32.80/9.15 YES 32.99/9.19 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 32.99/9.19 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 32.99/9.19 32.99/9.19 32.99/9.19 Termination w.r.t. Q of the given QTRS could be proven: 32.99/9.19 32.99/9.19 (0) QTRS 32.99/9.19 (1) QTRS Reverse [EQUIVALENT, 0 ms] 32.99/9.19 (2) QTRS 32.99/9.19 (3) Overlay + Local Confluence [EQUIVALENT, 0 ms] 32.99/9.19 (4) QTRS 32.99/9.19 (5) DependencyPairsProof [EQUIVALENT, 23 ms] 32.99/9.19 (6) QDP 32.99/9.19 (7) QDPOrderProof [EQUIVALENT, 141 ms] 32.99/9.19 (8) QDP 32.99/9.19 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 32.99/9.19 (10) QDP 32.99/9.19 (11) QDPOrderProof [EQUIVALENT, 0 ms] 32.99/9.19 (12) QDP 32.99/9.19 (13) PisEmptyProof [EQUIVALENT, 0 ms] 32.99/9.19 (14) YES 32.99/9.19 32.99/9.19 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (0) 32.99/9.19 Obligation: 32.99/9.19 Q restricted rewrite system: 32.99/9.19 The TRS R consists of the following rules: 32.99/9.19 32.99/9.19 a(a(b(x1))) -> c(b(a(a(a(x1))))) 32.99/9.19 a(c(x1)) -> b(a(x1)) 32.99/9.19 32.99/9.19 Q is empty. 32.99/9.19 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (1) QTRS Reverse (EQUIVALENT) 32.99/9.19 We applied the QTRS Reverse Processor [REVERSE]. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (2) 32.99/9.19 Obligation: 32.99/9.19 Q restricted rewrite system: 32.99/9.19 The TRS R consists of the following rules: 32.99/9.19 32.99/9.19 b(a(a(x1))) -> a(a(a(b(c(x1))))) 32.99/9.19 c(a(x1)) -> a(b(x1)) 32.99/9.19 32.99/9.19 Q is empty. 32.99/9.19 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (3) Overlay + Local Confluence (EQUIVALENT) 32.99/9.19 The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (4) 32.99/9.19 Obligation: 32.99/9.19 Q restricted rewrite system: 32.99/9.19 The TRS R consists of the following rules: 32.99/9.19 32.99/9.19 b(a(a(x1))) -> a(a(a(b(c(x1))))) 32.99/9.19 c(a(x1)) -> a(b(x1)) 32.99/9.19 32.99/9.19 The set Q consists of the following terms: 32.99/9.19 32.99/9.19 b(a(a(x0))) 32.99/9.19 c(a(x0)) 32.99/9.19 32.99/9.19 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (5) DependencyPairsProof (EQUIVALENT) 32.99/9.19 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (6) 32.99/9.19 Obligation: 32.99/9.19 Q DP problem: 32.99/9.19 The TRS P consists of the following rules: 32.99/9.19 32.99/9.19 B(a(a(x1))) -> B(c(x1)) 32.99/9.19 B(a(a(x1))) -> C(x1) 32.99/9.19 C(a(x1)) -> B(x1) 32.99/9.19 32.99/9.19 The TRS R consists of the following rules: 32.99/9.19 32.99/9.19 b(a(a(x1))) -> a(a(a(b(c(x1))))) 32.99/9.19 c(a(x1)) -> a(b(x1)) 32.99/9.19 32.99/9.19 The set Q consists of the following terms: 32.99/9.19 32.99/9.19 b(a(a(x0))) 32.99/9.19 c(a(x0)) 32.99/9.19 32.99/9.19 We have to consider all minimal (P,Q,R)-chains. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (7) QDPOrderProof (EQUIVALENT) 32.99/9.19 We use the reduction pair processor [LPAR04,JAR06]. 32.99/9.19 32.99/9.19 32.99/9.19 The following pairs can be oriented strictly and are deleted. 32.99/9.19 32.99/9.19 C(a(x1)) -> B(x1) 32.99/9.19 The remaining pairs can at least be oriented weakly. 32.99/9.19 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(a(x_1)) = [[1A], [-I], [-I]] + [[0A, 0A, 1A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(c(x_1)) = [[0A], [0A], [-I]] + [[1A, -I, -I], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(C(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, 0A, -I], [-I, 0A, -I]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 32.99/9.19 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 32.99/9.19 32.99/9.19 c(a(x1)) -> a(b(x1)) 32.99/9.19 b(a(a(x1))) -> a(a(a(b(c(x1))))) 32.99/9.19 32.99/9.19 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (8) 32.99/9.19 Obligation: 32.99/9.19 Q DP problem: 32.99/9.19 The TRS P consists of the following rules: 32.99/9.19 32.99/9.19 B(a(a(x1))) -> B(c(x1)) 32.99/9.19 B(a(a(x1))) -> C(x1) 32.99/9.19 32.99/9.19 The TRS R consists of the following rules: 32.99/9.19 32.99/9.19 b(a(a(x1))) -> a(a(a(b(c(x1))))) 32.99/9.19 c(a(x1)) -> a(b(x1)) 32.99/9.19 32.99/9.19 The set Q consists of the following terms: 32.99/9.19 32.99/9.19 b(a(a(x0))) 32.99/9.19 c(a(x0)) 32.99/9.19 32.99/9.19 We have to consider all minimal (P,Q,R)-chains. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (9) DependencyGraphProof (EQUIVALENT) 32.99/9.19 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (10) 32.99/9.19 Obligation: 32.99/9.19 Q DP problem: 32.99/9.19 The TRS P consists of the following rules: 32.99/9.19 32.99/9.19 B(a(a(x1))) -> B(c(x1)) 32.99/9.19 32.99/9.19 The TRS R consists of the following rules: 32.99/9.19 32.99/9.19 b(a(a(x1))) -> a(a(a(b(c(x1))))) 32.99/9.19 c(a(x1)) -> a(b(x1)) 32.99/9.19 32.99/9.19 The set Q consists of the following terms: 32.99/9.19 32.99/9.19 b(a(a(x0))) 32.99/9.19 c(a(x0)) 32.99/9.19 32.99/9.19 We have to consider all minimal (P,Q,R)-chains. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (11) QDPOrderProof (EQUIVALENT) 32.99/9.19 We use the reduction pair processor [LPAR04,JAR06]. 32.99/9.19 32.99/9.19 32.99/9.19 The following pairs can be oriented strictly and are deleted. 32.99/9.19 32.99/9.19 B(a(a(x1))) -> B(c(x1)) 32.99/9.19 The remaining pairs can at least be oriented weakly. 32.99/9.19 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(B(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(a(x_1)) = [[-I], [0A], [-I]] + [[0A, -I, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(c(x_1)) = [[-I], [0A], [-I]] + [[0A, -I, -I], [0A, 0A, 1A], [0A, -I, -I]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 <<< 32.99/9.19 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 32.99/9.19 >>> 32.99/9.19 32.99/9.19 32.99/9.19 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 32.99/9.19 32.99/9.19 c(a(x1)) -> a(b(x1)) 32.99/9.19 b(a(a(x1))) -> a(a(a(b(c(x1))))) 32.99/9.19 32.99/9.19 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (12) 32.99/9.19 Obligation: 32.99/9.19 Q DP problem: 32.99/9.19 P is empty. 32.99/9.19 The TRS R consists of the following rules: 32.99/9.19 32.99/9.19 b(a(a(x1))) -> a(a(a(b(c(x1))))) 32.99/9.19 c(a(x1)) -> a(b(x1)) 32.99/9.19 32.99/9.19 The set Q consists of the following terms: 32.99/9.19 32.99/9.19 b(a(a(x0))) 32.99/9.19 c(a(x0)) 32.99/9.19 32.99/9.19 We have to consider all minimal (P,Q,R)-chains. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (13) PisEmptyProof (EQUIVALENT) 32.99/9.19 The TRS P is empty. Hence, there is no (P,Q,R) chain. 32.99/9.19 ---------------------------------------- 32.99/9.19 32.99/9.19 (14) 32.99/9.19 YES 32.99/9.25 EOF