25.08/7.16 YES 26.69/7.72 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 26.69/7.72 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.69/7.72 26.69/7.72 26.69/7.72 Termination w.r.t. Q of the given QTRS could be proven: 26.69/7.72 26.69/7.72 (0) QTRS 26.69/7.72 (1) QTRS Reverse [EQUIVALENT, 0 ms] 26.69/7.72 (2) QTRS 26.69/7.72 (3) DependencyPairsProof [EQUIVALENT, 3 ms] 26.69/7.72 (4) QDP 26.69/7.72 (5) MRRProof [EQUIVALENT, 19 ms] 26.69/7.72 (6) QDP 26.69/7.72 (7) QDPOrderProof [EQUIVALENT, 50 ms] 26.69/7.72 (8) QDP 26.69/7.72 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 26.69/7.72 (10) TRUE 26.69/7.72 26.69/7.72 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (0) 26.69/7.72 Obligation: 26.69/7.72 Q restricted rewrite system: 26.69/7.72 The TRS R consists of the following rules: 26.69/7.72 26.69/7.72 a(x1) -> b(c(x1)) 26.69/7.72 a(b(x1)) -> c(a(x1)) 26.69/7.72 c(c(c(x1))) -> a(b(x1)) 26.69/7.72 26.69/7.72 Q is empty. 26.69/7.72 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (1) QTRS Reverse (EQUIVALENT) 26.69/7.72 We applied the QTRS Reverse Processor [REVERSE]. 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (2) 26.69/7.72 Obligation: 26.69/7.72 Q restricted rewrite system: 26.69/7.72 The TRS R consists of the following rules: 26.69/7.72 26.69/7.72 a(x1) -> c(b(x1)) 26.69/7.72 b(a(x1)) -> a(c(x1)) 26.69/7.72 c(c(c(x1))) -> b(a(x1)) 26.69/7.72 26.69/7.72 Q is empty. 26.69/7.72 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (3) DependencyPairsProof (EQUIVALENT) 26.69/7.72 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (4) 26.69/7.72 Obligation: 26.69/7.72 Q DP problem: 26.69/7.72 The TRS P consists of the following rules: 26.69/7.72 26.69/7.72 A(x1) -> C(b(x1)) 26.69/7.72 A(x1) -> B(x1) 26.69/7.72 B(a(x1)) -> A(c(x1)) 26.69/7.72 B(a(x1)) -> C(x1) 26.69/7.72 C(c(c(x1))) -> B(a(x1)) 26.69/7.72 C(c(c(x1))) -> A(x1) 26.69/7.72 26.69/7.72 The TRS R consists of the following rules: 26.69/7.72 26.69/7.72 a(x1) -> c(b(x1)) 26.69/7.72 b(a(x1)) -> a(c(x1)) 26.69/7.72 c(c(c(x1))) -> b(a(x1)) 26.69/7.72 26.69/7.72 Q is empty. 26.69/7.72 We have to consider all minimal (P,Q,R)-chains. 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (5) MRRProof (EQUIVALENT) 26.69/7.72 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 26.69/7.72 26.69/7.72 Strictly oriented dependency pairs: 26.69/7.72 26.69/7.72 A(x1) -> B(x1) 26.69/7.72 B(a(x1)) -> C(x1) 26.69/7.72 C(c(c(x1))) -> A(x1) 26.69/7.72 26.69/7.72 26.69/7.72 Used ordering: Polynomial interpretation [POLO]: 26.69/7.72 26.69/7.72 POL(A(x_1)) = 2 + 2*x_1 26.69/7.72 POL(B(x_1)) = 2*x_1 26.69/7.72 POL(C(x_1)) = 2*x_1 26.69/7.72 POL(a(x_1)) = 2 + x_1 26.69/7.72 POL(b(x_1)) = 1 + x_1 26.69/7.72 POL(c(x_1)) = 1 + x_1 26.69/7.72 26.69/7.72 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (6) 26.69/7.72 Obligation: 26.69/7.72 Q DP problem: 26.69/7.72 The TRS P consists of the following rules: 26.69/7.72 26.69/7.72 A(x1) -> C(b(x1)) 26.69/7.72 B(a(x1)) -> A(c(x1)) 26.69/7.72 C(c(c(x1))) -> B(a(x1)) 26.69/7.72 26.69/7.72 The TRS R consists of the following rules: 26.69/7.72 26.69/7.72 a(x1) -> c(b(x1)) 26.69/7.72 b(a(x1)) -> a(c(x1)) 26.69/7.72 c(c(c(x1))) -> b(a(x1)) 26.69/7.72 26.69/7.72 Q is empty. 26.69/7.72 We have to consider all minimal (P,Q,R)-chains. 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (7) QDPOrderProof (EQUIVALENT) 26.69/7.72 We use the reduction pair processor [LPAR04,JAR06]. 26.69/7.72 26.69/7.72 26.69/7.72 The following pairs can be oriented strictly and are deleted. 26.69/7.72 26.69/7.72 B(a(x1)) -> A(c(x1)) 26.69/7.72 The remaining pairs can at least be oriented weakly. 26.69/7.72 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 26.69/7.72 26.69/7.72 <<< 26.69/7.72 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 26.69/7.72 >>> 26.69/7.72 26.69/7.72 <<< 26.69/7.72 POL(C(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 26.69/7.72 >>> 26.69/7.72 26.69/7.72 <<< 26.69/7.72 POL(b(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [0A, -I, 1A]] * x_1 26.69/7.72 >>> 26.69/7.72 26.69/7.72 <<< 26.69/7.72 POL(B(x_1)) = [[-I]] + [[0A, 0A, 1A]] * x_1 26.69/7.72 >>> 26.69/7.72 26.69/7.72 <<< 26.69/7.72 POL(a(x_1)) = [[1A], [0A], [1A]] + [[0A, 0A, 1A], [0A, -I, 1A], [1A, 0A, 0A]] * x_1 26.69/7.72 >>> 26.69/7.72 26.69/7.72 <<< 26.69/7.72 POL(c(x_1)) = [[1A], [-I], [0A]] + [[1A, 0A, 0A], [0A, -I, 0A], [0A, -I, -I]] * x_1 26.69/7.72 >>> 26.69/7.72 26.69/7.72 26.69/7.72 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.69/7.72 26.69/7.72 c(c(c(x1))) -> b(a(x1)) 26.69/7.72 b(a(x1)) -> a(c(x1)) 26.69/7.72 a(x1) -> c(b(x1)) 26.69/7.72 26.69/7.72 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (8) 26.69/7.72 Obligation: 26.69/7.72 Q DP problem: 26.69/7.72 The TRS P consists of the following rules: 26.69/7.72 26.69/7.72 A(x1) -> C(b(x1)) 26.69/7.72 C(c(c(x1))) -> B(a(x1)) 26.69/7.72 26.69/7.72 The TRS R consists of the following rules: 26.69/7.72 26.69/7.72 a(x1) -> c(b(x1)) 26.69/7.72 b(a(x1)) -> a(c(x1)) 26.69/7.72 c(c(c(x1))) -> b(a(x1)) 26.69/7.72 26.69/7.72 Q is empty. 26.69/7.72 We have to consider all minimal (P,Q,R)-chains. 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (9) DependencyGraphProof (EQUIVALENT) 26.69/7.72 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 26.69/7.72 ---------------------------------------- 26.69/7.72 26.69/7.72 (10) 26.69/7.72 TRUE 26.91/7.87 EOF