28.80/8.28 YES 29.19/8.39 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 29.19/8.39 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.19/8.39 29.19/8.39 29.19/8.39 Termination w.r.t. Q of the given QTRS could be proven: 29.19/8.39 29.19/8.39 (0) QTRS 29.19/8.39 (1) QTRS Reverse [EQUIVALENT, 0 ms] 29.19/8.39 (2) QTRS 29.19/8.39 (3) DependencyPairsProof [EQUIVALENT, 2 ms] 29.19/8.39 (4) QDP 29.19/8.39 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 29.19/8.39 (6) QDP 29.19/8.39 (7) QDPOrderProof [EQUIVALENT, 155 ms] 29.19/8.39 (8) QDP 29.19/8.39 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 29.19/8.39 (10) QDP 29.19/8.39 (11) QDPOrderProof [EQUIVALENT, 118 ms] 29.19/8.39 (12) QDP 29.19/8.39 (13) PisEmptyProof [EQUIVALENT, 0 ms] 29.19/8.39 (14) YES 29.19/8.39 29.19/8.39 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (0) 29.19/8.39 Obligation: 29.19/8.39 Q restricted rewrite system: 29.19/8.39 The TRS R consists of the following rules: 29.19/8.39 29.19/8.39 a(x1) -> x1 29.19/8.39 a(b(x1)) -> c(x1) 29.19/8.39 a(c(c(x1))) -> c(b(a(c(a(x1))))) 29.19/8.39 29.19/8.39 Q is empty. 29.19/8.39 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (1) QTRS Reverse (EQUIVALENT) 29.19/8.39 We applied the QTRS Reverse Processor [REVERSE]. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (2) 29.19/8.39 Obligation: 29.19/8.39 Q restricted rewrite system: 29.19/8.39 The TRS R consists of the following rules: 29.19/8.39 29.19/8.39 a(x1) -> x1 29.19/8.39 b(a(x1)) -> c(x1) 29.19/8.39 c(c(a(x1))) -> a(c(a(b(c(x1))))) 29.19/8.39 29.19/8.39 Q is empty. 29.19/8.39 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (3) DependencyPairsProof (EQUIVALENT) 29.19/8.39 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (4) 29.19/8.39 Obligation: 29.19/8.39 Q DP problem: 29.19/8.39 The TRS P consists of the following rules: 29.19/8.39 29.19/8.39 B(a(x1)) -> C(x1) 29.19/8.39 C(c(a(x1))) -> A(c(a(b(c(x1))))) 29.19/8.39 C(c(a(x1))) -> C(a(b(c(x1)))) 29.19/8.39 C(c(a(x1))) -> A(b(c(x1))) 29.19/8.39 C(c(a(x1))) -> B(c(x1)) 29.19/8.39 C(c(a(x1))) -> C(x1) 29.19/8.39 29.19/8.39 The TRS R consists of the following rules: 29.19/8.39 29.19/8.39 a(x1) -> x1 29.19/8.39 b(a(x1)) -> c(x1) 29.19/8.39 c(c(a(x1))) -> a(c(a(b(c(x1))))) 29.19/8.39 29.19/8.39 Q is empty. 29.19/8.39 We have to consider all minimal (P,Q,R)-chains. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (5) DependencyGraphProof (EQUIVALENT) 29.19/8.39 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (6) 29.19/8.39 Obligation: 29.19/8.39 Q DP problem: 29.19/8.39 The TRS P consists of the following rules: 29.19/8.39 29.19/8.39 C(c(a(x1))) -> C(a(b(c(x1)))) 29.19/8.39 C(c(a(x1))) -> B(c(x1)) 29.19/8.39 B(a(x1)) -> C(x1) 29.19/8.39 C(c(a(x1))) -> C(x1) 29.19/8.39 29.19/8.39 The TRS R consists of the following rules: 29.19/8.39 29.19/8.39 a(x1) -> x1 29.19/8.39 b(a(x1)) -> c(x1) 29.19/8.39 c(c(a(x1))) -> a(c(a(b(c(x1))))) 29.19/8.39 29.19/8.39 Q is empty. 29.19/8.39 We have to consider all minimal (P,Q,R)-chains. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (7) QDPOrderProof (EQUIVALENT) 29.19/8.39 We use the reduction pair processor [LPAR04,JAR06]. 29.19/8.39 29.19/8.39 29.19/8.39 The following pairs can be oriented strictly and are deleted. 29.19/8.39 29.19/8.39 C(c(a(x1))) -> B(c(x1)) 29.19/8.39 The remaining pairs can at least be oriented weakly. 29.19/8.39 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(C(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(c(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 1A], [0A, 0A, 0A]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(a(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 1A], [0A, -I, 0A]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, -I, -I]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(B(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 29.19/8.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.19/8.39 29.19/8.39 c(c(a(x1))) -> a(c(a(b(c(x1))))) 29.19/8.39 b(a(x1)) -> c(x1) 29.19/8.39 a(x1) -> x1 29.19/8.39 29.19/8.39 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (8) 29.19/8.39 Obligation: 29.19/8.39 Q DP problem: 29.19/8.39 The TRS P consists of the following rules: 29.19/8.39 29.19/8.39 C(c(a(x1))) -> C(a(b(c(x1)))) 29.19/8.39 B(a(x1)) -> C(x1) 29.19/8.39 C(c(a(x1))) -> C(x1) 29.19/8.39 29.19/8.39 The TRS R consists of the following rules: 29.19/8.39 29.19/8.39 a(x1) -> x1 29.19/8.39 b(a(x1)) -> c(x1) 29.19/8.39 c(c(a(x1))) -> a(c(a(b(c(x1))))) 29.19/8.39 29.19/8.39 Q is empty. 29.19/8.39 We have to consider all minimal (P,Q,R)-chains. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (9) DependencyGraphProof (EQUIVALENT) 29.19/8.39 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (10) 29.19/8.39 Obligation: 29.19/8.39 Q DP problem: 29.19/8.39 The TRS P consists of the following rules: 29.19/8.39 29.19/8.39 C(c(a(x1))) -> C(x1) 29.19/8.39 C(c(a(x1))) -> C(a(b(c(x1)))) 29.19/8.39 29.19/8.39 The TRS R consists of the following rules: 29.19/8.39 29.19/8.39 a(x1) -> x1 29.19/8.39 b(a(x1)) -> c(x1) 29.19/8.39 c(c(a(x1))) -> a(c(a(b(c(x1))))) 29.19/8.39 29.19/8.39 Q is empty. 29.19/8.39 We have to consider all minimal (P,Q,R)-chains. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (11) QDPOrderProof (EQUIVALENT) 29.19/8.39 We use the reduction pair processor [LPAR04,JAR06]. 29.19/8.39 29.19/8.39 29.19/8.39 The following pairs can be oriented strictly and are deleted. 29.19/8.39 29.19/8.39 C(c(a(x1))) -> C(x1) 29.19/8.39 C(c(a(x1))) -> C(a(b(c(x1)))) 29.19/8.39 The remaining pairs can at least be oriented weakly. 29.19/8.39 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(C(x_1)) = [[0A]] + [[0A, 0A, 1A]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(c(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 1A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(a(x_1)) = [[0A], [0A], [-I]] + [[0A, 1A, 1A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 <<< 29.19/8.39 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 29.19/8.39 >>> 29.19/8.39 29.19/8.39 29.19/8.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.19/8.39 29.19/8.39 c(c(a(x1))) -> a(c(a(b(c(x1))))) 29.19/8.39 b(a(x1)) -> c(x1) 29.19/8.39 a(x1) -> x1 29.19/8.39 29.19/8.39 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (12) 29.19/8.39 Obligation: 29.19/8.39 Q DP problem: 29.19/8.39 P is empty. 29.19/8.39 The TRS R consists of the following rules: 29.19/8.39 29.19/8.39 a(x1) -> x1 29.19/8.39 b(a(x1)) -> c(x1) 29.19/8.39 c(c(a(x1))) -> a(c(a(b(c(x1))))) 29.19/8.39 29.19/8.39 Q is empty. 29.19/8.39 We have to consider all minimal (P,Q,R)-chains. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (13) PisEmptyProof (EQUIVALENT) 29.19/8.39 The TRS P is empty. Hence, there is no (P,Q,R) chain. 29.19/8.39 ---------------------------------------- 29.19/8.39 29.19/8.39 (14) 29.19/8.39 YES 29.42/8.48 EOF