39.63/11.15 YES 40.04/11.18 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 40.04/11.18 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 40.04/11.18 40.04/11.18 40.04/11.18 Termination w.r.t. Q of the given QTRS could be proven: 40.04/11.18 40.04/11.18 (0) QTRS 40.04/11.18 (1) QTRS Reverse [EQUIVALENT, 0 ms] 40.04/11.18 (2) QTRS 40.04/11.18 (3) DependencyPairsProof [EQUIVALENT, 22 ms] 40.04/11.18 (4) QDP 40.04/11.18 (5) DependencyGraphProof [EQUIVALENT, 1 ms] 40.04/11.18 (6) QDP 40.04/11.18 (7) QDPOrderProof [EQUIVALENT, 58 ms] 40.04/11.18 (8) QDP 40.04/11.18 (9) UsableRulesProof [EQUIVALENT, 0 ms] 40.04/11.18 (10) QDP 40.04/11.18 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 40.04/11.18 (12) YES 40.04/11.18 40.04/11.18 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (0) 40.04/11.18 Obligation: 40.04/11.18 Q restricted rewrite system: 40.04/11.18 The TRS R consists of the following rules: 40.04/11.18 40.04/11.18 a(x1) -> x1 40.04/11.18 a(b(x1)) -> b(b(a(c(x1)))) 40.04/11.18 b(b(x1)) -> x1 40.04/11.18 c(c(x1)) -> a(x1) 40.04/11.18 40.04/11.18 Q is empty. 40.04/11.18 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (1) QTRS Reverse (EQUIVALENT) 40.04/11.18 We applied the QTRS Reverse Processor [REVERSE]. 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (2) 40.04/11.18 Obligation: 40.04/11.18 Q restricted rewrite system: 40.04/11.18 The TRS R consists of the following rules: 40.04/11.18 40.04/11.18 a(x1) -> x1 40.04/11.18 b(a(x1)) -> c(a(b(b(x1)))) 40.04/11.18 b(b(x1)) -> x1 40.04/11.18 c(c(x1)) -> a(x1) 40.04/11.18 40.04/11.18 Q is empty. 40.04/11.18 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (3) DependencyPairsProof (EQUIVALENT) 40.04/11.18 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (4) 40.04/11.18 Obligation: 40.04/11.18 Q DP problem: 40.04/11.18 The TRS P consists of the following rules: 40.04/11.18 40.04/11.18 B(a(x1)) -> C(a(b(b(x1)))) 40.04/11.18 B(a(x1)) -> A(b(b(x1))) 40.04/11.18 B(a(x1)) -> B(b(x1)) 40.04/11.18 B(a(x1)) -> B(x1) 40.04/11.18 C(c(x1)) -> A(x1) 40.04/11.18 40.04/11.18 The TRS R consists of the following rules: 40.04/11.18 40.04/11.18 a(x1) -> x1 40.04/11.18 b(a(x1)) -> c(a(b(b(x1)))) 40.04/11.18 b(b(x1)) -> x1 40.04/11.18 c(c(x1)) -> a(x1) 40.04/11.18 40.04/11.18 Q is empty. 40.04/11.18 We have to consider all minimal (P,Q,R)-chains. 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (5) DependencyGraphProof (EQUIVALENT) 40.04/11.18 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (6) 40.04/11.18 Obligation: 40.04/11.18 Q DP problem: 40.04/11.18 The TRS P consists of the following rules: 40.04/11.18 40.04/11.18 B(a(x1)) -> B(x1) 40.04/11.18 B(a(x1)) -> B(b(x1)) 40.04/11.18 40.04/11.18 The TRS R consists of the following rules: 40.04/11.18 40.04/11.18 a(x1) -> x1 40.04/11.18 b(a(x1)) -> c(a(b(b(x1)))) 40.04/11.18 b(b(x1)) -> x1 40.04/11.18 c(c(x1)) -> a(x1) 40.04/11.18 40.04/11.18 Q is empty. 40.04/11.18 We have to consider all minimal (P,Q,R)-chains. 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (7) QDPOrderProof (EQUIVALENT) 40.04/11.18 We use the reduction pair processor [LPAR04,JAR06]. 40.04/11.18 40.04/11.18 40.04/11.18 The following pairs can be oriented strictly and are deleted. 40.04/11.18 40.04/11.18 B(a(x1)) -> B(b(x1)) 40.04/11.18 The remaining pairs can at least be oriented weakly. 40.04/11.18 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 40.04/11.18 40.04/11.18 <<< 40.04/11.18 POL(B(x_1)) = [[0A]] + [[0A, 1A, 0A]] * x_1 40.04/11.18 >>> 40.04/11.18 40.04/11.18 <<< 40.04/11.18 POL(a(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, -I], [1A, 0A, 0A], [-I, 0A, 0A]] * x_1 40.04/11.18 >>> 40.04/11.18 40.04/11.18 <<< 40.04/11.18 POL(b(x_1)) = [[0A], [-I], [-I]] + [[-I, 0A, -I], [0A, -I, -I], [0A, 0A, 0A]] * x_1 40.04/11.18 >>> 40.04/11.18 40.04/11.18 <<< 40.04/11.18 POL(c(x_1)) = [[1A], [0A], [0A]] + [[1A, 0A, 0A], [0A, -I, -I], [0A, -I, 0A]] * x_1 40.04/11.18 >>> 40.04/11.18 40.04/11.18 40.04/11.18 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 40.04/11.18 40.04/11.18 b(a(x1)) -> c(a(b(b(x1)))) 40.04/11.18 b(b(x1)) -> x1 40.04/11.18 a(x1) -> x1 40.04/11.18 c(c(x1)) -> a(x1) 40.04/11.18 40.04/11.18 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (8) 40.04/11.18 Obligation: 40.04/11.18 Q DP problem: 40.04/11.18 The TRS P consists of the following rules: 40.04/11.18 40.04/11.18 B(a(x1)) -> B(x1) 40.04/11.18 40.04/11.18 The TRS R consists of the following rules: 40.04/11.18 40.04/11.18 a(x1) -> x1 40.04/11.18 b(a(x1)) -> c(a(b(b(x1)))) 40.04/11.18 b(b(x1)) -> x1 40.04/11.18 c(c(x1)) -> a(x1) 40.04/11.18 40.04/11.18 Q is empty. 40.04/11.18 We have to consider all minimal (P,Q,R)-chains. 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (9) UsableRulesProof (EQUIVALENT) 40.04/11.18 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (10) 40.04/11.18 Obligation: 40.04/11.18 Q DP problem: 40.04/11.18 The TRS P consists of the following rules: 40.04/11.18 40.04/11.18 B(a(x1)) -> B(x1) 40.04/11.18 40.04/11.18 R is empty. 40.04/11.18 Q is empty. 40.04/11.18 We have to consider all minimal (P,Q,R)-chains. 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (11) QDPSizeChangeProof (EQUIVALENT) 40.04/11.18 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 40.04/11.18 40.04/11.18 From the DPs we obtained the following set of size-change graphs: 40.04/11.18 *B(a(x1)) -> B(x1) 40.04/11.18 The graph contains the following edges 1 > 1 40.04/11.18 40.04/11.18 40.04/11.18 ---------------------------------------- 40.04/11.18 40.04/11.18 (12) 40.04/11.18 YES 40.34/11.36 EOF