25.64/7.63 YES 26.38/7.81 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 26.38/7.81 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.38/7.81 26.38/7.81 26.38/7.81 Termination w.r.t. Q of the given QTRS could be proven: 26.38/7.81 26.38/7.81 (0) QTRS 26.38/7.81 (1) QTRS Reverse [EQUIVALENT, 0 ms] 26.38/7.81 (2) QTRS 26.38/7.81 (3) DependencyPairsProof [EQUIVALENT, 4 ms] 26.38/7.81 (4) QDP 26.38/7.81 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 26.38/7.81 (6) QDP 26.38/7.81 (7) QDPOrderProof [EQUIVALENT, 61 ms] 26.38/7.81 (8) QDP 26.38/7.81 (9) UsableRulesProof [EQUIVALENT, 0 ms] 26.38/7.81 (10) QDP 26.38/7.81 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 26.38/7.81 (12) YES 26.38/7.81 26.38/7.81 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (0) 26.38/7.81 Obligation: 26.38/7.81 Q restricted rewrite system: 26.38/7.81 The TRS R consists of the following rules: 26.38/7.81 26.38/7.81 a(x1) -> x1 26.38/7.81 a(a(b(x1))) -> b(a(b(a(x1)))) 26.38/7.81 b(b(x1)) -> c(a(x1)) 26.38/7.81 26.38/7.81 Q is empty. 26.38/7.81 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (1) QTRS Reverse (EQUIVALENT) 26.38/7.81 We applied the QTRS Reverse Processor [REVERSE]. 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (2) 26.38/7.81 Obligation: 26.38/7.81 Q restricted rewrite system: 26.38/7.81 The TRS R consists of the following rules: 26.38/7.81 26.38/7.81 a(x1) -> x1 26.38/7.81 b(a(a(x1))) -> a(b(a(b(x1)))) 26.38/7.81 b(b(x1)) -> a(c(x1)) 26.38/7.81 26.38/7.81 Q is empty. 26.38/7.81 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (3) DependencyPairsProof (EQUIVALENT) 26.38/7.81 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (4) 26.38/7.81 Obligation: 26.38/7.81 Q DP problem: 26.38/7.81 The TRS P consists of the following rules: 26.38/7.81 26.38/7.81 B(a(a(x1))) -> A(b(a(b(x1)))) 26.38/7.81 B(a(a(x1))) -> B(a(b(x1))) 26.38/7.81 B(a(a(x1))) -> A(b(x1)) 26.38/7.81 B(a(a(x1))) -> B(x1) 26.38/7.81 B(b(x1)) -> A(c(x1)) 26.38/7.81 26.38/7.81 The TRS R consists of the following rules: 26.38/7.81 26.38/7.81 a(x1) -> x1 26.38/7.81 b(a(a(x1))) -> a(b(a(b(x1)))) 26.38/7.81 b(b(x1)) -> a(c(x1)) 26.38/7.81 26.38/7.81 Q is empty. 26.38/7.81 We have to consider all minimal (P,Q,R)-chains. 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (5) DependencyGraphProof (EQUIVALENT) 26.38/7.81 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (6) 26.38/7.81 Obligation: 26.38/7.81 Q DP problem: 26.38/7.81 The TRS P consists of the following rules: 26.38/7.81 26.38/7.81 B(a(a(x1))) -> B(x1) 26.38/7.81 B(a(a(x1))) -> B(a(b(x1))) 26.38/7.81 26.38/7.81 The TRS R consists of the following rules: 26.38/7.81 26.38/7.81 a(x1) -> x1 26.38/7.81 b(a(a(x1))) -> a(b(a(b(x1)))) 26.38/7.81 b(b(x1)) -> a(c(x1)) 26.38/7.81 26.38/7.81 Q is empty. 26.38/7.81 We have to consider all minimal (P,Q,R)-chains. 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (7) QDPOrderProof (EQUIVALENT) 26.38/7.81 We use the reduction pair processor [LPAR04,JAR06]. 26.38/7.81 26.38/7.81 26.38/7.81 The following pairs can be oriented strictly and are deleted. 26.38/7.81 26.38/7.81 B(a(a(x1))) -> B(a(b(x1))) 26.38/7.81 The remaining pairs can at least be oriented weakly. 26.38/7.81 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 26.38/7.81 26.38/7.81 <<< 26.38/7.81 POL(B(x_1)) = [[0A]] + [[-I, 1A, -I]] * x_1 26.38/7.81 >>> 26.38/7.81 26.38/7.81 <<< 26.38/7.81 POL(a(x_1)) = [[1A], [0A], [0A]] + [[0A, -I, 1A], [1A, 0A, 0A], [1A, 0A, 0A]] * x_1 26.38/7.81 >>> 26.38/7.81 26.38/7.81 <<< 26.38/7.81 POL(b(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, 0A], [0A, -I, 0A], [-I, -I, 0A]] * x_1 26.38/7.81 >>> 26.38/7.81 26.38/7.81 <<< 26.38/7.81 POL(c(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, -I, 0A], [-I, -I, -I]] * x_1 26.38/7.81 >>> 26.38/7.81 26.38/7.81 26.38/7.81 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.38/7.81 26.38/7.81 b(a(a(x1))) -> a(b(a(b(x1)))) 26.38/7.81 b(b(x1)) -> a(c(x1)) 26.38/7.81 a(x1) -> x1 26.38/7.81 26.38/7.81 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (8) 26.38/7.81 Obligation: 26.38/7.81 Q DP problem: 26.38/7.81 The TRS P consists of the following rules: 26.38/7.81 26.38/7.81 B(a(a(x1))) -> B(x1) 26.38/7.81 26.38/7.81 The TRS R consists of the following rules: 26.38/7.81 26.38/7.81 a(x1) -> x1 26.38/7.81 b(a(a(x1))) -> a(b(a(b(x1)))) 26.38/7.81 b(b(x1)) -> a(c(x1)) 26.38/7.81 26.38/7.81 Q is empty. 26.38/7.81 We have to consider all minimal (P,Q,R)-chains. 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (9) UsableRulesProof (EQUIVALENT) 26.38/7.81 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. 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (10) 26.38/7.81 Obligation: 26.38/7.81 Q DP problem: 26.38/7.81 The TRS P consists of the following rules: 26.38/7.81 26.38/7.81 B(a(a(x1))) -> B(x1) 26.38/7.81 26.38/7.81 R is empty. 26.38/7.81 Q is empty. 26.38/7.81 We have to consider all minimal (P,Q,R)-chains. 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (11) QDPSizeChangeProof (EQUIVALENT) 26.38/7.81 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. 26.38/7.81 26.38/7.81 From the DPs we obtained the following set of size-change graphs: 26.38/7.81 *B(a(a(x1))) -> B(x1) 26.38/7.81 The graph contains the following edges 1 > 1 26.38/7.81 26.38/7.81 26.38/7.81 ---------------------------------------- 26.38/7.81 26.38/7.81 (12) 26.38/7.81 YES 26.38/7.89 EOF