123.14/32.73 YES 123.29/32.75 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 123.29/32.75 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 123.29/32.75 123.29/32.75 123.29/32.75 Termination w.r.t. Q of the given QTRS could be proven: 123.29/32.75 123.29/32.75 (0) QTRS 123.29/32.75 (1) QTRS Reverse [EQUIVALENT, 0 ms] 123.29/32.75 (2) QTRS 123.29/32.75 (3) DependencyPairsProof [EQUIVALENT, 1 ms] 123.29/32.75 (4) QDP 123.29/32.75 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 123.29/32.75 (6) QDP 123.29/32.75 (7) QDPOrderProof [EQUIVALENT, 7505 ms] 123.29/32.75 (8) QDP 123.29/32.75 (9) UsableRulesProof [EQUIVALENT, 0 ms] 123.29/32.75 (10) QDP 123.29/32.75 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 123.29/32.75 (12) YES 123.29/32.75 123.29/32.75 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (0) 123.29/32.75 Obligation: 123.29/32.75 Q restricted rewrite system: 123.29/32.75 The TRS R consists of the following rules: 123.29/32.75 123.29/32.75 b(a(b(a(b(b(a(b(a(b(a(x1))))))))))) -> a(b(a(b(a(b(b(a(b(a(b(b(a(x1))))))))))))) 123.29/32.75 123.29/32.75 Q is empty. 123.29/32.75 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (1) QTRS Reverse (EQUIVALENT) 123.29/32.75 We applied the QTRS Reverse Processor [REVERSE]. 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (2) 123.29/32.75 Obligation: 123.29/32.75 Q restricted rewrite system: 123.29/32.75 The TRS R consists of the following rules: 123.29/32.75 123.29/32.75 a(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> a(b(b(a(b(a(b(b(a(b(a(b(a(x1))))))))))))) 123.29/32.75 123.29/32.75 Q is empty. 123.29/32.75 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (3) DependencyPairsProof (EQUIVALENT) 123.29/32.75 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (4) 123.29/32.75 Obligation: 123.29/32.75 Q DP problem: 123.29/32.75 The TRS P consists of the following rules: 123.29/32.75 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(b(a(b(a(b(b(a(b(a(b(a(x1))))))))))))) 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(a(b(b(a(b(a(b(a(x1)))))))))) 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(b(a(b(a(b(a(x1)))))))) 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(a(b(a(x1))))) 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(a(x1))) 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(x1) 123.29/32.75 123.29/32.75 The TRS R consists of the following rules: 123.29/32.75 123.29/32.75 a(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> a(b(b(a(b(a(b(b(a(b(a(b(a(x1))))))))))))) 123.29/32.75 123.29/32.75 Q is empty. 123.29/32.75 We have to consider all minimal (P,Q,R)-chains. 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (5) DependencyGraphProof (EQUIVALENT) 123.29/32.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (6) 123.29/32.75 Obligation: 123.29/32.75 Q DP problem: 123.29/32.75 The TRS P consists of the following rules: 123.29/32.75 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(a(x1))) 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(a(b(a(x1))))) 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(x1) 123.29/32.75 123.29/32.75 The TRS R consists of the following rules: 123.29/32.75 123.29/32.75 a(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> a(b(b(a(b(a(b(b(a(b(a(b(a(x1))))))))))))) 123.29/32.75 123.29/32.75 Q is empty. 123.29/32.75 We have to consider all minimal (P,Q,R)-chains. 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (7) QDPOrderProof (EQUIVALENT) 123.29/32.75 We use the reduction pair processor [LPAR04,JAR06]. 123.29/32.75 123.29/32.75 123.29/32.75 The following pairs can be oriented strictly and are deleted. 123.29/32.75 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(a(x1))) 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(b(a(b(a(x1))))) 123.29/32.75 The remaining pairs can at least be oriented weakly. 123.29/32.75 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 123.29/32.75 123.29/32.75 <<< 123.29/32.75 POL(A(x_1)) = [[0A]] + [[1A, 0A, 0A]] * x_1 123.29/32.75 >>> 123.29/32.75 123.29/32.75 <<< 123.29/32.75 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, 1A, -I], [-I, 0A, 0A], [0A, 0A, -I]] * x_1 123.29/32.75 >>> 123.29/32.75 123.29/32.75 <<< 123.29/32.75 POL(a(x_1)) = [[1A], [-I], [0A]] + [[-I, 0A, 0A], [-I, -I, -I], [0A, -I, 0A]] * x_1 123.29/32.75 >>> 123.29/32.75 123.29/32.75 123.29/32.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 123.29/32.75 123.29/32.75 a(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> a(b(b(a(b(a(b(b(a(b(a(b(a(x1))))))))))))) 123.29/32.75 123.29/32.75 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (8) 123.29/32.75 Obligation: 123.29/32.75 Q DP problem: 123.29/32.75 The TRS P consists of the following rules: 123.29/32.75 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(x1) 123.29/32.75 123.29/32.75 The TRS R consists of the following rules: 123.29/32.75 123.29/32.75 a(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> a(b(b(a(b(a(b(b(a(b(a(b(a(x1))))))))))))) 123.29/32.75 123.29/32.75 Q is empty. 123.29/32.75 We have to consider all minimal (P,Q,R)-chains. 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (9) UsableRulesProof (EQUIVALENT) 123.29/32.75 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. 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (10) 123.29/32.75 Obligation: 123.29/32.75 Q DP problem: 123.29/32.75 The TRS P consists of the following rules: 123.29/32.75 123.29/32.75 A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(x1) 123.29/32.75 123.29/32.75 R is empty. 123.29/32.75 Q is empty. 123.29/32.75 We have to consider all minimal (P,Q,R)-chains. 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (11) QDPSizeChangeProof (EQUIVALENT) 123.29/32.75 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. 123.29/32.75 123.29/32.75 From the DPs we obtained the following set of size-change graphs: 123.29/32.75 *A(b(a(b(a(b(b(a(b(a(b(x1))))))))))) -> A(x1) 123.29/32.75 The graph contains the following edges 1 > 1 123.29/32.75 123.29/32.75 123.29/32.75 ---------------------------------------- 123.29/32.75 123.29/32.75 (12) 123.29/32.75 YES 123.29/32.82 EOF