28.84/8.28 YES 29.44/8.43 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 29.44/8.43 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.44/8.43 29.44/8.43 29.44/8.43 Termination w.r.t. Q of the given QTRS could be proven: 29.44/8.43 29.44/8.43 (0) QTRS 29.44/8.43 (1) DependencyPairsProof [EQUIVALENT, 23 ms] 29.44/8.43 (2) QDP 29.44/8.43 (3) DependencyGraphProof [EQUIVALENT, 8 ms] 29.44/8.43 (4) QDP 29.44/8.43 (5) QDPOrderProof [EQUIVALENT, 173 ms] 29.44/8.43 (6) QDP 29.44/8.43 (7) QDPOrderProof [EQUIVALENT, 71 ms] 29.44/8.43 (8) QDP 29.44/8.43 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 29.44/8.43 (10) QDP 29.44/8.43 (11) UsableRulesProof [EQUIVALENT, 0 ms] 29.44/8.43 (12) QDP 29.44/8.43 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 29.44/8.43 (14) YES 29.44/8.43 29.44/8.43 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (0) 29.44/8.43 Obligation: 29.44/8.43 Q restricted rewrite system: 29.44/8.43 The TRS R consists of the following rules: 29.44/8.43 29.44/8.43 a(x1) -> b(c(x1)) 29.44/8.43 b(b(x1)) -> x1 29.44/8.43 c(c(b(x1))) -> a(a(a(c(x1)))) 29.44/8.43 29.44/8.43 Q is empty. 29.44/8.43 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (1) DependencyPairsProof (EQUIVALENT) 29.44/8.43 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (2) 29.44/8.43 Obligation: 29.44/8.43 Q DP problem: 29.44/8.43 The TRS P consists of the following rules: 29.44/8.43 29.44/8.43 A(x1) -> B(c(x1)) 29.44/8.43 A(x1) -> C(x1) 29.44/8.43 C(c(b(x1))) -> A(a(a(c(x1)))) 29.44/8.43 C(c(b(x1))) -> A(a(c(x1))) 29.44/8.43 C(c(b(x1))) -> A(c(x1)) 29.44/8.43 C(c(b(x1))) -> C(x1) 29.44/8.43 29.44/8.43 The TRS R consists of the following rules: 29.44/8.43 29.44/8.43 a(x1) -> b(c(x1)) 29.44/8.43 b(b(x1)) -> x1 29.44/8.43 c(c(b(x1))) -> a(a(a(c(x1)))) 29.44/8.43 29.44/8.43 Q is empty. 29.44/8.43 We have to consider all minimal (P,Q,R)-chains. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (3) DependencyGraphProof (EQUIVALENT) 29.44/8.43 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (4) 29.44/8.43 Obligation: 29.44/8.43 Q DP problem: 29.44/8.43 The TRS P consists of the following rules: 29.44/8.43 29.44/8.43 A(x1) -> C(x1) 29.44/8.43 C(c(b(x1))) -> A(a(a(c(x1)))) 29.44/8.43 C(c(b(x1))) -> A(a(c(x1))) 29.44/8.43 C(c(b(x1))) -> A(c(x1)) 29.44/8.43 C(c(b(x1))) -> C(x1) 29.44/8.43 29.44/8.43 The TRS R consists of the following rules: 29.44/8.43 29.44/8.43 a(x1) -> b(c(x1)) 29.44/8.43 b(b(x1)) -> x1 29.44/8.43 c(c(b(x1))) -> a(a(a(c(x1)))) 29.44/8.43 29.44/8.43 Q is empty. 29.44/8.43 We have to consider all minimal (P,Q,R)-chains. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (5) QDPOrderProof (EQUIVALENT) 29.44/8.43 We use the reduction pair processor [LPAR04,JAR06]. 29.44/8.43 29.44/8.43 29.44/8.43 The following pairs can be oriented strictly and are deleted. 29.44/8.43 29.44/8.43 C(c(b(x1))) -> A(a(a(c(x1)))) 29.44/8.43 C(c(b(x1))) -> A(a(c(x1))) 29.44/8.43 The remaining pairs can at least be oriented weakly. 29.44/8.43 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(A(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(C(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(c(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [-I, -I, 0A], [-I, 0A, -I]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(b(x_1)) = [[0A], [-I], [1A]] + [[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(a(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, -I], [-I, 0A, -I], [0A, 1A, 0A]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 29.44/8.43 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.44/8.43 29.44/8.43 c(c(b(x1))) -> a(a(a(c(x1)))) 29.44/8.43 a(x1) -> b(c(x1)) 29.44/8.43 b(b(x1)) -> x1 29.44/8.43 29.44/8.43 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (6) 29.44/8.43 Obligation: 29.44/8.43 Q DP problem: 29.44/8.43 The TRS P consists of the following rules: 29.44/8.43 29.44/8.43 A(x1) -> C(x1) 29.44/8.43 C(c(b(x1))) -> A(c(x1)) 29.44/8.43 C(c(b(x1))) -> C(x1) 29.44/8.43 29.44/8.43 The TRS R consists of the following rules: 29.44/8.43 29.44/8.43 a(x1) -> b(c(x1)) 29.44/8.43 b(b(x1)) -> x1 29.44/8.43 c(c(b(x1))) -> a(a(a(c(x1)))) 29.44/8.43 29.44/8.43 Q is empty. 29.44/8.43 We have to consider all minimal (P,Q,R)-chains. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (7) QDPOrderProof (EQUIVALENT) 29.44/8.43 We use the reduction pair processor [LPAR04,JAR06]. 29.44/8.43 29.44/8.43 29.44/8.43 The following pairs can be oriented strictly and are deleted. 29.44/8.43 29.44/8.43 A(x1) -> C(x1) 29.44/8.43 The remaining pairs can at least be oriented weakly. 29.44/8.43 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(A(x_1)) = [[1A]] + [[1A, 0A, 1A]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(C(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(c(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(b(x_1)) = [[-I], [1A], [-I]] + [[0A, -I, -I], [1A, 1A, 0A], [0A, 0A, -I]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 <<< 29.44/8.43 POL(a(x_1)) = [[-I], [1A], [-I]] + [[0A, -I, -I], [1A, 0A, 1A], [0A, -I, 0A]] * x_1 29.44/8.43 >>> 29.44/8.43 29.44/8.43 29.44/8.43 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.44/8.43 29.44/8.43 c(c(b(x1))) -> a(a(a(c(x1)))) 29.44/8.43 a(x1) -> b(c(x1)) 29.44/8.43 b(b(x1)) -> x1 29.44/8.43 29.44/8.43 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (8) 29.44/8.43 Obligation: 29.44/8.43 Q DP problem: 29.44/8.43 The TRS P consists of the following rules: 29.44/8.43 29.44/8.43 C(c(b(x1))) -> A(c(x1)) 29.44/8.43 C(c(b(x1))) -> C(x1) 29.44/8.43 29.44/8.43 The TRS R consists of the following rules: 29.44/8.43 29.44/8.43 a(x1) -> b(c(x1)) 29.44/8.43 b(b(x1)) -> x1 29.44/8.43 c(c(b(x1))) -> a(a(a(c(x1)))) 29.44/8.43 29.44/8.43 Q is empty. 29.44/8.43 We have to consider all minimal (P,Q,R)-chains. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (9) DependencyGraphProof (EQUIVALENT) 29.44/8.43 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (10) 29.44/8.43 Obligation: 29.44/8.43 Q DP problem: 29.44/8.43 The TRS P consists of the following rules: 29.44/8.43 29.44/8.43 C(c(b(x1))) -> C(x1) 29.44/8.43 29.44/8.43 The TRS R consists of the following rules: 29.44/8.43 29.44/8.43 a(x1) -> b(c(x1)) 29.44/8.43 b(b(x1)) -> x1 29.44/8.43 c(c(b(x1))) -> a(a(a(c(x1)))) 29.44/8.43 29.44/8.43 Q is empty. 29.44/8.43 We have to consider all minimal (P,Q,R)-chains. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (11) UsableRulesProof (EQUIVALENT) 29.44/8.43 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. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (12) 29.44/8.43 Obligation: 29.44/8.43 Q DP problem: 29.44/8.43 The TRS P consists of the following rules: 29.44/8.43 29.44/8.43 C(c(b(x1))) -> C(x1) 29.44/8.43 29.44/8.43 R is empty. 29.44/8.43 Q is empty. 29.44/8.43 We have to consider all minimal (P,Q,R)-chains. 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (13) QDPSizeChangeProof (EQUIVALENT) 29.44/8.43 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. 29.44/8.43 29.44/8.43 From the DPs we obtained the following set of size-change graphs: 29.44/8.43 *C(c(b(x1))) -> C(x1) 29.44/8.43 The graph contains the following edges 1 > 1 29.44/8.43 29.44/8.43 29.44/8.43 ---------------------------------------- 29.44/8.43 29.44/8.43 (14) 29.44/8.43 YES 29.70/8.50 EOF