64.58/17.37 YES 64.93/17.44 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 64.93/17.44 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 64.93/17.44 64.93/17.44 64.93/17.44 Termination w.r.t. Q of the given QTRS could be proven: 64.93/17.44 64.93/17.44 (0) QTRS 64.93/17.44 (1) QTRS Reverse [EQUIVALENT, 0 ms] 64.93/17.44 (2) QTRS 64.93/17.44 (3) DependencyPairsProof [EQUIVALENT, 31 ms] 64.93/17.44 (4) QDP 64.93/17.44 (5) DependencyGraphProof [EQUIVALENT, 2 ms] 64.93/17.44 (6) AND 64.93/17.44 (7) QDP 64.93/17.44 (8) UsableRulesProof [EQUIVALENT, 3 ms] 64.93/17.44 (9) QDP 64.93/17.44 (10) MRRProof [EQUIVALENT, 34 ms] 64.93/17.44 (11) QDP 64.93/17.44 (12) DependencyGraphProof [EQUIVALENT, 0 ms] 64.93/17.44 (13) TRUE 64.93/17.44 (14) QDP 64.93/17.44 (15) QDPOrderProof [EQUIVALENT, 236 ms] 64.93/17.44 (16) QDP 64.93/17.44 (17) QDPOrderProof [EQUIVALENT, 790 ms] 64.93/17.44 (18) QDP 64.93/17.44 (19) DependencyGraphProof [EQUIVALENT, 0 ms] 64.93/17.44 (20) QDP 64.93/17.44 (21) UsableRulesProof [EQUIVALENT, 0 ms] 64.93/17.44 (22) QDP 64.93/17.44 (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] 64.93/17.44 (24) YES 64.93/17.44 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (0) 64.93/17.44 Obligation: 64.93/17.44 Q restricted rewrite system: 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 b(a(x1)) -> a(a(d(x1))) 64.93/17.44 a(c(x1)) -> b(b(x1)) 64.93/17.44 d(a(b(x1))) -> b(d(d(c(x1)))) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 b(a(c(a(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (1) QTRS Reverse (EQUIVALENT) 64.93/17.44 We applied the QTRS Reverse Processor [REVERSE]. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (2) 64.93/17.44 Obligation: 64.93/17.44 Q restricted rewrite system: 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (3) DependencyPairsProof (EQUIVALENT) 64.93/17.44 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (4) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 A(b(x1)) -> D(a(a(x1))) 64.93/17.44 A(b(x1)) -> A(a(x1)) 64.93/17.44 A(b(x1)) -> A(x1) 64.93/17.44 C(a(x1)) -> B(b(x1)) 64.93/17.44 C(a(x1)) -> B(x1) 64.93/17.44 B(a(d(x1))) -> C(d(d(b(x1)))) 64.93/17.44 B(a(d(x1))) -> D(d(b(x1))) 64.93/17.44 B(a(d(x1))) -> D(b(x1)) 64.93/17.44 B(a(d(x1))) -> B(x1) 64.93/17.44 D(x1) -> A(x1) 64.93/17.44 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (5) DependencyGraphProof (EQUIVALENT) 64.93/17.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (6) 64.93/17.44 Complex Obligation (AND) 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (7) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 D(x1) -> A(x1) 64.93/17.44 A(b(x1)) -> D(a(a(x1))) 64.93/17.44 A(b(x1)) -> A(a(x1)) 64.93/17.44 A(b(x1)) -> A(x1) 64.93/17.44 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (8) UsableRulesProof (EQUIVALENT) 64.93/17.44 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. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (9) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 D(x1) -> A(x1) 64.93/17.44 A(b(x1)) -> D(a(a(x1))) 64.93/17.44 A(b(x1)) -> A(a(x1)) 64.93/17.44 A(b(x1)) -> A(x1) 64.93/17.44 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (10) MRRProof (EQUIVALENT) 64.93/17.44 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. 64.93/17.44 64.93/17.44 Strictly oriented dependency pairs: 64.93/17.44 64.93/17.44 A(b(x1)) -> D(a(a(x1))) 64.93/17.44 A(b(x1)) -> A(a(x1)) 64.93/17.44 A(b(x1)) -> A(x1) 64.93/17.44 64.93/17.44 Strictly oriented rules of the TRS R: 64.93/17.44 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Used ordering: Polynomial interpretation [POLO]: 64.93/17.44 64.93/17.44 POL(A(x_1)) = 3*x_1 64.93/17.44 POL(D(x_1)) = 3*x_1 64.93/17.44 POL(a(x_1)) = x_1 64.93/17.44 POL(b(x_1)) = 2 + 3*x_1 64.93/17.44 POL(c(x_1)) = 3 + 3*x_1 64.93/17.44 POL(d(x_1)) = x_1 64.93/17.44 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (11) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 D(x1) -> A(x1) 64.93/17.44 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (12) DependencyGraphProof (EQUIVALENT) 64.93/17.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (13) 64.93/17.44 TRUE 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (14) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 B(a(d(x1))) -> C(d(d(b(x1)))) 64.93/17.44 C(a(x1)) -> B(b(x1)) 64.93/17.44 B(a(d(x1))) -> B(x1) 64.93/17.44 C(a(x1)) -> B(x1) 64.93/17.44 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (15) QDPOrderProof (EQUIVALENT) 64.93/17.44 We use the reduction pair processor [LPAR04,JAR06]. 64.93/17.44 64.93/17.44 64.93/17.44 The following pairs can be oriented strictly and are deleted. 64.93/17.44 64.93/17.44 C(a(x1)) -> B(b(x1)) 64.93/17.44 The remaining pairs can at least be oriented weakly. 64.93/17.44 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(a(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(d(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(C(x_1)) = [[1A]] + [[-I, 0A, 0A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(b(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(c(x_1)) = [[1A], [1A], [0A]] + [[-I, -I, 0A], [0A, -I, 0A], [-I, -I, -I]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 64.93/17.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 64.93/17.44 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (16) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 B(a(d(x1))) -> C(d(d(b(x1)))) 64.93/17.44 B(a(d(x1))) -> B(x1) 64.93/17.44 C(a(x1)) -> B(x1) 64.93/17.44 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (17) QDPOrderProof (EQUIVALENT) 64.93/17.44 We use the reduction pair processor [LPAR04,JAR06]. 64.93/17.44 64.93/17.44 64.93/17.44 The following pairs can be oriented strictly and are deleted. 64.93/17.44 64.93/17.44 C(a(x1)) -> B(x1) 64.93/17.44 The remaining pairs can at least be oriented weakly. 64.93/17.44 Used ordering: Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]: 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(B(x_1)) = [[0A]] + [[-1A, -1A, -1A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(a(x_1)) = [[0A], [0A], [-1A]] + [[-I, 1A, -1A], [-I, -1A, -1A], [-1A, -1A, -1A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(d(x_1)) = [[0A], [1A], [-1A]] + [[-I, 1A, -1A], [-1A, -1A, -1A], [-1A, -1A, -1A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(C(x_1)) = [[1A]] + [[-1A, -1A, 1A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(b(x_1)) = [[0A], [-1A], [2A]] + [[-1A, -1A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 <<< 64.93/17.44 POL(c(x_1)) = [[1A], [-1A], [2A]] + [[-1A, -1A, -1A], [-I, -I, -I], [-1A, -1A, 2A]] * x_1 64.93/17.44 >>> 64.93/17.44 64.93/17.44 64.93/17.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 64.93/17.44 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (18) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 B(a(d(x1))) -> C(d(d(b(x1)))) 64.93/17.44 B(a(d(x1))) -> B(x1) 64.93/17.44 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (19) DependencyGraphProof (EQUIVALENT) 64.93/17.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (20) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 B(a(d(x1))) -> B(x1) 64.93/17.44 64.93/17.44 The TRS R consists of the following rules: 64.93/17.44 64.93/17.44 a(b(x1)) -> d(a(a(x1))) 64.93/17.44 c(a(x1)) -> b(b(x1)) 64.93/17.44 b(a(d(x1))) -> c(d(d(b(x1)))) 64.93/17.44 d(x1) -> a(x1) 64.93/17.44 a(c(a(b(x1)))) -> x1 64.93/17.44 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (21) UsableRulesProof (EQUIVALENT) 64.93/17.44 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. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (22) 64.93/17.44 Obligation: 64.93/17.44 Q DP problem: 64.93/17.44 The TRS P consists of the following rules: 64.93/17.44 64.93/17.44 B(a(d(x1))) -> B(x1) 64.93/17.44 64.93/17.44 R is empty. 64.93/17.44 Q is empty. 64.93/17.44 We have to consider all minimal (P,Q,R)-chains. 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (23) QDPSizeChangeProof (EQUIVALENT) 64.93/17.44 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. 64.93/17.44 64.93/17.44 From the DPs we obtained the following set of size-change graphs: 64.93/17.44 *B(a(d(x1))) -> B(x1) 64.93/17.44 The graph contains the following edges 1 > 1 64.93/17.44 64.93/17.44 64.93/17.44 ---------------------------------------- 64.93/17.44 64.93/17.44 (24) 64.93/17.44 YES 65.01/17.58 EOF