140.03/36.41 YES 140.03/36.44 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 140.03/36.44 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 140.03/36.44 140.03/36.44 140.03/36.44 Termination w.r.t. Q of the given QTRS could be proven: 140.03/36.44 140.03/36.44 (0) QTRS 140.03/36.44 (1) QTRS Reverse [EQUIVALENT, 0 ms] 140.03/36.44 (2) QTRS 140.03/36.44 (3) DependencyPairsProof [EQUIVALENT, 4 ms] 140.03/36.44 (4) QDP 140.03/36.44 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 140.03/36.44 (6) AND 140.03/36.44 (7) QDP 140.03/36.44 (8) UsableRulesProof [EQUIVALENT, 0 ms] 140.03/36.44 (9) QDP 140.03/36.44 (10) MNOCProof [EQUIVALENT, 0 ms] 140.03/36.44 (11) QDP 140.03/36.44 (12) QDPOrderProof [EQUIVALENT, 679 ms] 140.03/36.44 (13) QDP 140.03/36.44 (14) QDPOrderProof [EQUIVALENT, 5862 ms] 140.03/36.44 (15) QDP 140.03/36.44 (16) PisEmptyProof [EQUIVALENT, 0 ms] 140.03/36.44 (17) YES 140.03/36.44 (18) QDP 140.03/36.44 (19) UsableRulesProof [EQUIVALENT, 1 ms] 140.03/36.44 (20) QDP 140.03/36.44 (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] 140.03/36.44 (22) YES 140.03/36.44 140.03/36.44 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (0) 140.03/36.44 Obligation: 140.03/36.44 Q restricted rewrite system: 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 a(a(b(b(x1)))) -> b(b(b(a(a(a(x1)))))) 140.03/36.44 a(c(x1)) -> c(a(x1)) 140.03/36.44 b(c(x1)) -> c(b(x1)) 140.03/36.44 140.03/36.44 Q is empty. 140.03/36.44 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (1) QTRS Reverse (EQUIVALENT) 140.03/36.44 We applied the QTRS Reverse Processor [REVERSE]. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (2) 140.03/36.44 Obligation: 140.03/36.44 Q restricted rewrite system: 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 c(a(x1)) -> a(c(x1)) 140.03/36.44 c(b(x1)) -> b(c(x1)) 140.03/36.44 140.03/36.44 Q is empty. 140.03/36.44 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (3) DependencyPairsProof (EQUIVALENT) 140.03/36.44 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (4) 140.03/36.44 Obligation: 140.03/36.44 Q DP problem: 140.03/36.44 The TRS P consists of the following rules: 140.03/36.44 140.03/36.44 B(b(a(a(x1)))) -> B(b(b(x1))) 140.03/36.44 B(b(a(a(x1)))) -> B(b(x1)) 140.03/36.44 B(b(a(a(x1)))) -> B(x1) 140.03/36.44 C(a(x1)) -> C(x1) 140.03/36.44 C(b(x1)) -> B(c(x1)) 140.03/36.44 C(b(x1)) -> C(x1) 140.03/36.44 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 c(a(x1)) -> a(c(x1)) 140.03/36.44 c(b(x1)) -> b(c(x1)) 140.03/36.44 140.03/36.44 Q is empty. 140.03/36.44 We have to consider all minimal (P,Q,R)-chains. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (5) DependencyGraphProof (EQUIVALENT) 140.03/36.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (6) 140.03/36.44 Complex Obligation (AND) 140.03/36.44 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (7) 140.03/36.44 Obligation: 140.03/36.44 Q DP problem: 140.03/36.44 The TRS P consists of the following rules: 140.03/36.44 140.03/36.44 B(b(a(a(x1)))) -> B(b(x1)) 140.03/36.44 B(b(a(a(x1)))) -> B(b(b(x1))) 140.03/36.44 B(b(a(a(x1)))) -> B(x1) 140.03/36.44 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 c(a(x1)) -> a(c(x1)) 140.03/36.44 c(b(x1)) -> b(c(x1)) 140.03/36.44 140.03/36.44 Q is empty. 140.03/36.44 We have to consider all minimal (P,Q,R)-chains. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (8) UsableRulesProof (EQUIVALENT) 140.03/36.44 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. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (9) 140.03/36.44 Obligation: 140.03/36.44 Q DP problem: 140.03/36.44 The TRS P consists of the following rules: 140.03/36.44 140.03/36.44 B(b(a(a(x1)))) -> B(b(x1)) 140.03/36.44 B(b(a(a(x1)))) -> B(b(b(x1))) 140.03/36.44 B(b(a(a(x1)))) -> B(x1) 140.03/36.44 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 140.03/36.44 Q is empty. 140.03/36.44 We have to consider all minimal (P,Q,R)-chains. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (10) MNOCProof (EQUIVALENT) 140.03/36.44 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (11) 140.03/36.44 Obligation: 140.03/36.44 Q DP problem: 140.03/36.44 The TRS P consists of the following rules: 140.03/36.44 140.03/36.44 B(b(a(a(x1)))) -> B(b(x1)) 140.03/36.44 B(b(a(a(x1)))) -> B(b(b(x1))) 140.03/36.44 B(b(a(a(x1)))) -> B(x1) 140.03/36.44 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 140.03/36.44 The set Q consists of the following terms: 140.03/36.44 140.03/36.44 b(b(a(a(x0)))) 140.03/36.44 140.03/36.44 We have to consider all minimal (P,Q,R)-chains. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (12) QDPOrderProof (EQUIVALENT) 140.03/36.44 We use the reduction pair processor [LPAR04,JAR06]. 140.03/36.44 140.03/36.44 140.03/36.44 The following pairs can be oriented strictly and are deleted. 140.03/36.44 140.03/36.44 B(b(a(a(x1)))) -> B(b(b(x1))) 140.03/36.44 B(b(a(a(x1)))) -> B(x1) 140.03/36.44 The remaining pairs can at least be oriented weakly. 140.03/36.44 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 140.03/36.44 140.03/36.44 <<< 140.03/36.44 POL(B(x_1)) = [[0A]] + [[0A, -I, 1A]] * x_1 140.03/36.44 >>> 140.03/36.44 140.03/36.44 <<< 140.03/36.44 POL(b(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, -I, 1A], [0A, -I, -I]] * x_1 140.03/36.44 >>> 140.03/36.44 140.03/36.44 <<< 140.03/36.44 POL(a(x_1)) = [[0A], [1A], [-I]] + [[-I, 0A, 0A], [0A, -I, 1A], [-I, -I, -I]] * x_1 140.03/36.44 >>> 140.03/36.44 140.03/36.44 140.03/36.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 140.03/36.44 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (13) 140.03/36.44 Obligation: 140.03/36.44 Q DP problem: 140.03/36.44 The TRS P consists of the following rules: 140.03/36.44 140.03/36.44 B(b(a(a(x1)))) -> B(b(x1)) 140.03/36.44 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 140.03/36.44 The set Q consists of the following terms: 140.03/36.44 140.03/36.44 b(b(a(a(x0)))) 140.03/36.44 140.03/36.44 We have to consider all minimal (P,Q,R)-chains. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (14) QDPOrderProof (EQUIVALENT) 140.03/36.44 We use the reduction pair processor [LPAR04,JAR06]. 140.03/36.44 140.03/36.44 140.03/36.44 The following pairs can be oriented strictly and are deleted. 140.03/36.44 140.03/36.44 B(b(a(a(x1)))) -> B(b(x1)) 140.03/36.44 The remaining pairs can at least be oriented weakly. 140.03/36.44 Used ordering: Matrix interpretation [MATRO] to (N^5, +, *, >=, >) : 140.03/36.44 140.03/36.44 <<< 140.03/36.44 POL(B(x_1)) = [[0]] + [[0, 0, 0, 0, 1]] * x_1 140.03/36.44 >>> 140.03/36.44 140.03/36.44 <<< 140.03/36.44 POL(b(x_1)) = [[0], [0], [0], [0], [0]] + [[0, 1, 0, 0, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 1], [0, 0, 0, 0, 0], [0, 1, 0, 0, 0]] * x_1 140.03/36.44 >>> 140.03/36.44 140.03/36.44 <<< 140.03/36.44 POL(a(x_1)) = [[0], [0], [1], [0], [0]] + [[0, 0, 0, 0, 0], [0, 0, 1, 1, 0], [0, 1, 0, 0, 1], [0, 0, 0, 0, 1], [0, 0, 0, 0, 0]] * x_1 140.03/36.44 >>> 140.03/36.44 140.03/36.44 140.03/36.44 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 140.03/36.44 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (15) 140.03/36.44 Obligation: 140.03/36.44 Q DP problem: 140.03/36.44 P is empty. 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 140.03/36.44 The set Q consists of the following terms: 140.03/36.44 140.03/36.44 b(b(a(a(x0)))) 140.03/36.44 140.03/36.44 We have to consider all minimal (P,Q,R)-chains. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (16) PisEmptyProof (EQUIVALENT) 140.03/36.44 The TRS P is empty. Hence, there is no (P,Q,R) chain. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (17) 140.03/36.44 YES 140.03/36.44 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (18) 140.03/36.44 Obligation: 140.03/36.44 Q DP problem: 140.03/36.44 The TRS P consists of the following rules: 140.03/36.44 140.03/36.44 C(b(x1)) -> C(x1) 140.03/36.44 C(a(x1)) -> C(x1) 140.03/36.44 140.03/36.44 The TRS R consists of the following rules: 140.03/36.44 140.03/36.44 b(b(a(a(x1)))) -> a(a(a(b(b(b(x1)))))) 140.03/36.44 c(a(x1)) -> a(c(x1)) 140.03/36.44 c(b(x1)) -> b(c(x1)) 140.03/36.44 140.03/36.44 Q is empty. 140.03/36.44 We have to consider all minimal (P,Q,R)-chains. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (19) UsableRulesProof (EQUIVALENT) 140.03/36.44 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. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (20) 140.03/36.44 Obligation: 140.03/36.44 Q DP problem: 140.03/36.44 The TRS P consists of the following rules: 140.03/36.44 140.03/36.44 C(b(x1)) -> C(x1) 140.03/36.44 C(a(x1)) -> C(x1) 140.03/36.44 140.03/36.44 R is empty. 140.03/36.44 Q is empty. 140.03/36.44 We have to consider all minimal (P,Q,R)-chains. 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (21) QDPSizeChangeProof (EQUIVALENT) 140.03/36.44 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. 140.03/36.44 140.03/36.44 From the DPs we obtained the following set of size-change graphs: 140.03/36.44 *C(b(x1)) -> C(x1) 140.03/36.44 The graph contains the following edges 1 > 1 140.03/36.44 140.03/36.44 140.03/36.44 *C(a(x1)) -> C(x1) 140.03/36.44 The graph contains the following edges 1 > 1 140.03/36.44 140.03/36.44 140.03/36.44 ---------------------------------------- 140.03/36.44 140.03/36.44 (22) 140.03/36.44 YES 140.34/36.53 EOF