29.23/8.42 YES 29.23/8.48 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 29.23/8.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.23/8.48 29.23/8.48 29.23/8.48 Termination w.r.t. Q of the given QTRS could be proven: 29.23/8.48 29.23/8.48 (0) QTRS 29.23/8.48 (1) QTRS Reverse [EQUIVALENT, 0 ms] 29.23/8.48 (2) QTRS 29.23/8.48 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 29.23/8.48 (4) QDP 29.23/8.48 (5) DependencyGraphProof [EQUIVALENT, 5 ms] 29.23/8.48 (6) QDP 29.23/8.48 (7) QDPOrderProof [EQUIVALENT, 144 ms] 29.23/8.48 (8) QDP 29.23/8.48 (9) UsableRulesProof [EQUIVALENT, 0 ms] 29.23/8.48 (10) QDP 29.23/8.48 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 29.23/8.48 (12) YES 29.23/8.48 29.23/8.48 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (0) 29.23/8.48 Obligation: 29.23/8.48 Q restricted rewrite system: 29.23/8.48 The TRS R consists of the following rules: 29.23/8.48 29.23/8.48 a(x1) -> x1 29.23/8.48 a(x1) -> b(x1) 29.23/8.48 b(x1) -> x1 29.23/8.48 b(a(c(x1))) -> c(c(a(a(b(x1))))) 29.23/8.48 29.23/8.48 Q is empty. 29.23/8.48 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (1) QTRS Reverse (EQUIVALENT) 29.23/8.48 We applied the QTRS Reverse Processor [REVERSE]. 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (2) 29.23/8.48 Obligation: 29.23/8.48 Q restricted rewrite system: 29.23/8.48 The TRS R consists of the following rules: 29.23/8.48 29.23/8.48 a(x1) -> x1 29.23/8.48 a(x1) -> b(x1) 29.23/8.48 b(x1) -> x1 29.23/8.48 c(a(b(x1))) -> b(a(a(c(c(x1))))) 29.23/8.48 29.23/8.48 Q is empty. 29.23/8.48 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (3) DependencyPairsProof (EQUIVALENT) 29.23/8.48 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (4) 29.23/8.48 Obligation: 29.23/8.48 Q DP problem: 29.23/8.48 The TRS P consists of the following rules: 29.23/8.48 29.23/8.48 A(x1) -> B(x1) 29.23/8.48 C(a(b(x1))) -> B(a(a(c(c(x1))))) 29.23/8.48 C(a(b(x1))) -> A(a(c(c(x1)))) 29.23/8.48 C(a(b(x1))) -> A(c(c(x1))) 29.23/8.48 C(a(b(x1))) -> C(c(x1)) 29.23/8.48 C(a(b(x1))) -> C(x1) 29.23/8.48 29.23/8.48 The TRS R consists of the following rules: 29.23/8.48 29.23/8.48 a(x1) -> x1 29.23/8.48 a(x1) -> b(x1) 29.23/8.48 b(x1) -> x1 29.23/8.48 c(a(b(x1))) -> b(a(a(c(c(x1))))) 29.23/8.48 29.23/8.48 Q is empty. 29.23/8.48 We have to consider all minimal (P,Q,R)-chains. 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (5) DependencyGraphProof (EQUIVALENT) 29.23/8.48 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (6) 29.23/8.48 Obligation: 29.23/8.48 Q DP problem: 29.23/8.48 The TRS P consists of the following rules: 29.23/8.48 29.23/8.48 C(a(b(x1))) -> C(x1) 29.23/8.48 C(a(b(x1))) -> C(c(x1)) 29.23/8.48 29.23/8.48 The TRS R consists of the following rules: 29.23/8.48 29.23/8.48 a(x1) -> x1 29.23/8.48 a(x1) -> b(x1) 29.23/8.48 b(x1) -> x1 29.23/8.48 c(a(b(x1))) -> b(a(a(c(c(x1))))) 29.23/8.48 29.23/8.48 Q is empty. 29.23/8.48 We have to consider all minimal (P,Q,R)-chains. 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (7) QDPOrderProof (EQUIVALENT) 29.23/8.48 We use the reduction pair processor [LPAR04,JAR06]. 29.23/8.48 29.23/8.48 29.23/8.48 The following pairs can be oriented strictly and are deleted. 29.23/8.48 29.23/8.48 C(a(b(x1))) -> C(c(x1)) 29.23/8.48 The remaining pairs can at least be oriented weakly. 29.23/8.48 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 29.23/8.48 29.23/8.48 <<< 29.23/8.48 POL(C(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 29.23/8.48 >>> 29.23/8.48 29.23/8.48 <<< 29.23/8.48 POL(a(x_1)) = [[0A], [1A], [0A]] + [[0A, 1A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 29.23/8.48 >>> 29.23/8.48 29.23/8.48 <<< 29.23/8.48 POL(b(x_1)) = [[-I], [-I], [0A]] + [[0A, 1A, 0A], [-I, 0A, -I], [-I, -I, 0A]] * x_1 29.23/8.48 >>> 29.23/8.48 29.23/8.48 <<< 29.23/8.48 POL(c(x_1)) = [[0A], [-I], [0A]] + [[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 29.23/8.48 >>> 29.23/8.48 29.23/8.48 29.23/8.48 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.23/8.48 29.23/8.48 c(a(b(x1))) -> b(a(a(c(c(x1))))) 29.23/8.48 a(x1) -> x1 29.23/8.48 a(x1) -> b(x1) 29.23/8.48 b(x1) -> x1 29.23/8.48 29.23/8.48 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (8) 29.23/8.48 Obligation: 29.23/8.48 Q DP problem: 29.23/8.48 The TRS P consists of the following rules: 29.23/8.48 29.23/8.48 C(a(b(x1))) -> C(x1) 29.23/8.48 29.23/8.48 The TRS R consists of the following rules: 29.23/8.48 29.23/8.48 a(x1) -> x1 29.23/8.48 a(x1) -> b(x1) 29.23/8.48 b(x1) -> x1 29.23/8.48 c(a(b(x1))) -> b(a(a(c(c(x1))))) 29.23/8.48 29.23/8.48 Q is empty. 29.23/8.48 We have to consider all minimal (P,Q,R)-chains. 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (9) UsableRulesProof (EQUIVALENT) 29.23/8.48 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. 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (10) 29.23/8.48 Obligation: 29.23/8.48 Q DP problem: 29.23/8.48 The TRS P consists of the following rules: 29.23/8.48 29.23/8.48 C(a(b(x1))) -> C(x1) 29.23/8.48 29.23/8.48 R is empty. 29.23/8.48 Q is empty. 29.23/8.48 We have to consider all minimal (P,Q,R)-chains. 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (11) QDPSizeChangeProof (EQUIVALENT) 29.23/8.48 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. 29.23/8.48 29.23/8.48 From the DPs we obtained the following set of size-change graphs: 29.23/8.48 *C(a(b(x1))) -> C(x1) 29.23/8.48 The graph contains the following edges 1 > 1 29.23/8.48 29.23/8.48 29.23/8.48 ---------------------------------------- 29.23/8.48 29.23/8.48 (12) 29.23/8.48 YES 29.76/8.56 EOF