35.46/9.90 YES 35.77/9.95 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 35.77/9.95 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 35.77/9.95 35.77/9.95 35.77/9.95 Termination w.r.t. Q of the given QTRS could be proven: 35.77/9.95 35.77/9.95 (0) QTRS 35.77/9.95 (1) DependencyPairsProof [EQUIVALENT, 2 ms] 35.77/9.95 (2) QDP 35.77/9.95 (3) DependencyGraphProof [EQUIVALENT, 1 ms] 35.77/9.95 (4) QDP 35.77/9.95 (5) QDPOrderProof [EQUIVALENT, 118 ms] 35.77/9.95 (6) QDP 35.77/9.95 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 35.77/9.95 (8) QDP 35.77/9.95 (9) UsableRulesProof [EQUIVALENT, 0 ms] 35.77/9.95 (10) QDP 35.77/9.95 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 35.77/9.95 (12) YES 35.77/9.95 35.77/9.95 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (0) 35.77/9.95 Obligation: 35.77/9.95 Q restricted rewrite system: 35.77/9.95 The TRS R consists of the following rules: 35.77/9.95 35.77/9.95 a(x1) -> x1 35.77/9.95 a(x1) -> b(c(x1)) 35.77/9.95 b(b(x1)) -> x1 35.77/9.95 c(c(b(x1))) -> a(a(c(x1))) 35.77/9.95 35.77/9.95 Q is empty. 35.77/9.95 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (1) DependencyPairsProof (EQUIVALENT) 35.77/9.95 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (2) 35.77/9.95 Obligation: 35.77/9.95 Q DP problem: 35.77/9.95 The TRS P consists of the following rules: 35.77/9.95 35.77/9.95 A(x1) -> B(c(x1)) 35.77/9.95 A(x1) -> C(x1) 35.77/9.95 C(c(b(x1))) -> A(a(c(x1))) 35.77/9.95 C(c(b(x1))) -> A(c(x1)) 35.77/9.95 C(c(b(x1))) -> C(x1) 35.77/9.95 35.77/9.95 The TRS R consists of the following rules: 35.77/9.95 35.77/9.95 a(x1) -> x1 35.77/9.95 a(x1) -> b(c(x1)) 35.77/9.95 b(b(x1)) -> x1 35.77/9.95 c(c(b(x1))) -> a(a(c(x1))) 35.77/9.95 35.77/9.95 Q is empty. 35.77/9.95 We have to consider all minimal (P,Q,R)-chains. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (3) DependencyGraphProof (EQUIVALENT) 35.77/9.95 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (4) 35.77/9.95 Obligation: 35.77/9.95 Q DP problem: 35.77/9.95 The TRS P consists of the following rules: 35.77/9.95 35.77/9.95 A(x1) -> C(x1) 35.77/9.95 C(c(b(x1))) -> A(a(c(x1))) 35.77/9.95 C(c(b(x1))) -> A(c(x1)) 35.77/9.95 C(c(b(x1))) -> C(x1) 35.77/9.95 35.77/9.95 The TRS R consists of the following rules: 35.77/9.95 35.77/9.95 a(x1) -> x1 35.77/9.95 a(x1) -> b(c(x1)) 35.77/9.95 b(b(x1)) -> x1 35.77/9.95 c(c(b(x1))) -> a(a(c(x1))) 35.77/9.95 35.77/9.95 Q is empty. 35.77/9.95 We have to consider all minimal (P,Q,R)-chains. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (5) QDPOrderProof (EQUIVALENT) 35.77/9.95 We use the reduction pair processor [LPAR04,JAR06]. 35.77/9.95 35.77/9.95 35.77/9.95 The following pairs can be oriented strictly and are deleted. 35.77/9.95 35.77/9.95 A(x1) -> C(x1) 35.77/9.95 The remaining pairs can at least be oriented weakly. 35.77/9.95 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 35.77/9.95 35.77/9.95 <<< 35.77/9.95 POL(A(x_1)) = [[1A]] + [[0A, 0A, 1A]] * x_1 35.77/9.95 >>> 35.77/9.95 35.77/9.95 <<< 35.77/9.95 POL(C(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 35.77/9.95 >>> 35.77/9.95 35.77/9.95 <<< 35.77/9.95 POL(c(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 35.77/9.95 >>> 35.77/9.95 35.77/9.95 <<< 35.77/9.95 POL(b(x_1)) = [[0A], [1A], [-I]] + [[0A, 0A, 0A], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 35.77/9.95 >>> 35.77/9.95 35.77/9.95 <<< 35.77/9.95 POL(a(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 1A], [-I, -I, 0A]] * x_1 35.77/9.95 >>> 35.77/9.95 35.77/9.95 35.77/9.95 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.77/9.95 35.77/9.95 c(c(b(x1))) -> a(a(c(x1))) 35.77/9.95 a(x1) -> x1 35.77/9.95 a(x1) -> b(c(x1)) 35.77/9.95 b(b(x1)) -> x1 35.77/9.95 35.77/9.95 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (6) 35.77/9.95 Obligation: 35.77/9.95 Q DP problem: 35.77/9.95 The TRS P consists of the following rules: 35.77/9.95 35.77/9.95 C(c(b(x1))) -> A(a(c(x1))) 35.77/9.95 C(c(b(x1))) -> A(c(x1)) 35.77/9.95 C(c(b(x1))) -> C(x1) 35.77/9.95 35.77/9.95 The TRS R consists of the following rules: 35.77/9.95 35.77/9.95 a(x1) -> x1 35.77/9.95 a(x1) -> b(c(x1)) 35.77/9.95 b(b(x1)) -> x1 35.77/9.95 c(c(b(x1))) -> a(a(c(x1))) 35.77/9.95 35.77/9.95 Q is empty. 35.77/9.95 We have to consider all minimal (P,Q,R)-chains. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (7) DependencyGraphProof (EQUIVALENT) 35.77/9.95 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (8) 35.77/9.95 Obligation: 35.77/9.95 Q DP problem: 35.77/9.95 The TRS P consists of the following rules: 35.77/9.95 35.77/9.95 C(c(b(x1))) -> C(x1) 35.77/9.95 35.77/9.95 The TRS R consists of the following rules: 35.77/9.95 35.77/9.95 a(x1) -> x1 35.77/9.95 a(x1) -> b(c(x1)) 35.77/9.95 b(b(x1)) -> x1 35.77/9.95 c(c(b(x1))) -> a(a(c(x1))) 35.77/9.95 35.77/9.95 Q is empty. 35.77/9.95 We have to consider all minimal (P,Q,R)-chains. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (9) UsableRulesProof (EQUIVALENT) 35.77/9.95 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. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (10) 35.77/9.95 Obligation: 35.77/9.95 Q DP problem: 35.77/9.95 The TRS P consists of the following rules: 35.77/9.95 35.77/9.95 C(c(b(x1))) -> C(x1) 35.77/9.95 35.77/9.95 R is empty. 35.77/9.95 Q is empty. 35.77/9.95 We have to consider all minimal (P,Q,R)-chains. 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (11) QDPSizeChangeProof (EQUIVALENT) 35.77/9.95 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. 35.77/9.95 35.77/9.95 From the DPs we obtained the following set of size-change graphs: 35.77/9.95 *C(c(b(x1))) -> C(x1) 35.77/9.95 The graph contains the following edges 1 > 1 35.77/9.95 35.77/9.95 35.77/9.95 ---------------------------------------- 35.77/9.95 35.77/9.95 (12) 35.77/9.95 YES 36.11/10.06 EOF