42.40/11.74 YES 42.82/11.81 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 42.82/11.81 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 42.82/11.81 42.82/11.81 42.82/11.81 Termination w.r.t. Q of the given QTRS could be proven: 42.82/11.81 42.82/11.81 (0) QTRS 42.82/11.81 (1) QTRS Reverse [EQUIVALENT, 0 ms] 42.82/11.81 (2) QTRS 42.82/11.81 (3) DependencyPairsProof [EQUIVALENT, 13 ms] 42.82/11.81 (4) QDP 42.82/11.81 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 42.82/11.81 (6) QDP 42.82/11.81 (7) QDPOrderProof [EQUIVALENT, 147 ms] 42.82/11.81 (8) QDP 42.82/11.81 (9) QDPOrderProof [EQUIVALENT, 67 ms] 42.82/11.81 (10) QDP 42.82/11.81 (11) QDPOrderProof [EQUIVALENT, 41 ms] 42.82/11.81 (12) QDP 42.82/11.81 (13) QDPOrderProof [EQUIVALENT, 22 ms] 42.82/11.81 (14) QDP 42.82/11.81 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 42.82/11.81 (16) QDP 42.82/11.81 (17) UsableRulesProof [EQUIVALENT, 1 ms] 42.82/11.81 (18) QDP 42.82/11.81 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.82/11.81 (20) YES 42.82/11.81 42.82/11.81 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (0) 42.82/11.81 Obligation: 42.82/11.81 Q restricted rewrite system: 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> a(a(b(x1))) 42.82/11.81 b(a(a(x1))) -> a(a(b(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (1) QTRS Reverse (EQUIVALENT) 42.82/11.81 We applied the QTRS Reverse Processor [REVERSE]. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (2) 42.82/11.81 Obligation: 42.82/11.81 Q restricted rewrite system: 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (3) DependencyPairsProof (EQUIVALENT) 42.82/11.81 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (4) 42.82/11.81 Obligation: 42.82/11.81 Q DP problem: 42.82/11.81 The TRS P consists of the following rules: 42.82/11.81 42.82/11.81 B(b(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> A(a(x1)) 42.82/11.81 B(b(b(x1))) -> A(x1) 42.82/11.81 A(a(b(x1))) -> B(a(a(x1))) 42.82/11.81 A(a(b(x1))) -> A(a(x1)) 42.82/11.81 A(a(b(x1))) -> A(x1) 42.82/11.81 A(a(x1)) -> B(a(b(x1))) 42.82/11.81 A(a(x1)) -> A(b(x1)) 42.82/11.81 A(a(x1)) -> B(x1) 42.82/11.81 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 We have to consider all minimal (P,Q,R)-chains. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (5) DependencyGraphProof (EQUIVALENT) 42.82/11.81 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (6) 42.82/11.81 Obligation: 42.82/11.81 Q DP problem: 42.82/11.81 The TRS P consists of the following rules: 42.82/11.81 42.82/11.81 B(b(b(x1))) -> A(a(x1)) 42.82/11.81 A(a(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> A(x1) 42.82/11.81 A(a(b(x1))) -> A(a(x1)) 42.82/11.81 A(a(b(x1))) -> A(x1) 42.82/11.81 A(a(x1)) -> B(a(b(x1))) 42.82/11.81 A(a(x1)) -> B(x1) 42.82/11.81 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 We have to consider all minimal (P,Q,R)-chains. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (7) QDPOrderProof (EQUIVALENT) 42.82/11.81 We use the reduction pair processor [LPAR04,JAR06]. 42.82/11.81 42.82/11.81 42.82/11.81 The following pairs can be oriented strictly and are deleted. 42.82/11.81 42.82/11.81 A(a(x1)) -> B(a(b(x1))) 42.82/11.81 The remaining pairs can at least be oriented weakly. 42.82/11.81 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(B(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, 0A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(A(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(a(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, -I], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 42.82/11.81 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.82/11.81 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 42.82/11.81 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (8) 42.82/11.81 Obligation: 42.82/11.81 Q DP problem: 42.82/11.81 The TRS P consists of the following rules: 42.82/11.81 42.82/11.81 B(b(b(x1))) -> A(a(x1)) 42.82/11.81 A(a(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> A(x1) 42.82/11.81 A(a(b(x1))) -> A(a(x1)) 42.82/11.81 A(a(b(x1))) -> A(x1) 42.82/11.81 A(a(x1)) -> B(x1) 42.82/11.81 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 We have to consider all minimal (P,Q,R)-chains. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (9) QDPOrderProof (EQUIVALENT) 42.82/11.81 We use the reduction pair processor [LPAR04,JAR06]. 42.82/11.81 42.82/11.81 42.82/11.81 The following pairs can be oriented strictly and are deleted. 42.82/11.81 42.82/11.81 A(a(x1)) -> B(x1) 42.82/11.81 The remaining pairs can at least be oriented weakly. 42.82/11.81 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(B(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(b(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(A(x_1)) = [[0A]] + [[0A, 0A, 1A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(a(x_1)) = [[1A], [0A], [1A]] + [[0A, 1A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 42.82/11.81 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.82/11.81 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 42.82/11.81 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (10) 42.82/11.81 Obligation: 42.82/11.81 Q DP problem: 42.82/11.81 The TRS P consists of the following rules: 42.82/11.81 42.82/11.81 B(b(b(x1))) -> A(a(x1)) 42.82/11.81 A(a(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> A(x1) 42.82/11.81 A(a(b(x1))) -> A(a(x1)) 42.82/11.81 A(a(b(x1))) -> A(x1) 42.82/11.81 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 We have to consider all minimal (P,Q,R)-chains. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (11) QDPOrderProof (EQUIVALENT) 42.82/11.81 We use the reduction pair processor [LPAR04,JAR06]. 42.82/11.81 42.82/11.81 42.82/11.81 The following pairs can be oriented strictly and are deleted. 42.82/11.81 42.82/11.81 B(b(b(x1))) -> A(a(x1)) 42.82/11.81 A(a(b(x1))) -> A(a(x1)) 42.82/11.81 The remaining pairs can at least be oriented weakly. 42.82/11.81 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(B(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(b(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, -I], [-I, 1A, 0A], [0A, 1A, -I]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(A(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(a(x_1)) = [[1A], [0A], [0A]] + [[-I, 1A, 0A], [0A, -I, -I], [1A, 0A, -I]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 42.82/11.81 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.82/11.81 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 42.82/11.81 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (12) 42.82/11.81 Obligation: 42.82/11.81 Q DP problem: 42.82/11.81 The TRS P consists of the following rules: 42.82/11.81 42.82/11.81 A(a(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> A(x1) 42.82/11.81 A(a(b(x1))) -> A(x1) 42.82/11.81 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 We have to consider all minimal (P,Q,R)-chains. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (13) QDPOrderProof (EQUIVALENT) 42.82/11.81 We use the reduction pair processor [LPAR04,JAR06]. 42.82/11.81 42.82/11.81 42.82/11.81 The following pairs can be oriented strictly and are deleted. 42.82/11.81 42.82/11.81 B(b(b(x1))) -> B(a(a(x1))) 42.82/11.81 B(b(b(x1))) -> A(x1) 42.82/11.81 The remaining pairs can at least be oriented weakly. 42.82/11.81 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(A(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(a(x_1)) = [[0A], [1A], [0A]] + [[-I, 1A, 0A], [0A, -I, 1A], [-I, 0A, -I]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(b(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 1A], [-I, -I, 0A], [0A, -I, 1A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 <<< 42.82/11.81 POL(B(x_1)) = [[1A]] + [[-I, -I, 0A]] * x_1 42.82/11.81 >>> 42.82/11.81 42.82/11.81 42.82/11.81 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.82/11.81 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 42.82/11.81 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (14) 42.82/11.81 Obligation: 42.82/11.81 Q DP problem: 42.82/11.81 The TRS P consists of the following rules: 42.82/11.81 42.82/11.81 A(a(b(x1))) -> B(a(a(x1))) 42.82/11.81 A(a(b(x1))) -> A(x1) 42.82/11.81 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 We have to consider all minimal (P,Q,R)-chains. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (15) DependencyGraphProof (EQUIVALENT) 42.82/11.81 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (16) 42.82/11.81 Obligation: 42.82/11.81 Q DP problem: 42.82/11.81 The TRS P consists of the following rules: 42.82/11.81 42.82/11.81 A(a(b(x1))) -> A(x1) 42.82/11.81 42.82/11.81 The TRS R consists of the following rules: 42.82/11.81 42.82/11.81 b(b(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(b(x1))) -> b(a(a(x1))) 42.82/11.81 a(a(x1)) -> b(a(b(x1))) 42.82/11.81 42.82/11.81 Q is empty. 42.82/11.81 We have to consider all minimal (P,Q,R)-chains. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (17) UsableRulesProof (EQUIVALENT) 42.82/11.81 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. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (18) 42.82/11.81 Obligation: 42.82/11.81 Q DP problem: 42.82/11.81 The TRS P consists of the following rules: 42.82/11.81 42.82/11.81 A(a(b(x1))) -> A(x1) 42.82/11.81 42.82/11.81 R is empty. 42.82/11.81 Q is empty. 42.82/11.81 We have to consider all minimal (P,Q,R)-chains. 42.82/11.81 ---------------------------------------- 42.82/11.81 42.82/11.81 (19) QDPSizeChangeProof (EQUIVALENT) 42.82/11.82 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. 42.82/11.82 42.82/11.82 From the DPs we obtained the following set of size-change graphs: 42.82/11.82 *A(a(b(x1))) -> A(x1) 42.82/11.82 The graph contains the following edges 1 > 1 42.82/11.82 42.82/11.82 42.82/11.82 ---------------------------------------- 42.82/11.82 42.82/11.82 (20) 42.82/11.82 YES 43.28/12.02 EOF