37.67/10.60 YES 37.88/10.61 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 37.88/10.61 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 37.88/10.61 37.88/10.61 37.88/10.61 Termination w.r.t. Q of the given QTRS could be proven: 37.88/10.61 37.88/10.61 (0) QTRS 37.88/10.61 (1) QTRS Reverse [EQUIVALENT, 0 ms] 37.88/10.61 (2) QTRS 37.88/10.61 (3) DependencyPairsProof [EQUIVALENT, 1 ms] 37.88/10.61 (4) QDP 37.88/10.61 (5) DependencyGraphProof [EQUIVALENT, 7 ms] 37.88/10.61 (6) AND 37.88/10.61 (7) QDP 37.88/10.61 (8) UsableRulesProof [EQUIVALENT, 0 ms] 37.88/10.61 (9) QDP 37.88/10.61 (10) MRRProof [EQUIVALENT, 21 ms] 37.88/10.61 (11) QDP 37.88/10.61 (12) PisEmptyProof [EQUIVALENT, 0 ms] 37.88/10.61 (13) YES 37.88/10.61 (14) QDP 37.88/10.61 (15) UsableRulesProof [EQUIVALENT, 1 ms] 37.88/10.61 (16) QDP 37.88/10.61 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 37.88/10.61 (18) YES 37.88/10.61 (19) QDP 37.88/10.61 (20) UsableRulesProof [EQUIVALENT, 1 ms] 37.88/10.61 (21) QDP 37.88/10.61 (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] 37.88/10.61 (23) YES 37.88/10.61 (24) QDP 37.88/10.61 (25) QDPOrderProof [EQUIVALENT, 688 ms] 37.88/10.61 (26) QDP 37.88/10.61 (27) QDPOrderProof [EQUIVALENT, 10 ms] 37.88/10.61 (28) QDP 37.88/10.61 (29) PisEmptyProof [EQUIVALENT, 0 ms] 37.88/10.61 (30) YES 37.88/10.61 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (0) 37.88/10.61 Obligation: 37.88/10.61 Q restricted rewrite system: 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 r1(a(x1)) -> a(a(a(r1(x1)))) 37.88/10.61 r2(a(x1)) -> a(a(a(r2(x1)))) 37.88/10.61 a(l1(x1)) -> l1(a(a(a(x1)))) 37.88/10.61 a(a(l2(x1))) -> l2(a(a(x1))) 37.88/10.61 r1(b(x1)) -> l1(b(x1)) 37.88/10.61 r2(b(x1)) -> l2(a(b(x1))) 37.88/10.61 b(l1(x1)) -> b(r2(x1)) 37.88/10.61 b(l2(x1)) -> b(r1(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (1) QTRS Reverse (EQUIVALENT) 37.88/10.61 We applied the QTRS Reverse Processor [REVERSE]. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (2) 37.88/10.61 Obligation: 37.88/10.61 Q restricted rewrite system: 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (3) DependencyPairsProof (EQUIVALENT) 37.88/10.61 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (4) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 A(r1(x1)) -> A(a(a(x1))) 37.88/10.61 A(r1(x1)) -> A(a(x1)) 37.88/10.61 A(r1(x1)) -> A(x1) 37.88/10.61 A(r2(x1)) -> A(a(a(x1))) 37.88/10.61 A(r2(x1)) -> A(a(x1)) 37.88/10.61 A(r2(x1)) -> A(x1) 37.88/10.61 L1(a(x1)) -> A(a(a(l1(x1)))) 37.88/10.61 L1(a(x1)) -> A(a(l1(x1))) 37.88/10.61 L1(a(x1)) -> A(l1(x1)) 37.88/10.61 L1(a(x1)) -> L1(x1) 37.88/10.61 L2(a(a(x1))) -> A(a(l2(x1))) 37.88/10.61 L2(a(a(x1))) -> A(l2(x1)) 37.88/10.61 L2(a(a(x1))) -> L2(x1) 37.88/10.61 B(r1(x1)) -> B(l1(x1)) 37.88/10.61 B(r1(x1)) -> L1(x1) 37.88/10.61 B(r2(x1)) -> B(a(l2(x1))) 37.88/10.61 B(r2(x1)) -> A(l2(x1)) 37.88/10.61 B(r2(x1)) -> L2(x1) 37.88/10.61 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (5) DependencyGraphProof (EQUIVALENT) 37.88/10.61 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 8 less nodes. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (6) 37.88/10.61 Complex Obligation (AND) 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (7) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 A(r1(x1)) -> A(a(x1)) 37.88/10.61 A(r1(x1)) -> A(a(a(x1))) 37.88/10.61 A(r1(x1)) -> A(x1) 37.88/10.61 A(r2(x1)) -> A(a(a(x1))) 37.88/10.61 A(r2(x1)) -> A(a(x1)) 37.88/10.61 A(r2(x1)) -> A(x1) 37.88/10.61 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (8) UsableRulesProof (EQUIVALENT) 37.88/10.61 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. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (9) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 A(r1(x1)) -> A(a(x1)) 37.88/10.61 A(r1(x1)) -> A(a(a(x1))) 37.88/10.61 A(r1(x1)) -> A(x1) 37.88/10.61 A(r2(x1)) -> A(a(a(x1))) 37.88/10.61 A(r2(x1)) -> A(a(x1)) 37.88/10.61 A(r2(x1)) -> A(x1) 37.88/10.61 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (10) MRRProof (EQUIVALENT) 37.88/10.61 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 37.88/10.61 37.88/10.61 Strictly oriented dependency pairs: 37.88/10.61 37.88/10.61 A(r1(x1)) -> A(a(x1)) 37.88/10.61 A(r1(x1)) -> A(a(a(x1))) 37.88/10.61 A(r1(x1)) -> A(x1) 37.88/10.61 A(r2(x1)) -> A(a(a(x1))) 37.88/10.61 A(r2(x1)) -> A(a(x1)) 37.88/10.61 A(r2(x1)) -> A(x1) 37.88/10.61 37.88/10.61 37.88/10.61 Used ordering: Polynomial interpretation [POLO]: 37.88/10.61 37.88/10.61 POL(A(x_1)) = x_1 37.88/10.61 POL(a(x_1)) = x_1 37.88/10.61 POL(r1(x_1)) = 1 + 2*x_1 37.88/10.61 POL(r2(x_1)) = 3 + 2*x_1 37.88/10.61 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (11) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 P is empty. 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (12) PisEmptyProof (EQUIVALENT) 37.88/10.61 The TRS P is empty. Hence, there is no (P,Q,R) chain. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (13) 37.88/10.61 YES 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (14) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 L2(a(a(x1))) -> L2(x1) 37.88/10.61 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (15) UsableRulesProof (EQUIVALENT) 37.88/10.61 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. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (16) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 L2(a(a(x1))) -> L2(x1) 37.88/10.61 37.88/10.61 R is empty. 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (17) QDPSizeChangeProof (EQUIVALENT) 37.88/10.61 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. 37.88/10.61 37.88/10.61 From the DPs we obtained the following set of size-change graphs: 37.88/10.61 *L2(a(a(x1))) -> L2(x1) 37.88/10.61 The graph contains the following edges 1 > 1 37.88/10.61 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (18) 37.88/10.61 YES 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (19) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 L1(a(x1)) -> L1(x1) 37.88/10.61 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (20) UsableRulesProof (EQUIVALENT) 37.88/10.61 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. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (21) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 L1(a(x1)) -> L1(x1) 37.88/10.61 37.88/10.61 R is empty. 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (22) QDPSizeChangeProof (EQUIVALENT) 37.88/10.61 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. 37.88/10.61 37.88/10.61 From the DPs we obtained the following set of size-change graphs: 37.88/10.61 *L1(a(x1)) -> L1(x1) 37.88/10.61 The graph contains the following edges 1 > 1 37.88/10.61 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (23) 37.88/10.61 YES 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (24) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 B(r2(x1)) -> B(a(l2(x1))) 37.88/10.61 B(r1(x1)) -> B(l1(x1)) 37.88/10.61 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (25) QDPOrderProof (EQUIVALENT) 37.88/10.61 We use the reduction pair processor [LPAR04,JAR06]. 37.88/10.61 37.88/10.61 37.88/10.61 The following pairs can be oriented strictly and are deleted. 37.88/10.61 37.88/10.61 B(r2(x1)) -> B(a(l2(x1))) 37.88/10.61 The remaining pairs can at least be oriented weakly. 37.88/10.61 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 37.88/10.61 37.88/10.61 <<< 37.88/10.61 POL(B(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 37.88/10.61 >>> 37.88/10.61 37.88/10.61 <<< 37.88/10.61 POL(r2(x_1)) = [[1A], [1A], [0A]] + [[1A, 0A, 0A], [0A, 1A, 0A], [0A, -I, 1A]] * x_1 37.88/10.61 >>> 37.88/10.61 37.88/10.61 <<< 37.88/10.61 POL(a(x_1)) = [[0A], [-I], [-I]] + [[-I, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] * x_1 37.88/10.61 >>> 37.88/10.61 37.88/10.61 <<< 37.88/10.61 POL(l2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [0A, -I, 1A], [0A, 1A, 0A]] * x_1 37.88/10.61 >>> 37.88/10.61 37.88/10.61 <<< 37.88/10.61 POL(r1(x_1)) = [[1A], [0A], [1A]] + [[1A, 0A, 0A], [0A, -I, 1A], [0A, 1A, 0A]] * x_1 37.88/10.61 >>> 37.88/10.61 37.88/10.61 <<< 37.88/10.61 POL(l1(x_1)) = [[0A], [0A], [0A]] + [[1A, -I, 0A], [0A, 1A, 0A], [0A, -I, 1A]] * x_1 37.88/10.61 >>> 37.88/10.61 37.88/10.61 <<< 37.88/10.61 POL(b(x_1)) = [[0A], [1A], [-I]] + [[-I, -I, -I], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 37.88/10.61 >>> 37.88/10.61 37.88/10.61 37.88/10.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 37.88/10.61 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (26) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 The TRS P consists of the following rules: 37.88/10.61 37.88/10.61 B(r1(x1)) -> B(l1(x1)) 37.88/10.61 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (27) QDPOrderProof (EQUIVALENT) 37.88/10.61 We use the reduction pair processor [LPAR04,JAR06]. 37.88/10.61 37.88/10.61 37.88/10.61 The following pairs can be oriented strictly and are deleted. 37.88/10.61 37.88/10.61 B(r1(x1)) -> B(l1(x1)) 37.88/10.61 The remaining pairs can at least be oriented weakly. 37.88/10.61 Used ordering: Polynomial interpretation [POLO]: 37.88/10.61 37.88/10.61 POL(B(x_1)) = x_1 37.88/10.61 POL(a(x_1)) = x_1 37.88/10.61 POL(b(x_1)) = 1 37.88/10.61 POL(l1(x_1)) = 0 37.88/10.61 POL(l2(x_1)) = 1 + x_1 37.88/10.61 POL(r1(x_1)) = 1 37.88/10.61 POL(r2(x_1)) = 0 37.88/10.61 37.88/10.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 37.88/10.61 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (28) 37.88/10.61 Obligation: 37.88/10.61 Q DP problem: 37.88/10.61 P is empty. 37.88/10.61 The TRS R consists of the following rules: 37.88/10.61 37.88/10.61 a(r1(x1)) -> r1(a(a(a(x1)))) 37.88/10.61 a(r2(x1)) -> r2(a(a(a(x1)))) 37.88/10.61 l1(a(x1)) -> a(a(a(l1(x1)))) 37.88/10.61 l2(a(a(x1))) -> a(a(l2(x1))) 37.88/10.61 b(r1(x1)) -> b(l1(x1)) 37.88/10.61 b(r2(x1)) -> b(a(l2(x1))) 37.88/10.61 l1(b(x1)) -> r2(b(x1)) 37.88/10.61 l2(b(x1)) -> r1(b(x1)) 37.88/10.61 a(a(x1)) -> x1 37.88/10.61 37.88/10.61 Q is empty. 37.88/10.61 We have to consider all minimal (P,Q,R)-chains. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (29) PisEmptyProof (EQUIVALENT) 37.88/10.61 The TRS P is empty. Hence, there is no (P,Q,R) chain. 37.88/10.61 ---------------------------------------- 37.88/10.61 37.88/10.61 (30) 37.88/10.61 YES 38.01/10.69 EOF