27.59/7.94 YES 28.08/7.96 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 28.08/7.96 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.08/7.96 28.08/7.96 28.08/7.96 Termination w.r.t. Q of the given QTRS could be proven: 28.08/7.96 28.08/7.96 (0) QTRS 28.08/7.96 (1) DependencyPairsProof [EQUIVALENT, 0 ms] 28.08/7.96 (2) QDP 28.08/7.96 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 28.08/7.96 (4) QDP 28.08/7.96 (5) QDPOrderProof [EQUIVALENT, 126 ms] 28.08/7.96 (6) QDP 28.08/7.96 (7) QDPOrderProof [EQUIVALENT, 38 ms] 28.08/7.96 (8) QDP 28.08/7.96 (9) PisEmptyProof [EQUIVALENT, 0 ms] 28.08/7.96 (10) YES 28.08/7.96 28.08/7.96 28.08/7.96 ---------------------------------------- 28.08/7.96 28.08/7.96 (0) 28.08/7.96 Obligation: 28.08/7.96 Q restricted rewrite system: 28.08/7.96 The TRS R consists of the following rules: 28.08/7.96 28.08/7.96 a(x1) -> x1 28.08/7.96 a(b(x1)) -> b(a(c(b(a(x1))))) 28.08/7.96 b(x1) -> x1 28.08/7.96 c(c(c(x1))) -> x1 28.08/7.96 28.08/7.96 Q is empty. 28.08/7.96 28.08/7.96 ---------------------------------------- 28.08/7.96 28.08/7.96 (1) DependencyPairsProof (EQUIVALENT) 28.08/7.96 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 28.08/7.96 ---------------------------------------- 28.08/7.96 28.08/7.96 (2) 28.08/7.96 Obligation: 28.08/7.96 Q DP problem: 28.08/7.96 The TRS P consists of the following rules: 28.08/7.96 28.08/7.96 A(b(x1)) -> B(a(c(b(a(x1))))) 28.08/7.96 A(b(x1)) -> A(c(b(a(x1)))) 28.08/7.97 A(b(x1)) -> C(b(a(x1))) 28.08/7.97 A(b(x1)) -> B(a(x1)) 28.08/7.97 A(b(x1)) -> A(x1) 28.08/7.97 28.08/7.97 The TRS R consists of the following rules: 28.08/7.97 28.08/7.97 a(x1) -> x1 28.08/7.97 a(b(x1)) -> b(a(c(b(a(x1))))) 28.08/7.97 b(x1) -> x1 28.08/7.97 c(c(c(x1))) -> x1 28.08/7.97 28.08/7.97 Q is empty. 28.08/7.97 We have to consider all minimal (P,Q,R)-chains. 28.08/7.97 ---------------------------------------- 28.08/7.97 28.08/7.97 (3) DependencyGraphProof (EQUIVALENT) 28.08/7.97 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 28.08/7.97 ---------------------------------------- 28.08/7.97 28.08/7.97 (4) 28.08/7.97 Obligation: 28.08/7.97 Q DP problem: 28.08/7.97 The TRS P consists of the following rules: 28.08/7.97 28.08/7.97 A(b(x1)) -> A(x1) 28.08/7.97 A(b(x1)) -> A(c(b(a(x1)))) 28.08/7.97 28.08/7.97 The TRS R consists of the following rules: 28.08/7.97 28.08/7.97 a(x1) -> x1 28.08/7.97 a(b(x1)) -> b(a(c(b(a(x1))))) 28.08/7.97 b(x1) -> x1 28.08/7.97 c(c(c(x1))) -> x1 28.08/7.97 28.08/7.97 Q is empty. 28.08/7.97 We have to consider all minimal (P,Q,R)-chains. 28.08/7.97 ---------------------------------------- 28.08/7.97 28.08/7.97 (5) QDPOrderProof (EQUIVALENT) 28.08/7.97 We use the reduction pair processor [LPAR04,JAR06]. 28.08/7.97 28.08/7.97 28.08/7.97 The following pairs can be oriented strictly and are deleted. 28.08/7.97 28.08/7.97 A(b(x1)) -> A(x1) 28.08/7.97 The remaining pairs can at least be oriented weakly. 28.08/7.97 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 28.08/7.97 28.08/7.97 <<< 28.08/7.97 POL(A(x_1)) = [[0A]] + [[0A, 0A, 1A]] * x_1 28.08/7.97 >>> 28.08/7.97 28.08/7.97 <<< 28.08/7.97 POL(b(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, -I], [-I, 0A, -I], [0A, 1A, 1A]] * x_1 28.08/7.97 >>> 28.08/7.97 28.08/7.97 <<< 28.08/7.97 POL(c(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [-I, 0A, -I]] * x_1 28.08/7.97 >>> 28.08/7.97 28.08/7.97 <<< 28.08/7.97 POL(a(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, 1A], [-I, 0A, 0A], [-I, 0A, 1A]] * x_1 28.08/7.97 >>> 28.08/7.97 28.08/7.97 28.08/7.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.08/7.97 28.08/7.97 a(x1) -> x1 28.08/7.97 a(b(x1)) -> b(a(c(b(a(x1))))) 28.08/7.97 b(x1) -> x1 28.08/7.97 c(c(c(x1))) -> x1 28.08/7.97 28.08/7.97 28.08/7.97 ---------------------------------------- 28.08/7.97 28.08/7.97 (6) 28.08/7.97 Obligation: 28.08/7.97 Q DP problem: 28.08/7.97 The TRS P consists of the following rules: 28.08/7.97 28.08/7.97 A(b(x1)) -> A(c(b(a(x1)))) 28.08/7.97 28.08/7.97 The TRS R consists of the following rules: 28.08/7.97 28.08/7.97 a(x1) -> x1 28.08/7.97 a(b(x1)) -> b(a(c(b(a(x1))))) 28.08/7.97 b(x1) -> x1 28.08/7.97 c(c(c(x1))) -> x1 28.08/7.97 28.08/7.97 Q is empty. 28.08/7.97 We have to consider all minimal (P,Q,R)-chains. 28.08/7.97 ---------------------------------------- 28.08/7.97 28.08/7.97 (7) QDPOrderProof (EQUIVALENT) 28.08/7.97 We use the reduction pair processor [LPAR04,JAR06]. 28.08/7.97 28.08/7.97 28.08/7.97 The following pairs can be oriented strictly and are deleted. 28.08/7.97 28.08/7.97 A(b(x1)) -> A(c(b(a(x1)))) 28.08/7.97 The remaining pairs can at least be oriented weakly. 28.08/7.97 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 28.08/7.97 28.08/7.97 <<< 28.08/7.97 POL(A(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 28.08/7.97 >>> 28.08/7.97 28.08/7.97 <<< 28.08/7.97 POL(b(x_1)) = [[1A], [-I], [-I]] + [[1A, 0A, 1A], [-I, 0A, 0A], [-I, -I, 0A]] * x_1 28.08/7.97 >>> 28.08/7.97 28.08/7.97 <<< 28.08/7.97 POL(c(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, 0A], [0A, 0A, -I], [-I, 0A, 1A]] * x_1 28.08/7.97 >>> 28.08/7.97 28.08/7.97 <<< 28.08/7.97 POL(a(x_1)) = [[0A], [-I], [-I]] + [[1A, -I, 0A], [1A, 0A, 0A], [0A, -I, 0A]] * x_1 28.08/7.97 >>> 28.08/7.97 28.08/7.97 28.08/7.97 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.08/7.97 28.08/7.97 a(x1) -> x1 28.08/7.97 a(b(x1)) -> b(a(c(b(a(x1))))) 28.08/7.97 b(x1) -> x1 28.08/7.97 c(c(c(x1))) -> x1 28.08/7.97 28.08/7.97 28.08/7.97 ---------------------------------------- 28.08/7.97 28.08/7.97 (8) 28.08/7.97 Obligation: 28.08/7.97 Q DP problem: 28.08/7.97 P is empty. 28.08/7.97 The TRS R consists of the following rules: 28.08/7.97 28.08/7.97 a(x1) -> x1 28.08/7.97 a(b(x1)) -> b(a(c(b(a(x1))))) 28.08/7.97 b(x1) -> x1 28.08/7.97 c(c(c(x1))) -> x1 28.08/7.97 28.08/7.97 Q is empty. 28.08/7.97 We have to consider all minimal (P,Q,R)-chains. 28.08/7.97 ---------------------------------------- 28.08/7.97 28.08/7.97 (9) PisEmptyProof (EQUIVALENT) 28.08/7.97 The TRS P is empty. Hence, there is no (P,Q,R) chain. 28.08/7.97 ---------------------------------------- 28.08/7.97 28.08/7.97 (10) 28.08/7.97 YES 28.36/8.08 EOF