21.31/6.29 YES 21.65/6.30 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 21.65/6.30 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 21.65/6.30 21.65/6.30 21.65/6.30 Termination w.r.t. Q of the given QTRS could be proven: 21.65/6.30 21.65/6.30 (0) QTRS 21.65/6.30 (1) QTRS Reverse [EQUIVALENT, 0 ms] 21.65/6.30 (2) QTRS 21.65/6.30 (3) DependencyPairsProof [EQUIVALENT, 13 ms] 21.65/6.30 (4) QDP 21.65/6.30 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 21.65/6.30 (6) QDP 21.65/6.30 (7) QDPOrderProof [EQUIVALENT, 208 ms] 21.65/6.30 (8) QDP 21.65/6.30 (9) UsableRulesProof [EQUIVALENT, 0 ms] 21.65/6.30 (10) QDP 21.65/6.30 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 21.65/6.30 (12) YES 21.65/6.30 21.65/6.30 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (0) 21.65/6.30 Obligation: 21.65/6.30 Q restricted rewrite system: 21.65/6.30 The TRS R consists of the following rules: 21.65/6.30 21.65/6.30 a(x1) -> b(x1) 21.65/6.30 a(c(x1)) -> c(b(c(b(a(a(x1)))))) 21.65/6.30 b(b(x1)) -> x1 21.65/6.30 21.65/6.30 Q is empty. 21.65/6.30 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (1) QTRS Reverse (EQUIVALENT) 21.65/6.30 We applied the QTRS Reverse Processor [REVERSE]. 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (2) 21.65/6.30 Obligation: 21.65/6.30 Q restricted rewrite system: 21.65/6.30 The TRS R consists of the following rules: 21.65/6.30 21.65/6.30 a(x1) -> b(x1) 21.65/6.30 c(a(x1)) -> a(a(b(c(b(c(x1)))))) 21.65/6.30 b(b(x1)) -> x1 21.65/6.30 21.65/6.30 Q is empty. 21.65/6.30 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (3) DependencyPairsProof (EQUIVALENT) 21.65/6.30 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (4) 21.65/6.30 Obligation: 21.65/6.30 Q DP problem: 21.65/6.30 The TRS P consists of the following rules: 21.65/6.30 21.65/6.30 A(x1) -> B(x1) 21.65/6.30 C(a(x1)) -> A(a(b(c(b(c(x1)))))) 21.65/6.30 C(a(x1)) -> A(b(c(b(c(x1))))) 21.65/6.30 C(a(x1)) -> B(c(b(c(x1)))) 21.65/6.30 C(a(x1)) -> C(b(c(x1))) 21.65/6.30 C(a(x1)) -> B(c(x1)) 21.65/6.30 C(a(x1)) -> C(x1) 21.65/6.30 21.65/6.30 The TRS R consists of the following rules: 21.65/6.30 21.65/6.30 a(x1) -> b(x1) 21.65/6.30 c(a(x1)) -> a(a(b(c(b(c(x1)))))) 21.65/6.30 b(b(x1)) -> x1 21.65/6.30 21.65/6.30 Q is empty. 21.65/6.30 We have to consider all minimal (P,Q,R)-chains. 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (5) DependencyGraphProof (EQUIVALENT) 21.65/6.30 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (6) 21.65/6.30 Obligation: 21.65/6.30 Q DP problem: 21.65/6.30 The TRS P consists of the following rules: 21.65/6.30 21.65/6.30 C(a(x1)) -> C(x1) 21.65/6.30 C(a(x1)) -> C(b(c(x1))) 21.65/6.30 21.65/6.30 The TRS R consists of the following rules: 21.65/6.30 21.65/6.30 a(x1) -> b(x1) 21.65/6.30 c(a(x1)) -> a(a(b(c(b(c(x1)))))) 21.65/6.30 b(b(x1)) -> x1 21.65/6.30 21.65/6.30 Q is empty. 21.65/6.30 We have to consider all minimal (P,Q,R)-chains. 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (7) QDPOrderProof (EQUIVALENT) 21.65/6.30 We use the reduction pair processor [LPAR04,JAR06]. 21.65/6.30 21.65/6.30 21.65/6.30 The following pairs can be oriented strictly and are deleted. 21.65/6.30 21.65/6.30 C(a(x1)) -> C(b(c(x1))) 21.65/6.30 The remaining pairs can at least be oriented weakly. 21.65/6.30 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 21.65/6.30 21.65/6.30 <<< 21.65/6.30 POL(C(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 21.65/6.30 >>> 21.65/6.30 21.65/6.30 <<< 21.65/6.30 POL(a(x_1)) = [[1A], [0A], [0A]] + [[1A, 0A, 0A], [0A, -I, 0A], [-I, 0A, -I]] * x_1 21.65/6.30 >>> 21.65/6.30 21.65/6.30 <<< 21.65/6.30 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [0A, -I, 0A], [-I, 0A, -I]] * x_1 21.65/6.30 >>> 21.65/6.30 21.65/6.30 <<< 21.65/6.30 POL(c(x_1)) = [[0A], [-I], [0A]] + [[1A, 0A, 0A], [0A, -I, -I], [-I, 0A, 0A]] * x_1 21.65/6.30 >>> 21.65/6.30 21.65/6.30 21.65/6.30 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 21.65/6.30 21.65/6.30 c(a(x1)) -> a(a(b(c(b(c(x1)))))) 21.65/6.30 b(b(x1)) -> x1 21.65/6.30 a(x1) -> b(x1) 21.65/6.30 21.65/6.30 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (8) 21.65/6.30 Obligation: 21.65/6.30 Q DP problem: 21.65/6.30 The TRS P consists of the following rules: 21.65/6.30 21.65/6.30 C(a(x1)) -> C(x1) 21.65/6.30 21.65/6.30 The TRS R consists of the following rules: 21.65/6.30 21.65/6.30 a(x1) -> b(x1) 21.65/6.30 c(a(x1)) -> a(a(b(c(b(c(x1)))))) 21.65/6.30 b(b(x1)) -> x1 21.65/6.30 21.65/6.30 Q is empty. 21.65/6.30 We have to consider all minimal (P,Q,R)-chains. 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (9) UsableRulesProof (EQUIVALENT) 21.65/6.30 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. 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (10) 21.65/6.30 Obligation: 21.65/6.30 Q DP problem: 21.65/6.30 The TRS P consists of the following rules: 21.65/6.30 21.65/6.30 C(a(x1)) -> C(x1) 21.65/6.30 21.65/6.30 R is empty. 21.65/6.30 Q is empty. 21.65/6.30 We have to consider all minimal (P,Q,R)-chains. 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (11) QDPSizeChangeProof (EQUIVALENT) 21.65/6.30 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. 21.65/6.30 21.65/6.30 From the DPs we obtained the following set of size-change graphs: 21.65/6.30 *C(a(x1)) -> C(x1) 21.65/6.30 The graph contains the following edges 1 > 1 21.65/6.30 21.65/6.30 21.65/6.30 ---------------------------------------- 21.65/6.30 21.65/6.30 (12) 21.65/6.30 YES 21.74/6.34 EOF