20.98/6.22 YES 22.76/6.99 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 22.76/6.99 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 22.76/6.99 22.76/6.99 22.76/6.99 Termination w.r.t. Q of the given QTRS could be proven: 22.76/6.99 22.76/6.99 (0) QTRS 22.76/6.99 (1) QTRS Reverse [EQUIVALENT, 0 ms] 22.76/6.99 (2) QTRS 22.76/6.99 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 22.76/6.99 (4) QDP 22.76/6.99 (5) DependencyGraphProof [EQUIVALENT, 2 ms] 22.76/6.99 (6) QDP 22.76/6.99 (7) QDPOrderProof [EQUIVALENT, 128 ms] 22.76/6.99 (8) QDP 22.76/6.99 (9) UsableRulesProof [EQUIVALENT, 0 ms] 22.76/6.99 (10) QDP 22.76/6.99 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 22.76/6.99 (12) YES 22.76/6.99 22.76/6.99 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (0) 22.76/6.99 Obligation: 22.76/6.99 Q restricted rewrite system: 22.76/6.99 The TRS R consists of the following rules: 22.76/6.99 22.76/6.99 a(x1) -> b(x1) 22.76/6.99 b(b(x1)) -> x1 22.76/6.99 b(c(c(x1))) -> c(c(b(c(a(x1))))) 22.76/6.99 22.76/6.99 Q is empty. 22.76/6.99 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (1) QTRS Reverse (EQUIVALENT) 22.76/6.99 We applied the QTRS Reverse Processor [REVERSE]. 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (2) 22.76/6.99 Obligation: 22.76/6.99 Q restricted rewrite system: 22.76/6.99 The TRS R consists of the following rules: 22.76/6.99 22.76/6.99 a(x1) -> b(x1) 22.76/6.99 b(b(x1)) -> x1 22.76/6.99 c(c(b(x1))) -> a(c(b(c(c(x1))))) 22.76/6.99 22.76/6.99 Q is empty. 22.76/6.99 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (3) DependencyPairsProof (EQUIVALENT) 22.76/6.99 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (4) 22.76/6.99 Obligation: 22.76/6.99 Q DP problem: 22.76/6.99 The TRS P consists of the following rules: 22.76/6.99 22.76/6.99 A(x1) -> B(x1) 22.76/6.99 C(c(b(x1))) -> A(c(b(c(c(x1))))) 22.76/6.99 C(c(b(x1))) -> C(b(c(c(x1)))) 22.76/6.99 C(c(b(x1))) -> B(c(c(x1))) 22.76/6.99 C(c(b(x1))) -> C(c(x1)) 22.76/6.99 C(c(b(x1))) -> C(x1) 22.76/6.99 22.76/6.99 The TRS R consists of the following rules: 22.76/6.99 22.76/6.99 a(x1) -> b(x1) 22.76/6.99 b(b(x1)) -> x1 22.76/6.99 c(c(b(x1))) -> a(c(b(c(c(x1))))) 22.76/6.99 22.76/6.99 Q is empty. 22.76/6.99 We have to consider all minimal (P,Q,R)-chains. 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (5) DependencyGraphProof (EQUIVALENT) 22.76/6.99 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (6) 22.76/6.99 Obligation: 22.76/6.99 Q DP problem: 22.76/6.99 The TRS P consists of the following rules: 22.76/6.99 22.76/6.99 C(c(b(x1))) -> C(c(x1)) 22.76/6.99 C(c(b(x1))) -> C(b(c(c(x1)))) 22.76/6.99 C(c(b(x1))) -> C(x1) 22.76/6.99 22.76/6.99 The TRS R consists of the following rules: 22.76/6.99 22.76/6.99 a(x1) -> b(x1) 22.76/6.99 b(b(x1)) -> x1 22.76/6.99 c(c(b(x1))) -> a(c(b(c(c(x1))))) 22.76/6.99 22.76/6.99 Q is empty. 22.76/6.99 We have to consider all minimal (P,Q,R)-chains. 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (7) QDPOrderProof (EQUIVALENT) 22.76/6.99 We use the reduction pair processor [LPAR04,JAR06]. 22.76/6.99 22.76/6.99 22.76/6.99 The following pairs can be oriented strictly and are deleted. 22.76/6.99 22.76/6.99 C(c(b(x1))) -> C(c(x1)) 22.76/6.99 C(c(b(x1))) -> C(b(c(c(x1)))) 22.76/6.99 The remaining pairs can at least be oriented weakly. 22.76/6.99 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 22.76/6.99 22.76/6.99 <<< 22.76/6.99 POL(C(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 22.76/6.99 >>> 22.76/6.99 22.76/6.99 <<< 22.76/6.99 POL(c(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 22.76/6.99 >>> 22.76/6.99 22.76/6.99 <<< 22.76/6.99 POL(b(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 22.76/6.99 >>> 22.76/6.99 22.76/6.99 <<< 22.76/6.99 POL(a(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, -I], [0A, 1A, 0A], [0A, 0A, -I]] * x_1 22.76/6.99 >>> 22.76/6.99 22.76/6.99 22.76/6.99 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 22.76/6.99 22.76/6.99 c(c(b(x1))) -> a(c(b(c(c(x1))))) 22.76/6.99 b(b(x1)) -> x1 22.76/6.99 a(x1) -> b(x1) 22.76/6.99 22.76/6.99 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (8) 22.76/6.99 Obligation: 22.76/6.99 Q DP problem: 22.76/6.99 The TRS P consists of the following rules: 22.76/6.99 22.76/6.99 C(c(b(x1))) -> C(x1) 22.76/6.99 22.76/6.99 The TRS R consists of the following rules: 22.76/6.99 22.76/6.99 a(x1) -> b(x1) 22.76/6.99 b(b(x1)) -> x1 22.76/6.99 c(c(b(x1))) -> a(c(b(c(c(x1))))) 22.76/6.99 22.76/6.99 Q is empty. 22.76/6.99 We have to consider all minimal (P,Q,R)-chains. 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (9) UsableRulesProof (EQUIVALENT) 22.76/6.99 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. 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (10) 22.76/6.99 Obligation: 22.76/6.99 Q DP problem: 22.76/6.99 The TRS P consists of the following rules: 22.76/6.99 22.76/6.99 C(c(b(x1))) -> C(x1) 22.76/6.99 22.76/6.99 R is empty. 22.76/6.99 Q is empty. 22.76/6.99 We have to consider all minimal (P,Q,R)-chains. 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (11) QDPSizeChangeProof (EQUIVALENT) 22.76/6.99 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. 22.76/6.99 22.76/6.99 From the DPs we obtained the following set of size-change graphs: 22.76/6.99 *C(c(b(x1))) -> C(x1) 22.76/6.99 The graph contains the following edges 1 > 1 22.76/6.99 22.76/6.99 22.76/6.99 ---------------------------------------- 22.76/6.99 22.76/6.99 (12) 22.76/6.99 YES 24.34/11.04 EOF