22.17/6.53 YES 24.03/7.15 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.03/7.15 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.03/7.15 24.03/7.15 24.03/7.15 Termination w.r.t. Q of the given QTRS could be proven: 24.03/7.15 24.03/7.15 (0) QTRS 24.03/7.15 (1) QTRS Reverse [EQUIVALENT, 0 ms] 24.03/7.15 (2) QTRS 24.03/7.15 (3) DependencyPairsProof [EQUIVALENT, 2 ms] 24.03/7.15 (4) QDP 24.03/7.15 (5) DependencyGraphProof [EQUIVALENT, 1 ms] 24.03/7.15 (6) QDP 24.03/7.15 (7) QDPOrderProof [EQUIVALENT, 92 ms] 24.03/7.15 (8) QDP 24.03/7.15 (9) UsableRulesProof [EQUIVALENT, 0 ms] 24.03/7.15 (10) QDP 24.03/7.15 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 24.03/7.15 (12) YES 24.03/7.15 24.03/7.15 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (0) 24.03/7.15 Obligation: 24.03/7.15 Q restricted rewrite system: 24.03/7.15 The TRS R consists of the following rules: 24.03/7.15 24.03/7.15 a(x1) -> b(x1) 24.03/7.15 a(c(x1)) -> c(c(a(a(b(x1))))) 24.03/7.15 b(b(x1)) -> x1 24.03/7.15 24.03/7.15 Q is empty. 24.03/7.15 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (1) QTRS Reverse (EQUIVALENT) 24.03/7.15 We applied the QTRS Reverse Processor [REVERSE]. 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (2) 24.03/7.15 Obligation: 24.03/7.15 Q restricted rewrite system: 24.03/7.15 The TRS R consists of the following rules: 24.03/7.15 24.03/7.15 a(x1) -> b(x1) 24.03/7.15 c(a(x1)) -> b(a(a(c(c(x1))))) 24.03/7.15 b(b(x1)) -> x1 24.03/7.15 24.03/7.15 Q is empty. 24.03/7.15 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (3) DependencyPairsProof (EQUIVALENT) 24.03/7.15 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (4) 24.03/7.15 Obligation: 24.03/7.15 Q DP problem: 24.03/7.15 The TRS P consists of the following rules: 24.03/7.15 24.03/7.15 A(x1) -> B(x1) 24.03/7.15 C(a(x1)) -> B(a(a(c(c(x1))))) 24.03/7.15 C(a(x1)) -> A(a(c(c(x1)))) 24.03/7.15 C(a(x1)) -> A(c(c(x1))) 24.03/7.15 C(a(x1)) -> C(c(x1)) 24.03/7.15 C(a(x1)) -> C(x1) 24.03/7.15 24.03/7.15 The TRS R consists of the following rules: 24.03/7.15 24.03/7.15 a(x1) -> b(x1) 24.03/7.15 c(a(x1)) -> b(a(a(c(c(x1))))) 24.03/7.15 b(b(x1)) -> x1 24.03/7.15 24.03/7.15 Q is empty. 24.03/7.15 We have to consider all minimal (P,Q,R)-chains. 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (5) DependencyGraphProof (EQUIVALENT) 24.03/7.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (6) 24.03/7.15 Obligation: 24.03/7.15 Q DP problem: 24.03/7.15 The TRS P consists of the following rules: 24.03/7.15 24.03/7.15 C(a(x1)) -> C(x1) 24.03/7.15 C(a(x1)) -> C(c(x1)) 24.03/7.15 24.03/7.15 The TRS R consists of the following rules: 24.03/7.15 24.03/7.15 a(x1) -> b(x1) 24.03/7.15 c(a(x1)) -> b(a(a(c(c(x1))))) 24.03/7.15 b(b(x1)) -> x1 24.03/7.15 24.03/7.15 Q is empty. 24.03/7.15 We have to consider all minimal (P,Q,R)-chains. 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (7) QDPOrderProof (EQUIVALENT) 24.03/7.15 We use the reduction pair processor [LPAR04,JAR06]. 24.03/7.15 24.03/7.15 24.03/7.15 The following pairs can be oriented strictly and are deleted. 24.03/7.15 24.03/7.15 C(a(x1)) -> C(c(x1)) 24.03/7.15 The remaining pairs can at least be oriented weakly. 24.03/7.15 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 24.03/7.15 24.03/7.15 <<< 24.03/7.15 POL(C(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 24.03/7.15 >>> 24.03/7.15 24.03/7.15 <<< 24.03/7.15 POL(a(x_1)) = [[0A], [0A], [0A]] + [[1A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 24.03/7.15 >>> 24.03/7.15 24.03/7.15 <<< 24.03/7.15 POL(c(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [1A, 0A, 0A], [0A, -I, -I]] * x_1 24.03/7.15 >>> 24.03/7.15 24.03/7.15 <<< 24.03/7.15 POL(b(x_1)) = [[0A], [-I], [-I]] + [[-I, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 24.03/7.15 >>> 24.03/7.15 24.03/7.15 24.03/7.15 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.03/7.15 24.03/7.15 c(a(x1)) -> b(a(a(c(c(x1))))) 24.03/7.15 a(x1) -> b(x1) 24.03/7.15 b(b(x1)) -> x1 24.03/7.15 24.03/7.15 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (8) 24.03/7.15 Obligation: 24.03/7.15 Q DP problem: 24.03/7.15 The TRS P consists of the following rules: 24.03/7.15 24.03/7.15 C(a(x1)) -> C(x1) 24.03/7.15 24.03/7.15 The TRS R consists of the following rules: 24.03/7.15 24.03/7.15 a(x1) -> b(x1) 24.03/7.15 c(a(x1)) -> b(a(a(c(c(x1))))) 24.03/7.15 b(b(x1)) -> x1 24.03/7.15 24.03/7.15 Q is empty. 24.03/7.15 We have to consider all minimal (P,Q,R)-chains. 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (9) UsableRulesProof (EQUIVALENT) 24.03/7.15 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. 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (10) 24.03/7.15 Obligation: 24.03/7.15 Q DP problem: 24.03/7.15 The TRS P consists of the following rules: 24.03/7.15 24.03/7.15 C(a(x1)) -> C(x1) 24.03/7.15 24.03/7.15 R is empty. 24.03/7.15 Q is empty. 24.03/7.15 We have to consider all minimal (P,Q,R)-chains. 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (11) QDPSizeChangeProof (EQUIVALENT) 24.03/7.15 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. 24.03/7.15 24.03/7.15 From the DPs we obtained the following set of size-change graphs: 24.03/7.15 *C(a(x1)) -> C(x1) 24.03/7.15 The graph contains the following edges 1 > 1 24.03/7.15 24.03/7.15 24.03/7.15 ---------------------------------------- 24.03/7.15 24.03/7.15 (12) 24.03/7.15 YES 24.55/9.41 EOF