20.64/6.17 YES 20.64/6.19 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 20.64/6.19 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 20.64/6.19 20.64/6.19 20.64/6.19 Termination w.r.t. Q of the given QTRS could be proven: 20.64/6.19 20.64/6.19 (0) QTRS 20.64/6.19 (1) QTRS Reverse [EQUIVALENT, 0 ms] 20.64/6.19 (2) QTRS 20.64/6.19 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 20.64/6.19 (4) QDP 20.64/6.19 (5) DependencyGraphProof [EQUIVALENT, 1 ms] 20.64/6.19 (6) QDP 20.64/6.19 (7) QDPOrderProof [EQUIVALENT, 164 ms] 20.64/6.19 (8) QDP 20.64/6.19 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 20.64/6.19 (10) QDP 20.64/6.19 (11) UsableRulesProof [EQUIVALENT, 0 ms] 20.64/6.19 (12) QDP 20.64/6.19 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 20.64/6.19 (14) YES 20.64/6.19 20.64/6.19 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (0) 20.64/6.19 Obligation: 20.64/6.19 Q restricted rewrite system: 20.64/6.19 The TRS R consists of the following rules: 20.64/6.19 20.64/6.19 a(x1) -> x1 20.64/6.19 a(a(x1)) -> a(b(c(b(a(x1))))) 20.64/6.19 b(c(x1)) -> c(a(x1)) 20.64/6.19 20.64/6.19 Q is empty. 20.64/6.19 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (1) QTRS Reverse (EQUIVALENT) 20.64/6.19 We applied the QTRS Reverse Processor [REVERSE]. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (2) 20.64/6.19 Obligation: 20.64/6.19 Q restricted rewrite system: 20.64/6.19 The TRS R consists of the following rules: 20.64/6.19 20.64/6.19 a(x1) -> x1 20.64/6.19 a(a(x1)) -> a(b(c(b(a(x1))))) 20.64/6.19 c(b(x1)) -> a(c(x1)) 20.64/6.19 20.64/6.19 Q is empty. 20.64/6.19 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (3) DependencyPairsProof (EQUIVALENT) 20.64/6.19 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (4) 20.64/6.19 Obligation: 20.64/6.19 Q DP problem: 20.64/6.19 The TRS P consists of the following rules: 20.64/6.19 20.64/6.19 A(a(x1)) -> A(b(c(b(a(x1))))) 20.64/6.19 A(a(x1)) -> C(b(a(x1))) 20.64/6.19 C(b(x1)) -> A(c(x1)) 20.64/6.19 C(b(x1)) -> C(x1) 20.64/6.19 20.64/6.19 The TRS R consists of the following rules: 20.64/6.19 20.64/6.19 a(x1) -> x1 20.64/6.19 a(a(x1)) -> a(b(c(b(a(x1))))) 20.64/6.19 c(b(x1)) -> a(c(x1)) 20.64/6.19 20.64/6.19 Q is empty. 20.64/6.19 We have to consider all minimal (P,Q,R)-chains. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (5) DependencyGraphProof (EQUIVALENT) 20.64/6.19 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (6) 20.64/6.19 Obligation: 20.64/6.19 Q DP problem: 20.64/6.19 The TRS P consists of the following rules: 20.64/6.19 20.64/6.19 A(a(x1)) -> C(b(a(x1))) 20.64/6.19 C(b(x1)) -> A(c(x1)) 20.64/6.19 C(b(x1)) -> C(x1) 20.64/6.19 20.64/6.19 The TRS R consists of the following rules: 20.64/6.19 20.64/6.19 a(x1) -> x1 20.64/6.19 a(a(x1)) -> a(b(c(b(a(x1))))) 20.64/6.19 c(b(x1)) -> a(c(x1)) 20.64/6.19 20.64/6.19 Q is empty. 20.64/6.19 We have to consider all minimal (P,Q,R)-chains. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (7) QDPOrderProof (EQUIVALENT) 20.64/6.19 We use the reduction pair processor [LPAR04,JAR06]. 20.64/6.19 20.64/6.19 20.64/6.19 The following pairs can be oriented strictly and are deleted. 20.64/6.19 20.64/6.19 A(a(x1)) -> C(b(a(x1))) 20.64/6.19 The remaining pairs can at least be oriented weakly. 20.64/6.19 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 20.64/6.19 20.64/6.19 <<< 20.64/6.19 POL(A(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 20.64/6.19 >>> 20.64/6.19 20.64/6.19 <<< 20.64/6.19 POL(a(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [1A, 1A, 1A], [0A, 0A, 0A]] * x_1 20.64/6.19 >>> 20.64/6.19 20.64/6.19 <<< 20.64/6.19 POL(C(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 20.64/6.19 >>> 20.64/6.19 20.64/6.19 <<< 20.64/6.19 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [1A, 0A, 1A]] * x_1 20.64/6.19 >>> 20.64/6.19 20.64/6.19 <<< 20.64/6.19 POL(c(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [0A, -I, -I]] * x_1 20.64/6.19 >>> 20.64/6.19 20.64/6.19 20.64/6.19 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 20.64/6.19 20.64/6.19 a(x1) -> x1 20.64/6.19 a(a(x1)) -> a(b(c(b(a(x1))))) 20.64/6.19 c(b(x1)) -> a(c(x1)) 20.64/6.19 20.64/6.19 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (8) 20.64/6.19 Obligation: 20.64/6.19 Q DP problem: 20.64/6.19 The TRS P consists of the following rules: 20.64/6.19 20.64/6.19 C(b(x1)) -> A(c(x1)) 20.64/6.19 C(b(x1)) -> C(x1) 20.64/6.19 20.64/6.19 The TRS R consists of the following rules: 20.64/6.19 20.64/6.19 a(x1) -> x1 20.64/6.19 a(a(x1)) -> a(b(c(b(a(x1))))) 20.64/6.19 c(b(x1)) -> a(c(x1)) 20.64/6.19 20.64/6.19 Q is empty. 20.64/6.19 We have to consider all minimal (P,Q,R)-chains. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (9) DependencyGraphProof (EQUIVALENT) 20.64/6.19 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (10) 20.64/6.19 Obligation: 20.64/6.19 Q DP problem: 20.64/6.19 The TRS P consists of the following rules: 20.64/6.19 20.64/6.19 C(b(x1)) -> C(x1) 20.64/6.19 20.64/6.19 The TRS R consists of the following rules: 20.64/6.19 20.64/6.19 a(x1) -> x1 20.64/6.19 a(a(x1)) -> a(b(c(b(a(x1))))) 20.64/6.19 c(b(x1)) -> a(c(x1)) 20.64/6.19 20.64/6.19 Q is empty. 20.64/6.19 We have to consider all minimal (P,Q,R)-chains. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (11) UsableRulesProof (EQUIVALENT) 20.64/6.19 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. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (12) 20.64/6.19 Obligation: 20.64/6.19 Q DP problem: 20.64/6.19 The TRS P consists of the following rules: 20.64/6.19 20.64/6.19 C(b(x1)) -> C(x1) 20.64/6.19 20.64/6.19 R is empty. 20.64/6.19 Q is empty. 20.64/6.19 We have to consider all minimal (P,Q,R)-chains. 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (13) QDPSizeChangeProof (EQUIVALENT) 20.64/6.19 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. 20.64/6.19 20.64/6.19 From the DPs we obtained the following set of size-change graphs: 20.64/6.19 *C(b(x1)) -> C(x1) 20.64/6.19 The graph contains the following edges 1 > 1 20.64/6.19 20.64/6.19 20.64/6.19 ---------------------------------------- 20.64/6.19 20.64/6.19 (14) 20.64/6.19 YES 21.11/6.29 EOF