23.57/6.83 YES 23.57/6.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 23.57/6.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 23.57/6.85 23.57/6.85 23.57/6.85 Termination w.r.t. Q of the given QTRS could be proven: 23.57/6.85 23.57/6.85 (0) QTRS 23.57/6.85 (1) QTRS Reverse [EQUIVALENT, 0 ms] 23.57/6.85 (2) QTRS 23.57/6.85 (3) DependencyPairsProof [EQUIVALENT, 2 ms] 23.57/6.85 (4) QDP 23.57/6.85 (5) MRRProof [EQUIVALENT, 44 ms] 23.57/6.85 (6) QDP 23.57/6.85 (7) QDPOrderProof [EQUIVALENT, 23 ms] 23.57/6.85 (8) QDP 23.57/6.85 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 23.57/6.85 (10) TRUE 23.57/6.85 23.57/6.85 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (0) 23.57/6.85 Obligation: 23.57/6.85 Q restricted rewrite system: 23.57/6.85 The TRS R consists of the following rules: 23.57/6.85 23.57/6.85 a(a(x1)) -> b(b(x1)) 23.57/6.85 a(b(x1)) -> c(c(x1)) 23.57/6.85 b(c(x1)) -> a(a(x1)) 23.57/6.85 23.57/6.85 Q is empty. 23.57/6.85 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (1) QTRS Reverse (EQUIVALENT) 23.57/6.85 We applied the QTRS Reverse Processor [REVERSE]. 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (2) 23.57/6.85 Obligation: 23.57/6.85 Q restricted rewrite system: 23.57/6.85 The TRS R consists of the following rules: 23.57/6.85 23.57/6.85 a(a(x1)) -> b(b(x1)) 23.57/6.85 b(a(x1)) -> c(c(x1)) 23.57/6.85 c(b(x1)) -> a(a(x1)) 23.57/6.85 23.57/6.85 Q is empty. 23.57/6.85 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (3) DependencyPairsProof (EQUIVALENT) 23.57/6.85 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (4) 23.57/6.85 Obligation: 23.57/6.85 Q DP problem: 23.57/6.85 The TRS P consists of the following rules: 23.57/6.85 23.57/6.85 A(a(x1)) -> B(b(x1)) 23.57/6.85 A(a(x1)) -> B(x1) 23.57/6.85 B(a(x1)) -> C(c(x1)) 23.57/6.85 B(a(x1)) -> C(x1) 23.57/6.85 C(b(x1)) -> A(a(x1)) 23.57/6.85 C(b(x1)) -> A(x1) 23.57/6.85 23.57/6.85 The TRS R consists of the following rules: 23.57/6.85 23.57/6.85 a(a(x1)) -> b(b(x1)) 23.57/6.85 b(a(x1)) -> c(c(x1)) 23.57/6.85 c(b(x1)) -> a(a(x1)) 23.57/6.85 23.57/6.85 Q is empty. 23.57/6.85 We have to consider all minimal (P,Q,R)-chains. 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (5) MRRProof (EQUIVALENT) 23.57/6.85 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. 23.57/6.85 23.57/6.85 Strictly oriented dependency pairs: 23.57/6.85 23.57/6.85 A(a(x1)) -> B(x1) 23.57/6.85 B(a(x1)) -> C(x1) 23.57/6.85 C(b(x1)) -> A(x1) 23.57/6.85 23.57/6.85 23.57/6.85 Used ordering: Polynomial interpretation [POLO]: 23.57/6.85 23.57/6.85 POL(A(x_1)) = 2*x_1 23.57/6.85 POL(B(x_1)) = 2*x_1 23.57/6.85 POL(C(x_1)) = 2*x_1 23.57/6.85 POL(a(x_1)) = 2 + 2*x_1 23.57/6.85 POL(b(x_1)) = 2 + 2*x_1 23.57/6.85 POL(c(x_1)) = 2 + 2*x_1 23.57/6.85 23.57/6.85 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (6) 23.57/6.85 Obligation: 23.57/6.85 Q DP problem: 23.57/6.85 The TRS P consists of the following rules: 23.57/6.85 23.57/6.85 A(a(x1)) -> B(b(x1)) 23.57/6.85 B(a(x1)) -> C(c(x1)) 23.57/6.85 C(b(x1)) -> A(a(x1)) 23.57/6.85 23.57/6.85 The TRS R consists of the following rules: 23.57/6.85 23.57/6.85 a(a(x1)) -> b(b(x1)) 23.57/6.85 b(a(x1)) -> c(c(x1)) 23.57/6.85 c(b(x1)) -> a(a(x1)) 23.57/6.85 23.57/6.85 Q is empty. 23.57/6.85 We have to consider all minimal (P,Q,R)-chains. 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (7) QDPOrderProof (EQUIVALENT) 23.57/6.85 We use the reduction pair processor [LPAR04,JAR06]. 23.57/6.85 23.57/6.85 23.57/6.85 The following pairs can be oriented strictly and are deleted. 23.57/6.85 23.57/6.85 C(b(x1)) -> A(a(x1)) 23.57/6.85 The remaining pairs can at least be oriented weakly. 23.57/6.85 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 23.57/6.85 23.57/6.85 <<< 23.57/6.85 POL(A(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 23.57/6.85 >>> 23.57/6.85 23.57/6.85 <<< 23.57/6.85 POL(a(x_1)) = [[1A], [-I], [-I]] + [[-I, 0A, 1A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 23.57/6.85 >>> 23.57/6.85 23.57/6.85 <<< 23.57/6.85 POL(B(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 23.57/6.85 >>> 23.57/6.85 23.57/6.85 <<< 23.57/6.85 POL(b(x_1)) = [[1A], [-I], [-I]] + [[0A, 1A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 23.57/6.85 >>> 23.57/6.85 23.57/6.85 <<< 23.57/6.85 POL(C(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 23.57/6.85 >>> 23.57/6.85 23.57/6.85 <<< 23.57/6.85 POL(c(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 23.57/6.85 >>> 23.57/6.85 23.57/6.85 23.57/6.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 23.57/6.85 23.57/6.85 b(a(x1)) -> c(c(x1)) 23.57/6.85 c(b(x1)) -> a(a(x1)) 23.57/6.85 a(a(x1)) -> b(b(x1)) 23.57/6.85 23.57/6.85 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (8) 23.57/6.85 Obligation: 23.57/6.85 Q DP problem: 23.57/6.85 The TRS P consists of the following rules: 23.57/6.85 23.57/6.85 A(a(x1)) -> B(b(x1)) 23.57/6.85 B(a(x1)) -> C(c(x1)) 23.57/6.85 23.57/6.85 The TRS R consists of the following rules: 23.57/6.85 23.57/6.85 a(a(x1)) -> b(b(x1)) 23.57/6.85 b(a(x1)) -> c(c(x1)) 23.57/6.85 c(b(x1)) -> a(a(x1)) 23.57/6.85 23.57/6.85 Q is empty. 23.57/6.85 We have to consider all minimal (P,Q,R)-chains. 23.57/6.85 ---------------------------------------- 23.57/6.85 23.57/6.85 (9) DependencyGraphProof (EQUIVALENT) 23.57/6.85 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 23.64/6.85 ---------------------------------------- 23.64/6.85 23.64/6.85 (10) 23.64/6.85 TRUE 23.64/6.89 EOF