35.85/10.05 YES 35.94/10.06 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 35.94/10.06 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 35.94/10.06 35.94/10.06 35.94/10.06 Termination w.r.t. Q of the given QTRS could be proven: 35.94/10.06 35.94/10.06 (0) QTRS 35.94/10.06 (1) DependencyPairsProof [EQUIVALENT, 0 ms] 35.94/10.06 (2) QDP 35.94/10.06 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 35.94/10.06 (4) QDP 35.94/10.06 (5) QDPOrderProof [EQUIVALENT, 281 ms] 35.94/10.06 (6) QDP 35.94/10.06 (7) DependencyGraphProof [EQUIVALENT, 1 ms] 35.94/10.06 (8) AND 35.94/10.06 (9) QDP 35.94/10.06 (10) QDPOrderProof [EQUIVALENT, 0 ms] 35.94/10.06 (11) QDP 35.94/10.06 (12) PisEmptyProof [EQUIVALENT, 0 ms] 35.94/10.06 (13) YES 35.94/10.06 (14) QDP 35.94/10.06 (15) UsableRulesProof [EQUIVALENT, 0 ms] 35.94/10.06 (16) QDP 35.94/10.06 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 35.94/10.06 (18) YES 35.94/10.06 35.94/10.06 35.94/10.06 ---------------------------------------- 35.94/10.06 35.94/10.06 (0) 35.94/10.06 Obligation: 35.94/10.06 Q restricted rewrite system: 35.94/10.06 The TRS R consists of the following rules: 35.94/10.06 35.94/10.06 a(x1) -> x1 35.94/10.06 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.06 b(x1) -> x1 35.94/10.06 c(b(x1)) -> a(c(x1)) 35.94/10.06 35.94/10.06 Q is empty. 35.94/10.06 35.94/10.06 ---------------------------------------- 35.94/10.06 35.94/10.06 (1) DependencyPairsProof (EQUIVALENT) 35.94/10.06 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 35.94/10.06 ---------------------------------------- 35.94/10.06 35.94/10.06 (2) 35.94/10.06 Obligation: 35.94/10.06 Q DP problem: 35.94/10.06 The TRS P consists of the following rules: 35.94/10.06 35.94/10.06 A(a(x1)) -> B(a(b(c(x1)))) 35.94/10.06 A(a(x1)) -> A(b(c(x1))) 35.94/10.06 A(a(x1)) -> B(c(x1)) 35.94/10.06 A(a(x1)) -> C(x1) 35.94/10.06 C(b(x1)) -> A(c(x1)) 35.94/10.06 C(b(x1)) -> C(x1) 35.94/10.06 35.94/10.06 The TRS R consists of the following rules: 35.94/10.06 35.94/10.06 a(x1) -> x1 35.94/10.06 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.06 b(x1) -> x1 35.94/10.06 c(b(x1)) -> a(c(x1)) 35.94/10.06 35.94/10.06 Q is empty. 35.94/10.06 We have to consider all minimal (P,Q,R)-chains. 35.94/10.06 ---------------------------------------- 35.94/10.06 35.94/10.06 (3) DependencyGraphProof (EQUIVALENT) 35.94/10.06 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 35.94/10.06 ---------------------------------------- 35.94/10.06 35.94/10.06 (4) 35.94/10.06 Obligation: 35.94/10.06 Q DP problem: 35.94/10.06 The TRS P consists of the following rules: 35.94/10.06 35.94/10.06 A(a(x1)) -> C(x1) 35.94/10.06 C(b(x1)) -> A(c(x1)) 35.94/10.06 A(a(x1)) -> A(b(c(x1))) 35.94/10.06 C(b(x1)) -> C(x1) 35.94/10.06 35.94/10.06 The TRS R consists of the following rules: 35.94/10.06 35.94/10.06 a(x1) -> x1 35.94/10.06 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.06 b(x1) -> x1 35.94/10.07 c(b(x1)) -> a(c(x1)) 35.94/10.07 35.94/10.07 Q is empty. 35.94/10.07 We have to consider all minimal (P,Q,R)-chains. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (5) QDPOrderProof (EQUIVALENT) 35.94/10.07 We use the reduction pair processor [LPAR04,JAR06]. 35.94/10.07 35.94/10.07 35.94/10.07 The following pairs can be oriented strictly and are deleted. 35.94/10.07 35.94/10.07 A(a(x1)) -> C(x1) 35.94/10.07 The remaining pairs can at least be oriented weakly. 35.94/10.07 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(a(x_1)) = [[1A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 1A, 1A], [-I, 1A, 0A]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(C(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(b(x_1)) = [[1A], [-I], [0A]] + [[0A, -I, 0A], [-I, 0A, 0A], [0A, 0A, 1A]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(c(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, -I], [-I, -I, 0A], [-I, -I, 0A]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 35.94/10.07 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.94/10.07 35.94/10.07 c(b(x1)) -> a(c(x1)) 35.94/10.07 b(x1) -> x1 35.94/10.07 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.07 a(x1) -> x1 35.94/10.07 35.94/10.07 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (6) 35.94/10.07 Obligation: 35.94/10.07 Q DP problem: 35.94/10.07 The TRS P consists of the following rules: 35.94/10.07 35.94/10.07 C(b(x1)) -> A(c(x1)) 35.94/10.07 A(a(x1)) -> A(b(c(x1))) 35.94/10.07 C(b(x1)) -> C(x1) 35.94/10.07 35.94/10.07 The TRS R consists of the following rules: 35.94/10.07 35.94/10.07 a(x1) -> x1 35.94/10.07 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.07 b(x1) -> x1 35.94/10.07 c(b(x1)) -> a(c(x1)) 35.94/10.07 35.94/10.07 Q is empty. 35.94/10.07 We have to consider all minimal (P,Q,R)-chains. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (7) DependencyGraphProof (EQUIVALENT) 35.94/10.07 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (8) 35.94/10.07 Complex Obligation (AND) 35.94/10.07 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (9) 35.94/10.07 Obligation: 35.94/10.07 Q DP problem: 35.94/10.07 The TRS P consists of the following rules: 35.94/10.07 35.94/10.07 A(a(x1)) -> A(b(c(x1))) 35.94/10.07 35.94/10.07 The TRS R consists of the following rules: 35.94/10.07 35.94/10.07 a(x1) -> x1 35.94/10.07 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.07 b(x1) -> x1 35.94/10.07 c(b(x1)) -> a(c(x1)) 35.94/10.07 35.94/10.07 Q is empty. 35.94/10.07 We have to consider all minimal (P,Q,R)-chains. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (10) QDPOrderProof (EQUIVALENT) 35.94/10.07 We use the reduction pair processor [LPAR04,JAR06]. 35.94/10.07 35.94/10.07 35.94/10.07 The following pairs can be oriented strictly and are deleted. 35.94/10.07 35.94/10.07 A(a(x1)) -> A(b(c(x1))) 35.94/10.07 The remaining pairs can at least be oriented weakly. 35.94/10.07 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(A(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(a(x_1)) = [[1A], [0A], [0A]] + [[1A, 1A, 0A], [1A, 0A, 0A], [0A, -I, 0A]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(b(x_1)) = [[0A], [1A], [-I]] + [[0A, 0A, -I], [0A, 1A, 0A], [-I, -I, 0A]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 <<< 35.94/10.07 POL(c(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 35.94/10.07 >>> 35.94/10.07 35.94/10.07 35.94/10.07 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.94/10.07 35.94/10.07 c(b(x1)) -> a(c(x1)) 35.94/10.07 b(x1) -> x1 35.94/10.07 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.07 a(x1) -> x1 35.94/10.07 35.94/10.07 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (11) 35.94/10.07 Obligation: 35.94/10.07 Q DP problem: 35.94/10.07 P is empty. 35.94/10.07 The TRS R consists of the following rules: 35.94/10.07 35.94/10.07 a(x1) -> x1 35.94/10.07 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.07 b(x1) -> x1 35.94/10.07 c(b(x1)) -> a(c(x1)) 35.94/10.07 35.94/10.07 Q is empty. 35.94/10.07 We have to consider all minimal (P,Q,R)-chains. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (12) PisEmptyProof (EQUIVALENT) 35.94/10.07 The TRS P is empty. Hence, there is no (P,Q,R) chain. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (13) 35.94/10.07 YES 35.94/10.07 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (14) 35.94/10.07 Obligation: 35.94/10.07 Q DP problem: 35.94/10.07 The TRS P consists of the following rules: 35.94/10.07 35.94/10.07 C(b(x1)) -> C(x1) 35.94/10.07 35.94/10.07 The TRS R consists of the following rules: 35.94/10.07 35.94/10.07 a(x1) -> x1 35.94/10.07 a(a(x1)) -> b(a(b(c(x1)))) 35.94/10.07 b(x1) -> x1 35.94/10.07 c(b(x1)) -> a(c(x1)) 35.94/10.07 35.94/10.07 Q is empty. 35.94/10.07 We have to consider all minimal (P,Q,R)-chains. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (15) UsableRulesProof (EQUIVALENT) 35.94/10.07 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. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (16) 35.94/10.07 Obligation: 35.94/10.07 Q DP problem: 35.94/10.07 The TRS P consists of the following rules: 35.94/10.07 35.94/10.07 C(b(x1)) -> C(x1) 35.94/10.07 35.94/10.07 R is empty. 35.94/10.07 Q is empty. 35.94/10.07 We have to consider all minimal (P,Q,R)-chains. 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (17) QDPSizeChangeProof (EQUIVALENT) 35.94/10.07 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. 35.94/10.07 35.94/10.07 From the DPs we obtained the following set of size-change graphs: 35.94/10.07 *C(b(x1)) -> C(x1) 35.94/10.07 The graph contains the following edges 1 > 1 35.94/10.07 35.94/10.07 35.94/10.07 ---------------------------------------- 35.94/10.07 35.94/10.07 (18) 35.94/10.07 YES 36.08/10.17 EOF