41.61/11.53 YES 61.90/16.70 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 61.90/16.70 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 61.90/16.70 61.90/16.70 61.90/16.70 Termination w.r.t. Q of the given QTRS could be proven: 61.90/16.70 61.90/16.70 (0) QTRS 61.90/16.70 (1) DependencyPairsProof [EQUIVALENT, 32 ms] 61.90/16.70 (2) QDP 61.90/16.70 (3) DependencyGraphProof [EQUIVALENT, 1 ms] 61.90/16.70 (4) AND 61.90/16.70 (5) QDP 61.90/16.70 (6) UsableRulesProof [EQUIVALENT, 0 ms] 61.90/16.70 (7) QDP 61.90/16.70 (8) QDPOrderProof [EQUIVALENT, 65 ms] 61.90/16.70 (9) QDP 61.90/16.70 (10) QDPOrderProof [EQUIVALENT, 134 ms] 61.90/16.70 (11) QDP 61.90/16.70 (12) QDPOrderProof [EQUIVALENT, 63 ms] 61.90/16.70 (13) QDP 61.90/16.70 (14) QDPOrderProof [EQUIVALENT, 57 ms] 61.90/16.70 (15) QDP 61.90/16.70 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 61.90/16.70 (17) TRUE 61.90/16.70 (18) QDP 61.90/16.70 (19) UsableRulesProof [EQUIVALENT, 0 ms] 61.90/16.70 (20) QDP 61.90/16.70 (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] 61.90/16.70 (22) YES 61.90/16.70 61.90/16.70 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (0) 61.90/16.70 Obligation: 61.90/16.70 Q restricted rewrite system: 61.90/16.70 The TRS R consists of the following rules: 61.90/16.70 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 e(d(x1)) -> a(b(c(d(e(x1))))) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 e(c(x1)) -> b(a(a(e(x1)))) 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 61.90/16.70 Q is empty. 61.90/16.70 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (1) DependencyPairsProof (EQUIVALENT) 61.90/16.70 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (2) 61.90/16.70 Obligation: 61.90/16.70 Q DP problem: 61.90/16.70 The TRS P consists of the following rules: 61.90/16.70 61.90/16.70 A(a(x1)) -> B(b(b(x1))) 61.90/16.70 A(a(x1)) -> B(b(x1)) 61.90/16.70 A(a(x1)) -> B(x1) 61.90/16.70 A(x1) -> C(d(x1)) 61.90/16.70 B(b(x1)) -> C(c(c(x1))) 61.90/16.70 B(b(x1)) -> C(c(x1)) 61.90/16.70 B(b(x1)) -> C(x1) 61.90/16.70 E(d(x1)) -> A(b(c(d(e(x1))))) 61.90/16.70 E(d(x1)) -> B(c(d(e(x1)))) 61.90/16.70 E(d(x1)) -> C(d(e(x1))) 61.90/16.70 E(d(x1)) -> E(x1) 61.90/16.70 E(c(x1)) -> B(a(a(e(x1)))) 61.90/16.70 E(c(x1)) -> A(a(e(x1))) 61.90/16.70 E(c(x1)) -> A(e(x1)) 61.90/16.70 E(c(x1)) -> E(x1) 61.90/16.70 C(d(d(x1))) -> A(x1) 61.90/16.70 61.90/16.70 The TRS R consists of the following rules: 61.90/16.70 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 e(d(x1)) -> a(b(c(d(e(x1))))) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 e(c(x1)) -> b(a(a(e(x1)))) 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 61.90/16.70 Q is empty. 61.90/16.70 We have to consider all minimal (P,Q,R)-chains. 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (3) DependencyGraphProof (EQUIVALENT) 61.90/16.70 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes. 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (4) 61.90/16.70 Complex Obligation (AND) 61.90/16.70 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (5) 61.90/16.70 Obligation: 61.90/16.70 Q DP problem: 61.90/16.70 The TRS P consists of the following rules: 61.90/16.70 61.90/16.70 B(b(x1)) -> C(c(c(x1))) 61.90/16.70 C(d(d(x1))) -> A(x1) 61.90/16.70 A(a(x1)) -> B(b(b(x1))) 61.90/16.70 B(b(x1)) -> C(c(x1)) 61.90/16.70 B(b(x1)) -> C(x1) 61.90/16.70 A(a(x1)) -> B(b(x1)) 61.90/16.70 A(a(x1)) -> B(x1) 61.90/16.70 A(x1) -> C(d(x1)) 61.90/16.70 61.90/16.70 The TRS R consists of the following rules: 61.90/16.70 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 e(d(x1)) -> a(b(c(d(e(x1))))) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 e(c(x1)) -> b(a(a(e(x1)))) 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 61.90/16.70 Q is empty. 61.90/16.70 We have to consider all minimal (P,Q,R)-chains. 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (6) UsableRulesProof (EQUIVALENT) 61.90/16.70 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. 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (7) 61.90/16.70 Obligation: 61.90/16.70 Q DP problem: 61.90/16.70 The TRS P consists of the following rules: 61.90/16.70 61.90/16.70 B(b(x1)) -> C(c(c(x1))) 61.90/16.70 C(d(d(x1))) -> A(x1) 61.90/16.70 A(a(x1)) -> B(b(b(x1))) 61.90/16.70 B(b(x1)) -> C(c(x1)) 61.90/16.70 B(b(x1)) -> C(x1) 61.90/16.70 A(a(x1)) -> B(b(x1)) 61.90/16.70 A(a(x1)) -> B(x1) 61.90/16.70 A(x1) -> C(d(x1)) 61.90/16.70 61.90/16.70 The TRS R consists of the following rules: 61.90/16.70 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 61.90/16.70 Q is empty. 61.90/16.70 We have to consider all minimal (P,Q,R)-chains. 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (8) QDPOrderProof (EQUIVALENT) 61.90/16.70 We use the reduction pair processor [LPAR04,JAR06]. 61.90/16.70 61.90/16.70 61.90/16.70 The following pairs can be oriented strictly and are deleted. 61.90/16.70 61.90/16.70 A(a(x1)) -> B(b(x1)) 61.90/16.70 A(a(x1)) -> B(x1) 61.90/16.70 The remaining pairs can at least be oriented weakly. 61.90/16.70 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(B(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 1A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(C(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(c(x_1)) = [[0A], [-I], [0A]] + [[-I, 0A, 1A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(d(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(a(x_1)) = [[1A], [0A], [0A]] + [[1A, 1A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 61.90/16.70 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 61.90/16.70 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 61.90/16.70 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (9) 61.90/16.70 Obligation: 61.90/16.70 Q DP problem: 61.90/16.70 The TRS P consists of the following rules: 61.90/16.70 61.90/16.70 B(b(x1)) -> C(c(c(x1))) 61.90/16.70 C(d(d(x1))) -> A(x1) 61.90/16.70 A(a(x1)) -> B(b(b(x1))) 61.90/16.70 B(b(x1)) -> C(c(x1)) 61.90/16.70 B(b(x1)) -> C(x1) 61.90/16.70 A(x1) -> C(d(x1)) 61.90/16.70 61.90/16.70 The TRS R consists of the following rules: 61.90/16.70 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 61.90/16.70 Q is empty. 61.90/16.70 We have to consider all minimal (P,Q,R)-chains. 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (10) QDPOrderProof (EQUIVALENT) 61.90/16.70 We use the reduction pair processor [LPAR04,JAR06]. 61.90/16.70 61.90/16.70 61.90/16.70 The following pairs can be oriented strictly and are deleted. 61.90/16.70 61.90/16.70 B(b(x1)) -> C(x1) 61.90/16.70 The remaining pairs can at least be oriented weakly. 61.90/16.70 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(B(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(C(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(c(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, 1A, -I]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(d(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(a(x_1)) = [[0A], [-I], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 1A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 61.90/16.70 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 61.90/16.70 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 61.90/16.70 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (11) 61.90/16.70 Obligation: 61.90/16.70 Q DP problem: 61.90/16.70 The TRS P consists of the following rules: 61.90/16.70 61.90/16.70 B(b(x1)) -> C(c(c(x1))) 61.90/16.70 C(d(d(x1))) -> A(x1) 61.90/16.70 A(a(x1)) -> B(b(b(x1))) 61.90/16.70 B(b(x1)) -> C(c(x1)) 61.90/16.70 A(x1) -> C(d(x1)) 61.90/16.70 61.90/16.70 The TRS R consists of the following rules: 61.90/16.70 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 61.90/16.70 Q is empty. 61.90/16.70 We have to consider all minimal (P,Q,R)-chains. 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (12) QDPOrderProof (EQUIVALENT) 61.90/16.70 We use the reduction pair processor [LPAR04,JAR06]. 61.90/16.70 61.90/16.70 61.90/16.70 The following pairs can be oriented strictly and are deleted. 61.90/16.70 61.90/16.70 B(b(x1)) -> C(c(x1)) 61.90/16.70 The remaining pairs can at least be oriented weakly. 61.90/16.70 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(B(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 1A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(C(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(c(x_1)) = [[1A], [-I], [-I]] + [[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(d(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 <<< 61.90/16.70 POL(a(x_1)) = [[1A], [-I], [-I]] + [[1A, 1A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 61.90/16.70 >>> 61.90/16.70 61.90/16.70 61.90/16.70 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 61.90/16.70 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 61.90/16.70 61.90/16.70 ---------------------------------------- 61.90/16.70 61.90/16.70 (13) 61.90/16.70 Obligation: 61.90/16.70 Q DP problem: 61.90/16.70 The TRS P consists of the following rules: 61.90/16.70 61.90/16.70 B(b(x1)) -> C(c(c(x1))) 61.90/16.70 C(d(d(x1))) -> A(x1) 61.90/16.70 A(a(x1)) -> B(b(b(x1))) 61.90/16.70 A(x1) -> C(d(x1)) 61.90/16.70 61.90/16.70 The TRS R consists of the following rules: 61.90/16.70 61.90/16.70 b(b(x1)) -> c(c(c(x1))) 61.90/16.70 c(d(d(x1))) -> a(x1) 61.90/16.70 a(a(x1)) -> b(b(b(x1))) 61.90/16.70 a(x1) -> c(d(x1)) 61.90/16.70 b(x1) -> d(d(x1)) 61.90/16.70 c(c(x1)) -> d(d(d(x1))) 61.90/16.70 61.90/16.70 Q is empty. 61.90/16.70 We have to consider all minimal (P,Q,R)-chains. 61.90/16.70 ---------------------------------------- 61.90/16.71 61.90/16.71 (14) QDPOrderProof (EQUIVALENT) 61.90/16.71 We use the reduction pair processor [LPAR04,JAR06]. 61.90/16.71 61.90/16.71 61.90/16.71 The following pairs can be oriented strictly and are deleted. 61.90/16.71 61.90/16.71 C(d(d(x1))) -> A(x1) 61.90/16.71 A(a(x1)) -> B(b(b(x1))) 61.90/16.71 A(x1) -> C(d(x1)) 61.90/16.71 The remaining pairs can at least be oriented weakly. 61.90/16.71 Used ordering: Polynomial interpretation [POLO,RATPOLO]: 61.90/16.71 61.90/16.71 POL(A(x_1)) = [15/4] + [2]x_1 61.90/16.71 POL(B(x_1)) = [3/2] + [2]x_1 61.90/16.71 POL(C(x_1)) = [2]x_1 61.90/16.71 POL(a(x_1)) = [7/2] + x_1 61.90/16.71 POL(b(x_1)) = [9/4] + x_1 61.90/16.71 POL(c(x_1)) = [3/2] + x_1 61.90/16.71 POL(d(x_1)) = [1] + x_1 61.90/16.71 The value of delta used in the strict ordering is 1/4. 61.90/16.71 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 61.90/16.71 61.90/16.71 c(d(d(x1))) -> a(x1) 61.90/16.71 a(a(x1)) -> b(b(b(x1))) 61.90/16.71 b(b(x1)) -> c(c(c(x1))) 61.90/16.71 a(x1) -> c(d(x1)) 61.90/16.71 c(c(x1)) -> d(d(d(x1))) 61.90/16.71 b(x1) -> d(d(x1)) 61.90/16.71 61.90/16.71 61.90/16.71 ---------------------------------------- 61.90/16.71 61.90/16.71 (15) 61.90/16.71 Obligation: 61.90/16.71 Q DP problem: 61.90/16.71 The TRS P consists of the following rules: 61.90/16.71 61.90/16.71 B(b(x1)) -> C(c(c(x1))) 61.90/16.71 61.90/16.71 The TRS R consists of the following rules: 61.90/16.71 61.90/16.71 b(b(x1)) -> c(c(c(x1))) 61.90/16.71 c(d(d(x1))) -> a(x1) 61.90/16.71 a(a(x1)) -> b(b(b(x1))) 61.90/16.71 a(x1) -> c(d(x1)) 61.90/16.71 b(x1) -> d(d(x1)) 61.90/16.71 c(c(x1)) -> d(d(d(x1))) 61.90/16.71 61.90/16.71 Q is empty. 61.90/16.71 We have to consider all minimal (P,Q,R)-chains. 61.90/16.71 ---------------------------------------- 61.90/16.71 61.90/16.71 (16) DependencyGraphProof (EQUIVALENT) 61.90/16.71 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 61.90/16.71 ---------------------------------------- 61.90/16.71 61.90/16.71 (17) 61.90/16.71 TRUE 61.90/16.71 61.90/16.71 ---------------------------------------- 61.90/16.71 61.90/16.71 (18) 61.90/16.71 Obligation: 61.90/16.71 Q DP problem: 61.90/16.71 The TRS P consists of the following rules: 61.90/16.71 61.90/16.71 E(c(x1)) -> E(x1) 61.90/16.71 E(d(x1)) -> E(x1) 61.90/16.71 61.90/16.71 The TRS R consists of the following rules: 61.90/16.71 61.90/16.71 a(a(x1)) -> b(b(b(x1))) 61.90/16.71 a(x1) -> c(d(x1)) 61.90/16.71 b(b(x1)) -> c(c(c(x1))) 61.90/16.71 c(c(x1)) -> d(d(d(x1))) 61.90/16.71 e(d(x1)) -> a(b(c(d(e(x1))))) 61.90/16.71 b(x1) -> d(d(x1)) 61.90/16.71 e(c(x1)) -> b(a(a(e(x1)))) 61.90/16.71 c(d(d(x1))) -> a(x1) 61.90/16.71 61.90/16.71 Q is empty. 61.90/16.71 We have to consider all minimal (P,Q,R)-chains. 61.90/16.71 ---------------------------------------- 61.90/16.71 61.90/16.71 (19) UsableRulesProof (EQUIVALENT) 61.90/16.71 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. 61.90/16.71 ---------------------------------------- 61.90/16.71 61.90/16.71 (20) 61.90/16.71 Obligation: 61.90/16.71 Q DP problem: 61.90/16.71 The TRS P consists of the following rules: 61.90/16.71 61.90/16.71 E(c(x1)) -> E(x1) 61.90/16.71 E(d(x1)) -> E(x1) 61.90/16.71 61.90/16.71 R is empty. 61.90/16.71 Q is empty. 61.90/16.71 We have to consider all minimal (P,Q,R)-chains. 61.90/16.71 ---------------------------------------- 61.90/16.71 61.90/16.71 (21) QDPSizeChangeProof (EQUIVALENT) 61.90/16.71 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. 61.90/16.71 61.90/16.71 From the DPs we obtained the following set of size-change graphs: 61.90/16.71 *E(c(x1)) -> E(x1) 61.90/16.71 The graph contains the following edges 1 > 1 61.90/16.71 61.90/16.71 61.90/16.71 *E(d(x1)) -> E(x1) 61.90/16.71 The graph contains the following edges 1 > 1 61.90/16.71 61.90/16.71 61.90/16.71 ---------------------------------------- 61.90/16.71 61.90/16.71 (22) 61.90/16.71 YES 62.14/16.76 EOF