24.09/7.10 YES 24.09/7.11 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.09/7.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.09/7.11 24.09/7.11 24.09/7.11 Termination w.r.t. Q of the given QTRS could be proven: 24.09/7.11 24.09/7.11 (0) QTRS 24.09/7.11 (1) QTRS Reverse [EQUIVALENT, 0 ms] 24.09/7.11 (2) QTRS 24.09/7.11 (3) DependencyPairsProof [EQUIVALENT, 12 ms] 24.09/7.11 (4) QDP 24.09/7.11 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 24.09/7.11 (6) AND 24.09/7.11 (7) QDP 24.09/7.11 (8) UsableRulesProof [EQUIVALENT, 0 ms] 24.09/7.11 (9) QDP 24.09/7.11 (10) MRRProof [EQUIVALENT, 62 ms] 24.09/7.11 (11) QDP 24.09/7.11 (12) DependencyGraphProof [EQUIVALENT, 0 ms] 24.09/7.11 (13) AND 24.09/7.11 (14) QDP 24.09/7.11 (15) QDPOrderProof [EQUIVALENT, 0 ms] 24.09/7.11 (16) QDP 24.09/7.11 (17) PisEmptyProof [EQUIVALENT, 0 ms] 24.09/7.11 (18) YES 24.09/7.11 (19) QDP 24.09/7.11 (20) QDPOrderProof [EQUIVALENT, 0 ms] 24.09/7.11 (21) QDP 24.09/7.11 (22) PisEmptyProof [EQUIVALENT, 0 ms] 24.09/7.11 (23) YES 24.09/7.11 (24) QDP 24.09/7.11 (25) UsableRulesProof [EQUIVALENT, 0 ms] 24.09/7.11 (26) QDP 24.09/7.11 (27) QDPOrderProof [EQUIVALENT, 9 ms] 24.09/7.11 (28) QDP 24.09/7.11 (29) QDPOrderProof [EQUIVALENT, 0 ms] 24.09/7.11 (30) QDP 24.09/7.11 (31) PisEmptyProof [EQUIVALENT, 0 ms] 24.09/7.11 (32) YES 24.09/7.11 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (0) 24.09/7.11 Obligation: 24.09/7.11 Q restricted rewrite system: 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 a(s(x1)) -> s(a(x1)) 24.09/7.11 b(a(b(s(x1)))) -> a(b(s(a(x1)))) 24.09/7.11 b(a(b(b(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 a(b(a(a(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (1) QTRS Reverse (EQUIVALENT) 24.09/7.11 We applied the QTRS Reverse Processor [REVERSE]. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (2) 24.09/7.11 Obligation: 24.09/7.11 Q restricted rewrite system: 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 s(a(x1)) -> a(s(x1)) 24.09/7.11 s(b(a(b(x1)))) -> a(s(b(a(x1)))) 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (3) DependencyPairsProof (EQUIVALENT) 24.09/7.11 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (4) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 The TRS P consists of the following rules: 24.09/7.11 24.09/7.11 S(a(x1)) -> A(s(x1)) 24.09/7.11 S(a(x1)) -> S(x1) 24.09/7.11 S(b(a(b(x1)))) -> A(s(b(a(x1)))) 24.09/7.11 S(b(a(b(x1)))) -> S(b(a(x1))) 24.09/7.11 S(b(a(b(x1)))) -> B(a(x1)) 24.09/7.11 S(b(a(b(x1)))) -> A(x1) 24.09/7.11 B(b(a(b(x1)))) -> B(a(b(a(x1)))) 24.09/7.11 B(b(a(b(x1)))) -> A(b(a(x1))) 24.09/7.11 B(b(a(b(x1)))) -> B(a(x1)) 24.09/7.11 B(b(a(b(x1)))) -> A(x1) 24.09/7.11 A(a(b(a(x1)))) -> A(b(a(b(x1)))) 24.09/7.11 A(a(b(a(x1)))) -> B(a(b(x1))) 24.09/7.11 A(a(b(a(x1)))) -> A(b(x1)) 24.09/7.11 A(a(b(a(x1)))) -> B(x1) 24.09/7.11 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 s(a(x1)) -> a(s(x1)) 24.09/7.11 s(b(a(b(x1)))) -> a(s(b(a(x1)))) 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (5) DependencyGraphProof (EQUIVALENT) 24.09/7.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (6) 24.09/7.11 Complex Obligation (AND) 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (7) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 The TRS P consists of the following rules: 24.09/7.11 24.09/7.11 B(b(a(b(x1)))) -> A(b(a(x1))) 24.09/7.11 A(a(b(a(x1)))) -> A(b(a(b(x1)))) 24.09/7.11 A(a(b(a(x1)))) -> B(a(b(x1))) 24.09/7.11 B(b(a(b(x1)))) -> B(a(b(a(x1)))) 24.09/7.11 B(b(a(b(x1)))) -> B(a(x1)) 24.09/7.11 B(b(a(b(x1)))) -> A(x1) 24.09/7.11 A(a(b(a(x1)))) -> A(b(x1)) 24.09/7.11 A(a(b(a(x1)))) -> B(x1) 24.09/7.11 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 s(a(x1)) -> a(s(x1)) 24.09/7.11 s(b(a(b(x1)))) -> a(s(b(a(x1)))) 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (8) UsableRulesProof (EQUIVALENT) 24.09/7.11 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. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (9) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 The TRS P consists of the following rules: 24.09/7.11 24.09/7.11 B(b(a(b(x1)))) -> A(b(a(x1))) 24.09/7.11 A(a(b(a(x1)))) -> A(b(a(b(x1)))) 24.09/7.11 A(a(b(a(x1)))) -> B(a(b(x1))) 24.09/7.11 B(b(a(b(x1)))) -> B(a(b(a(x1)))) 24.09/7.11 B(b(a(b(x1)))) -> B(a(x1)) 24.09/7.11 B(b(a(b(x1)))) -> A(x1) 24.09/7.11 A(a(b(a(x1)))) -> A(b(x1)) 24.09/7.11 A(a(b(a(x1)))) -> B(x1) 24.09/7.11 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (10) MRRProof (EQUIVALENT) 24.09/7.11 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. 24.09/7.11 24.09/7.11 Strictly oriented dependency pairs: 24.09/7.11 24.09/7.11 B(b(a(b(x1)))) -> A(b(a(x1))) 24.09/7.11 A(a(b(a(x1)))) -> B(a(b(x1))) 24.09/7.11 B(b(a(b(x1)))) -> B(a(x1)) 24.09/7.11 B(b(a(b(x1)))) -> A(x1) 24.09/7.11 A(a(b(a(x1)))) -> A(b(x1)) 24.09/7.11 A(a(b(a(x1)))) -> B(x1) 24.09/7.11 24.09/7.11 24.09/7.11 Used ordering: Polynomial interpretation [POLO]: 24.09/7.11 24.09/7.11 POL(A(x_1)) = 1 + 2*x_1 24.09/7.11 POL(B(x_1)) = 2 + 2*x_1 24.09/7.11 POL(a(x_1)) = 2 + x_1 24.09/7.11 POL(b(x_1)) = 2 + x_1 24.09/7.11 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (11) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 The TRS P consists of the following rules: 24.09/7.11 24.09/7.11 A(a(b(a(x1)))) -> A(b(a(b(x1)))) 24.09/7.11 B(b(a(b(x1)))) -> B(a(b(a(x1)))) 24.09/7.11 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (12) DependencyGraphProof (EQUIVALENT) 24.09/7.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (13) 24.09/7.11 Complex Obligation (AND) 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (14) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 The TRS P consists of the following rules: 24.09/7.11 24.09/7.11 B(b(a(b(x1)))) -> B(a(b(a(x1)))) 24.09/7.11 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (15) QDPOrderProof (EQUIVALENT) 24.09/7.11 We use the reduction pair processor [LPAR04,JAR06]. 24.09/7.11 24.09/7.11 24.09/7.11 The following pairs can be oriented strictly and are deleted. 24.09/7.11 24.09/7.11 B(b(a(b(x1)))) -> B(a(b(a(x1)))) 24.09/7.11 The remaining pairs can at least be oriented weakly. 24.09/7.11 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 24.09/7.11 24.09/7.11 POL( B_1(x_1) ) = 2x_1 + 2 24.09/7.11 POL( a_1(x_1) ) = 0 24.09/7.11 POL( b_1(x_1) ) = 2 24.09/7.11 24.09/7.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.09/7.11 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (16) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 P is empty. 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (17) PisEmptyProof (EQUIVALENT) 24.09/7.11 The TRS P is empty. Hence, there is no (P,Q,R) chain. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (18) 24.09/7.11 YES 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (19) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 The TRS P consists of the following rules: 24.09/7.11 24.09/7.11 A(a(b(a(x1)))) -> A(b(a(b(x1)))) 24.09/7.11 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (20) QDPOrderProof (EQUIVALENT) 24.09/7.11 We use the reduction pair processor [LPAR04,JAR06]. 24.09/7.11 24.09/7.11 24.09/7.11 The following pairs can be oriented strictly and are deleted. 24.09/7.11 24.09/7.11 A(a(b(a(x1)))) -> A(b(a(b(x1)))) 24.09/7.11 The remaining pairs can at least be oriented weakly. 24.09/7.11 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 24.09/7.11 24.09/7.11 POL( A_1(x_1) ) = 2x_1 + 2 24.09/7.11 POL( b_1(x_1) ) = 0 24.09/7.11 POL( a_1(x_1) ) = 2 24.09/7.11 24.09/7.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.09/7.11 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (21) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 P is empty. 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (22) PisEmptyProof (EQUIVALENT) 24.09/7.11 The TRS P is empty. Hence, there is no (P,Q,R) chain. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (23) 24.09/7.11 YES 24.09/7.11 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (24) 24.09/7.11 Obligation: 24.09/7.11 Q DP problem: 24.09/7.11 The TRS P consists of the following rules: 24.09/7.11 24.09/7.11 S(b(a(b(x1)))) -> S(b(a(x1))) 24.09/7.11 S(a(x1)) -> S(x1) 24.09/7.11 24.09/7.11 The TRS R consists of the following rules: 24.09/7.11 24.09/7.11 s(a(x1)) -> a(s(x1)) 24.09/7.11 s(b(a(b(x1)))) -> a(s(b(a(x1)))) 24.09/7.11 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.11 24.09/7.11 Q is empty. 24.09/7.11 We have to consider all minimal (P,Q,R)-chains. 24.09/7.11 ---------------------------------------- 24.09/7.11 24.09/7.11 (25) UsableRulesProof (EQUIVALENT) 24.09/7.11 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. 24.09/7.11 ---------------------------------------- 24.09/7.12 24.09/7.12 (26) 24.09/7.12 Obligation: 24.09/7.12 Q DP problem: 24.09/7.12 The TRS P consists of the following rules: 24.09/7.12 24.09/7.12 S(b(a(b(x1)))) -> S(b(a(x1))) 24.09/7.12 S(a(x1)) -> S(x1) 24.09/7.12 24.09/7.12 The TRS R consists of the following rules: 24.09/7.12 24.09/7.12 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.12 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.12 24.09/7.12 Q is empty. 24.09/7.12 We have to consider all minimal (P,Q,R)-chains. 24.09/7.12 ---------------------------------------- 24.09/7.12 24.09/7.12 (27) QDPOrderProof (EQUIVALENT) 24.09/7.12 We use the reduction pair processor [LPAR04,JAR06]. 24.09/7.12 24.09/7.12 24.09/7.12 The following pairs can be oriented strictly and are deleted. 24.09/7.12 24.09/7.12 S(a(x1)) -> S(x1) 24.09/7.12 The remaining pairs can at least be oriented weakly. 24.09/7.12 Used ordering: Polynomial interpretation [POLO]: 24.09/7.12 24.09/7.12 POL(S(x_1)) = x_1 24.09/7.12 POL(a(x_1)) = 1 + x_1 24.09/7.12 POL(b(x_1)) = 1 24.09/7.12 24.09/7.12 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.09/7.12 24.09/7.12 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.12 24.09/7.12 24.09/7.12 ---------------------------------------- 24.09/7.12 24.09/7.12 (28) 24.09/7.12 Obligation: 24.09/7.12 Q DP problem: 24.09/7.12 The TRS P consists of the following rules: 24.09/7.12 24.09/7.12 S(b(a(b(x1)))) -> S(b(a(x1))) 24.09/7.12 24.09/7.12 The TRS R consists of the following rules: 24.09/7.12 24.09/7.12 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.12 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.12 24.09/7.12 Q is empty. 24.09/7.12 We have to consider all minimal (P,Q,R)-chains. 24.09/7.12 ---------------------------------------- 24.09/7.12 24.09/7.12 (29) QDPOrderProof (EQUIVALENT) 24.09/7.12 We use the reduction pair processor [LPAR04,JAR06]. 24.09/7.12 24.09/7.12 24.09/7.12 The following pairs can be oriented strictly and are deleted. 24.09/7.12 24.09/7.12 S(b(a(b(x1)))) -> S(b(a(x1))) 24.09/7.12 The remaining pairs can at least be oriented weakly. 24.09/7.12 Used ordering: Polynomial interpretation [POLO]: 24.09/7.12 24.09/7.12 POL(S(x_1)) = x_1 24.09/7.12 POL(a(x_1)) = 1 + x_1 24.09/7.12 POL(b(x_1)) = 1 + x_1 24.09/7.12 24.09/7.12 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.09/7.12 24.09/7.12 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.12 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.12 24.09/7.12 24.09/7.12 ---------------------------------------- 24.09/7.12 24.09/7.12 (30) 24.09/7.12 Obligation: 24.09/7.12 Q DP problem: 24.09/7.12 P is empty. 24.09/7.12 The TRS R consists of the following rules: 24.09/7.12 24.09/7.12 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 24.09/7.12 b(b(a(b(x1)))) -> b(a(b(a(x1)))) 24.09/7.12 24.09/7.12 Q is empty. 24.09/7.12 We have to consider all minimal (P,Q,R)-chains. 24.09/7.12 ---------------------------------------- 24.09/7.12 24.09/7.12 (31) PisEmptyProof (EQUIVALENT) 24.09/7.12 The TRS P is empty. Hence, there is no (P,Q,R) chain. 24.09/7.12 ---------------------------------------- 24.09/7.12 24.09/7.12 (32) 24.09/7.12 YES 24.26/7.23 EOF