25.16/7.26 YES 26.05/7.46 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 26.05/7.46 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.05/7.46 26.05/7.46 26.05/7.46 Termination w.r.t. Q of the given QTRS could be proven: 26.05/7.46 26.05/7.46 (0) QTRS 26.05/7.46 (1) DependencyPairsProof [EQUIVALENT, 0 ms] 26.05/7.46 (2) QDP 26.05/7.46 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 26.05/7.46 (4) AND 26.05/7.46 (5) QDP 26.05/7.46 (6) UsableRulesProof [EQUIVALENT, 0 ms] 26.05/7.46 (7) QDP 26.05/7.46 (8) QDPSizeChangeProof [EQUIVALENT, 2 ms] 26.05/7.46 (9) YES 26.05/7.46 (10) QDP 26.05/7.46 (11) QDPOrderProof [EQUIVALENT, 144 ms] 26.05/7.46 (12) QDP 26.05/7.46 (13) UsableRulesProof [EQUIVALENT, 0 ms] 26.05/7.46 (14) QDP 26.05/7.46 (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] 26.05/7.46 (16) YES 26.05/7.46 26.05/7.46 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (0) 26.05/7.46 Obligation: 26.05/7.46 Q restricted rewrite system: 26.05/7.46 The TRS R consists of the following rules: 26.05/7.46 26.05/7.46 a(a(x1)) -> b(x1) 26.05/7.46 b(c(x1)) -> a(x1) 26.05/7.46 c(b(x1)) -> a(b(c(c(x1)))) 26.05/7.46 26.05/7.46 Q is empty. 26.05/7.46 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (1) DependencyPairsProof (EQUIVALENT) 26.05/7.46 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (2) 26.05/7.46 Obligation: 26.05/7.46 Q DP problem: 26.05/7.46 The TRS P consists of the following rules: 26.05/7.46 26.05/7.46 A(a(x1)) -> B(x1) 26.05/7.46 B(c(x1)) -> A(x1) 26.05/7.46 C(b(x1)) -> A(b(c(c(x1)))) 26.05/7.46 C(b(x1)) -> B(c(c(x1))) 26.05/7.46 C(b(x1)) -> C(c(x1)) 26.05/7.46 C(b(x1)) -> C(x1) 26.05/7.46 26.05/7.46 The TRS R consists of the following rules: 26.05/7.46 26.05/7.46 a(a(x1)) -> b(x1) 26.05/7.46 b(c(x1)) -> a(x1) 26.05/7.46 c(b(x1)) -> a(b(c(c(x1)))) 26.05/7.46 26.05/7.46 Q is empty. 26.05/7.46 We have to consider all minimal (P,Q,R)-chains. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (3) DependencyGraphProof (EQUIVALENT) 26.05/7.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (4) 26.05/7.46 Complex Obligation (AND) 26.05/7.46 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (5) 26.05/7.46 Obligation: 26.05/7.46 Q DP problem: 26.05/7.46 The TRS P consists of the following rules: 26.05/7.46 26.05/7.46 B(c(x1)) -> A(x1) 26.05/7.46 A(a(x1)) -> B(x1) 26.05/7.46 26.05/7.46 The TRS R consists of the following rules: 26.05/7.46 26.05/7.46 a(a(x1)) -> b(x1) 26.05/7.46 b(c(x1)) -> a(x1) 26.05/7.46 c(b(x1)) -> a(b(c(c(x1)))) 26.05/7.46 26.05/7.46 Q is empty. 26.05/7.46 We have to consider all minimal (P,Q,R)-chains. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (6) UsableRulesProof (EQUIVALENT) 26.05/7.46 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. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (7) 26.05/7.46 Obligation: 26.05/7.46 Q DP problem: 26.05/7.46 The TRS P consists of the following rules: 26.05/7.46 26.05/7.46 B(c(x1)) -> A(x1) 26.05/7.46 A(a(x1)) -> B(x1) 26.05/7.46 26.05/7.46 R is empty. 26.05/7.46 Q is empty. 26.05/7.46 We have to consider all minimal (P,Q,R)-chains. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (8) QDPSizeChangeProof (EQUIVALENT) 26.05/7.46 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. 26.05/7.46 26.05/7.46 From the DPs we obtained the following set of size-change graphs: 26.05/7.46 *A(a(x1)) -> B(x1) 26.05/7.46 The graph contains the following edges 1 > 1 26.05/7.46 26.05/7.46 26.05/7.46 *B(c(x1)) -> A(x1) 26.05/7.46 The graph contains the following edges 1 > 1 26.05/7.46 26.05/7.46 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (9) 26.05/7.46 YES 26.05/7.46 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (10) 26.05/7.46 Obligation: 26.05/7.46 Q DP problem: 26.05/7.46 The TRS P consists of the following rules: 26.05/7.46 26.05/7.46 C(b(x1)) -> C(x1) 26.05/7.46 C(b(x1)) -> C(c(x1)) 26.05/7.46 26.05/7.46 The TRS R consists of the following rules: 26.05/7.46 26.05/7.46 a(a(x1)) -> b(x1) 26.05/7.46 b(c(x1)) -> a(x1) 26.05/7.46 c(b(x1)) -> a(b(c(c(x1)))) 26.05/7.46 26.05/7.46 Q is empty. 26.05/7.46 We have to consider all minimal (P,Q,R)-chains. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (11) QDPOrderProof (EQUIVALENT) 26.05/7.46 We use the reduction pair processor [LPAR04,JAR06]. 26.05/7.46 26.05/7.46 26.05/7.46 The following pairs can be oriented strictly and are deleted. 26.05/7.46 26.05/7.46 C(b(x1)) -> C(c(x1)) 26.05/7.46 The remaining pairs can at least be oriented weakly. 26.05/7.46 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 26.05/7.46 26.05/7.46 <<< 26.05/7.46 POL(C(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 26.05/7.46 >>> 26.05/7.46 26.05/7.46 <<< 26.05/7.46 POL(b(x_1)) = [[1A], [-I], [1A]] + [[1A, 1A, 1A], [0A, 0A, 0A], [1A, 0A, 0A]] * x_1 26.05/7.46 >>> 26.05/7.46 26.05/7.46 <<< 26.05/7.46 POL(c(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] * x_1 26.05/7.46 >>> 26.05/7.46 26.05/7.46 <<< 26.05/7.46 POL(a(x_1)) = [[0A], [0A], [-I]] + [[0A, 1A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 26.05/7.46 >>> 26.05/7.46 26.05/7.46 26.05/7.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.05/7.46 26.05/7.46 c(b(x1)) -> a(b(c(c(x1)))) 26.05/7.46 b(c(x1)) -> a(x1) 26.05/7.46 a(a(x1)) -> b(x1) 26.05/7.46 26.05/7.46 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (12) 26.05/7.46 Obligation: 26.05/7.46 Q DP problem: 26.05/7.46 The TRS P consists of the following rules: 26.05/7.46 26.05/7.46 C(b(x1)) -> C(x1) 26.05/7.46 26.05/7.46 The TRS R consists of the following rules: 26.05/7.46 26.05/7.46 a(a(x1)) -> b(x1) 26.05/7.46 b(c(x1)) -> a(x1) 26.05/7.46 c(b(x1)) -> a(b(c(c(x1)))) 26.05/7.46 26.05/7.46 Q is empty. 26.05/7.46 We have to consider all minimal (P,Q,R)-chains. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (13) UsableRulesProof (EQUIVALENT) 26.05/7.46 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. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (14) 26.05/7.46 Obligation: 26.05/7.46 Q DP problem: 26.05/7.46 The TRS P consists of the following rules: 26.05/7.46 26.05/7.46 C(b(x1)) -> C(x1) 26.05/7.46 26.05/7.46 R is empty. 26.05/7.46 Q is empty. 26.05/7.46 We have to consider all minimal (P,Q,R)-chains. 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (15) QDPSizeChangeProof (EQUIVALENT) 26.05/7.46 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. 26.05/7.46 26.05/7.46 From the DPs we obtained the following set of size-change graphs: 26.05/7.46 *C(b(x1)) -> C(x1) 26.05/7.46 The graph contains the following edges 1 > 1 26.05/7.46 26.05/7.46 26.05/7.46 ---------------------------------------- 26.05/7.46 26.05/7.46 (16) 26.05/7.46 YES 26.31/8.75 EOF