42.73/12.20 YES 42.91/12.25 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 42.91/12.25 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 42.91/12.25 42.91/12.25 42.91/12.25 Termination w.r.t. Q of the given QTRS could be proven: 42.91/12.25 42.91/12.25 (0) QTRS 42.91/12.25 (1) QTRS Reverse [EQUIVALENT, 0 ms] 42.91/12.25 (2) QTRS 42.91/12.25 (3) DependencyPairsProof [EQUIVALENT, 6 ms] 42.91/12.25 (4) QDP 42.91/12.25 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 42.91/12.25 (6) QDP 42.91/12.25 (7) QDPOrderProof [EQUIVALENT, 98 ms] 42.91/12.25 (8) QDP 42.91/12.25 (9) QDPOrderProof [EQUIVALENT, 17 ms] 42.91/12.25 (10) QDP 42.91/12.25 (11) UsableRulesProof [EQUIVALENT, 0 ms] 42.91/12.25 (12) QDP 42.91/12.25 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.91/12.25 (14) YES 42.91/12.25 42.91/12.25 42.91/12.25 ---------------------------------------- 42.91/12.25 42.91/12.25 (0) 42.91/12.25 Obligation: 42.91/12.25 Q restricted rewrite system: 42.91/12.25 The TRS R consists of the following rules: 42.91/12.25 42.91/12.25 a(a(x1)) -> x1 42.91/12.25 a(b(x1)) -> x1 42.91/12.25 a(b(b(x1))) -> b(b(a(b(a(x1))))) 42.91/12.25 42.91/12.25 Q is empty. 42.91/12.25 42.91/12.25 ---------------------------------------- 42.91/12.25 42.91/12.25 (1) QTRS Reverse (EQUIVALENT) 42.91/12.25 We applied the QTRS Reverse Processor [REVERSE]. 42.91/12.25 ---------------------------------------- 42.91/12.25 42.91/12.25 (2) 42.91/12.25 Obligation: 42.91/12.25 Q restricted rewrite system: 42.91/12.25 The TRS R consists of the following rules: 42.91/12.26 42.91/12.26 a(a(x1)) -> x1 42.91/12.26 b(a(x1)) -> x1 42.91/12.26 b(b(a(x1))) -> a(b(a(b(b(x1))))) 42.91/12.26 42.91/12.26 Q is empty. 42.91/12.26 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (3) DependencyPairsProof (EQUIVALENT) 42.91/12.26 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (4) 42.91/12.26 Obligation: 42.91/12.26 Q DP problem: 42.91/12.26 The TRS P consists of the following rules: 42.91/12.26 42.91/12.26 B(b(a(x1))) -> A(b(a(b(b(x1))))) 42.91/12.26 B(b(a(x1))) -> B(a(b(b(x1)))) 42.91/12.26 B(b(a(x1))) -> A(b(b(x1))) 42.91/12.26 B(b(a(x1))) -> B(b(x1)) 42.91/12.26 B(b(a(x1))) -> B(x1) 42.91/12.26 42.91/12.26 The TRS R consists of the following rules: 42.91/12.26 42.91/12.26 a(a(x1)) -> x1 42.91/12.26 b(a(x1)) -> x1 42.91/12.26 b(b(a(x1))) -> a(b(a(b(b(x1))))) 42.91/12.26 42.91/12.26 Q is empty. 42.91/12.26 We have to consider all minimal (P,Q,R)-chains. 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (5) DependencyGraphProof (EQUIVALENT) 42.91/12.26 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (6) 42.91/12.26 Obligation: 42.91/12.26 Q DP problem: 42.91/12.26 The TRS P consists of the following rules: 42.91/12.26 42.91/12.26 B(b(a(x1))) -> B(b(x1)) 42.91/12.26 B(b(a(x1))) -> B(a(b(b(x1)))) 42.91/12.26 B(b(a(x1))) -> B(x1) 42.91/12.26 42.91/12.26 The TRS R consists of the following rules: 42.91/12.26 42.91/12.26 a(a(x1)) -> x1 42.91/12.26 b(a(x1)) -> x1 42.91/12.26 b(b(a(x1))) -> a(b(a(b(b(x1))))) 42.91/12.26 42.91/12.26 Q is empty. 42.91/12.26 We have to consider all minimal (P,Q,R)-chains. 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (7) QDPOrderProof (EQUIVALENT) 42.91/12.26 We use the reduction pair processor [LPAR04,JAR06]. 42.91/12.26 42.91/12.26 42.91/12.26 The following pairs can be oriented strictly and are deleted. 42.91/12.26 42.91/12.26 B(b(a(x1))) -> B(a(b(b(x1)))) 42.91/12.26 The remaining pairs can at least be oriented weakly. 42.91/12.26 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 42.91/12.26 42.91/12.26 <<< 42.91/12.26 POL(B(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 42.91/12.26 >>> 42.91/12.26 42.91/12.26 <<< 42.91/12.26 POL(b(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [0A, -I, -I]] * x_1 42.91/12.26 >>> 42.91/12.26 42.91/12.26 <<< 42.91/12.26 POL(a(x_1)) = [[0A], [-I], [1A]] + [[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 42.91/12.26 >>> 42.91/12.26 42.91/12.26 42.91/12.26 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.91/12.26 42.91/12.26 b(a(x1)) -> x1 42.91/12.26 b(b(a(x1))) -> a(b(a(b(b(x1))))) 42.91/12.26 a(a(x1)) -> x1 42.91/12.26 42.91/12.26 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (8) 42.91/12.26 Obligation: 42.91/12.26 Q DP problem: 42.91/12.26 The TRS P consists of the following rules: 42.91/12.26 42.91/12.26 B(b(a(x1))) -> B(b(x1)) 42.91/12.26 B(b(a(x1))) -> B(x1) 42.91/12.26 42.91/12.26 The TRS R consists of the following rules: 42.91/12.26 42.91/12.26 a(a(x1)) -> x1 42.91/12.26 b(a(x1)) -> x1 42.91/12.26 b(b(a(x1))) -> a(b(a(b(b(x1))))) 42.91/12.26 42.91/12.26 Q is empty. 42.91/12.26 We have to consider all minimal (P,Q,R)-chains. 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (9) QDPOrderProof (EQUIVALENT) 42.91/12.26 We use the reduction pair processor [LPAR04,JAR06]. 42.91/12.26 42.91/12.26 42.91/12.26 The following pairs can be oriented strictly and are deleted. 42.91/12.26 42.91/12.26 B(b(a(x1))) -> B(b(x1)) 42.91/12.26 The remaining pairs can at least be oriented weakly. 42.91/12.26 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 42.91/12.26 42.91/12.26 <<< 42.91/12.26 POL(B(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 42.91/12.26 >>> 42.91/12.26 42.91/12.26 <<< 42.91/12.26 POL(b(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, -I], [-I, -I, 0A], [-I, 0A, -I]] * x_1 42.91/12.26 >>> 42.91/12.26 42.91/12.26 <<< 42.91/12.26 POL(a(x_1)) = [[-I], [-I], [1A]] + [[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 42.91/12.26 >>> 42.91/12.26 42.91/12.26 42.91/12.26 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.91/12.26 42.91/12.26 b(a(x1)) -> x1 42.91/12.26 b(b(a(x1))) -> a(b(a(b(b(x1))))) 42.91/12.26 a(a(x1)) -> x1 42.91/12.26 42.91/12.26 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (10) 42.91/12.26 Obligation: 42.91/12.26 Q DP problem: 42.91/12.26 The TRS P consists of the following rules: 42.91/12.26 42.91/12.26 B(b(a(x1))) -> B(x1) 42.91/12.26 42.91/12.26 The TRS R consists of the following rules: 42.91/12.26 42.91/12.26 a(a(x1)) -> x1 42.91/12.26 b(a(x1)) -> x1 42.91/12.26 b(b(a(x1))) -> a(b(a(b(b(x1))))) 42.91/12.26 42.91/12.26 Q is empty. 42.91/12.26 We have to consider all minimal (P,Q,R)-chains. 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (11) UsableRulesProof (EQUIVALENT) 42.91/12.26 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. 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (12) 42.91/12.26 Obligation: 42.91/12.26 Q DP problem: 42.91/12.26 The TRS P consists of the following rules: 42.91/12.26 42.91/12.26 B(b(a(x1))) -> B(x1) 42.91/12.26 42.91/12.26 R is empty. 42.91/12.26 Q is empty. 42.91/12.26 We have to consider all minimal (P,Q,R)-chains. 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (13) QDPSizeChangeProof (EQUIVALENT) 42.91/12.26 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. 42.91/12.26 42.91/12.26 From the DPs we obtained the following set of size-change graphs: 42.91/12.26 *B(b(a(x1))) -> B(x1) 42.91/12.26 The graph contains the following edges 1 > 1 42.91/12.26 42.91/12.26 42.91/12.26 ---------------------------------------- 42.91/12.26 42.91/12.26 (14) 42.91/12.26 YES 43.11/12.35 EOF