28.27/8.21 YES 28.31/8.23 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 28.31/8.23 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.31/8.23 28.31/8.23 28.31/8.23 Termination w.r.t. Q of the given QTRS could be proven: 28.31/8.23 28.31/8.23 (0) QTRS 28.31/8.23 (1) QTRS Reverse [EQUIVALENT, 0 ms] 28.31/8.23 (2) QTRS 28.31/8.23 (3) DependencyPairsProof [EQUIVALENT, 29 ms] 28.31/8.23 (4) QDP 28.31/8.23 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 28.31/8.23 (6) QDP 28.31/8.23 (7) QDPOrderProof [EQUIVALENT, 253 ms] 28.31/8.23 (8) QDP 28.31/8.23 (9) UsableRulesProof [EQUIVALENT, 0 ms] 28.31/8.23 (10) QDP 28.31/8.23 (11) MRRProof [EQUIVALENT, 0 ms] 28.31/8.23 (12) QDP 28.31/8.23 (13) PisEmptyProof [EQUIVALENT, 0 ms] 28.31/8.23 (14) YES 28.31/8.23 28.31/8.23 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (0) 28.31/8.23 Obligation: 28.31/8.23 Q restricted rewrite system: 28.31/8.23 The TRS R consists of the following rules: 28.31/8.23 28.31/8.23 a(b(x1)) -> x1 28.31/8.23 a(b(c(x1))) -> b(c(b(c(a(a(b(x1))))))) 28.31/8.23 28.31/8.23 Q is empty. 28.31/8.23 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (1) QTRS Reverse (EQUIVALENT) 28.31/8.23 We applied the QTRS Reverse Processor [REVERSE]. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (2) 28.31/8.23 Obligation: 28.31/8.23 Q restricted rewrite system: 28.31/8.23 The TRS R consists of the following rules: 28.31/8.23 28.31/8.23 b(a(x1)) -> x1 28.31/8.23 c(b(a(x1))) -> b(a(a(c(b(c(b(x1))))))) 28.31/8.23 28.31/8.23 Q is empty. 28.31/8.23 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (3) DependencyPairsProof (EQUIVALENT) 28.31/8.23 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (4) 28.31/8.23 Obligation: 28.31/8.23 Q DP problem: 28.31/8.23 The TRS P consists of the following rules: 28.31/8.23 28.31/8.23 C(b(a(x1))) -> B(a(a(c(b(c(b(x1))))))) 28.31/8.23 C(b(a(x1))) -> C(b(c(b(x1)))) 28.31/8.23 C(b(a(x1))) -> B(c(b(x1))) 28.31/8.23 C(b(a(x1))) -> C(b(x1)) 28.31/8.23 C(b(a(x1))) -> B(x1) 28.31/8.23 28.31/8.23 The TRS R consists of the following rules: 28.31/8.23 28.31/8.23 b(a(x1)) -> x1 28.31/8.23 c(b(a(x1))) -> b(a(a(c(b(c(b(x1))))))) 28.31/8.23 28.31/8.23 Q is empty. 28.31/8.23 We have to consider all minimal (P,Q,R)-chains. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (5) DependencyGraphProof (EQUIVALENT) 28.31/8.23 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (6) 28.31/8.23 Obligation: 28.31/8.23 Q DP problem: 28.31/8.23 The TRS P consists of the following rules: 28.31/8.23 28.31/8.23 C(b(a(x1))) -> C(b(x1)) 28.31/8.23 C(b(a(x1))) -> C(b(c(b(x1)))) 28.31/8.23 28.31/8.23 The TRS R consists of the following rules: 28.31/8.23 28.31/8.23 b(a(x1)) -> x1 28.31/8.23 c(b(a(x1))) -> b(a(a(c(b(c(b(x1))))))) 28.31/8.23 28.31/8.23 Q is empty. 28.31/8.23 We have to consider all minimal (P,Q,R)-chains. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (7) QDPOrderProof (EQUIVALENT) 28.31/8.23 We use the reduction pair processor [LPAR04,JAR06]. 28.31/8.23 28.31/8.23 28.31/8.23 The following pairs can be oriented strictly and are deleted. 28.31/8.23 28.31/8.23 C(b(a(x1))) -> C(b(c(b(x1)))) 28.31/8.23 The remaining pairs can at least be oriented weakly. 28.31/8.23 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 28.31/8.23 28.31/8.23 <<< 28.31/8.23 POL(C(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 28.31/8.23 >>> 28.31/8.23 28.31/8.23 <<< 28.31/8.23 POL(b(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, -I], [-I, 0A, -I]] * x_1 28.31/8.23 >>> 28.31/8.23 28.31/8.23 <<< 28.31/8.23 POL(a(x_1)) = [[-I], [1A], [-I]] + [[0A, 0A, 0A], [0A, 1A, 0A], [0A, 0A, 0A]] * x_1 28.31/8.23 >>> 28.31/8.23 28.31/8.23 <<< 28.31/8.23 POL(c(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 28.31/8.23 >>> 28.31/8.23 28.31/8.23 28.31/8.23 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.31/8.23 28.31/8.23 b(a(x1)) -> x1 28.31/8.23 c(b(a(x1))) -> b(a(a(c(b(c(b(x1))))))) 28.31/8.23 28.31/8.23 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (8) 28.31/8.23 Obligation: 28.31/8.23 Q DP problem: 28.31/8.23 The TRS P consists of the following rules: 28.31/8.23 28.31/8.23 C(b(a(x1))) -> C(b(x1)) 28.31/8.23 28.31/8.23 The TRS R consists of the following rules: 28.31/8.23 28.31/8.23 b(a(x1)) -> x1 28.31/8.23 c(b(a(x1))) -> b(a(a(c(b(c(b(x1))))))) 28.31/8.23 28.31/8.23 Q is empty. 28.31/8.23 We have to consider all minimal (P,Q,R)-chains. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (9) UsableRulesProof (EQUIVALENT) 28.31/8.23 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. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (10) 28.31/8.23 Obligation: 28.31/8.23 Q DP problem: 28.31/8.23 The TRS P consists of the following rules: 28.31/8.23 28.31/8.23 C(b(a(x1))) -> C(b(x1)) 28.31/8.23 28.31/8.23 The TRS R consists of the following rules: 28.31/8.23 28.31/8.23 b(a(x1)) -> x1 28.31/8.23 28.31/8.23 Q is empty. 28.31/8.23 We have to consider all minimal (P,Q,R)-chains. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (11) MRRProof (EQUIVALENT) 28.31/8.23 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. 28.31/8.23 28.31/8.23 Strictly oriented dependency pairs: 28.31/8.23 28.31/8.23 C(b(a(x1))) -> C(b(x1)) 28.31/8.23 28.31/8.23 Strictly oriented rules of the TRS R: 28.31/8.23 28.31/8.23 b(a(x1)) -> x1 28.31/8.23 28.31/8.23 Used ordering: Polynomial interpretation [POLO]: 28.31/8.23 28.31/8.23 POL(C(x_1)) = x_1 28.31/8.23 POL(a(x_1)) = 3 + x_1 28.31/8.23 POL(b(x_1)) = x_1 28.31/8.23 28.31/8.23 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (12) 28.31/8.23 Obligation: 28.31/8.23 Q DP problem: 28.31/8.23 P is empty. 28.31/8.23 R is empty. 28.31/8.23 Q is empty. 28.31/8.23 We have to consider all minimal (P,Q,R)-chains. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (13) PisEmptyProof (EQUIVALENT) 28.31/8.23 The TRS P is empty. Hence, there is no (P,Q,R) chain. 28.31/8.23 ---------------------------------------- 28.31/8.23 28.31/8.23 (14) 28.31/8.23 YES 28.63/8.34 EOF