27.08/7.80 YES 28.15/8.13 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 28.15/8.13 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.15/8.13 28.15/8.13 28.15/8.13 Termination w.r.t. Q of the given QTRS could be proven: 28.15/8.13 28.15/8.13 (0) QTRS 28.15/8.13 (1) DependencyPairsProof [EQUIVALENT, 21 ms] 28.15/8.13 (2) QDP 28.15/8.13 (3) DependencyGraphProof [EQUIVALENT, 3 ms] 28.15/8.13 (4) QDP 28.15/8.13 (5) QDPOrderProof [EQUIVALENT, 79 ms] 28.15/8.13 (6) QDP 28.15/8.13 (7) UsableRulesProof [EQUIVALENT, 0 ms] 28.15/8.13 (8) QDP 28.15/8.13 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.15/8.13 (10) YES 28.15/8.13 28.15/8.13 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (0) 28.15/8.13 Obligation: 28.15/8.13 Q restricted rewrite system: 28.15/8.13 The TRS R consists of the following rules: 28.15/8.13 28.15/8.13 a(x1) -> b(x1) 28.15/8.13 b(b(c(x1))) -> c(b(c(b(a(x1))))) 28.15/8.13 c(c(x1)) -> x1 28.15/8.13 28.15/8.13 Q is empty. 28.15/8.13 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (1) DependencyPairsProof (EQUIVALENT) 28.15/8.13 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (2) 28.15/8.13 Obligation: 28.15/8.13 Q DP problem: 28.15/8.13 The TRS P consists of the following rules: 28.15/8.13 28.15/8.13 A(x1) -> B(x1) 28.15/8.13 B(b(c(x1))) -> C(b(c(b(a(x1))))) 28.15/8.13 B(b(c(x1))) -> B(c(b(a(x1)))) 28.15/8.13 B(b(c(x1))) -> C(b(a(x1))) 28.15/8.13 B(b(c(x1))) -> B(a(x1)) 28.15/8.13 B(b(c(x1))) -> A(x1) 28.15/8.13 28.15/8.13 The TRS R consists of the following rules: 28.15/8.13 28.15/8.13 a(x1) -> b(x1) 28.15/8.13 b(b(c(x1))) -> c(b(c(b(a(x1))))) 28.15/8.13 c(c(x1)) -> x1 28.15/8.13 28.15/8.13 Q is empty. 28.15/8.13 We have to consider all minimal (P,Q,R)-chains. 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (3) DependencyGraphProof (EQUIVALENT) 28.15/8.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (4) 28.15/8.13 Obligation: 28.15/8.13 Q DP problem: 28.15/8.13 The TRS P consists of the following rules: 28.15/8.13 28.15/8.13 B(b(c(x1))) -> B(c(b(a(x1)))) 28.15/8.13 B(b(c(x1))) -> B(a(x1)) 28.15/8.13 B(b(c(x1))) -> A(x1) 28.15/8.13 A(x1) -> B(x1) 28.15/8.13 28.15/8.13 The TRS R consists of the following rules: 28.15/8.13 28.15/8.13 a(x1) -> b(x1) 28.15/8.13 b(b(c(x1))) -> c(b(c(b(a(x1))))) 28.15/8.13 c(c(x1)) -> x1 28.15/8.13 28.15/8.13 Q is empty. 28.15/8.13 We have to consider all minimal (P,Q,R)-chains. 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (5) QDPOrderProof (EQUIVALENT) 28.15/8.13 We use the reduction pair processor [LPAR04,JAR06]. 28.15/8.13 28.15/8.13 28.15/8.13 The following pairs can be oriented strictly and are deleted. 28.15/8.13 28.15/8.13 B(b(c(x1))) -> B(c(b(a(x1)))) 28.15/8.13 B(b(c(x1))) -> B(a(x1)) 28.15/8.13 The remaining pairs can at least be oriented weakly. 28.15/8.13 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 28.15/8.13 28.15/8.13 <<< 28.15/8.13 POL(B(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 28.15/8.13 >>> 28.15/8.13 28.15/8.13 <<< 28.15/8.13 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [0A, -I, -I]] * x_1 28.15/8.13 >>> 28.15/8.13 28.15/8.13 <<< 28.15/8.13 POL(c(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] * x_1 28.15/8.13 >>> 28.15/8.13 28.15/8.13 <<< 28.15/8.13 POL(a(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 28.15/8.13 >>> 28.15/8.13 28.15/8.13 <<< 28.15/8.13 POL(A(x_1)) = [[1A]] + [[0A, 0A, 1A]] * x_1 28.15/8.13 >>> 28.15/8.13 28.15/8.13 28.15/8.13 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.15/8.13 28.15/8.13 a(x1) -> b(x1) 28.15/8.13 b(b(c(x1))) -> c(b(c(b(a(x1))))) 28.15/8.13 c(c(x1)) -> x1 28.15/8.13 28.15/8.13 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (6) 28.15/8.13 Obligation: 28.15/8.13 Q DP problem: 28.15/8.13 The TRS P consists of the following rules: 28.15/8.13 28.15/8.13 B(b(c(x1))) -> A(x1) 28.15/8.13 A(x1) -> B(x1) 28.15/8.13 28.15/8.13 The TRS R consists of the following rules: 28.15/8.13 28.15/8.13 a(x1) -> b(x1) 28.15/8.13 b(b(c(x1))) -> c(b(c(b(a(x1))))) 28.15/8.13 c(c(x1)) -> x1 28.15/8.13 28.15/8.13 Q is empty. 28.15/8.13 We have to consider all minimal (P,Q,R)-chains. 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (7) UsableRulesProof (EQUIVALENT) 28.15/8.13 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. 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (8) 28.15/8.13 Obligation: 28.15/8.13 Q DP problem: 28.15/8.13 The TRS P consists of the following rules: 28.15/8.13 28.15/8.13 B(b(c(x1))) -> A(x1) 28.15/8.13 A(x1) -> B(x1) 28.15/8.13 28.15/8.13 R is empty. 28.15/8.13 Q is empty. 28.15/8.13 We have to consider all minimal (P,Q,R)-chains. 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (9) QDPSizeChangeProof (EQUIVALENT) 28.15/8.13 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. 28.15/8.13 28.15/8.13 From the DPs we obtained the following set of size-change graphs: 28.15/8.13 *A(x1) -> B(x1) 28.15/8.13 The graph contains the following edges 1 >= 1 28.15/8.13 28.15/8.13 28.15/8.13 *B(b(c(x1))) -> A(x1) 28.15/8.13 The graph contains the following edges 1 > 1 28.15/8.13 28.15/8.13 28.15/8.13 ---------------------------------------- 28.15/8.13 28.15/8.13 (10) 28.15/8.13 YES 28.57/9.81 EOF