17.70/5.99 YES 18.01/6.05 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 18.01/6.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 18.01/6.05 18.01/6.05 18.01/6.05 Termination w.r.t. Q of the given QTRS could be proven: 18.01/6.05 18.01/6.05 (0) QTRS 18.01/6.05 (1) DependencyPairsProof [EQUIVALENT, 0 ms] 18.01/6.05 (2) QDP 18.01/6.05 (3) DependencyGraphProof [EQUIVALENT, 1 ms] 18.01/6.05 (4) QDP 18.01/6.05 (5) QDPOrderProof [EQUIVALENT, 146 ms] 18.01/6.05 (6) QDP 18.01/6.05 (7) QDPOrderProof [EQUIVALENT, 49 ms] 18.01/6.05 (8) QDP 18.01/6.05 (9) UsableRulesProof [EQUIVALENT, 2 ms] 18.01/6.05 (10) QDP 18.01/6.05 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 18.01/6.05 (12) YES 18.01/6.05 18.01/6.05 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (0) 18.01/6.05 Obligation: 18.01/6.05 Q restricted rewrite system: 18.01/6.05 The TRS R consists of the following rules: 18.01/6.05 18.01/6.05 a(x1) -> b(x1) 18.01/6.05 a(b(c(x1))) -> c(a(c(a(a(x1))))) 18.01/6.05 c(c(x1)) -> x1 18.01/6.05 18.01/6.05 Q is empty. 18.01/6.05 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (1) DependencyPairsProof (EQUIVALENT) 18.01/6.05 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (2) 18.01/6.05 Obligation: 18.01/6.05 Q DP problem: 18.01/6.05 The TRS P consists of the following rules: 18.01/6.05 18.01/6.05 A(b(c(x1))) -> C(a(c(a(a(x1))))) 18.01/6.05 A(b(c(x1))) -> A(c(a(a(x1)))) 18.01/6.05 A(b(c(x1))) -> C(a(a(x1))) 18.01/6.05 A(b(c(x1))) -> A(a(x1)) 18.01/6.05 A(b(c(x1))) -> A(x1) 18.01/6.05 18.01/6.05 The TRS R consists of the following rules: 18.01/6.05 18.01/6.05 a(x1) -> b(x1) 18.01/6.05 a(b(c(x1))) -> c(a(c(a(a(x1))))) 18.01/6.05 c(c(x1)) -> x1 18.01/6.05 18.01/6.05 Q is empty. 18.01/6.05 We have to consider all minimal (P,Q,R)-chains. 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (3) DependencyGraphProof (EQUIVALENT) 18.01/6.05 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (4) 18.01/6.05 Obligation: 18.01/6.05 Q DP problem: 18.01/6.05 The TRS P consists of the following rules: 18.01/6.05 18.01/6.05 A(b(c(x1))) -> A(a(x1)) 18.01/6.05 A(b(c(x1))) -> A(c(a(a(x1)))) 18.01/6.05 A(b(c(x1))) -> A(x1) 18.01/6.05 18.01/6.05 The TRS R consists of the following rules: 18.01/6.05 18.01/6.05 a(x1) -> b(x1) 18.01/6.05 a(b(c(x1))) -> c(a(c(a(a(x1))))) 18.01/6.05 c(c(x1)) -> x1 18.01/6.05 18.01/6.05 Q is empty. 18.01/6.05 We have to consider all minimal (P,Q,R)-chains. 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (5) QDPOrderProof (EQUIVALENT) 18.01/6.05 We use the reduction pair processor [LPAR04,JAR06]. 18.01/6.05 18.01/6.05 18.01/6.05 The following pairs can be oriented strictly and are deleted. 18.01/6.05 18.01/6.05 A(b(c(x1))) -> A(c(a(a(x1)))) 18.01/6.05 The remaining pairs can at least be oriented weakly. 18.01/6.05 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 18.01/6.05 18.01/6.05 <<< 18.01/6.05 POL(A(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 18.01/6.05 >>> 18.01/6.05 18.01/6.05 <<< 18.01/6.05 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [-I, -I, 0A], [-I, 0A, -I]] * x_1 18.01/6.05 >>> 18.01/6.05 18.01/6.05 <<< 18.01/6.05 POL(c(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 18.01/6.05 >>> 18.01/6.05 18.01/6.05 <<< 18.01/6.05 POL(a(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 18.01/6.05 >>> 18.01/6.05 18.01/6.05 18.01/6.05 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 18.01/6.05 18.01/6.05 a(x1) -> b(x1) 18.01/6.05 a(b(c(x1))) -> c(a(c(a(a(x1))))) 18.01/6.05 c(c(x1)) -> x1 18.01/6.05 18.01/6.05 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (6) 18.01/6.05 Obligation: 18.01/6.05 Q DP problem: 18.01/6.05 The TRS P consists of the following rules: 18.01/6.05 18.01/6.05 A(b(c(x1))) -> A(a(x1)) 18.01/6.05 A(b(c(x1))) -> A(x1) 18.01/6.05 18.01/6.05 The TRS R consists of the following rules: 18.01/6.05 18.01/6.05 a(x1) -> b(x1) 18.01/6.05 a(b(c(x1))) -> c(a(c(a(a(x1))))) 18.01/6.05 c(c(x1)) -> x1 18.01/6.05 18.01/6.05 Q is empty. 18.01/6.05 We have to consider all minimal (P,Q,R)-chains. 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (7) QDPOrderProof (EQUIVALENT) 18.01/6.05 We use the reduction pair processor [LPAR04,JAR06]. 18.01/6.05 18.01/6.05 18.01/6.05 The following pairs can be oriented strictly and are deleted. 18.01/6.05 18.01/6.05 A(b(c(x1))) -> A(a(x1)) 18.01/6.05 The remaining pairs can at least be oriented weakly. 18.01/6.05 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 18.01/6.05 18.01/6.05 <<< 18.01/6.05 POL(A(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 18.01/6.05 >>> 18.01/6.05 18.01/6.05 <<< 18.01/6.05 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [-I, 0A, -I]] * x_1 18.01/6.05 >>> 18.01/6.05 18.01/6.05 <<< 18.01/6.05 POL(c(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, -I], [1A, 1A, 0A], [0A, 0A, -I]] * x_1 18.01/6.05 >>> 18.01/6.05 18.01/6.05 <<< 18.01/6.05 POL(a(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 18.01/6.05 >>> 18.01/6.05 18.01/6.05 18.01/6.05 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 18.01/6.05 18.01/6.05 a(x1) -> b(x1) 18.01/6.05 a(b(c(x1))) -> c(a(c(a(a(x1))))) 18.01/6.05 c(c(x1)) -> x1 18.01/6.05 18.01/6.05 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (8) 18.01/6.05 Obligation: 18.01/6.05 Q DP problem: 18.01/6.05 The TRS P consists of the following rules: 18.01/6.05 18.01/6.05 A(b(c(x1))) -> A(x1) 18.01/6.05 18.01/6.05 The TRS R consists of the following rules: 18.01/6.05 18.01/6.05 a(x1) -> b(x1) 18.01/6.05 a(b(c(x1))) -> c(a(c(a(a(x1))))) 18.01/6.05 c(c(x1)) -> x1 18.01/6.05 18.01/6.05 Q is empty. 18.01/6.05 We have to consider all minimal (P,Q,R)-chains. 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (9) UsableRulesProof (EQUIVALENT) 18.01/6.05 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. 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (10) 18.01/6.05 Obligation: 18.01/6.05 Q DP problem: 18.01/6.05 The TRS P consists of the following rules: 18.01/6.05 18.01/6.05 A(b(c(x1))) -> A(x1) 18.01/6.05 18.01/6.05 R is empty. 18.01/6.05 Q is empty. 18.01/6.05 We have to consider all minimal (P,Q,R)-chains. 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (11) QDPSizeChangeProof (EQUIVALENT) 18.01/6.05 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. 18.01/6.05 18.01/6.05 From the DPs we obtained the following set of size-change graphs: 18.01/6.05 *A(b(c(x1))) -> A(x1) 18.01/6.05 The graph contains the following edges 1 > 1 18.01/6.05 18.01/6.05 18.01/6.05 ---------------------------------------- 18.01/6.05 18.01/6.05 (12) 18.01/6.05 YES 18.12/6.11 EOF