7.32/7.67 YES 7.32/7.67 7.32/7.67 Problem 1: 7.32/7.67 7.32/7.67 (VAR v_NonEmpty:S x1:S) 7.32/7.67 (RULES 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 ) 7.32/7.67 7.32/7.67 Problem 1: 7.32/7.67 7.32/7.67 Dependency Pairs Processor: 7.32/7.67 -> Pairs: 7.32/7.67 A(a(l2(x1:S))) -> A(a(x1:S)) 7.32/7.67 A(a(l2(x1:S))) -> A(x1:S) 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 B(l1(x1:S)) -> B(r2(x1:S)) 7.32/7.67 B(l1(x1:S)) -> R2(x1:S) 7.32/7.67 B(l2(x1:S)) -> B(r1(x1:S)) 7.32/7.67 B(l2(x1:S)) -> R1(x1:S) 7.32/7.67 R1(a(x1:S)) -> A(a(a(r1(x1:S)))) 7.32/7.67 R1(a(x1:S)) -> A(a(r1(x1:S))) 7.32/7.67 R1(a(x1:S)) -> A(r1(x1:S)) 7.32/7.67 R1(a(x1:S)) -> R1(x1:S) 7.32/7.67 R2(a(x1:S)) -> A(a(a(r2(x1:S)))) 7.32/7.67 R2(a(x1:S)) -> A(a(r2(x1:S))) 7.32/7.67 R2(a(x1:S)) -> A(r2(x1:S)) 7.32/7.67 R2(a(x1:S)) -> R2(x1:S) 7.32/7.67 R2(b(x1:S)) -> A(b(x1:S)) 7.32/7.67 -> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 7.32/7.67 Problem 1: 7.32/7.67 7.32/7.67 SCC Processor: 7.32/7.67 -> Pairs: 7.32/7.67 A(a(l2(x1:S))) -> A(a(x1:S)) 7.32/7.67 A(a(l2(x1:S))) -> A(x1:S) 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 B(l1(x1:S)) -> B(r2(x1:S)) 7.32/7.67 B(l1(x1:S)) -> R2(x1:S) 7.32/7.67 B(l2(x1:S)) -> B(r1(x1:S)) 7.32/7.67 B(l2(x1:S)) -> R1(x1:S) 7.32/7.67 R1(a(x1:S)) -> A(a(a(r1(x1:S)))) 7.32/7.67 R1(a(x1:S)) -> A(a(r1(x1:S))) 7.32/7.67 R1(a(x1:S)) -> A(r1(x1:S)) 7.32/7.67 R1(a(x1:S)) -> R1(x1:S) 7.32/7.67 R2(a(x1:S)) -> A(a(a(r2(x1:S)))) 7.32/7.67 R2(a(x1:S)) -> A(a(r2(x1:S))) 7.32/7.67 R2(a(x1:S)) -> A(r2(x1:S)) 7.32/7.67 R2(a(x1:S)) -> R2(x1:S) 7.32/7.67 R2(b(x1:S)) -> A(b(x1:S)) 7.32/7.67 -> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 ->Strongly Connected Components: 7.32/7.67 ->->Cycle: 7.32/7.67 ->->-> Pairs: 7.32/7.67 A(a(l2(x1:S))) -> A(a(x1:S)) 7.32/7.67 A(a(l2(x1:S))) -> A(x1:S) 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 ->->-> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 ->->Cycle: 7.32/7.67 ->->-> Pairs: 7.32/7.67 R1(a(x1:S)) -> R1(x1:S) 7.32/7.67 ->->-> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 ->->Cycle: 7.32/7.67 ->->-> Pairs: 7.32/7.67 R2(a(x1:S)) -> R2(x1:S) 7.32/7.67 ->->-> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 ->->Cycle: 7.32/7.67 ->->-> Pairs: 7.32/7.67 B(l1(x1:S)) -> B(r2(x1:S)) 7.32/7.67 B(l2(x1:S)) -> B(r1(x1:S)) 7.32/7.67 ->->-> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 7.32/7.67 7.32/7.67 The problem is decomposed in 4 subproblems. 7.32/7.67 7.32/7.67 Problem 1.1: 7.32/7.67 7.32/7.67 Reduction Pair Processor: 7.32/7.67 -> Pairs: 7.32/7.67 A(a(l2(x1:S))) -> A(a(x1:S)) 7.32/7.67 A(a(l2(x1:S))) -> A(x1:S) 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 -> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 -> Usable rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 ->Interpretation type: 7.32/7.67 Linear 7.32/7.67 ->Coefficients: 7.32/7.67 Natural Numbers 7.32/7.67 ->Dimension: 7.32/7.67 1 7.32/7.67 ->Bound: 7.32/7.67 2 7.32/7.67 ->Interpretation: 7.32/7.67 7.32/7.67 [a](X) = X 7.32/7.67 [l1](X) = 2.X 7.32/7.67 [l2](X) = 2.X + 2 7.32/7.67 [A](X) = 2.X 7.32/7.67 7.32/7.67 Problem 1.1: 7.32/7.67 7.32/7.67 SCC Processor: 7.32/7.67 -> Pairs: 7.32/7.67 A(a(l2(x1:S))) -> A(x1:S) 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 -> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 ->Strongly Connected Components: 7.32/7.67 ->->Cycle: 7.32/7.67 ->->-> Pairs: 7.32/7.67 A(a(l2(x1:S))) -> A(x1:S) 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 ->->-> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 7.32/7.67 Problem 1.1: 7.32/7.67 7.32/7.67 Reduction Pair Processor: 7.32/7.67 -> Pairs: 7.32/7.67 A(a(l2(x1:S))) -> A(x1:S) 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 -> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 -> Usable rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 ->Interpretation type: 7.32/7.67 Linear 7.32/7.67 ->Coefficients: 7.32/7.67 Natural Numbers 7.32/7.67 ->Dimension: 7.32/7.67 1 7.32/7.67 ->Bound: 7.32/7.67 2 7.32/7.67 ->Interpretation: 7.32/7.67 7.32/7.67 [a](X) = X 7.32/7.67 [l1](X) = 2.X + 1 7.32/7.67 [l2](X) = 2.X + 2 7.32/7.67 [A](X) = 2.X 7.32/7.67 7.32/7.67 Problem 1.1: 7.32/7.67 7.32/7.67 SCC Processor: 7.32/7.67 -> Pairs: 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 -> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 ->Strongly Connected Components: 7.32/7.67 ->->Cycle: 7.32/7.67 ->->-> Pairs: 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 ->->-> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 7.32/7.67 Problem 1.1: 7.32/7.67 7.32/7.67 Reduction Pair Processor: 7.32/7.67 -> Pairs: 7.32/7.67 A(l1(x1:S)) -> A(a(a(x1:S))) 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 -> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 -> Usable rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 ->Interpretation type: 7.32/7.67 Linear 7.32/7.67 ->Coefficients: 7.32/7.67 Natural Numbers 7.32/7.67 ->Dimension: 7.32/7.67 1 7.32/7.67 ->Bound: 7.32/7.67 2 7.32/7.67 ->Interpretation: 7.32/7.67 7.32/7.67 [a](X) = X 7.32/7.67 [l1](X) = 2.X + 2 7.32/7.67 [l2](X) = 0 7.32/7.67 [A](X) = 2.X 7.32/7.67 7.32/7.67 Problem 1.1: 7.32/7.67 7.32/7.67 SCC Processor: 7.32/7.67 -> Pairs: 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 -> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.67 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.67 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.67 ->Strongly Connected Components: 7.32/7.67 ->->Cycle: 7.32/7.67 ->->-> Pairs: 7.32/7.67 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.67 A(l1(x1:S)) -> A(x1:S) 7.32/7.67 ->->-> Rules: 7.32/7.67 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.67 a(a(x1:S)) -> x1:S 7.32/7.67 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.67 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.67 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.67 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.67 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 7.32/7.68 Problem 1.1: 7.32/7.68 7.32/7.68 Reduction Pair Processor: 7.32/7.68 -> Pairs: 7.32/7.68 A(l1(x1:S)) -> A(a(x1:S)) 7.32/7.68 A(l1(x1:S)) -> A(x1:S) 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 -> Usable rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 ->Interpretation type: 7.32/7.68 Linear 7.32/7.68 ->Coefficients: 7.32/7.68 Natural Numbers 7.32/7.68 ->Dimension: 7.32/7.68 1 7.32/7.68 ->Bound: 7.32/7.68 2 7.32/7.68 ->Interpretation: 7.32/7.68 7.32/7.68 [a](X) = X 7.32/7.68 [l1](X) = X + 2 7.32/7.68 [l2](X) = 0 7.32/7.68 [A](X) = 2.X 7.32/7.68 7.32/7.68 Problem 1.1: 7.32/7.68 7.32/7.68 SCC Processor: 7.32/7.68 -> Pairs: 7.32/7.68 A(l1(x1:S)) -> A(x1:S) 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Strongly Connected Components: 7.32/7.68 ->->Cycle: 7.32/7.68 ->->-> Pairs: 7.32/7.68 A(l1(x1:S)) -> A(x1:S) 7.32/7.68 ->->-> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 7.32/7.68 Problem 1.1: 7.32/7.68 7.32/7.68 Subterm Processor: 7.32/7.68 -> Pairs: 7.32/7.68 A(l1(x1:S)) -> A(x1:S) 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Projection: 7.32/7.68 pi(A) = 1 7.32/7.68 7.32/7.68 Problem 1.1: 7.32/7.68 7.32/7.68 SCC Processor: 7.32/7.68 -> Pairs: 7.32/7.68 Empty 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Strongly Connected Components: 7.32/7.68 There is no strongly connected component 7.32/7.68 7.32/7.68 The problem is finite. 7.32/7.68 7.32/7.68 Problem 1.2: 7.32/7.68 7.32/7.68 Subterm Processor: 7.32/7.68 -> Pairs: 7.32/7.68 R1(a(x1:S)) -> R1(x1:S) 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Projection: 7.32/7.68 pi(R1) = 1 7.32/7.68 7.32/7.68 Problem 1.2: 7.32/7.68 7.32/7.68 SCC Processor: 7.32/7.68 -> Pairs: 7.32/7.68 Empty 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Strongly Connected Components: 7.32/7.68 There is no strongly connected component 7.32/7.68 7.32/7.68 The problem is finite. 7.32/7.68 7.32/7.68 Problem 1.3: 7.32/7.68 7.32/7.68 Subterm Processor: 7.32/7.68 -> Pairs: 7.32/7.68 R2(a(x1:S)) -> R2(x1:S) 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Projection: 7.32/7.68 pi(R2) = 1 7.32/7.68 7.32/7.68 Problem 1.3: 7.32/7.68 7.32/7.68 SCC Processor: 7.32/7.68 -> Pairs: 7.32/7.68 Empty 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Strongly Connected Components: 7.32/7.68 There is no strongly connected component 7.32/7.68 7.32/7.68 The problem is finite. 7.32/7.68 7.32/7.68 Problem 1.4: 7.32/7.68 7.32/7.68 Reduction Pair Processor: 7.32/7.68 -> Pairs: 7.32/7.68 B(l1(x1:S)) -> B(r2(x1:S)) 7.32/7.68 B(l2(x1:S)) -> B(r1(x1:S)) 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 -> Usable rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Interpretation type: 7.32/7.68 Linear 7.32/7.68 ->Coefficients: 7.32/7.68 Natural Numbers 7.32/7.68 ->Dimension: 7.32/7.68 2 7.32/7.68 ->Bound: 7.32/7.68 1 7.32/7.68 ->Interpretation: 7.32/7.68 7.32/7.68 [a](X) = [0 1;1 0].X 7.32/7.68 [b](X) = [1 0;1 0].X + [0;1] 7.32/7.68 [r1](X) = [0 1;1 0].X + [1;1] 7.32/7.68 [r2](X) = [0 1;1 0].X 7.32/7.68 [l1](X) = [0 1;1 0].X + [1;1] 7.32/7.68 [l2](X) = [0 1;0 0].X + [1;0] 7.32/7.68 [B](X) = [1 0;1 0].X 7.32/7.68 7.32/7.68 Problem 1.4: 7.32/7.68 7.32/7.68 SCC Processor: 7.32/7.68 -> Pairs: 7.32/7.68 B(l2(x1:S)) -> B(r1(x1:S)) 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Strongly Connected Components: 7.32/7.68 ->->Cycle: 7.32/7.68 ->->-> Pairs: 7.32/7.68 B(l2(x1:S)) -> B(r1(x1:S)) 7.32/7.68 ->->-> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 7.32/7.68 Problem 1.4: 7.32/7.68 7.32/7.68 Reduction Pair Processor: 7.32/7.68 -> Pairs: 7.32/7.68 B(l2(x1:S)) -> B(r1(x1:S)) 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 -> Usable rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Interpretation type: 7.32/7.68 Linear 7.32/7.68 ->Coefficients: 7.32/7.68 Natural Numbers 7.32/7.68 ->Dimension: 7.32/7.68 1 7.32/7.68 ->Bound: 7.32/7.68 2 7.32/7.68 ->Interpretation: 7.32/7.68 7.32/7.68 [a](X) = X 7.32/7.68 [b](X) = 0 7.32/7.68 [r1](X) = 2.X 7.32/7.68 [r2](X) = 2 7.32/7.68 [l1](X) = 2.X 7.32/7.68 [l2](X) = 2.X + 2 7.32/7.68 [B](X) = 2.X 7.32/7.68 7.32/7.68 Problem 1.4: 7.32/7.68 7.32/7.68 SCC Processor: 7.32/7.68 -> Pairs: 7.32/7.68 Empty 7.32/7.68 -> Rules: 7.32/7.68 a(a(l2(x1:S))) -> l2(a(a(x1:S))) 7.32/7.68 a(a(x1:S)) -> x1:S 7.32/7.68 a(l1(x1:S)) -> l1(a(a(a(x1:S)))) 7.32/7.68 b(l1(x1:S)) -> b(r2(x1:S)) 7.32/7.68 b(l2(x1:S)) -> b(r1(x1:S)) 7.32/7.68 r1(a(x1:S)) -> a(a(a(r1(x1:S)))) 7.32/7.68 r1(b(x1:S)) -> l1(b(x1:S)) 7.32/7.68 r2(a(x1:S)) -> a(a(a(r2(x1:S)))) 7.32/7.68 r2(b(x1:S)) -> l2(a(b(x1:S))) 7.32/7.68 ->Strongly Connected Components: 7.32/7.68 There is no strongly connected component 7.32/7.68 7.32/7.68 The problem is finite. 7.32/7.68 EOF