25.36/7.36 YES 25.36/7.40 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 25.36/7.40 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.36/7.40 25.36/7.40 25.36/7.40 Termination w.r.t. Q of the given QTRS could be proven: 25.36/7.40 25.36/7.40 (0) QTRS 25.36/7.40 (1) DependencyPairsProof [EQUIVALENT, 5 ms] 25.36/7.40 (2) QDP 25.36/7.40 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 25.36/7.40 (4) QDP 25.36/7.40 (5) QDPOrderProof [EQUIVALENT, 124 ms] 25.36/7.40 (6) QDP 25.36/7.40 (7) UsableRulesProof [EQUIVALENT, 0 ms] 25.36/7.40 (8) QDP 25.36/7.40 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 25.36/7.40 (10) YES 25.36/7.40 25.36/7.40 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (0) 25.36/7.40 Obligation: 25.36/7.40 Q restricted rewrite system: 25.36/7.40 The TRS R consists of the following rules: 25.36/7.40 25.36/7.40 a(x1) -> b(x1) 25.36/7.40 b(b(c(x1))) -> c(a(c(a(a(x1))))) 25.36/7.40 c(c(x1)) -> x1 25.36/7.40 25.36/7.40 Q is empty. 25.36/7.40 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (1) DependencyPairsProof (EQUIVALENT) 25.36/7.40 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (2) 25.36/7.40 Obligation: 25.36/7.40 Q DP problem: 25.36/7.40 The TRS P consists of the following rules: 25.36/7.40 25.36/7.40 A(x1) -> B(x1) 25.36/7.40 B(b(c(x1))) -> C(a(c(a(a(x1))))) 25.36/7.40 B(b(c(x1))) -> A(c(a(a(x1)))) 25.36/7.40 B(b(c(x1))) -> C(a(a(x1))) 25.36/7.40 B(b(c(x1))) -> A(a(x1)) 25.36/7.40 B(b(c(x1))) -> A(x1) 25.36/7.40 25.36/7.40 The TRS R consists of the following rules: 25.36/7.40 25.36/7.40 a(x1) -> b(x1) 25.36/7.40 b(b(c(x1))) -> c(a(c(a(a(x1))))) 25.36/7.40 c(c(x1)) -> x1 25.36/7.40 25.36/7.40 Q is empty. 25.36/7.40 We have to consider all minimal (P,Q,R)-chains. 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (3) DependencyGraphProof (EQUIVALENT) 25.36/7.40 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (4) 25.36/7.40 Obligation: 25.36/7.40 Q DP problem: 25.36/7.40 The TRS P consists of the following rules: 25.36/7.40 25.36/7.40 B(b(c(x1))) -> A(c(a(a(x1)))) 25.36/7.40 A(x1) -> B(x1) 25.36/7.40 B(b(c(x1))) -> A(a(x1)) 25.36/7.40 B(b(c(x1))) -> A(x1) 25.36/7.40 25.36/7.40 The TRS R consists of the following rules: 25.36/7.40 25.36/7.40 a(x1) -> b(x1) 25.36/7.40 b(b(c(x1))) -> c(a(c(a(a(x1))))) 25.36/7.40 c(c(x1)) -> x1 25.36/7.40 25.36/7.40 Q is empty. 25.36/7.40 We have to consider all minimal (P,Q,R)-chains. 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (5) QDPOrderProof (EQUIVALENT) 25.36/7.40 We use the reduction pair processor [LPAR04,JAR06]. 25.36/7.40 25.36/7.40 25.36/7.40 The following pairs can be oriented strictly and are deleted. 25.36/7.40 25.36/7.40 B(b(c(x1))) -> A(c(a(a(x1)))) 25.36/7.40 B(b(c(x1))) -> A(a(x1)) 25.36/7.40 The remaining pairs can at least be oriented weakly. 25.36/7.40 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.36/7.40 25.36/7.40 <<< 25.36/7.40 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 25.36/7.40 >>> 25.36/7.40 25.36/7.40 <<< 25.36/7.40 POL(b(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 25.36/7.40 >>> 25.36/7.40 25.36/7.40 <<< 25.36/7.40 POL(c(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, 0A], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 25.36/7.40 >>> 25.36/7.40 25.36/7.40 <<< 25.36/7.40 POL(A(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 25.36/7.40 >>> 25.36/7.40 25.36/7.40 <<< 25.36/7.40 POL(a(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 25.36/7.40 >>> 25.36/7.40 25.36/7.40 25.36/7.40 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.36/7.40 25.36/7.40 a(x1) -> b(x1) 25.36/7.40 c(c(x1)) -> x1 25.36/7.40 b(b(c(x1))) -> c(a(c(a(a(x1))))) 25.36/7.40 25.36/7.40 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (6) 25.36/7.40 Obligation: 25.36/7.40 Q DP problem: 25.36/7.40 The TRS P consists of the following rules: 25.36/7.40 25.36/7.40 A(x1) -> B(x1) 25.36/7.40 B(b(c(x1))) -> A(x1) 25.36/7.40 25.36/7.40 The TRS R consists of the following rules: 25.36/7.40 25.36/7.40 a(x1) -> b(x1) 25.36/7.40 b(b(c(x1))) -> c(a(c(a(a(x1))))) 25.36/7.40 c(c(x1)) -> x1 25.36/7.40 25.36/7.40 Q is empty. 25.36/7.40 We have to consider all minimal (P,Q,R)-chains. 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (7) UsableRulesProof (EQUIVALENT) 25.36/7.40 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. 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (8) 25.36/7.40 Obligation: 25.36/7.40 Q DP problem: 25.36/7.40 The TRS P consists of the following rules: 25.36/7.40 25.36/7.40 A(x1) -> B(x1) 25.36/7.40 B(b(c(x1))) -> A(x1) 25.36/7.40 25.36/7.40 R is empty. 25.36/7.40 Q is empty. 25.36/7.40 We have to consider all minimal (P,Q,R)-chains. 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (9) QDPSizeChangeProof (EQUIVALENT) 25.36/7.40 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. 25.36/7.40 25.36/7.40 From the DPs we obtained the following set of size-change graphs: 25.36/7.40 *B(b(c(x1))) -> A(x1) 25.36/7.40 The graph contains the following edges 1 > 1 25.36/7.40 25.36/7.40 25.36/7.40 *A(x1) -> B(x1) 25.36/7.40 The graph contains the following edges 1 >= 1 25.36/7.40 25.36/7.40 25.36/7.40 ---------------------------------------- 25.36/7.40 25.36/7.40 (10) 25.36/7.40 YES 25.73/7.45 EOF