253.50/65.30 YES 253.80/65.35 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 253.80/65.35 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 253.80/65.35 253.80/65.35 253.80/65.35 Termination w.r.t. Q of the given QTRS could be proven: 253.80/65.35 253.80/65.35 (0) QTRS 253.80/65.35 (1) QTRS Reverse [EQUIVALENT, 0 ms] 253.80/65.35 (2) QTRS 253.80/65.35 (3) DependencyPairsProof [EQUIVALENT, 2 ms] 253.80/65.35 (4) QDP 253.80/65.35 (5) MRRProof [EQUIVALENT, 96 ms] 253.80/65.35 (6) QDP 253.80/65.35 (7) QDPOrderProof [EQUIVALENT, 3249 ms] 253.80/65.35 (8) QDP 253.80/65.35 (9) PisEmptyProof [EQUIVALENT, 0 ms] 253.80/65.35 (10) YES 253.80/65.35 253.80/65.35 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (0) 253.80/65.35 Obligation: 253.80/65.35 Q restricted rewrite system: 253.80/65.35 The TRS R consists of the following rules: 253.80/65.35 253.80/65.35 b(b(b(a(x1)))) -> b(a(a(b(x1)))) 253.80/65.35 b(a(b(a(x1)))) -> a(a(a(b(x1)))) 253.80/65.35 a(a(b(b(x1)))) -> b(b(b(a(x1)))) 253.80/65.35 253.80/65.35 Q is empty. 253.80/65.35 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (1) QTRS Reverse (EQUIVALENT) 253.80/65.35 We applied the QTRS Reverse Processor [REVERSE]. 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (2) 253.80/65.35 Obligation: 253.80/65.35 Q restricted rewrite system: 253.80/65.35 The TRS R consists of the following rules: 253.80/65.35 253.80/65.35 a(b(b(b(x1)))) -> b(a(a(b(x1)))) 253.80/65.35 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 253.80/65.35 b(b(a(a(x1)))) -> a(b(b(b(x1)))) 253.80/65.35 253.80/65.35 Q is empty. 253.80/65.35 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (3) DependencyPairsProof (EQUIVALENT) 253.80/65.35 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (4) 253.80/65.35 Obligation: 253.80/65.35 Q DP problem: 253.80/65.35 The TRS P consists of the following rules: 253.80/65.35 253.80/65.35 A(b(b(b(x1)))) -> B(a(a(b(x1)))) 253.80/65.35 A(b(b(b(x1)))) -> A(a(b(x1))) 253.80/65.35 A(b(b(b(x1)))) -> A(b(x1)) 253.80/65.35 A(b(a(b(x1)))) -> B(a(a(a(x1)))) 253.80/65.35 A(b(a(b(x1)))) -> A(a(a(x1))) 253.80/65.35 A(b(a(b(x1)))) -> A(a(x1)) 253.80/65.35 A(b(a(b(x1)))) -> A(x1) 253.80/65.35 B(b(a(a(x1)))) -> A(b(b(b(x1)))) 253.80/65.35 B(b(a(a(x1)))) -> B(b(b(x1))) 253.80/65.35 B(b(a(a(x1)))) -> B(b(x1)) 253.80/65.35 B(b(a(a(x1)))) -> B(x1) 253.80/65.35 253.80/65.35 The TRS R consists of the following rules: 253.80/65.35 253.80/65.35 a(b(b(b(x1)))) -> b(a(a(b(x1)))) 253.80/65.35 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 253.80/65.35 b(b(a(a(x1)))) -> a(b(b(b(x1)))) 253.80/65.35 253.80/65.35 Q is empty. 253.80/65.35 We have to consider all minimal (P,Q,R)-chains. 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (5) MRRProof (EQUIVALENT) 253.80/65.35 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 253.80/65.35 253.80/65.35 Strictly oriented dependency pairs: 253.80/65.35 253.80/65.35 A(b(b(b(x1)))) -> A(a(b(x1))) 253.80/65.35 A(b(b(b(x1)))) -> A(b(x1)) 253.80/65.35 A(b(a(b(x1)))) -> A(a(a(x1))) 253.80/65.35 A(b(a(b(x1)))) -> A(a(x1)) 253.80/65.35 A(b(a(b(x1)))) -> A(x1) 253.80/65.35 B(b(a(a(x1)))) -> B(b(b(x1))) 253.80/65.35 B(b(a(a(x1)))) -> B(b(x1)) 253.80/65.35 B(b(a(a(x1)))) -> B(x1) 253.80/65.35 253.80/65.35 253.80/65.35 Used ordering: Polynomial interpretation [POLO]: 253.80/65.35 253.80/65.35 POL(A(x_1)) = x_1 253.80/65.35 POL(B(x_1)) = x_1 253.80/65.35 POL(a(x_1)) = 1 + x_1 253.80/65.35 POL(b(x_1)) = 1 + x_1 253.80/65.35 253.80/65.35 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (6) 253.80/65.35 Obligation: 253.80/65.35 Q DP problem: 253.80/65.35 The TRS P consists of the following rules: 253.80/65.35 253.80/65.35 A(b(b(b(x1)))) -> B(a(a(b(x1)))) 253.80/65.35 A(b(a(b(x1)))) -> B(a(a(a(x1)))) 253.80/65.35 B(b(a(a(x1)))) -> A(b(b(b(x1)))) 253.80/65.35 253.80/65.35 The TRS R consists of the following rules: 253.80/65.35 253.80/65.35 a(b(b(b(x1)))) -> b(a(a(b(x1)))) 253.80/65.35 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 253.80/65.35 b(b(a(a(x1)))) -> a(b(b(b(x1)))) 253.80/65.35 253.80/65.35 Q is empty. 253.80/65.35 We have to consider all minimal (P,Q,R)-chains. 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (7) QDPOrderProof (EQUIVALENT) 253.80/65.35 We use the reduction pair processor [LPAR04,JAR06]. 253.80/65.35 253.80/65.35 253.80/65.35 The following pairs can be oriented strictly and are deleted. 253.80/65.35 253.80/65.35 A(b(b(b(x1)))) -> B(a(a(b(x1)))) 253.80/65.35 A(b(a(b(x1)))) -> B(a(a(a(x1)))) 253.80/65.35 B(b(a(a(x1)))) -> A(b(b(b(x1)))) 253.80/65.35 The remaining pairs can at least be oriented weakly. 253.80/65.35 Used ordering: Polynomial interpretation [POLO,RATPOLO]: 253.80/65.35 253.80/65.35 POL(A(x_1)) = [5/2] + [4]x_1 253.80/65.35 POL(B(x_1)) = [4] + [4]x_1 253.80/65.35 POL(a(x_1)) = [3/4]x_1 253.80/65.35 POL(b(x_1)) = [1/4] + [3/4]x_1 253.80/65.35 The value of delta used in the strict ordering is 1/16. 253.80/65.35 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 253.80/65.35 253.80/65.35 b(b(a(a(x1)))) -> a(b(b(b(x1)))) 253.80/65.35 a(b(b(b(x1)))) -> b(a(a(b(x1)))) 253.80/65.35 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 253.80/65.35 253.80/65.35 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (8) 253.80/65.35 Obligation: 253.80/65.35 Q DP problem: 253.80/65.35 P is empty. 253.80/65.35 The TRS R consists of the following rules: 253.80/65.35 253.80/65.35 a(b(b(b(x1)))) -> b(a(a(b(x1)))) 253.80/65.35 a(b(a(b(x1)))) -> b(a(a(a(x1)))) 253.80/65.35 b(b(a(a(x1)))) -> a(b(b(b(x1)))) 253.80/65.35 253.80/65.35 Q is empty. 253.80/65.35 We have to consider all minimal (P,Q,R)-chains. 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (9) PisEmptyProof (EQUIVALENT) 253.80/65.35 The TRS P is empty. Hence, there is no (P,Q,R) chain. 253.80/65.35 ---------------------------------------- 253.80/65.35 253.80/65.35 (10) 253.80/65.35 YES 254.10/65.40 EOF