25.29/7.40 YES 25.57/7.44 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 25.57/7.44 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.57/7.44 25.57/7.44 25.57/7.44 Termination w.r.t. Q of the given QTRS could be proven: 25.57/7.44 25.57/7.44 (0) QTRS 25.57/7.44 (1) QTRS Reverse [EQUIVALENT, 0 ms] 25.57/7.44 (2) QTRS 25.57/7.44 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 25.57/7.44 (4) QDP 25.57/7.44 (5) QDPOrderProof [EQUIVALENT, 193 ms] 25.57/7.44 (6) QDP 25.57/7.44 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 25.57/7.44 (8) QDP 25.57/7.44 (9) QDPOrderProof [EQUIVALENT, 0 ms] 25.57/7.44 (10) QDP 25.57/7.44 (11) PisEmptyProof [EQUIVALENT, 0 ms] 25.57/7.44 (12) YES 25.57/7.44 25.57/7.44 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (0) 25.57/7.44 Obligation: 25.57/7.44 Q restricted rewrite system: 25.57/7.44 The TRS R consists of the following rules: 25.57/7.44 25.57/7.44 a(a(x1)) -> a(b(a(c(c(x1))))) 25.57/7.44 c(a(x1)) -> x1 25.57/7.44 c(b(x1)) -> a(x1) 25.57/7.44 25.57/7.44 Q is empty. 25.57/7.44 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (1) QTRS Reverse (EQUIVALENT) 25.57/7.44 We applied the QTRS Reverse Processor [REVERSE]. 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (2) 25.57/7.44 Obligation: 25.57/7.44 Q restricted rewrite system: 25.57/7.44 The TRS R consists of the following rules: 25.57/7.44 25.57/7.44 a(a(x1)) -> c(c(a(b(a(x1))))) 25.57/7.44 a(c(x1)) -> x1 25.57/7.44 b(c(x1)) -> a(x1) 25.57/7.44 25.57/7.44 Q is empty. 25.57/7.44 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (3) DependencyPairsProof (EQUIVALENT) 25.57/7.44 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (4) 25.57/7.44 Obligation: 25.57/7.44 Q DP problem: 25.57/7.44 The TRS P consists of the following rules: 25.57/7.44 25.57/7.44 A(a(x1)) -> A(b(a(x1))) 25.57/7.44 A(a(x1)) -> B(a(x1)) 25.57/7.44 B(c(x1)) -> A(x1) 25.57/7.44 25.57/7.44 The TRS R consists of the following rules: 25.57/7.44 25.57/7.44 a(a(x1)) -> c(c(a(b(a(x1))))) 25.57/7.44 a(c(x1)) -> x1 25.57/7.44 b(c(x1)) -> a(x1) 25.57/7.44 25.57/7.44 Q is empty. 25.57/7.44 We have to consider all minimal (P,Q,R)-chains. 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (5) QDPOrderProof (EQUIVALENT) 25.57/7.44 We use the reduction pair processor [LPAR04,JAR06]. 25.57/7.44 25.57/7.44 25.57/7.44 The following pairs can be oriented strictly and are deleted. 25.57/7.44 25.57/7.44 B(c(x1)) -> A(x1) 25.57/7.44 The remaining pairs can at least be oriented weakly. 25.57/7.44 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(A(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(a(x_1)) = [[1A], [0A], [0A]] + [[1A, 0A, -I], [0A, -I, 0A], [0A, -I, -I]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(b(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, 0A], [0A, -I, 0A], [-I, 0A, 0A]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(B(x_1)) = [[0A]] + [[0A, 0A, 1A]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(c(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, 0A], [0A, -I, -I], [1A, 0A, -I]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 25.57/7.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.57/7.44 25.57/7.44 a(a(x1)) -> c(c(a(b(a(x1))))) 25.57/7.44 a(c(x1)) -> x1 25.57/7.44 b(c(x1)) -> a(x1) 25.57/7.44 25.57/7.44 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (6) 25.57/7.44 Obligation: 25.57/7.44 Q DP problem: 25.57/7.44 The TRS P consists of the following rules: 25.57/7.44 25.57/7.44 A(a(x1)) -> A(b(a(x1))) 25.57/7.44 A(a(x1)) -> B(a(x1)) 25.57/7.44 25.57/7.44 The TRS R consists of the following rules: 25.57/7.44 25.57/7.44 a(a(x1)) -> c(c(a(b(a(x1))))) 25.57/7.44 a(c(x1)) -> x1 25.57/7.44 b(c(x1)) -> a(x1) 25.57/7.44 25.57/7.44 Q is empty. 25.57/7.44 We have to consider all minimal (P,Q,R)-chains. 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (7) DependencyGraphProof (EQUIVALENT) 25.57/7.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (8) 25.57/7.44 Obligation: 25.57/7.44 Q DP problem: 25.57/7.44 The TRS P consists of the following rules: 25.57/7.44 25.57/7.44 A(a(x1)) -> A(b(a(x1))) 25.57/7.44 25.57/7.44 The TRS R consists of the following rules: 25.57/7.44 25.57/7.44 a(a(x1)) -> c(c(a(b(a(x1))))) 25.57/7.44 a(c(x1)) -> x1 25.57/7.44 b(c(x1)) -> a(x1) 25.57/7.44 25.57/7.44 Q is empty. 25.57/7.44 We have to consider all minimal (P,Q,R)-chains. 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (9) QDPOrderProof (EQUIVALENT) 25.57/7.44 We use the reduction pair processor [LPAR04,JAR06]. 25.57/7.44 25.57/7.44 25.57/7.44 The following pairs can be oriented strictly and are deleted. 25.57/7.44 25.57/7.44 A(a(x1)) -> A(b(a(x1))) 25.57/7.44 The remaining pairs can at least be oriented weakly. 25.57/7.44 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(A(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(a(x_1)) = [[1A], [0A], [-I]] + [[1A, 0A, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(b(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [-I, -I, 0A], [-I, -I, 0A]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 <<< 25.57/7.44 POL(c(x_1)) = [[0A], [1A], [-I]] + [[-I, 0A, 0A], [0A, 0A, -I], [1A, -I, -I]] * x_1 25.57/7.44 >>> 25.57/7.44 25.57/7.44 25.57/7.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.57/7.44 25.57/7.44 a(a(x1)) -> c(c(a(b(a(x1))))) 25.57/7.44 a(c(x1)) -> x1 25.57/7.44 b(c(x1)) -> a(x1) 25.57/7.44 25.57/7.44 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (10) 25.57/7.44 Obligation: 25.57/7.44 Q DP problem: 25.57/7.44 P is empty. 25.57/7.44 The TRS R consists of the following rules: 25.57/7.44 25.57/7.44 a(a(x1)) -> c(c(a(b(a(x1))))) 25.57/7.44 a(c(x1)) -> x1 25.57/7.44 b(c(x1)) -> a(x1) 25.57/7.44 25.57/7.44 Q is empty. 25.57/7.44 We have to consider all minimal (P,Q,R)-chains. 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (11) PisEmptyProof (EQUIVALENT) 25.57/7.44 The TRS P is empty. Hence, there is no (P,Q,R) chain. 25.57/7.44 ---------------------------------------- 25.57/7.44 25.57/7.44 (12) 25.57/7.44 YES 25.87/7.59 EOF