35.89/10.10 YES 35.89/10.11 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 35.89/10.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 35.89/10.11 35.89/10.11 35.89/10.11 Termination w.r.t. Q of the given QTRS could be proven: 35.89/10.11 35.89/10.11 (0) QTRS 35.89/10.11 (1) QTRS Reverse [EQUIVALENT, 0 ms] 35.89/10.11 (2) QTRS 35.89/10.11 (3) DependencyPairsProof [EQUIVALENT, 2 ms] 35.89/10.11 (4) QDP 35.89/10.11 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 35.89/10.11 (6) QDP 35.89/10.11 (7) QDPOrderProof [EQUIVALENT, 110 ms] 35.89/10.11 (8) QDP 35.89/10.11 (9) QDPOrderProof [EQUIVALENT, 0 ms] 35.89/10.11 (10) QDP 35.89/10.11 (11) UsableRulesProof [EQUIVALENT, 0 ms] 35.89/10.11 (12) QDP 35.89/10.11 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 35.89/10.11 (14) YES 35.89/10.11 35.89/10.11 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (0) 35.89/10.11 Obligation: 35.89/10.11 Q restricted rewrite system: 35.89/10.11 The TRS R consists of the following rules: 35.89/10.11 35.89/10.11 a(a(x1)) -> x1 35.89/10.11 a(b(a(b(x1)))) -> b(a(b(b(a(a(x1)))))) 35.89/10.11 35.89/10.11 Q is empty. 35.89/10.11 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (1) QTRS Reverse (EQUIVALENT) 35.89/10.11 We applied the QTRS Reverse Processor [REVERSE]. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (2) 35.89/10.11 Obligation: 35.89/10.11 Q restricted rewrite system: 35.89/10.11 The TRS R consists of the following rules: 35.89/10.11 35.89/10.11 a(a(x1)) -> x1 35.89/10.11 b(a(b(a(x1)))) -> a(a(b(b(a(b(x1)))))) 35.89/10.11 35.89/10.11 Q is empty. 35.89/10.11 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (3) DependencyPairsProof (EQUIVALENT) 35.89/10.11 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (4) 35.89/10.11 Obligation: 35.89/10.11 Q DP problem: 35.89/10.11 The TRS P consists of the following rules: 35.89/10.11 35.89/10.11 B(a(b(a(x1)))) -> A(a(b(b(a(b(x1)))))) 35.89/10.11 B(a(b(a(x1)))) -> A(b(b(a(b(x1))))) 35.89/10.11 B(a(b(a(x1)))) -> B(b(a(b(x1)))) 35.89/10.11 B(a(b(a(x1)))) -> B(a(b(x1))) 35.89/10.11 B(a(b(a(x1)))) -> A(b(x1)) 35.89/10.11 B(a(b(a(x1)))) -> B(x1) 35.89/10.11 35.89/10.11 The TRS R consists of the following rules: 35.89/10.11 35.89/10.11 a(a(x1)) -> x1 35.89/10.11 b(a(b(a(x1)))) -> a(a(b(b(a(b(x1)))))) 35.89/10.11 35.89/10.11 Q is empty. 35.89/10.11 We have to consider all minimal (P,Q,R)-chains. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (5) DependencyGraphProof (EQUIVALENT) 35.89/10.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (6) 35.89/10.11 Obligation: 35.89/10.11 Q DP problem: 35.89/10.11 The TRS P consists of the following rules: 35.89/10.11 35.89/10.11 B(a(b(a(x1)))) -> B(a(b(x1))) 35.89/10.11 B(a(b(a(x1)))) -> B(b(a(b(x1)))) 35.89/10.11 B(a(b(a(x1)))) -> B(x1) 35.89/10.11 35.89/10.11 The TRS R consists of the following rules: 35.89/10.11 35.89/10.11 a(a(x1)) -> x1 35.89/10.11 b(a(b(a(x1)))) -> a(a(b(b(a(b(x1)))))) 35.89/10.11 35.89/10.11 Q is empty. 35.89/10.11 We have to consider all minimal (P,Q,R)-chains. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (7) QDPOrderProof (EQUIVALENT) 35.89/10.11 We use the reduction pair processor [LPAR04,JAR06]. 35.89/10.11 35.89/10.11 35.89/10.11 The following pairs can be oriented strictly and are deleted. 35.89/10.11 35.89/10.11 B(a(b(a(x1)))) -> B(b(a(b(x1)))) 35.89/10.11 The remaining pairs can at least be oriented weakly. 35.89/10.11 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 35.89/10.11 35.89/10.11 <<< 35.89/10.11 POL(B(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 35.89/10.11 >>> 35.89/10.11 35.89/10.11 <<< 35.89/10.11 POL(a(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [0A, -I, 0A], [-I, -I, 0A]] * x_1 35.89/10.11 >>> 35.89/10.11 35.89/10.11 <<< 35.89/10.11 POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [-I, -I, -I]] * x_1 35.89/10.11 >>> 35.89/10.11 35.89/10.11 35.89/10.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.89/10.11 35.89/10.11 b(a(b(a(x1)))) -> a(a(b(b(a(b(x1)))))) 35.89/10.11 a(a(x1)) -> x1 35.89/10.11 35.89/10.11 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (8) 35.89/10.11 Obligation: 35.89/10.11 Q DP problem: 35.89/10.11 The TRS P consists of the following rules: 35.89/10.11 35.89/10.11 B(a(b(a(x1)))) -> B(a(b(x1))) 35.89/10.11 B(a(b(a(x1)))) -> B(x1) 35.89/10.11 35.89/10.11 The TRS R consists of the following rules: 35.89/10.11 35.89/10.11 a(a(x1)) -> x1 35.89/10.11 b(a(b(a(x1)))) -> a(a(b(b(a(b(x1)))))) 35.89/10.11 35.89/10.11 Q is empty. 35.89/10.11 We have to consider all minimal (P,Q,R)-chains. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (9) QDPOrderProof (EQUIVALENT) 35.89/10.11 We use the reduction pair processor [LPAR04,JAR06]. 35.89/10.11 35.89/10.11 35.89/10.11 The following pairs can be oriented strictly and are deleted. 35.89/10.11 35.89/10.11 B(a(b(a(x1)))) -> B(a(b(x1))) 35.89/10.11 The remaining pairs can at least be oriented weakly. 35.89/10.11 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 35.89/10.11 35.89/10.11 <<< 35.89/10.11 POL(B(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 35.89/10.11 >>> 35.89/10.11 35.89/10.11 <<< 35.89/10.11 POL(a(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, 0A], [0A, -I, -I], [-I, 0A, 1A]] * x_1 35.89/10.11 >>> 35.89/10.11 35.89/10.11 <<< 35.89/10.11 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 1A], [-I, -I, -I], [-I, 0A, -I]] * x_1 35.89/10.11 >>> 35.89/10.11 35.89/10.11 35.89/10.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.89/10.11 35.89/10.11 b(a(b(a(x1)))) -> a(a(b(b(a(b(x1)))))) 35.89/10.11 a(a(x1)) -> x1 35.89/10.11 35.89/10.11 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (10) 35.89/10.11 Obligation: 35.89/10.11 Q DP problem: 35.89/10.11 The TRS P consists of the following rules: 35.89/10.11 35.89/10.11 B(a(b(a(x1)))) -> B(x1) 35.89/10.11 35.89/10.11 The TRS R consists of the following rules: 35.89/10.11 35.89/10.11 a(a(x1)) -> x1 35.89/10.11 b(a(b(a(x1)))) -> a(a(b(b(a(b(x1)))))) 35.89/10.11 35.89/10.11 Q is empty. 35.89/10.11 We have to consider all minimal (P,Q,R)-chains. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (11) UsableRulesProof (EQUIVALENT) 35.89/10.11 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. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (12) 35.89/10.11 Obligation: 35.89/10.11 Q DP problem: 35.89/10.11 The TRS P consists of the following rules: 35.89/10.11 35.89/10.11 B(a(b(a(x1)))) -> B(x1) 35.89/10.11 35.89/10.11 R is empty. 35.89/10.11 Q is empty. 35.89/10.11 We have to consider all minimal (P,Q,R)-chains. 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (13) QDPSizeChangeProof (EQUIVALENT) 35.89/10.11 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. 35.89/10.11 35.89/10.11 From the DPs we obtained the following set of size-change graphs: 35.89/10.11 *B(a(b(a(x1)))) -> B(x1) 35.89/10.11 The graph contains the following edges 1 > 1 35.89/10.11 35.89/10.11 35.89/10.11 ---------------------------------------- 35.89/10.11 35.89/10.11 (14) 35.89/10.11 YES 36.07/10.20 EOF