25.99/7.48 YES 26.63/7.60 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 26.63/7.60 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.63/7.60 26.63/7.60 26.63/7.60 Termination w.r.t. Q of the given QTRS could be proven: 26.63/7.60 26.63/7.60 (0) QTRS 26.63/7.60 (1) DependencyPairsProof [EQUIVALENT, 2 ms] 26.63/7.60 (2) QDP 26.63/7.60 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 26.63/7.60 (4) QDP 26.63/7.60 (5) QDPOrderProof [EQUIVALENT, 109 ms] 26.63/7.60 (6) QDP 26.63/7.60 (7) QDPOrderProof [EQUIVALENT, 27 ms] 26.63/7.60 (8) QDP 26.63/7.60 (9) UsableRulesProof [EQUIVALENT, 0 ms] 26.63/7.60 (10) QDP 26.63/7.60 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 26.63/7.60 (12) YES 26.63/7.60 26.63/7.60 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (0) 26.63/7.60 Obligation: 26.63/7.60 Q restricted rewrite system: 26.63/7.60 The TRS R consists of the following rules: 26.63/7.60 26.63/7.60 a(x1) -> b(b(x1)) 26.63/7.60 b(b(c(x1))) -> c(b(c(a(x1)))) 26.63/7.60 c(c(x1)) -> x1 26.63/7.60 26.63/7.60 Q is empty. 26.63/7.60 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (1) DependencyPairsProof (EQUIVALENT) 26.63/7.60 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (2) 26.63/7.60 Obligation: 26.63/7.60 Q DP problem: 26.63/7.60 The TRS P consists of the following rules: 26.63/7.60 26.63/7.60 A(x1) -> B(b(x1)) 26.63/7.60 A(x1) -> B(x1) 26.63/7.60 B(b(c(x1))) -> C(b(c(a(x1)))) 26.63/7.60 B(b(c(x1))) -> B(c(a(x1))) 26.63/7.60 B(b(c(x1))) -> C(a(x1)) 26.63/7.60 B(b(c(x1))) -> A(x1) 26.63/7.60 26.63/7.60 The TRS R consists of the following rules: 26.63/7.60 26.63/7.60 a(x1) -> b(b(x1)) 26.63/7.60 b(b(c(x1))) -> c(b(c(a(x1)))) 26.63/7.60 c(c(x1)) -> x1 26.63/7.60 26.63/7.60 Q is empty. 26.63/7.60 We have to consider all minimal (P,Q,R)-chains. 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (3) DependencyGraphProof (EQUIVALENT) 26.63/7.60 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (4) 26.63/7.60 Obligation: 26.63/7.60 Q DP problem: 26.63/7.60 The TRS P consists of the following rules: 26.63/7.60 26.63/7.60 B(b(c(x1))) -> B(c(a(x1))) 26.63/7.60 B(b(c(x1))) -> A(x1) 26.63/7.60 A(x1) -> B(b(x1)) 26.63/7.60 A(x1) -> B(x1) 26.63/7.60 26.63/7.60 The TRS R consists of the following rules: 26.63/7.60 26.63/7.60 a(x1) -> b(b(x1)) 26.63/7.60 b(b(c(x1))) -> c(b(c(a(x1)))) 26.63/7.60 c(c(x1)) -> x1 26.63/7.60 26.63/7.60 Q is empty. 26.63/7.60 We have to consider all minimal (P,Q,R)-chains. 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (5) QDPOrderProof (EQUIVALENT) 26.63/7.60 We use the reduction pair processor [LPAR04,JAR06]. 26.63/7.60 26.63/7.60 26.63/7.60 The following pairs can be oriented strictly and are deleted. 26.63/7.60 26.63/7.60 B(b(c(x1))) -> B(c(a(x1))) 26.63/7.60 The remaining pairs can at least be oriented weakly. 26.63/7.60 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(b(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(c(x_1)) = [[-I], [1A], [-I]] + [[0A, 0A, 0A], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(a(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 26.63/7.60 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.63/7.60 26.63/7.60 a(x1) -> b(b(x1)) 26.63/7.60 c(c(x1)) -> x1 26.63/7.60 b(b(c(x1))) -> c(b(c(a(x1)))) 26.63/7.60 26.63/7.60 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (6) 26.63/7.60 Obligation: 26.63/7.60 Q DP problem: 26.63/7.60 The TRS P consists of the following rules: 26.63/7.60 26.63/7.60 B(b(c(x1))) -> A(x1) 26.63/7.60 A(x1) -> B(b(x1)) 26.63/7.60 A(x1) -> B(x1) 26.63/7.60 26.63/7.60 The TRS R consists of the following rules: 26.63/7.60 26.63/7.60 a(x1) -> b(b(x1)) 26.63/7.60 b(b(c(x1))) -> c(b(c(a(x1)))) 26.63/7.60 c(c(x1)) -> x1 26.63/7.60 26.63/7.60 Q is empty. 26.63/7.60 We have to consider all minimal (P,Q,R)-chains. 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (7) QDPOrderProof (EQUIVALENT) 26.63/7.60 We use the reduction pair processor [LPAR04,JAR06]. 26.63/7.60 26.63/7.60 26.63/7.60 The following pairs can be oriented strictly and are deleted. 26.63/7.60 26.63/7.60 A(x1) -> B(b(x1)) 26.63/7.60 The remaining pairs can at least be oriented weakly. 26.63/7.60 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(c(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(A(x_1)) = [[1A]] + [[0A, 1A, 0A]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 <<< 26.63/7.60 POL(a(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 26.63/7.60 >>> 26.63/7.60 26.63/7.60 26.63/7.60 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.63/7.60 26.63/7.60 b(b(c(x1))) -> c(b(c(a(x1)))) 26.63/7.60 a(x1) -> b(b(x1)) 26.63/7.60 c(c(x1)) -> x1 26.63/7.60 26.63/7.60 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (8) 26.63/7.60 Obligation: 26.63/7.60 Q DP problem: 26.63/7.60 The TRS P consists of the following rules: 26.63/7.60 26.63/7.60 B(b(c(x1))) -> A(x1) 26.63/7.60 A(x1) -> B(x1) 26.63/7.60 26.63/7.60 The TRS R consists of the following rules: 26.63/7.60 26.63/7.60 a(x1) -> b(b(x1)) 26.63/7.60 b(b(c(x1))) -> c(b(c(a(x1)))) 26.63/7.60 c(c(x1)) -> x1 26.63/7.60 26.63/7.60 Q is empty. 26.63/7.60 We have to consider all minimal (P,Q,R)-chains. 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (9) UsableRulesProof (EQUIVALENT) 26.63/7.60 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. 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (10) 26.63/7.60 Obligation: 26.63/7.60 Q DP problem: 26.63/7.60 The TRS P consists of the following rules: 26.63/7.60 26.63/7.60 B(b(c(x1))) -> A(x1) 26.63/7.60 A(x1) -> B(x1) 26.63/7.60 26.63/7.60 R is empty. 26.63/7.60 Q is empty. 26.63/7.60 We have to consider all minimal (P,Q,R)-chains. 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (11) QDPSizeChangeProof (EQUIVALENT) 26.63/7.60 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. 26.63/7.60 26.63/7.60 From the DPs we obtained the following set of size-change graphs: 26.63/7.60 *A(x1) -> B(x1) 26.63/7.60 The graph contains the following edges 1 >= 1 26.63/7.60 26.63/7.60 26.63/7.60 *B(b(c(x1))) -> A(x1) 26.63/7.60 The graph contains the following edges 1 > 1 26.63/7.60 26.63/7.60 26.63/7.60 ---------------------------------------- 26.63/7.60 26.63/7.60 (12) 26.63/7.60 YES 26.89/7.69 EOF