20.25/5.95 YES 22.44/8.63 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 22.44/8.63 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 22.44/8.63 22.44/8.63 22.44/8.63 Termination w.r.t. Q of the given QTRS could be proven: 22.44/8.63 22.44/8.63 (0) QTRS 22.44/8.63 (1) QTRS Reverse [EQUIVALENT, 0 ms] 22.44/8.63 (2) QTRS 22.44/8.63 (3) DependencyPairsProof [EQUIVALENT, 24 ms] 22.44/8.63 (4) QDP 22.44/8.63 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 22.44/8.63 (6) QDP 22.44/8.63 (7) QDPOrderProof [EQUIVALENT, 133 ms] 22.44/8.63 (8) QDP 22.44/8.63 (9) UsableRulesProof [EQUIVALENT, 0 ms] 22.44/8.63 (10) QDP 22.44/8.63 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 22.44/8.63 (12) YES 22.44/8.63 22.44/8.63 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (0) 22.44/8.63 Obligation: 22.44/8.63 Q restricted rewrite system: 22.44/8.63 The TRS R consists of the following rules: 22.44/8.63 22.44/8.63 a(x1) -> x1 22.44/8.63 a(a(b(x1))) -> b(b(c(a(a(a(x1)))))) 22.44/8.63 c(a(x1)) -> x1 22.44/8.63 22.44/8.63 Q is empty. 22.44/8.63 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (1) QTRS Reverse (EQUIVALENT) 22.44/8.63 We applied the QTRS Reverse Processor [REVERSE]. 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (2) 22.44/8.63 Obligation: 22.44/8.63 Q restricted rewrite system: 22.44/8.63 The TRS R consists of the following rules: 22.44/8.63 22.44/8.63 a(x1) -> x1 22.44/8.63 b(a(a(x1))) -> a(a(a(c(b(b(x1)))))) 22.44/8.63 a(c(x1)) -> x1 22.44/8.63 22.44/8.63 Q is empty. 22.44/8.63 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (3) DependencyPairsProof (EQUIVALENT) 22.44/8.63 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (4) 22.44/8.63 Obligation: 22.44/8.63 Q DP problem: 22.44/8.63 The TRS P consists of the following rules: 22.44/8.63 22.44/8.63 B(a(a(x1))) -> A(a(a(c(b(b(x1)))))) 22.44/8.63 B(a(a(x1))) -> A(a(c(b(b(x1))))) 22.44/8.63 B(a(a(x1))) -> A(c(b(b(x1)))) 22.44/8.63 B(a(a(x1))) -> B(b(x1)) 22.44/8.63 B(a(a(x1))) -> B(x1) 22.44/8.63 22.44/8.63 The TRS R consists of the following rules: 22.44/8.63 22.44/8.63 a(x1) -> x1 22.44/8.63 b(a(a(x1))) -> a(a(a(c(b(b(x1)))))) 22.44/8.63 a(c(x1)) -> x1 22.44/8.63 22.44/8.63 Q is empty. 22.44/8.63 We have to consider all minimal (P,Q,R)-chains. 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (5) DependencyGraphProof (EQUIVALENT) 22.44/8.63 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (6) 22.44/8.63 Obligation: 22.44/8.63 Q DP problem: 22.44/8.63 The TRS P consists of the following rules: 22.44/8.63 22.44/8.63 B(a(a(x1))) -> B(x1) 22.44/8.63 B(a(a(x1))) -> B(b(x1)) 22.44/8.63 22.44/8.63 The TRS R consists of the following rules: 22.44/8.63 22.44/8.63 a(x1) -> x1 22.44/8.63 b(a(a(x1))) -> a(a(a(c(b(b(x1)))))) 22.44/8.63 a(c(x1)) -> x1 22.44/8.63 22.44/8.63 Q is empty. 22.44/8.63 We have to consider all minimal (P,Q,R)-chains. 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (7) QDPOrderProof (EQUIVALENT) 22.44/8.63 We use the reduction pair processor [LPAR04,JAR06]. 22.44/8.63 22.44/8.63 22.44/8.63 The following pairs can be oriented strictly and are deleted. 22.44/8.63 22.44/8.63 B(a(a(x1))) -> B(b(x1)) 22.44/8.63 The remaining pairs can at least be oriented weakly. 22.44/8.63 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 22.44/8.63 22.44/8.63 <<< 22.44/8.63 POL(B(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 22.44/8.63 >>> 22.44/8.63 22.44/8.63 <<< 22.44/8.63 POL(a(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 1A], [-I, 0A, 0A], [0A, -I, 0A]] * x_1 22.44/8.63 >>> 22.44/8.63 22.44/8.63 <<< 22.44/8.63 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 22.44/8.63 >>> 22.44/8.63 22.44/8.63 <<< 22.44/8.63 POL(c(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, 0A, 0A], [-I, -I, -I]] * x_1 22.44/8.63 >>> 22.44/8.63 22.44/8.63 22.44/8.63 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 22.44/8.63 22.44/8.63 b(a(a(x1))) -> a(a(a(c(b(b(x1)))))) 22.44/8.63 a(x1) -> x1 22.44/8.63 a(c(x1)) -> x1 22.44/8.63 22.44/8.63 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (8) 22.44/8.63 Obligation: 22.44/8.63 Q DP problem: 22.44/8.63 The TRS P consists of the following rules: 22.44/8.63 22.44/8.63 B(a(a(x1))) -> B(x1) 22.44/8.63 22.44/8.63 The TRS R consists of the following rules: 22.44/8.63 22.44/8.63 a(x1) -> x1 22.44/8.63 b(a(a(x1))) -> a(a(a(c(b(b(x1)))))) 22.44/8.63 a(c(x1)) -> x1 22.44/8.63 22.44/8.63 Q is empty. 22.44/8.63 We have to consider all minimal (P,Q,R)-chains. 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (9) UsableRulesProof (EQUIVALENT) 22.44/8.63 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. 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (10) 22.44/8.63 Obligation: 22.44/8.63 Q DP problem: 22.44/8.63 The TRS P consists of the following rules: 22.44/8.63 22.44/8.63 B(a(a(x1))) -> B(x1) 22.44/8.63 22.44/8.63 R is empty. 22.44/8.63 Q is empty. 22.44/8.63 We have to consider all minimal (P,Q,R)-chains. 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (11) QDPSizeChangeProof (EQUIVALENT) 22.44/8.63 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. 22.44/8.63 22.44/8.63 From the DPs we obtained the following set of size-change graphs: 22.44/8.63 *B(a(a(x1))) -> B(x1) 22.44/8.63 The graph contains the following edges 1 > 1 22.44/8.63 22.44/8.63 22.44/8.63 ---------------------------------------- 22.44/8.63 22.44/8.63 (12) 22.44/8.63 YES 22.46/8.67 EOF