35.18/10.02 YES 35.18/10.03 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 35.18/10.03 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 35.18/10.03 35.18/10.03 35.18/10.03 Termination w.r.t. Q of the given QTRS could be proven: 35.18/10.03 35.18/10.03 (0) QTRS 35.18/10.03 (1) QTRS Reverse [EQUIVALENT, 0 ms] 35.18/10.03 (2) QTRS 35.18/10.03 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 35.18/10.03 (4) QDP 35.18/10.03 (5) QDPOrderProof [EQUIVALENT, 0 ms] 35.18/10.03 (6) QDP 35.18/10.03 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 35.18/10.03 (8) QDP 35.18/10.03 (9) UsableRulesProof [EQUIVALENT, 0 ms] 35.18/10.03 (10) QDP 35.18/10.03 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 35.18/10.03 (12) YES 35.18/10.03 35.18/10.03 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (0) 35.18/10.03 Obligation: 35.18/10.03 Q restricted rewrite system: 35.18/10.03 The TRS R consists of the following rules: 35.18/10.03 35.18/10.03 b(b(x1)) -> a(a(a(x1))) 35.18/10.03 a(a(b(x1))) -> b(x1) 35.18/10.03 a(b(a(x1))) -> a(b(b(x1))) 35.18/10.03 35.18/10.03 Q is empty. 35.18/10.03 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (1) QTRS Reverse (EQUIVALENT) 35.18/10.03 We applied the QTRS Reverse Processor [REVERSE]. 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (2) 35.18/10.03 Obligation: 35.18/10.03 Q restricted rewrite system: 35.18/10.03 The TRS R consists of the following rules: 35.18/10.03 35.18/10.03 b(b(x1)) -> a(a(a(x1))) 35.18/10.03 b(a(a(x1))) -> b(x1) 35.18/10.03 a(b(a(x1))) -> b(b(a(x1))) 35.18/10.03 35.18/10.03 Q is empty. 35.18/10.03 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (3) DependencyPairsProof (EQUIVALENT) 35.18/10.03 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (4) 35.18/10.03 Obligation: 35.18/10.03 Q DP problem: 35.18/10.03 The TRS P consists of the following rules: 35.18/10.03 35.18/10.03 B(b(x1)) -> A(a(a(x1))) 35.18/10.03 B(b(x1)) -> A(a(x1)) 35.18/10.03 B(b(x1)) -> A(x1) 35.18/10.03 B(a(a(x1))) -> B(x1) 35.18/10.03 A(b(a(x1))) -> B(b(a(x1))) 35.18/10.03 35.18/10.03 The TRS R consists of the following rules: 35.18/10.03 35.18/10.03 b(b(x1)) -> a(a(a(x1))) 35.18/10.03 b(a(a(x1))) -> b(x1) 35.18/10.03 a(b(a(x1))) -> b(b(a(x1))) 35.18/10.03 35.18/10.03 Q is empty. 35.18/10.03 We have to consider all minimal (P,Q,R)-chains. 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (5) QDPOrderProof (EQUIVALENT) 35.18/10.03 We use the reduction pair processor [LPAR04,JAR06]. 35.18/10.03 35.18/10.03 35.18/10.03 The following pairs can be oriented strictly and are deleted. 35.18/10.03 35.18/10.03 A(b(a(x1))) -> B(b(a(x1))) 35.18/10.03 The remaining pairs can at least be oriented weakly. 35.18/10.03 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 35.18/10.03 35.18/10.03 <<< 35.18/10.03 POL(B(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 35.18/10.03 >>> 35.18/10.03 35.18/10.03 <<< 35.18/10.03 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [1A, 0A, -I], [0A, 0A, 0A]] * x_1 35.18/10.03 >>> 35.18/10.03 35.18/10.03 <<< 35.18/10.03 POL(A(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 35.18/10.03 >>> 35.18/10.03 35.18/10.03 <<< 35.18/10.03 POL(a(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 35.18/10.03 >>> 35.18/10.03 35.18/10.03 35.18/10.03 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.18/10.03 35.18/10.03 a(b(a(x1))) -> b(b(a(x1))) 35.18/10.03 b(b(x1)) -> a(a(a(x1))) 35.18/10.03 b(a(a(x1))) -> b(x1) 35.18/10.03 35.18/10.03 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (6) 35.18/10.03 Obligation: 35.18/10.03 Q DP problem: 35.18/10.03 The TRS P consists of the following rules: 35.18/10.03 35.18/10.03 B(b(x1)) -> A(a(a(x1))) 35.18/10.03 B(b(x1)) -> A(a(x1)) 35.18/10.03 B(b(x1)) -> A(x1) 35.18/10.03 B(a(a(x1))) -> B(x1) 35.18/10.03 35.18/10.03 The TRS R consists of the following rules: 35.18/10.03 35.18/10.03 b(b(x1)) -> a(a(a(x1))) 35.18/10.03 b(a(a(x1))) -> b(x1) 35.18/10.03 a(b(a(x1))) -> b(b(a(x1))) 35.18/10.03 35.18/10.03 Q is empty. 35.18/10.03 We have to consider all minimal (P,Q,R)-chains. 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (7) DependencyGraphProof (EQUIVALENT) 35.18/10.03 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (8) 35.18/10.03 Obligation: 35.18/10.03 Q DP problem: 35.18/10.03 The TRS P consists of the following rules: 35.18/10.03 35.18/10.03 B(a(a(x1))) -> B(x1) 35.18/10.03 35.18/10.03 The TRS R consists of the following rules: 35.18/10.03 35.18/10.03 b(b(x1)) -> a(a(a(x1))) 35.18/10.03 b(a(a(x1))) -> b(x1) 35.18/10.03 a(b(a(x1))) -> b(b(a(x1))) 35.18/10.03 35.18/10.03 Q is empty. 35.18/10.03 We have to consider all minimal (P,Q,R)-chains. 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (9) UsableRulesProof (EQUIVALENT) 35.18/10.03 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. 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (10) 35.18/10.03 Obligation: 35.18/10.03 Q DP problem: 35.18/10.03 The TRS P consists of the following rules: 35.18/10.03 35.18/10.03 B(a(a(x1))) -> B(x1) 35.18/10.03 35.18/10.03 R is empty. 35.18/10.03 Q is empty. 35.18/10.03 We have to consider all minimal (P,Q,R)-chains. 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (11) QDPSizeChangeProof (EQUIVALENT) 35.18/10.03 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. 35.18/10.03 35.18/10.03 From the DPs we obtained the following set of size-change graphs: 35.18/10.03 *B(a(a(x1))) -> B(x1) 35.18/10.03 The graph contains the following edges 1 > 1 35.18/10.03 35.18/10.03 35.18/10.03 ---------------------------------------- 35.18/10.03 35.18/10.03 (12) 35.18/10.03 YES 35.56/10.11 EOF