24.65/7.28 YES 24.65/7.32 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 24.65/7.32 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.65/7.32 24.65/7.32 24.65/7.32 Termination w.r.t. Q of the given QTRS could be proven: 24.65/7.32 24.65/7.32 (0) QTRS 24.65/7.32 (1) QTRS Reverse [EQUIVALENT, 0 ms] 24.65/7.32 (2) QTRS 24.65/7.32 (3) DependencyPairsProof [EQUIVALENT, 29 ms] 24.65/7.32 (4) QDP 24.65/7.32 (5) DependencyGraphProof [EQUIVALENT, 3 ms] 24.65/7.32 (6) AND 24.65/7.32 (7) QDP 24.65/7.32 (8) UsableRulesProof [EQUIVALENT, 3 ms] 24.65/7.32 (9) QDP 24.65/7.32 (10) MRRProof [EQUIVALENT, 8 ms] 24.65/7.32 (11) QDP 24.65/7.32 (12) PisEmptyProof [EQUIVALENT, 0 ms] 24.65/7.32 (13) YES 24.65/7.32 (14) QDP 24.65/7.32 (15) UsableRulesProof [EQUIVALENT, 0 ms] 24.65/7.32 (16) QDP 24.65/7.32 (17) QDPOrderProof [EQUIVALENT, 5 ms] 24.65/7.32 (18) QDP 24.65/7.32 (19) PisEmptyProof [EQUIVALENT, 0 ms] 24.65/7.32 (20) YES 24.65/7.32 (21) QDP 24.65/7.32 (22) UsableRulesProof [EQUIVALENT, 0 ms] 24.65/7.32 (23) QDP 24.65/7.32 (24) QDPOrderProof [EQUIVALENT, 19 ms] 24.65/7.32 (25) QDP 24.65/7.32 (26) PisEmptyProof [EQUIVALENT, 0 ms] 24.65/7.32 (27) YES 24.65/7.32 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (0) 24.65/7.32 Obligation: 24.65/7.32 Q restricted rewrite system: 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 a(b(x1)) -> b(b(b(x1))) 24.65/7.32 a(c(x1)) -> c(b(b(x1))) 24.65/7.32 b(a(x1)) -> a(a(a(x1))) 24.65/7.32 b(c(x1)) -> c(c(c(x1))) 24.65/7.32 a(x1) -> x1 24.65/7.32 b(x1) -> x1 24.65/7.32 c(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (1) QTRS Reverse (EQUIVALENT) 24.65/7.32 We applied the QTRS Reverse Processor [REVERSE]. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (2) 24.65/7.32 Obligation: 24.65/7.32 Q restricted rewrite system: 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 c(a(x1)) -> b(b(c(x1))) 24.65/7.32 a(b(x1)) -> a(a(a(x1))) 24.65/7.32 c(b(x1)) -> c(c(c(x1))) 24.65/7.32 a(x1) -> x1 24.65/7.32 b(x1) -> x1 24.65/7.32 c(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (3) DependencyPairsProof (EQUIVALENT) 24.65/7.32 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (4) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 The TRS P consists of the following rules: 24.65/7.32 24.65/7.32 B(a(x1)) -> B(b(b(x1))) 24.65/7.32 B(a(x1)) -> B(b(x1)) 24.65/7.32 B(a(x1)) -> B(x1) 24.65/7.32 C(a(x1)) -> B(b(c(x1))) 24.65/7.32 C(a(x1)) -> B(c(x1)) 24.65/7.32 C(a(x1)) -> C(x1) 24.65/7.32 A(b(x1)) -> A(a(a(x1))) 24.65/7.32 A(b(x1)) -> A(a(x1)) 24.65/7.32 A(b(x1)) -> A(x1) 24.65/7.32 C(b(x1)) -> C(c(c(x1))) 24.65/7.32 C(b(x1)) -> C(c(x1)) 24.65/7.32 C(b(x1)) -> C(x1) 24.65/7.32 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 c(a(x1)) -> b(b(c(x1))) 24.65/7.32 a(b(x1)) -> a(a(a(x1))) 24.65/7.32 c(b(x1)) -> c(c(c(x1))) 24.65/7.32 a(x1) -> x1 24.65/7.32 b(x1) -> x1 24.65/7.32 c(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (5) DependencyGraphProof (EQUIVALENT) 24.65/7.32 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (6) 24.65/7.32 Complex Obligation (AND) 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (7) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 The TRS P consists of the following rules: 24.65/7.32 24.65/7.32 A(b(x1)) -> A(a(x1)) 24.65/7.32 A(b(x1)) -> A(a(a(x1))) 24.65/7.32 A(b(x1)) -> A(x1) 24.65/7.32 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 c(a(x1)) -> b(b(c(x1))) 24.65/7.32 a(b(x1)) -> a(a(a(x1))) 24.65/7.32 c(b(x1)) -> c(c(c(x1))) 24.65/7.32 a(x1) -> x1 24.65/7.32 b(x1) -> x1 24.65/7.32 c(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (8) UsableRulesProof (EQUIVALENT) 24.65/7.32 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. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (9) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 The TRS P consists of the following rules: 24.65/7.32 24.65/7.32 A(b(x1)) -> A(a(x1)) 24.65/7.32 A(b(x1)) -> A(a(a(x1))) 24.65/7.32 A(b(x1)) -> A(x1) 24.65/7.32 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 a(b(x1)) -> a(a(a(x1))) 24.65/7.32 a(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (10) MRRProof (EQUIVALENT) 24.65/7.32 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. 24.65/7.32 24.65/7.32 Strictly oriented dependency pairs: 24.65/7.32 24.65/7.32 A(b(x1)) -> A(a(x1)) 24.65/7.32 A(b(x1)) -> A(a(a(x1))) 24.65/7.32 A(b(x1)) -> A(x1) 24.65/7.32 24.65/7.32 Strictly oriented rules of the TRS R: 24.65/7.32 24.65/7.32 a(b(x1)) -> a(a(a(x1))) 24.65/7.32 a(x1) -> x1 24.65/7.32 24.65/7.32 Used ordering: Polynomial interpretation [POLO]: 24.65/7.32 24.65/7.32 POL(A(x_1)) = 3*x_1 24.65/7.32 POL(a(x_1)) = 1 + x_1 24.65/7.32 POL(b(x_1)) = 3 + x_1 24.65/7.32 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (11) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 P is empty. 24.65/7.32 R is empty. 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (12) PisEmptyProof (EQUIVALENT) 24.65/7.32 The TRS P is empty. Hence, there is no (P,Q,R) chain. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (13) 24.65/7.32 YES 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (14) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 The TRS P consists of the following rules: 24.65/7.32 24.65/7.32 B(a(x1)) -> B(b(x1)) 24.65/7.32 B(a(x1)) -> B(b(b(x1))) 24.65/7.32 B(a(x1)) -> B(x1) 24.65/7.32 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 c(a(x1)) -> b(b(c(x1))) 24.65/7.32 a(b(x1)) -> a(a(a(x1))) 24.65/7.32 c(b(x1)) -> c(c(c(x1))) 24.65/7.32 a(x1) -> x1 24.65/7.32 b(x1) -> x1 24.65/7.32 c(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (15) UsableRulesProof (EQUIVALENT) 24.65/7.32 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. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (16) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 The TRS P consists of the following rules: 24.65/7.32 24.65/7.32 B(a(x1)) -> B(b(x1)) 24.65/7.32 B(a(x1)) -> B(b(b(x1))) 24.65/7.32 B(a(x1)) -> B(x1) 24.65/7.32 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 b(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (17) QDPOrderProof (EQUIVALENT) 24.65/7.32 We use the reduction pair processor [LPAR04,JAR06]. 24.65/7.32 24.65/7.32 24.65/7.32 The following pairs can be oriented strictly and are deleted. 24.65/7.32 24.65/7.32 B(a(x1)) -> B(b(x1)) 24.65/7.32 B(a(x1)) -> B(b(b(x1))) 24.65/7.32 B(a(x1)) -> B(x1) 24.65/7.32 The remaining pairs can at least be oriented weakly. 24.65/7.32 Used ordering: Polynomial interpretation [POLO]: 24.65/7.32 24.65/7.32 POL(B(x_1)) = x_1 24.65/7.32 POL(a(x_1)) = 1 + x_1 24.65/7.32 POL(b(x_1)) = x_1 24.65/7.32 24.65/7.32 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.65/7.32 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 b(x1) -> x1 24.65/7.32 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (18) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 P is empty. 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 b(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (19) PisEmptyProof (EQUIVALENT) 24.65/7.32 The TRS P is empty. Hence, there is no (P,Q,R) chain. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (20) 24.65/7.32 YES 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (21) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 The TRS P consists of the following rules: 24.65/7.32 24.65/7.32 C(b(x1)) -> C(c(c(x1))) 24.65/7.32 C(a(x1)) -> C(x1) 24.65/7.32 C(b(x1)) -> C(c(x1)) 24.65/7.32 C(b(x1)) -> C(x1) 24.65/7.32 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 c(a(x1)) -> b(b(c(x1))) 24.65/7.32 a(b(x1)) -> a(a(a(x1))) 24.65/7.32 c(b(x1)) -> c(c(c(x1))) 24.65/7.32 a(x1) -> x1 24.65/7.32 b(x1) -> x1 24.65/7.32 c(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (22) UsableRulesProof (EQUIVALENT) 24.65/7.32 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. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (23) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 The TRS P consists of the following rules: 24.65/7.32 24.65/7.32 C(b(x1)) -> C(c(c(x1))) 24.65/7.32 C(a(x1)) -> C(x1) 24.65/7.32 C(b(x1)) -> C(c(x1)) 24.65/7.32 C(b(x1)) -> C(x1) 24.65/7.32 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 c(a(x1)) -> b(b(c(x1))) 24.65/7.32 c(b(x1)) -> c(c(c(x1))) 24.65/7.32 c(x1) -> x1 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 b(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (24) QDPOrderProof (EQUIVALENT) 24.65/7.32 We use the reduction pair processor [LPAR04,JAR06]. 24.65/7.32 24.65/7.32 24.65/7.32 The following pairs can be oriented strictly and are deleted. 24.65/7.32 24.65/7.32 C(b(x1)) -> C(c(c(x1))) 24.65/7.32 C(a(x1)) -> C(x1) 24.65/7.32 C(b(x1)) -> C(c(x1)) 24.65/7.32 C(b(x1)) -> C(x1) 24.65/7.32 The remaining pairs can at least be oriented weakly. 24.65/7.32 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 24.65/7.32 24.65/7.32 POL( C_1(x_1) ) = x_1 + 1 24.65/7.32 POL( c_1(x_1) ) = x_1 24.65/7.32 POL( a_1(x_1) ) = 2x_1 + 2 24.65/7.32 POL( b_1(x_1) ) = x_1 + 1 24.65/7.32 24.65/7.32 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.65/7.32 24.65/7.32 c(a(x1)) -> b(b(c(x1))) 24.65/7.32 c(b(x1)) -> c(c(c(x1))) 24.65/7.32 c(x1) -> x1 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 b(x1) -> x1 24.65/7.32 24.65/7.32 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (25) 24.65/7.32 Obligation: 24.65/7.32 Q DP problem: 24.65/7.32 P is empty. 24.65/7.32 The TRS R consists of the following rules: 24.65/7.32 24.65/7.32 c(a(x1)) -> b(b(c(x1))) 24.65/7.32 c(b(x1)) -> c(c(c(x1))) 24.65/7.32 c(x1) -> x1 24.65/7.32 b(a(x1)) -> b(b(b(x1))) 24.65/7.32 b(x1) -> x1 24.65/7.32 24.65/7.32 Q is empty. 24.65/7.32 We have to consider all minimal (P,Q,R)-chains. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (26) PisEmptyProof (EQUIVALENT) 24.65/7.32 The TRS P is empty. Hence, there is no (P,Q,R) chain. 24.65/7.32 ---------------------------------------- 24.65/7.32 24.65/7.32 (27) 24.65/7.32 YES 25.02/7.46 EOF