49.71/13.52 YES 49.71/13.58 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 49.71/13.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 49.71/13.58 49.71/13.58 49.71/13.58 Termination w.r.t. Q of the given QTRS could be proven: 49.71/13.58 49.71/13.58 (0) QTRS 49.71/13.58 (1) QTRS Reverse [EQUIVALENT, 0 ms] 49.71/13.58 (2) QTRS 49.71/13.58 (3) DependencyPairsProof [EQUIVALENT, 32 ms] 49.71/13.58 (4) QDP 49.71/13.58 (5) QDPOrderProof [EQUIVALENT, 296 ms] 49.71/13.58 (6) QDP 49.71/13.58 (7) QDPOrderProof [EQUIVALENT, 46 ms] 49.71/13.58 (8) QDP 49.71/13.58 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 49.71/13.58 (10) AND 49.71/13.58 (11) QDP 49.71/13.58 (12) QDPOrderProof [EQUIVALENT, 453 ms] 49.71/13.58 (13) QDP 49.71/13.58 (14) PisEmptyProof [EQUIVALENT, 0 ms] 49.71/13.58 (15) YES 49.71/13.58 (16) QDP 49.71/13.58 (17) QDPOrderProof [EQUIVALENT, 0 ms] 49.71/13.58 (18) QDP 49.71/13.58 (19) QDPOrderProof [EQUIVALENT, 101 ms] 49.71/13.58 (20) QDP 49.71/13.58 (21) UsableRulesProof [EQUIVALENT, 0 ms] 49.71/13.58 (22) QDP 49.71/13.58 (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] 49.71/13.58 (24) YES 49.71/13.58 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (0) 49.71/13.58 Obligation: 49.71/13.58 Q restricted rewrite system: 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 a(a(b(x1))) -> b(b(a(a(x1)))) 49.71/13.58 b(a(b(a(x1)))) -> a(a(a(b(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (1) QTRS Reverse (EQUIVALENT) 49.71/13.58 We applied the QTRS Reverse Processor [REVERSE]. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (2) 49.71/13.58 Obligation: 49.71/13.58 Q restricted rewrite system: 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (3) DependencyPairsProof (EQUIVALENT) 49.71/13.58 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (4) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 The TRS P consists of the following rules: 49.71/13.58 49.71/13.58 B(a(a(x1))) -> A(a(b(b(x1)))) 49.71/13.58 B(a(a(x1))) -> A(b(b(x1))) 49.71/13.58 B(a(a(x1))) -> B(b(x1)) 49.71/13.58 B(a(a(x1))) -> B(x1) 49.71/13.58 A(b(a(b(x1)))) -> B(a(a(a(x1)))) 49.71/13.58 A(b(a(b(x1)))) -> A(a(a(x1))) 49.71/13.58 A(b(a(b(x1)))) -> A(a(x1)) 49.71/13.58 A(b(a(b(x1)))) -> A(x1) 49.71/13.58 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (5) QDPOrderProof (EQUIVALENT) 49.71/13.58 We use the reduction pair processor [LPAR04,JAR06]. 49.71/13.58 49.71/13.58 49.71/13.58 The following pairs can be oriented strictly and are deleted. 49.71/13.58 49.71/13.58 B(a(a(x1))) -> A(b(b(x1))) 49.71/13.58 The remaining pairs can at least be oriented weakly. 49.71/13.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(B(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(a(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(A(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(b(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 49.71/13.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.71/13.58 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (6) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 The TRS P consists of the following rules: 49.71/13.58 49.71/13.58 B(a(a(x1))) -> A(a(b(b(x1)))) 49.71/13.58 B(a(a(x1))) -> B(b(x1)) 49.71/13.58 B(a(a(x1))) -> B(x1) 49.71/13.58 A(b(a(b(x1)))) -> B(a(a(a(x1)))) 49.71/13.58 A(b(a(b(x1)))) -> A(a(a(x1))) 49.71/13.58 A(b(a(b(x1)))) -> A(a(x1)) 49.71/13.58 A(b(a(b(x1)))) -> A(x1) 49.71/13.58 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (7) QDPOrderProof (EQUIVALENT) 49.71/13.58 We use the reduction pair processor [LPAR04,JAR06]. 49.71/13.58 49.71/13.58 49.71/13.58 The following pairs can be oriented strictly and are deleted. 49.71/13.58 49.71/13.58 B(a(a(x1))) -> A(a(b(b(x1)))) 49.71/13.58 The remaining pairs can at least be oriented weakly. 49.71/13.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(B(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(a(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(A(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(b(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 49.71/13.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.71/13.58 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (8) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 The TRS P consists of the following rules: 49.71/13.58 49.71/13.58 B(a(a(x1))) -> B(b(x1)) 49.71/13.58 B(a(a(x1))) -> B(x1) 49.71/13.58 A(b(a(b(x1)))) -> B(a(a(a(x1)))) 49.71/13.58 A(b(a(b(x1)))) -> A(a(a(x1))) 49.71/13.58 A(b(a(b(x1)))) -> A(a(x1)) 49.71/13.58 A(b(a(b(x1)))) -> A(x1) 49.71/13.58 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (9) DependencyGraphProof (EQUIVALENT) 49.71/13.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (10) 49.71/13.58 Complex Obligation (AND) 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (11) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 The TRS P consists of the following rules: 49.71/13.58 49.71/13.58 B(a(a(x1))) -> B(x1) 49.71/13.58 B(a(a(x1))) -> B(b(x1)) 49.71/13.58 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (12) QDPOrderProof (EQUIVALENT) 49.71/13.58 We use the reduction pair processor [LPAR04,JAR06]. 49.71/13.58 49.71/13.58 49.71/13.58 The following pairs can be oriented strictly and are deleted. 49.71/13.58 49.71/13.58 B(a(a(x1))) -> B(x1) 49.71/13.58 B(a(a(x1))) -> B(b(x1)) 49.71/13.58 The remaining pairs can at least be oriented weakly. 49.71/13.58 Used ordering: Matrix interpretation [MATRO] to (N^5, +, *, >=, >) : 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(B(x_1)) = [[0]] + [[0, 0, 0, 0, 1]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(a(x_1)) = [[0], [0], [0], [0], [1]] + [[0, 0, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 0], [0, 0, 0, 1, 1]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(b(x_1)) = [[0], [0], [1], [0], [0]] + [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 1, 1, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 1]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 49.71/13.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.71/13.58 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (13) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 P is empty. 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (14) PisEmptyProof (EQUIVALENT) 49.71/13.58 The TRS P is empty. Hence, there is no (P,Q,R) chain. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (15) 49.71/13.58 YES 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (16) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 The TRS P consists of the following rules: 49.71/13.58 49.71/13.58 A(b(a(b(x1)))) -> A(a(x1)) 49.71/13.58 A(b(a(b(x1)))) -> A(a(a(x1))) 49.71/13.58 A(b(a(b(x1)))) -> A(x1) 49.71/13.58 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (17) QDPOrderProof (EQUIVALENT) 49.71/13.58 We use the reduction pair processor [LPAR04,JAR06]. 49.71/13.58 49.71/13.58 49.71/13.58 The following pairs can be oriented strictly and are deleted. 49.71/13.58 49.71/13.58 A(b(a(b(x1)))) -> A(a(a(x1))) 49.71/13.58 The remaining pairs can at least be oriented weakly. 49.71/13.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [0A, -I, -I], [0A, 1A, 0A]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(a(x_1)) = [[0A], [-I], [0A]] + [[-I, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 49.71/13.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.71/13.58 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (18) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 The TRS P consists of the following rules: 49.71/13.58 49.71/13.58 A(b(a(b(x1)))) -> A(a(x1)) 49.71/13.58 A(b(a(b(x1)))) -> A(x1) 49.71/13.58 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (19) QDPOrderProof (EQUIVALENT) 49.71/13.58 We use the reduction pair processor [LPAR04,JAR06]. 49.71/13.58 49.71/13.58 49.71/13.58 The following pairs can be oriented strictly and are deleted. 49.71/13.58 49.71/13.58 A(b(a(b(x1)))) -> A(a(x1)) 49.71/13.58 The remaining pairs can at least be oriented weakly. 49.71/13.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(A(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(b(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, -I, -I]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 <<< 49.71/13.58 POL(a(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, 0A], [-I, -I, -I]] * x_1 49.71/13.58 >>> 49.71/13.58 49.71/13.58 49.71/13.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 49.71/13.58 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (20) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 The TRS P consists of the following rules: 49.71/13.58 49.71/13.58 A(b(a(b(x1)))) -> A(x1) 49.71/13.58 49.71/13.58 The TRS R consists of the following rules: 49.71/13.58 49.71/13.58 b(a(a(x1))) -> a(a(b(b(x1)))) 49.71/13.58 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 49.71/13.58 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (21) UsableRulesProof (EQUIVALENT) 49.71/13.58 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. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (22) 49.71/13.58 Obligation: 49.71/13.58 Q DP problem: 49.71/13.58 The TRS P consists of the following rules: 49.71/13.58 49.71/13.58 A(b(a(b(x1)))) -> A(x1) 49.71/13.58 49.71/13.58 R is empty. 49.71/13.58 Q is empty. 49.71/13.58 We have to consider all minimal (P,Q,R)-chains. 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (23) QDPSizeChangeProof (EQUIVALENT) 49.71/13.58 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. 49.71/13.58 49.71/13.58 From the DPs we obtained the following set of size-change graphs: 49.71/13.58 *A(b(a(b(x1)))) -> A(x1) 49.71/13.58 The graph contains the following edges 1 > 1 49.71/13.58 49.71/13.58 49.71/13.58 ---------------------------------------- 49.71/13.58 49.71/13.58 (24) 49.71/13.58 YES 50.23/13.70 EOF