28.76/8.29 YES 29.03/8.32 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 29.03/8.32 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.03/8.32 29.03/8.32 29.03/8.32 Termination w.r.t. Q of the given QTRS could be proven: 29.03/8.32 29.03/8.32 (0) QTRS 29.03/8.32 (1) QTRS Reverse [EQUIVALENT, 0 ms] 29.03/8.32 (2) QTRS 29.03/8.32 (3) DependencyPairsProof [EQUIVALENT, 25 ms] 29.03/8.32 (4) QDP 29.03/8.32 (5) QDPOrderProof [EQUIVALENT, 26 ms] 29.03/8.32 (6) QDP 29.03/8.32 (7) QDPOrderProof [EQUIVALENT, 728 ms] 29.03/8.32 (8) QDP 29.03/8.32 (9) UsableRulesProof [EQUIVALENT, 0 ms] 29.03/8.32 (10) QDP 29.03/8.32 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 29.03/8.32 (12) YES 29.03/8.32 29.03/8.32 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (0) 29.03/8.32 Obligation: 29.03/8.32 Q restricted rewrite system: 29.03/8.32 The TRS R consists of the following rules: 29.03/8.32 29.03/8.32 a(a(a(b(x1)))) -> b(a(a(a(x1)))) 29.03/8.32 b(b(x1)) -> a(b(a(b(x1)))) 29.03/8.32 29.03/8.32 Q is empty. 29.03/8.32 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (1) QTRS Reverse (EQUIVALENT) 29.03/8.32 We applied the QTRS Reverse Processor [REVERSE]. 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (2) 29.03/8.32 Obligation: 29.03/8.32 Q restricted rewrite system: 29.03/8.32 The TRS R consists of the following rules: 29.03/8.32 29.03/8.32 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 29.03/8.32 b(b(x1)) -> b(a(b(a(x1)))) 29.03/8.32 29.03/8.32 Q is empty. 29.03/8.32 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (3) DependencyPairsProof (EQUIVALENT) 29.03/8.32 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (4) 29.03/8.32 Obligation: 29.03/8.32 Q DP problem: 29.03/8.32 The TRS P consists of the following rules: 29.03/8.32 29.03/8.32 B(a(a(a(x1)))) -> B(x1) 29.03/8.32 B(b(x1)) -> B(a(b(a(x1)))) 29.03/8.32 B(b(x1)) -> B(a(x1)) 29.03/8.32 29.03/8.32 The TRS R consists of the following rules: 29.03/8.32 29.03/8.32 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 29.03/8.32 b(b(x1)) -> b(a(b(a(x1)))) 29.03/8.32 29.03/8.32 Q is empty. 29.03/8.32 We have to consider all minimal (P,Q,R)-chains. 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (5) QDPOrderProof (EQUIVALENT) 29.03/8.32 We use the reduction pair processor [LPAR04,JAR06]. 29.03/8.32 29.03/8.32 29.03/8.32 The following pairs can be oriented strictly and are deleted. 29.03/8.32 29.03/8.32 B(b(x1)) -> B(a(x1)) 29.03/8.32 The remaining pairs can at least be oriented weakly. 29.03/8.32 Used ordering: Polynomial interpretation [POLO]: 29.03/8.32 29.03/8.32 POL(B(x_1)) = x_1 29.03/8.32 POL(a(x_1)) = x_1 29.03/8.32 POL(b(x_1)) = 1 + x_1 29.03/8.32 29.03/8.32 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.03/8.32 29.03/8.32 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 29.03/8.32 b(b(x1)) -> b(a(b(a(x1)))) 29.03/8.32 29.03/8.32 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (6) 29.03/8.32 Obligation: 29.03/8.32 Q DP problem: 29.03/8.32 The TRS P consists of the following rules: 29.03/8.32 29.03/8.32 B(a(a(a(x1)))) -> B(x1) 29.03/8.32 B(b(x1)) -> B(a(b(a(x1)))) 29.03/8.32 29.03/8.32 The TRS R consists of the following rules: 29.03/8.32 29.03/8.32 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 29.03/8.32 b(b(x1)) -> b(a(b(a(x1)))) 29.03/8.32 29.03/8.32 Q is empty. 29.03/8.32 We have to consider all minimal (P,Q,R)-chains. 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (7) QDPOrderProof (EQUIVALENT) 29.03/8.32 We use the reduction pair processor [LPAR04,JAR06]. 29.03/8.32 29.03/8.32 29.03/8.32 The following pairs can be oriented strictly and are deleted. 29.03/8.32 29.03/8.32 B(b(x1)) -> B(a(b(a(x1)))) 29.03/8.32 The remaining pairs can at least be oriented weakly. 29.03/8.32 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 29.03/8.32 29.03/8.32 <<< 29.03/8.32 POL(B(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 29.03/8.32 >>> 29.03/8.32 29.03/8.32 <<< 29.03/8.32 POL(a(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, -I, 0A], [0A, -I, -I]] * x_1 29.03/8.32 >>> 29.03/8.32 29.03/8.32 <<< 29.03/8.32 POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, 0A]] * x_1 29.03/8.32 >>> 29.03/8.32 29.03/8.32 29.03/8.32 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.03/8.32 29.03/8.32 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 29.03/8.32 b(b(x1)) -> b(a(b(a(x1)))) 29.03/8.32 29.03/8.32 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (8) 29.03/8.32 Obligation: 29.03/8.32 Q DP problem: 29.03/8.32 The TRS P consists of the following rules: 29.03/8.32 29.03/8.32 B(a(a(a(x1)))) -> B(x1) 29.03/8.32 29.03/8.32 The TRS R consists of the following rules: 29.03/8.32 29.03/8.32 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 29.03/8.32 b(b(x1)) -> b(a(b(a(x1)))) 29.03/8.32 29.03/8.32 Q is empty. 29.03/8.32 We have to consider all minimal (P,Q,R)-chains. 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (9) UsableRulesProof (EQUIVALENT) 29.03/8.32 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. 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (10) 29.03/8.32 Obligation: 29.03/8.32 Q DP problem: 29.03/8.32 The TRS P consists of the following rules: 29.03/8.32 29.03/8.32 B(a(a(a(x1)))) -> B(x1) 29.03/8.32 29.03/8.32 R is empty. 29.03/8.32 Q is empty. 29.03/8.32 We have to consider all minimal (P,Q,R)-chains. 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (11) QDPSizeChangeProof (EQUIVALENT) 29.03/8.32 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. 29.03/8.32 29.03/8.32 From the DPs we obtained the following set of size-change graphs: 29.03/8.32 *B(a(a(a(x1)))) -> B(x1) 29.03/8.32 The graph contains the following edges 1 > 1 29.03/8.32 29.03/8.32 29.03/8.32 ---------------------------------------- 29.03/8.32 29.03/8.32 (12) 29.03/8.32 YES 29.31/8.40 EOF