25.45/7.37 YES 25.74/7.44 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 25.74/7.44 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.74/7.44 25.74/7.44 25.74/7.44 Termination w.r.t. Q of the given QTRS could be proven: 25.74/7.44 25.74/7.44 (0) QTRS 25.74/7.44 (1) DependencyPairsProof [EQUIVALENT, 22 ms] 25.74/7.44 (2) QDP 25.74/7.44 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 25.74/7.44 (4) QDP 25.74/7.44 (5) QDPOrderProof [EQUIVALENT, 170 ms] 25.74/7.44 (6) QDP 25.74/7.44 (7) UsableRulesProof [EQUIVALENT, 2 ms] 25.74/7.44 (8) QDP 25.74/7.44 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 25.74/7.44 (10) YES 25.74/7.44 25.74/7.44 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (0) 25.74/7.44 Obligation: 25.74/7.44 Q restricted rewrite system: 25.74/7.44 The TRS R consists of the following rules: 25.74/7.44 25.74/7.44 a(x1) -> x1 25.74/7.44 a(a(b(x1))) -> b(a(b(a(c(a(x1)))))) 25.74/7.44 c(b(x1)) -> x1 25.74/7.44 25.74/7.44 Q is empty. 25.74/7.44 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (1) DependencyPairsProof (EQUIVALENT) 25.74/7.44 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (2) 25.74/7.44 Obligation: 25.74/7.44 Q DP problem: 25.74/7.44 The TRS P consists of the following rules: 25.74/7.44 25.74/7.44 A(a(b(x1))) -> A(b(a(c(a(x1))))) 25.74/7.44 A(a(b(x1))) -> A(c(a(x1))) 25.74/7.44 A(a(b(x1))) -> C(a(x1)) 25.74/7.44 A(a(b(x1))) -> A(x1) 25.74/7.44 25.74/7.44 The TRS R consists of the following rules: 25.74/7.44 25.74/7.44 a(x1) -> x1 25.74/7.44 a(a(b(x1))) -> b(a(b(a(c(a(x1)))))) 25.74/7.44 c(b(x1)) -> x1 25.74/7.44 25.74/7.44 Q is empty. 25.74/7.44 We have to consider all minimal (P,Q,R)-chains. 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (3) DependencyGraphProof (EQUIVALENT) 25.74/7.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (4) 25.74/7.44 Obligation: 25.74/7.44 Q DP problem: 25.74/7.44 The TRS P consists of the following rules: 25.74/7.44 25.74/7.44 A(a(b(x1))) -> A(x1) 25.74/7.44 A(a(b(x1))) -> A(c(a(x1))) 25.74/7.44 25.74/7.44 The TRS R consists of the following rules: 25.74/7.44 25.74/7.44 a(x1) -> x1 25.74/7.44 a(a(b(x1))) -> b(a(b(a(c(a(x1)))))) 25.74/7.44 c(b(x1)) -> x1 25.74/7.44 25.74/7.44 Q is empty. 25.74/7.44 We have to consider all minimal (P,Q,R)-chains. 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (5) QDPOrderProof (EQUIVALENT) 25.74/7.44 We use the reduction pair processor [LPAR04,JAR06]. 25.74/7.44 25.74/7.44 25.74/7.44 The following pairs can be oriented strictly and are deleted. 25.74/7.44 25.74/7.44 A(a(b(x1))) -> A(c(a(x1))) 25.74/7.44 The remaining pairs can at least be oriented weakly. 25.74/7.44 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.74/7.44 25.74/7.44 <<< 25.74/7.44 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 25.74/7.44 >>> 25.74/7.44 25.74/7.44 <<< 25.74/7.44 POL(a(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [1A, 0A, 0A], [0A, -I, 0A]] * x_1 25.74/7.44 >>> 25.74/7.44 25.74/7.44 <<< 25.74/7.44 POL(b(x_1)) = [[-I], [1A], [-I]] + [[0A, -I, 0A], [1A, -I, 1A], [0A, 0A, 0A]] * x_1 25.74/7.44 >>> 25.74/7.44 25.74/7.44 <<< 25.74/7.44 POL(c(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, 0A], [-I, -I, 0A], [-I, -I, 0A]] * x_1 25.74/7.44 >>> 25.74/7.44 25.74/7.44 25.74/7.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.74/7.44 25.74/7.44 a(x1) -> x1 25.74/7.44 a(a(b(x1))) -> b(a(b(a(c(a(x1)))))) 25.74/7.44 c(b(x1)) -> x1 25.74/7.44 25.74/7.44 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (6) 25.74/7.44 Obligation: 25.74/7.44 Q DP problem: 25.74/7.44 The TRS P consists of the following rules: 25.74/7.44 25.74/7.44 A(a(b(x1))) -> A(x1) 25.74/7.44 25.74/7.44 The TRS R consists of the following rules: 25.74/7.44 25.74/7.44 a(x1) -> x1 25.74/7.44 a(a(b(x1))) -> b(a(b(a(c(a(x1)))))) 25.74/7.44 c(b(x1)) -> x1 25.74/7.44 25.74/7.44 Q is empty. 25.74/7.44 We have to consider all minimal (P,Q,R)-chains. 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (7) UsableRulesProof (EQUIVALENT) 25.74/7.44 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. 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (8) 25.74/7.44 Obligation: 25.74/7.44 Q DP problem: 25.74/7.44 The TRS P consists of the following rules: 25.74/7.44 25.74/7.44 A(a(b(x1))) -> A(x1) 25.74/7.44 25.74/7.44 R is empty. 25.74/7.44 Q is empty. 25.74/7.44 We have to consider all minimal (P,Q,R)-chains. 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (9) QDPSizeChangeProof (EQUIVALENT) 25.74/7.44 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. 25.74/7.44 25.74/7.44 From the DPs we obtained the following set of size-change graphs: 25.74/7.44 *A(a(b(x1))) -> A(x1) 25.74/7.44 The graph contains the following edges 1 > 1 25.74/7.44 25.74/7.44 25.74/7.44 ---------------------------------------- 25.74/7.44 25.74/7.44 (10) 25.74/7.44 YES 26.14/7.53 EOF