30.50/8.78 YES 31.05/8.81 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 31.05/8.81 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 31.05/8.81 31.05/8.81 31.05/8.81 Termination w.r.t. Q of the given QTRS could be proven: 31.05/8.81 31.05/8.81 (0) QTRS 31.05/8.81 (1) DependencyPairsProof [EQUIVALENT, 0 ms] 31.05/8.81 (2) QDP 31.05/8.81 (3) QDPOrderProof [EQUIVALENT, 156 ms] 31.05/8.81 (4) QDP 31.05/8.81 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 31.05/8.81 (6) QDP 31.05/8.81 (7) UsableRulesProof [EQUIVALENT, 0 ms] 31.05/8.81 (8) QDP 31.05/8.81 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 31.05/8.81 (10) YES 31.05/8.81 31.05/8.81 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (0) 31.05/8.81 Obligation: 31.05/8.81 Q restricted rewrite system: 31.05/8.81 The TRS R consists of the following rules: 31.05/8.81 31.05/8.81 a(x1) -> x1 31.05/8.81 a(b(x1)) -> c(b(x1)) 31.05/8.81 b(x1) -> a(a(c(x1))) 31.05/8.81 c(c(x1)) -> b(x1) 31.05/8.81 31.05/8.81 Q is empty. 31.05/8.81 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (1) DependencyPairsProof (EQUIVALENT) 31.05/8.81 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (2) 31.05/8.81 Obligation: 31.05/8.81 Q DP problem: 31.05/8.81 The TRS P consists of the following rules: 31.05/8.81 31.05/8.81 A(b(x1)) -> C(b(x1)) 31.05/8.81 B(x1) -> A(a(c(x1))) 31.05/8.81 B(x1) -> A(c(x1)) 31.05/8.81 B(x1) -> C(x1) 31.05/8.81 C(c(x1)) -> B(x1) 31.05/8.81 31.05/8.81 The TRS R consists of the following rules: 31.05/8.81 31.05/8.81 a(x1) -> x1 31.05/8.81 a(b(x1)) -> c(b(x1)) 31.05/8.81 b(x1) -> a(a(c(x1))) 31.05/8.81 c(c(x1)) -> b(x1) 31.05/8.81 31.05/8.81 Q is empty. 31.05/8.81 We have to consider all minimal (P,Q,R)-chains. 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (3) QDPOrderProof (EQUIVALENT) 31.05/8.81 We use the reduction pair processor [LPAR04,JAR06]. 31.05/8.81 31.05/8.81 31.05/8.81 The following pairs can be oriented strictly and are deleted. 31.05/8.81 31.05/8.81 A(b(x1)) -> C(b(x1)) 31.05/8.81 The remaining pairs can at least be oriented weakly. 31.05/8.81 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 31.05/8.81 31.05/8.81 <<< 31.05/8.81 POL(A(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 31.05/8.81 >>> 31.05/8.81 31.05/8.81 <<< 31.05/8.81 POL(b(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [1A, 1A, 1A], [1A, 1A, 1A]] * x_1 31.05/8.81 >>> 31.05/8.81 31.05/8.81 <<< 31.05/8.81 POL(C(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 31.05/8.81 >>> 31.05/8.81 31.05/8.81 <<< 31.05/8.81 POL(B(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 31.05/8.81 >>> 31.05/8.81 31.05/8.81 <<< 31.05/8.81 POL(a(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 1A, 0A]] * x_1 31.05/8.81 >>> 31.05/8.81 31.05/8.81 <<< 31.05/8.81 POL(c(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 1A]] * x_1 31.05/8.81 >>> 31.05/8.81 31.05/8.81 31.05/8.81 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 31.05/8.81 31.05/8.81 a(b(x1)) -> c(b(x1)) 31.05/8.81 c(c(x1)) -> b(x1) 31.05/8.81 b(x1) -> a(a(c(x1))) 31.05/8.81 a(x1) -> x1 31.05/8.81 31.05/8.81 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (4) 31.05/8.81 Obligation: 31.05/8.81 Q DP problem: 31.05/8.81 The TRS P consists of the following rules: 31.05/8.81 31.05/8.81 B(x1) -> A(a(c(x1))) 31.05/8.81 B(x1) -> A(c(x1)) 31.05/8.81 B(x1) -> C(x1) 31.05/8.81 C(c(x1)) -> B(x1) 31.05/8.81 31.05/8.81 The TRS R consists of the following rules: 31.05/8.81 31.05/8.81 a(x1) -> x1 31.05/8.81 a(b(x1)) -> c(b(x1)) 31.05/8.81 b(x1) -> a(a(c(x1))) 31.05/8.81 c(c(x1)) -> b(x1) 31.05/8.81 31.05/8.81 Q is empty. 31.05/8.81 We have to consider all minimal (P,Q,R)-chains. 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (5) DependencyGraphProof (EQUIVALENT) 31.05/8.81 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (6) 31.05/8.81 Obligation: 31.05/8.81 Q DP problem: 31.05/8.81 The TRS P consists of the following rules: 31.05/8.81 31.05/8.81 B(x1) -> C(x1) 31.05/8.81 C(c(x1)) -> B(x1) 31.05/8.81 31.05/8.81 The TRS R consists of the following rules: 31.05/8.81 31.05/8.81 a(x1) -> x1 31.05/8.81 a(b(x1)) -> c(b(x1)) 31.05/8.81 b(x1) -> a(a(c(x1))) 31.05/8.81 c(c(x1)) -> b(x1) 31.05/8.81 31.05/8.81 Q is empty. 31.05/8.81 We have to consider all minimal (P,Q,R)-chains. 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (7) UsableRulesProof (EQUIVALENT) 31.05/8.81 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. 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (8) 31.05/8.81 Obligation: 31.05/8.81 Q DP problem: 31.05/8.81 The TRS P consists of the following rules: 31.05/8.81 31.05/8.81 B(x1) -> C(x1) 31.05/8.81 C(c(x1)) -> B(x1) 31.05/8.81 31.05/8.81 R is empty. 31.05/8.81 Q is empty. 31.05/8.81 We have to consider all minimal (P,Q,R)-chains. 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (9) QDPSizeChangeProof (EQUIVALENT) 31.05/8.81 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. 31.05/8.81 31.05/8.81 From the DPs we obtained the following set of size-change graphs: 31.05/8.81 *C(c(x1)) -> B(x1) 31.05/8.81 The graph contains the following edges 1 > 1 31.05/8.81 31.05/8.81 31.05/8.81 *B(x1) -> C(x1) 31.05/8.81 The graph contains the following edges 1 >= 1 31.05/8.81 31.05/8.81 31.05/8.81 ---------------------------------------- 31.05/8.81 31.05/8.81 (10) 31.05/8.81 YES 31.27/8.93 EOF