25.90/7.52 YES 25.90/7.56 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 25.90/7.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.90/7.56 25.90/7.56 25.90/7.56 Termination w.r.t. Q of the given QTRS could be proven: 25.90/7.56 25.90/7.56 (0) QTRS 25.90/7.56 (1) DependencyPairsProof [EQUIVALENT, 0 ms] 25.90/7.56 (2) QDP 25.90/7.56 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 25.90/7.56 (4) QDP 25.90/7.56 (5) QDPOrderProof [EQUIVALENT, 176 ms] 25.90/7.56 (6) QDP 25.90/7.56 (7) UsableRulesProof [EQUIVALENT, 1 ms] 25.90/7.56 (8) QDP 25.90/7.56 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 25.90/7.56 (10) YES 25.90/7.56 25.90/7.56 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (0) 25.90/7.56 Obligation: 25.90/7.56 Q restricted rewrite system: 25.90/7.56 The TRS R consists of the following rules: 25.90/7.56 25.90/7.56 a(a(x1)) -> x1 25.90/7.56 a(a(b(x1))) -> b(a(b(a(a(x1))))) 25.90/7.56 b(b(x1)) -> x1 25.90/7.56 25.90/7.56 Q is empty. 25.90/7.56 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (1) DependencyPairsProof (EQUIVALENT) 25.90/7.56 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (2) 25.90/7.56 Obligation: 25.90/7.56 Q DP problem: 25.90/7.56 The TRS P consists of the following rules: 25.90/7.56 25.90/7.56 A(a(b(x1))) -> B(a(b(a(a(x1))))) 25.90/7.56 A(a(b(x1))) -> A(b(a(a(x1)))) 25.90/7.56 A(a(b(x1))) -> B(a(a(x1))) 25.90/7.56 A(a(b(x1))) -> A(a(x1)) 25.90/7.56 A(a(b(x1))) -> A(x1) 25.90/7.56 25.90/7.56 The TRS R consists of the following rules: 25.90/7.56 25.90/7.56 a(a(x1)) -> x1 25.90/7.56 a(a(b(x1))) -> b(a(b(a(a(x1))))) 25.90/7.56 b(b(x1)) -> x1 25.90/7.56 25.90/7.56 Q is empty. 25.90/7.56 We have to consider all minimal (P,Q,R)-chains. 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (3) DependencyGraphProof (EQUIVALENT) 25.90/7.56 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (4) 25.90/7.56 Obligation: 25.90/7.56 Q DP problem: 25.90/7.56 The TRS P consists of the following rules: 25.90/7.56 25.90/7.56 A(a(b(x1))) -> A(a(x1)) 25.90/7.56 A(a(b(x1))) -> A(b(a(a(x1)))) 25.90/7.56 A(a(b(x1))) -> A(x1) 25.90/7.56 25.90/7.56 The TRS R consists of the following rules: 25.90/7.56 25.90/7.56 a(a(x1)) -> x1 25.90/7.56 a(a(b(x1))) -> b(a(b(a(a(x1))))) 25.90/7.56 b(b(x1)) -> x1 25.90/7.56 25.90/7.56 Q is empty. 25.90/7.56 We have to consider all minimal (P,Q,R)-chains. 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (5) QDPOrderProof (EQUIVALENT) 25.90/7.56 We use the reduction pair processor [LPAR04,JAR06]. 25.90/7.56 25.90/7.56 25.90/7.56 The following pairs can be oriented strictly and are deleted. 25.90/7.56 25.90/7.56 A(a(b(x1))) -> A(a(x1)) 25.90/7.56 A(a(b(x1))) -> A(b(a(a(x1)))) 25.90/7.56 The remaining pairs can at least be oriented weakly. 25.90/7.56 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.90/7.56 25.90/7.56 <<< 25.90/7.56 POL(A(x_1)) = [[0A]] + [[0A, 1A, 0A]] * x_1 25.90/7.56 >>> 25.90/7.56 25.90/7.56 <<< 25.90/7.56 POL(a(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 25.90/7.56 >>> 25.90/7.56 25.90/7.56 <<< 25.90/7.56 POL(b(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 25.90/7.56 >>> 25.90/7.56 25.90/7.56 25.90/7.56 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.90/7.56 25.90/7.56 a(a(x1)) -> x1 25.90/7.56 a(a(b(x1))) -> b(a(b(a(a(x1))))) 25.90/7.56 b(b(x1)) -> x1 25.90/7.56 25.90/7.56 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (6) 25.90/7.56 Obligation: 25.90/7.56 Q DP problem: 25.90/7.56 The TRS P consists of the following rules: 25.90/7.56 25.90/7.56 A(a(b(x1))) -> A(x1) 25.90/7.56 25.90/7.56 The TRS R consists of the following rules: 25.90/7.56 25.90/7.56 a(a(x1)) -> x1 25.90/7.56 a(a(b(x1))) -> b(a(b(a(a(x1))))) 25.90/7.56 b(b(x1)) -> x1 25.90/7.56 25.90/7.56 Q is empty. 25.90/7.56 We have to consider all minimal (P,Q,R)-chains. 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (7) UsableRulesProof (EQUIVALENT) 25.90/7.56 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. 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (8) 25.90/7.56 Obligation: 25.90/7.56 Q DP problem: 25.90/7.56 The TRS P consists of the following rules: 25.90/7.56 25.90/7.56 A(a(b(x1))) -> A(x1) 25.90/7.56 25.90/7.56 R is empty. 25.90/7.56 Q is empty. 25.90/7.56 We have to consider all minimal (P,Q,R)-chains. 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (9) QDPSizeChangeProof (EQUIVALENT) 25.90/7.56 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. 25.90/7.56 25.90/7.56 From the DPs we obtained the following set of size-change graphs: 25.90/7.56 *A(a(b(x1))) -> A(x1) 25.90/7.56 The graph contains the following edges 1 > 1 25.90/7.56 25.90/7.56 25.90/7.56 ---------------------------------------- 25.90/7.56 25.90/7.56 (10) 25.90/7.56 YES 26.35/7.71 EOF