32.13/9.15 YES 35.77/10.66 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 35.77/10.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 35.77/10.66 35.77/10.66 35.77/10.66 Termination w.r.t. Q of the given QTRS could be proven: 35.77/10.66 35.77/10.66 (0) QTRS 35.77/10.66 (1) DependencyPairsProof [EQUIVALENT, 5 ms] 35.77/10.66 (2) QDP 35.77/10.66 (3) DependencyGraphProof [EQUIVALENT, 3 ms] 35.77/10.66 (4) QDP 35.77/10.66 (5) QDPOrderProof [EQUIVALENT, 136 ms] 35.77/10.66 (6) QDP 35.77/10.66 (7) QDPOrderProof [EQUIVALENT, 0 ms] 35.77/10.66 (8) QDP 35.77/10.66 (9) PisEmptyProof [EQUIVALENT, 0 ms] 35.77/10.66 (10) YES 35.77/10.66 35.77/10.66 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (0) 35.77/10.66 Obligation: 35.77/10.66 Q restricted rewrite system: 35.77/10.66 The TRS R consists of the following rules: 35.77/10.66 35.77/10.66 a(x1) -> x1 35.77/10.66 a(b(x1)) -> c(b(a(a(c(x1))))) 35.77/10.66 b(x1) -> x1 35.77/10.66 c(c(x1)) -> b(x1) 35.77/10.66 35.77/10.66 Q is empty. 35.77/10.66 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (1) DependencyPairsProof (EQUIVALENT) 35.77/10.66 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (2) 35.77/10.66 Obligation: 35.77/10.66 Q DP problem: 35.77/10.66 The TRS P consists of the following rules: 35.77/10.66 35.77/10.66 A(b(x1)) -> C(b(a(a(c(x1))))) 35.77/10.66 A(b(x1)) -> B(a(a(c(x1)))) 35.77/10.66 A(b(x1)) -> A(a(c(x1))) 35.77/10.66 A(b(x1)) -> A(c(x1)) 35.77/10.66 A(b(x1)) -> C(x1) 35.77/10.66 C(c(x1)) -> B(x1) 35.77/10.66 35.77/10.66 The TRS R consists of the following rules: 35.77/10.66 35.77/10.66 a(x1) -> x1 35.77/10.66 a(b(x1)) -> c(b(a(a(c(x1))))) 35.77/10.66 b(x1) -> x1 35.77/10.66 c(c(x1)) -> b(x1) 35.77/10.66 35.77/10.66 Q is empty. 35.77/10.66 We have to consider all minimal (P,Q,R)-chains. 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (3) DependencyGraphProof (EQUIVALENT) 35.77/10.66 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (4) 35.77/10.66 Obligation: 35.77/10.66 Q DP problem: 35.77/10.66 The TRS P consists of the following rules: 35.77/10.66 35.77/10.66 A(b(x1)) -> A(c(x1)) 35.77/10.66 A(b(x1)) -> A(a(c(x1))) 35.77/10.66 35.77/10.66 The TRS R consists of the following rules: 35.77/10.66 35.77/10.66 a(x1) -> x1 35.77/10.66 a(b(x1)) -> c(b(a(a(c(x1))))) 35.77/10.66 b(x1) -> x1 35.77/10.66 c(c(x1)) -> b(x1) 35.77/10.66 35.77/10.66 Q is empty. 35.77/10.66 We have to consider all minimal (P,Q,R)-chains. 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (5) QDPOrderProof (EQUIVALENT) 35.77/10.66 We use the reduction pair processor [LPAR04,JAR06]. 35.77/10.66 35.77/10.66 35.77/10.66 The following pairs can be oriented strictly and are deleted. 35.77/10.66 35.77/10.66 A(b(x1)) -> A(c(x1)) 35.77/10.66 The remaining pairs can at least be oriented weakly. 35.77/10.66 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 35.77/10.66 35.77/10.66 <<< 35.77/10.66 POL(A(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 35.77/10.66 >>> 35.77/10.66 35.77/10.66 <<< 35.77/10.66 POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 1A, 1A], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 35.77/10.66 >>> 35.77/10.66 35.77/10.66 <<< 35.77/10.66 POL(c(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, 0A], [-I, 0A, -I], [0A, 1A, 1A]] * x_1 35.77/10.66 >>> 35.77/10.66 35.77/10.66 <<< 35.77/10.66 POL(a(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, -I], [-I, 0A, -I], [1A, 0A, 0A]] * x_1 35.77/10.66 >>> 35.77/10.66 35.77/10.66 35.77/10.66 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.77/10.66 35.77/10.66 c(c(x1)) -> b(x1) 35.77/10.66 a(x1) -> x1 35.77/10.66 a(b(x1)) -> c(b(a(a(c(x1))))) 35.77/10.66 b(x1) -> x1 35.77/10.66 35.77/10.66 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (6) 35.77/10.66 Obligation: 35.77/10.66 Q DP problem: 35.77/10.66 The TRS P consists of the following rules: 35.77/10.66 35.77/10.66 A(b(x1)) -> A(a(c(x1))) 35.77/10.66 35.77/10.66 The TRS R consists of the following rules: 35.77/10.66 35.77/10.66 a(x1) -> x1 35.77/10.66 a(b(x1)) -> c(b(a(a(c(x1))))) 35.77/10.66 b(x1) -> x1 35.77/10.66 c(c(x1)) -> b(x1) 35.77/10.66 35.77/10.66 Q is empty. 35.77/10.66 We have to consider all minimal (P,Q,R)-chains. 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (7) QDPOrderProof (EQUIVALENT) 35.77/10.66 We use the reduction pair processor [LPAR04,JAR06]. 35.77/10.66 35.77/10.66 35.77/10.66 The following pairs can be oriented strictly and are deleted. 35.77/10.66 35.77/10.66 A(b(x1)) -> A(a(c(x1))) 35.77/10.66 The remaining pairs can at least be oriented weakly. 35.77/10.66 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 35.77/10.66 35.77/10.66 <<< 35.77/10.66 POL(A(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 35.77/10.66 >>> 35.77/10.66 35.77/10.66 <<< 35.77/10.66 POL(b(x_1)) = [[-I], [1A], [0A]] + [[0A, 0A, -I], [1A, 0A, 1A], [0A, 0A, 0A]] * x_1 35.77/10.66 >>> 35.77/10.66 35.77/10.66 <<< 35.77/10.66 POL(a(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 1A, 0A]] * x_1 35.77/10.66 >>> 35.77/10.66 35.77/10.66 <<< 35.77/10.66 POL(c(x_1)) = [[-I], [-I], [1A]] + [[0A, -I, 0A], [0A, -I, 0A], [1A, 0A, 1A]] * x_1 35.77/10.66 >>> 35.77/10.66 35.77/10.66 35.77/10.66 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.77/10.66 35.77/10.66 c(c(x1)) -> b(x1) 35.77/10.66 a(x1) -> x1 35.77/10.66 a(b(x1)) -> c(b(a(a(c(x1))))) 35.77/10.66 b(x1) -> x1 35.77/10.66 35.77/10.66 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (8) 35.77/10.66 Obligation: 35.77/10.66 Q DP problem: 35.77/10.66 P is empty. 35.77/10.66 The TRS R consists of the following rules: 35.77/10.66 35.77/10.66 a(x1) -> x1 35.77/10.66 a(b(x1)) -> c(b(a(a(c(x1))))) 35.77/10.66 b(x1) -> x1 35.77/10.66 c(c(x1)) -> b(x1) 35.77/10.66 35.77/10.66 Q is empty. 35.77/10.66 We have to consider all minimal (P,Q,R)-chains. 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (9) PisEmptyProof (EQUIVALENT) 35.77/10.66 The TRS P is empty. Hence, there is no (P,Q,R) chain. 35.77/10.66 ---------------------------------------- 35.77/10.66 35.77/10.66 (10) 35.77/10.66 YES 35.92/10.88 EOF