23.98/7.08 YES 24.29/7.09 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 24.29/7.09 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.29/7.09 24.29/7.09 24.29/7.09 Termination w.r.t. Q of the given QTRS could be proven: 24.29/7.09 24.29/7.09 (0) QTRS 24.29/7.09 (1) QTRS Reverse [EQUIVALENT, 0 ms] 24.29/7.09 (2) QTRS 24.29/7.09 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 24.29/7.09 (4) QDP 24.29/7.09 (5) QDPOrderProof [EQUIVALENT, 33 ms] 24.29/7.09 (6) QDP 24.29/7.09 (7) DependencyGraphProof [EQUIVALENT, 3 ms] 24.29/7.09 (8) AND 24.29/7.09 (9) QDP 24.29/7.09 (10) UsableRulesProof [EQUIVALENT, 0 ms] 24.29/7.09 (11) QDP 24.29/7.09 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 24.29/7.09 (13) YES 24.29/7.09 (14) QDP 24.29/7.09 (15) UsableRulesProof [EQUIVALENT, 0 ms] 24.29/7.09 (16) QDP 24.29/7.09 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 24.29/7.09 (18) YES 24.29/7.09 24.29/7.09 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (0) 24.29/7.09 Obligation: 24.29/7.09 Q restricted rewrite system: 24.29/7.09 The TRS R consists of the following rules: 24.29/7.09 24.29/7.09 a(b(x1)) -> b(c(a(x1))) 24.29/7.09 b(c(x1)) -> c(b(b(x1))) 24.29/7.09 c(a(x1)) -> a(c(x1)) 24.29/7.09 24.29/7.09 Q is empty. 24.29/7.09 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (1) QTRS Reverse (EQUIVALENT) 24.29/7.09 We applied the QTRS Reverse Processor [REVERSE]. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (2) 24.29/7.09 Obligation: 24.29/7.09 Q restricted rewrite system: 24.29/7.09 The TRS R consists of the following rules: 24.29/7.09 24.29/7.09 b(a(x1)) -> a(c(b(x1))) 24.29/7.09 c(b(x1)) -> b(b(c(x1))) 24.29/7.09 a(c(x1)) -> c(a(x1)) 24.29/7.09 24.29/7.09 Q is empty. 24.29/7.09 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (3) DependencyPairsProof (EQUIVALENT) 24.29/7.09 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (4) 24.29/7.09 Obligation: 24.29/7.09 Q DP problem: 24.29/7.09 The TRS P consists of the following rules: 24.29/7.09 24.29/7.09 B(a(x1)) -> A(c(b(x1))) 24.29/7.09 B(a(x1)) -> C(b(x1)) 24.29/7.09 B(a(x1)) -> B(x1) 24.29/7.09 C(b(x1)) -> B(b(c(x1))) 24.29/7.09 C(b(x1)) -> B(c(x1)) 24.29/7.09 C(b(x1)) -> C(x1) 24.29/7.09 A(c(x1)) -> C(a(x1)) 24.29/7.09 A(c(x1)) -> A(x1) 24.29/7.09 24.29/7.09 The TRS R consists of the following rules: 24.29/7.09 24.29/7.09 b(a(x1)) -> a(c(b(x1))) 24.29/7.09 c(b(x1)) -> b(b(c(x1))) 24.29/7.09 a(c(x1)) -> c(a(x1)) 24.29/7.09 24.29/7.09 Q is empty. 24.29/7.09 We have to consider all minimal (P,Q,R)-chains. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (5) QDPOrderProof (EQUIVALENT) 24.29/7.09 We use the reduction pair processor [LPAR04,JAR06]. 24.29/7.09 24.29/7.09 24.29/7.09 The following pairs can be oriented strictly and are deleted. 24.29/7.09 24.29/7.09 B(a(x1)) -> C(b(x1)) 24.29/7.09 B(a(x1)) -> B(x1) 24.29/7.09 A(c(x1)) -> C(a(x1)) 24.29/7.09 The remaining pairs can at least be oriented weakly. 24.29/7.09 Used ordering: Polynomial interpretation [POLO]: 24.29/7.09 24.29/7.09 POL(A(x_1)) = 1 24.29/7.09 POL(B(x_1)) = x_1 24.29/7.09 POL(C(x_1)) = 0 24.29/7.09 POL(a(x_1)) = 1 + x_1 24.29/7.09 POL(b(x_1)) = x_1 24.29/7.09 POL(c(x_1)) = 0 24.29/7.09 24.29/7.09 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.29/7.09 24.29/7.09 a(c(x1)) -> c(a(x1)) 24.29/7.09 c(b(x1)) -> b(b(c(x1))) 24.29/7.09 b(a(x1)) -> a(c(b(x1))) 24.29/7.09 24.29/7.09 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (6) 24.29/7.09 Obligation: 24.29/7.09 Q DP problem: 24.29/7.09 The TRS P consists of the following rules: 24.29/7.09 24.29/7.09 B(a(x1)) -> A(c(b(x1))) 24.29/7.09 C(b(x1)) -> B(b(c(x1))) 24.29/7.09 C(b(x1)) -> B(c(x1)) 24.29/7.09 C(b(x1)) -> C(x1) 24.29/7.09 A(c(x1)) -> A(x1) 24.29/7.09 24.29/7.09 The TRS R consists of the following rules: 24.29/7.09 24.29/7.09 b(a(x1)) -> a(c(b(x1))) 24.29/7.09 c(b(x1)) -> b(b(c(x1))) 24.29/7.09 a(c(x1)) -> c(a(x1)) 24.29/7.09 24.29/7.09 Q is empty. 24.29/7.09 We have to consider all minimal (P,Q,R)-chains. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (7) DependencyGraphProof (EQUIVALENT) 24.29/7.09 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (8) 24.29/7.09 Complex Obligation (AND) 24.29/7.09 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (9) 24.29/7.09 Obligation: 24.29/7.09 Q DP problem: 24.29/7.09 The TRS P consists of the following rules: 24.29/7.09 24.29/7.09 A(c(x1)) -> A(x1) 24.29/7.09 24.29/7.09 The TRS R consists of the following rules: 24.29/7.09 24.29/7.09 b(a(x1)) -> a(c(b(x1))) 24.29/7.09 c(b(x1)) -> b(b(c(x1))) 24.29/7.09 a(c(x1)) -> c(a(x1)) 24.29/7.09 24.29/7.09 Q is empty. 24.29/7.09 We have to consider all minimal (P,Q,R)-chains. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (10) UsableRulesProof (EQUIVALENT) 24.29/7.09 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. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (11) 24.29/7.09 Obligation: 24.29/7.09 Q DP problem: 24.29/7.09 The TRS P consists of the following rules: 24.29/7.09 24.29/7.09 A(c(x1)) -> A(x1) 24.29/7.09 24.29/7.09 R is empty. 24.29/7.09 Q is empty. 24.29/7.09 We have to consider all minimal (P,Q,R)-chains. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (12) QDPSizeChangeProof (EQUIVALENT) 24.29/7.09 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. 24.29/7.09 24.29/7.09 From the DPs we obtained the following set of size-change graphs: 24.29/7.09 *A(c(x1)) -> A(x1) 24.29/7.09 The graph contains the following edges 1 > 1 24.29/7.09 24.29/7.09 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (13) 24.29/7.09 YES 24.29/7.09 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (14) 24.29/7.09 Obligation: 24.29/7.09 Q DP problem: 24.29/7.09 The TRS P consists of the following rules: 24.29/7.09 24.29/7.09 C(b(x1)) -> C(x1) 24.29/7.09 24.29/7.09 The TRS R consists of the following rules: 24.29/7.09 24.29/7.09 b(a(x1)) -> a(c(b(x1))) 24.29/7.09 c(b(x1)) -> b(b(c(x1))) 24.29/7.09 a(c(x1)) -> c(a(x1)) 24.29/7.09 24.29/7.09 Q is empty. 24.29/7.09 We have to consider all minimal (P,Q,R)-chains. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (15) UsableRulesProof (EQUIVALENT) 24.29/7.09 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. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (16) 24.29/7.09 Obligation: 24.29/7.09 Q DP problem: 24.29/7.09 The TRS P consists of the following rules: 24.29/7.09 24.29/7.09 C(b(x1)) -> C(x1) 24.29/7.09 24.29/7.09 R is empty. 24.29/7.09 Q is empty. 24.29/7.09 We have to consider all minimal (P,Q,R)-chains. 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (17) QDPSizeChangeProof (EQUIVALENT) 24.29/7.09 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. 24.29/7.09 24.29/7.09 From the DPs we obtained the following set of size-change graphs: 24.29/7.09 *C(b(x1)) -> C(x1) 24.29/7.09 The graph contains the following edges 1 > 1 24.29/7.09 24.29/7.09 24.29/7.09 ---------------------------------------- 24.29/7.09 24.29/7.09 (18) 24.29/7.09 YES 24.39/7.17 EOF