40.98/11.30 YES 41.18/11.41 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 41.18/11.41 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 41.18/11.41 41.18/11.41 41.18/11.41 Termination w.r.t. Q of the given QTRS could be proven: 41.18/11.41 41.18/11.41 (0) QTRS 41.18/11.41 (1) QTRS Reverse [EQUIVALENT, 0 ms] 41.18/11.41 (2) QTRS 41.18/11.41 (3) DependencyPairsProof [EQUIVALENT, 13 ms] 41.18/11.41 (4) QDP 41.18/11.41 (5) QDPOrderProof [EQUIVALENT, 78 ms] 41.18/11.41 (6) QDP 41.18/11.41 (7) QDPOrderProof [EQUIVALENT, 0 ms] 41.18/11.41 (8) QDP 41.18/11.41 (9) QDPOrderProof [EQUIVALENT, 26 ms] 41.18/11.41 (10) QDP 41.18/11.41 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 41.18/11.41 (12) QDP 41.18/11.41 (13) UsableRulesProof [EQUIVALENT, 0 ms] 41.18/11.41 (14) QDP 41.18/11.41 (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] 41.18/11.41 (16) YES 41.18/11.41 41.18/11.41 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (0) 41.18/11.41 Obligation: 41.18/11.41 Q restricted rewrite system: 41.18/11.41 The TRS R consists of the following rules: 41.18/11.41 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 a(b(x1)) -> b(a(c(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 41.18/11.41 Q is empty. 41.18/11.41 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (1) QTRS Reverse (EQUIVALENT) 41.18/11.41 We applied the QTRS Reverse Processor [REVERSE]. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (2) 41.18/11.41 Obligation: 41.18/11.41 Q restricted rewrite system: 41.18/11.41 The TRS R consists of the following rules: 41.18/11.41 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 41.18/11.41 Q is empty. 41.18/11.41 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (3) DependencyPairsProof (EQUIVALENT) 41.18/11.41 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (4) 41.18/11.41 Obligation: 41.18/11.41 Q DP problem: 41.18/11.41 The TRS P consists of the following rules: 41.18/11.41 41.18/11.41 A(x1) -> B(x1) 41.18/11.41 A(x1) -> C(x1) 41.18/11.41 B(a(x1)) -> C(a(b(x1))) 41.18/11.41 B(a(x1)) -> A(b(x1)) 41.18/11.41 B(a(x1)) -> B(x1) 41.18/11.41 C(c(x1)) -> A(x1) 41.18/11.41 41.18/11.41 The TRS R consists of the following rules: 41.18/11.41 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 41.18/11.41 Q is empty. 41.18/11.41 We have to consider all minimal (P,Q,R)-chains. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (5) QDPOrderProof (EQUIVALENT) 41.18/11.41 We use the reduction pair processor [LPAR04,JAR06]. 41.18/11.41 41.18/11.41 41.18/11.41 The following pairs can be oriented strictly and are deleted. 41.18/11.41 41.18/11.41 B(a(x1)) -> B(x1) 41.18/11.41 The remaining pairs can at least be oriented weakly. 41.18/11.41 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(A(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(B(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(C(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(a(x_1)) = [[-I], [0A], [0A]] + [[1A, 0A, 1A], [1A, 0A, 1A], [-I, -I, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [1A, 0A, 1A], [-I, -I, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(c(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [1A, -I, 1A], [-I, -I, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 41.18/11.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 41.18/11.41 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 41.18/11.41 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (6) 41.18/11.41 Obligation: 41.18/11.41 Q DP problem: 41.18/11.41 The TRS P consists of the following rules: 41.18/11.41 41.18/11.41 A(x1) -> B(x1) 41.18/11.41 A(x1) -> C(x1) 41.18/11.41 B(a(x1)) -> C(a(b(x1))) 41.18/11.41 B(a(x1)) -> A(b(x1)) 41.18/11.41 C(c(x1)) -> A(x1) 41.18/11.41 41.18/11.41 The TRS R consists of the following rules: 41.18/11.41 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 41.18/11.41 Q is empty. 41.18/11.41 We have to consider all minimal (P,Q,R)-chains. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (7) QDPOrderProof (EQUIVALENT) 41.18/11.41 We use the reduction pair processor [LPAR04,JAR06]. 41.18/11.41 41.18/11.41 41.18/11.41 The following pairs can be oriented strictly and are deleted. 41.18/11.41 41.18/11.41 B(a(x1)) -> A(b(x1)) 41.18/11.41 The remaining pairs can at least be oriented weakly. 41.18/11.41 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(B(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(C(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(a(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 1A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(b(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(c(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [1A, 0A, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 41.18/11.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 41.18/11.41 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 41.18/11.41 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (8) 41.18/11.41 Obligation: 41.18/11.41 Q DP problem: 41.18/11.41 The TRS P consists of the following rules: 41.18/11.41 41.18/11.41 A(x1) -> B(x1) 41.18/11.41 A(x1) -> C(x1) 41.18/11.41 B(a(x1)) -> C(a(b(x1))) 41.18/11.41 C(c(x1)) -> A(x1) 41.18/11.41 41.18/11.41 The TRS R consists of the following rules: 41.18/11.41 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 41.18/11.41 Q is empty. 41.18/11.41 We have to consider all minimal (P,Q,R)-chains. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (9) QDPOrderProof (EQUIVALENT) 41.18/11.41 We use the reduction pair processor [LPAR04,JAR06]. 41.18/11.41 41.18/11.41 41.18/11.41 The following pairs can be oriented strictly and are deleted. 41.18/11.41 41.18/11.41 A(x1) -> B(x1) 41.18/11.41 The remaining pairs can at least be oriented weakly. 41.18/11.41 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(A(x_1)) = [[0A]] + [[1A, 0A, 1A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(B(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(C(x_1)) = [[0A]] + [[0A, 0A, 1A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(a(x_1)) = [[0A], [0A], [-I]] + [[1A, -I, 1A], [-I, 0A, -I], [0A, -I, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(b(x_1)) = [[-I], [0A], [-I]] + [[0A, -I, 0A], [-I, -I, -I], [0A, -I, 0A]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 <<< 41.18/11.41 POL(c(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, 1A], [-I, 0A, -I], [0A, -I, -I]] * x_1 41.18/11.41 >>> 41.18/11.41 41.18/11.41 41.18/11.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 41.18/11.41 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 41.18/11.41 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (10) 41.18/11.41 Obligation: 41.18/11.41 Q DP problem: 41.18/11.41 The TRS P consists of the following rules: 41.18/11.41 41.18/11.41 A(x1) -> C(x1) 41.18/11.41 B(a(x1)) -> C(a(b(x1))) 41.18/11.41 C(c(x1)) -> A(x1) 41.18/11.41 41.18/11.41 The TRS R consists of the following rules: 41.18/11.41 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 41.18/11.41 Q is empty. 41.18/11.41 We have to consider all minimal (P,Q,R)-chains. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (11) DependencyGraphProof (EQUIVALENT) 41.18/11.41 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (12) 41.18/11.41 Obligation: 41.18/11.41 Q DP problem: 41.18/11.41 The TRS P consists of the following rules: 41.18/11.41 41.18/11.41 C(c(x1)) -> A(x1) 41.18/11.41 A(x1) -> C(x1) 41.18/11.41 41.18/11.41 The TRS R consists of the following rules: 41.18/11.41 41.18/11.41 a(x1) -> b(x1) 41.18/11.41 a(x1) -> c(x1) 41.18/11.41 b(a(x1)) -> c(a(b(x1))) 41.18/11.41 c(c(x1)) -> a(x1) 41.18/11.41 41.18/11.41 Q is empty. 41.18/11.41 We have to consider all minimal (P,Q,R)-chains. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (13) UsableRulesProof (EQUIVALENT) 41.18/11.41 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. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (14) 41.18/11.41 Obligation: 41.18/11.41 Q DP problem: 41.18/11.41 The TRS P consists of the following rules: 41.18/11.41 41.18/11.41 C(c(x1)) -> A(x1) 41.18/11.41 A(x1) -> C(x1) 41.18/11.41 41.18/11.41 R is empty. 41.18/11.41 Q is empty. 41.18/11.41 We have to consider all minimal (P,Q,R)-chains. 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (15) QDPSizeChangeProof (EQUIVALENT) 41.18/11.41 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. 41.18/11.41 41.18/11.41 From the DPs we obtained the following set of size-change graphs: 41.18/11.41 *A(x1) -> C(x1) 41.18/11.41 The graph contains the following edges 1 >= 1 41.18/11.41 41.18/11.41 41.18/11.41 *C(c(x1)) -> A(x1) 41.18/11.41 The graph contains the following edges 1 > 1 41.18/11.41 41.18/11.41 41.18/11.41 ---------------------------------------- 41.18/11.41 41.18/11.41 (16) 41.18/11.41 YES 41.56/11.45 EOF