0A, 1A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(a(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [0A, -I, 0A], [0A, 1A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(B(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(c(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, 0A, -I], [-I, 1A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(C(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 1A, 0A], [0A, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 49.03/13.28 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.03/13.28 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 49.03/13.28 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (10) 49.03/13.28 Obligation: 49.03/13.28 Q DP problem: 49.03/13.28 The TRS P consists of the following rules: 49.03/13.28 49.03/13.28 B(a(x1)) -> A(c(x1)) 49.03/13.28 B(a(x1)) -> C(x1) 49.03/13.28 C(b(x1)) -> B(c(x1)) 49.03/13.28 49.03/13.28 The TRS R consists of the following rules: 49.03/13.28 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 49.03/13.28 Q is empty. 49.03/13.28 We have to consider all minimal (P,Q,R)-chains. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (11) DependencyGraphProof (EQUIVALENT) 49.03/13.28 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (12) 49.03/13.28 Obligation: 49.03/13.28 Q DP problem: 49.03/13.28 The TRS P consists of the following rules: 49.03/13.28 49.03/13.28 B(a(x1)) -> C(x1) 49.03/13.28 C(b(x1)) -> B(c(x1)) 49.03/13.28 49.03/13.28 The TRS R consists of the following rules: 49.03/13.28 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 49.03/13.28 Q is empty. 49.03/13.28 We have to consider all minimal (P,Q,R)-chains. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (13) QDPOrderProof (EQUIVALENT) 49.03/13.28 We use the reduction pair processor [LPAR04,JAR06]. 49.03/13.28 49.03/13.28 49.03/13.28 The following pairs can be oriented strictly and are deleted. 49.03/13.28 49.03/13.28 C(b(x1)) -> B(c(x1)) 49.03/13.28 The remaining pairs can at least be oriented weakly. 49.03/13.28 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(B(x_1)) = [[0A]] + [[0A, 0A, 1A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(a(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [-I, -I, 1A], [-I, 0A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(C(x_1)) = [[0A]] + [[0A, 1A, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(b(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, -I], [-I, 0A, 1A], [-I, 0A, 1A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 <<< 49.03/13.28 POL(c(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, -I], [-I, 0A, 1A], [-I, -I, 0A]] * x_1 49.03/13.28 >>> 49.03/13.28 49.03/13.28 49.03/13.28 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.03/13.28 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 49.03/13.28 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (14) 49.03/13.28 Obligation: 49.03/13.28 Q DP problem: 49.03/13.28 The TRS P consists of the following rules: 49.03/13.28 49.03/13.28 B(a(x1)) -> C(x1) 49.03/13.28 49.03/13.28 The TRS R consists of the following rules: 49.03/13.28 49.03/13.28 a(a(x1)) -> b(x1) 49.03/13.28 b(a(x1)) -> a(c(x1)) 49.03/13.28 c(b(x1)) -> a(b(c(x1))) 49.03/13.28 49.03/13.28 Q is empty. 49.03/13.28 We have to consider all minimal (P,Q,R)-chains. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (15) DependencyGraphProof (EQUIVALENT) 49.03/13.28 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 49.03/13.28 ---------------------------------------- 49.03/13.28 49.03/13.28 (16) 49.03/13.28 TRUE 49.22/13.38 EOF