23.22/6.80 YES 26.88/8.76 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 26.88/8.76 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.88/8.76 26.88/8.76 26.88/8.76 Termination w.r.t. Q of the given QTRS could be proven: 26.88/8.76 26.88/8.76 (0) QTRS 26.88/8.76 (1) QTRS Reverse [EQUIVALENT, 0 ms] 26.88/8.76 (2) QTRS 26.88/8.76 (3) DependencyPairsProof [EQUIVALENT, 7 ms] 26.88/8.76 (4) QDP 26.88/8.76 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 26.88/8.76 (6) QDP 26.88/8.76 (7) QDPOrderProof [EQUIVALENT, 100 ms] 26.88/8.76 (8) QDP 26.88/8.76 (9) UsableRulesProof [EQUIVALENT, 0 ms] 26.88/8.76 (10) QDP 26.88/8.76 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 26.88/8.76 (12) YES 26.88/8.76 26.88/8.76 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (0) 26.88/8.76 Obligation: 26.88/8.76 Q restricted rewrite system: 26.88/8.76 The TRS R consists of the following rules: 26.88/8.76 26.88/8.76 a(x1) -> x1 26.88/8.76 a(b(x1)) -> b(b(a(a(c(a(x1)))))) 26.88/8.76 c(c(a(x1))) -> x1 26.88/8.76 26.88/8.76 Q is empty. 26.88/8.76 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (1) QTRS Reverse (EQUIVALENT) 26.88/8.76 We applied the QTRS Reverse Processor [REVERSE]. 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (2) 26.88/8.76 Obligation: 26.88/8.76 Q restricted rewrite system: 26.88/8.76 The TRS R consists of the following rules: 26.88/8.76 26.88/8.76 a(x1) -> x1 26.88/8.76 b(a(x1)) -> a(c(a(a(b(b(x1)))))) 26.88/8.76 a(c(c(x1))) -> x1 26.88/8.76 26.88/8.76 Q is empty. 26.88/8.76 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (3) DependencyPairsProof (EQUIVALENT) 26.88/8.76 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (4) 26.88/8.76 Obligation: 26.88/8.76 Q DP problem: 26.88/8.76 The TRS P consists of the following rules: 26.88/8.76 26.88/8.76 B(a(x1)) -> A(c(a(a(b(b(x1)))))) 26.88/8.76 B(a(x1)) -> A(a(b(b(x1)))) 26.88/8.76 B(a(x1)) -> A(b(b(x1))) 26.88/8.76 B(a(x1)) -> B(b(x1)) 26.88/8.76 B(a(x1)) -> B(x1) 26.88/8.76 26.88/8.76 The TRS R consists of the following rules: 26.88/8.76 26.88/8.76 a(x1) -> x1 26.88/8.76 b(a(x1)) -> a(c(a(a(b(b(x1)))))) 26.88/8.76 a(c(c(x1))) -> x1 26.88/8.76 26.88/8.76 Q is empty. 26.88/8.76 We have to consider all minimal (P,Q,R)-chains. 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (5) DependencyGraphProof (EQUIVALENT) 26.88/8.76 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (6) 26.88/8.76 Obligation: 26.88/8.76 Q DP problem: 26.88/8.76 The TRS P consists of the following rules: 26.88/8.76 26.88/8.76 B(a(x1)) -> B(x1) 26.88/8.76 B(a(x1)) -> B(b(x1)) 26.88/8.76 26.88/8.76 The TRS R consists of the following rules: 26.88/8.76 26.88/8.76 a(x1) -> x1 26.88/8.76 b(a(x1)) -> a(c(a(a(b(b(x1)))))) 26.88/8.76 a(c(c(x1))) -> x1 26.88/8.76 26.88/8.76 Q is empty. 26.88/8.76 We have to consider all minimal (P,Q,R)-chains. 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (7) QDPOrderProof (EQUIVALENT) 26.88/8.76 We use the reduction pair processor [LPAR04,JAR06]. 26.88/8.76 26.88/8.76 26.88/8.76 The following pairs can be oriented strictly and are deleted. 26.88/8.76 26.88/8.76 B(a(x1)) -> B(b(x1)) 26.88/8.76 The remaining pairs can at least be oriented weakly. 26.88/8.76 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 26.88/8.76 26.88/8.76 <<< 26.88/8.76 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 26.88/8.76 >>> 26.88/8.76 26.88/8.76 <<< 26.88/8.76 POL(a(x_1)) = [[1A], [0A], [1A]] + [[1A, -I, 0A], [0A, 0A, 0A], [1A, -I, 0A]] * x_1 26.88/8.76 >>> 26.88/8.76 26.88/8.76 <<< 26.88/8.76 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, -I, 1A], [0A, -I, -I]] * x_1 26.88/8.76 >>> 26.88/8.76 26.88/8.76 <<< 26.88/8.76 POL(c(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 26.88/8.76 >>> 26.88/8.76 26.88/8.76 26.88/8.76 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.88/8.76 26.88/8.76 b(a(x1)) -> a(c(a(a(b(b(x1)))))) 26.88/8.76 a(x1) -> x1 26.88/8.76 a(c(c(x1))) -> x1 26.88/8.76 26.88/8.76 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (8) 26.88/8.76 Obligation: 26.88/8.76 Q DP problem: 26.88/8.76 The TRS P consists of the following rules: 26.88/8.76 26.88/8.76 B(a(x1)) -> B(x1) 26.88/8.76 26.88/8.76 The TRS R consists of the following rules: 26.88/8.76 26.88/8.76 a(x1) -> x1 26.88/8.76 b(a(x1)) -> a(c(a(a(b(b(x1)))))) 26.88/8.76 a(c(c(x1))) -> x1 26.88/8.76 26.88/8.76 Q is empty. 26.88/8.76 We have to consider all minimal (P,Q,R)-chains. 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (9) UsableRulesProof (EQUIVALENT) 26.88/8.76 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. 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (10) 26.88/8.76 Obligation: 26.88/8.76 Q DP problem: 26.88/8.76 The TRS P consists of the following rules: 26.88/8.76 26.88/8.76 B(a(x1)) -> B(x1) 26.88/8.76 26.88/8.76 R is empty. 26.88/8.76 Q is empty. 26.88/8.76 We have to consider all minimal (P,Q,R)-chains. 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (11) QDPSizeChangeProof (EQUIVALENT) 26.88/8.76 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. 26.88/8.76 26.88/8.76 From the DPs we obtained the following set of size-change graphs: 26.88/8.76 *B(a(x1)) -> B(x1) 26.88/8.76 The graph contains the following edges 1 > 1 26.88/8.76 26.88/8.76 26.88/8.76 ---------------------------------------- 26.88/8.76 26.88/8.76 (12) 26.88/8.76 YES 27.08/9.03 EOF