YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ) Problem 1: Dependency Pairs Processor: -> Pairs: A#(A(A(A(x1:S)))) -> a#(a(a(x1:S))) A#(A(A(A(x1:S)))) -> a#(a(x1:S)) A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: SCC Processor: -> Pairs: A#(A(A(A(x1:S)))) -> a#(a(a(x1:S))) A#(A(A(A(x1:S)))) -> a#(a(x1:S)) A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(A(A(A(x1:S)))) -> a#(a(a(x1:S))) A#(A(A(A(x1:S)))) -> a#(a(x1:S)) A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(A(A(A(x1:S)))) -> a#(a(a(x1:S))) A#(A(A(A(x1:S)))) -> a#(a(x1:S)) A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 1 [b#](X) = 2.X + 2 [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A#(A(A(A(x1:S)))) -> a#(a(x1:S)) A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(A(A(A(x1:S)))) -> a#(a(x1:S)) A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(A(A(A(x1:S)))) -> a#(a(x1:S)) A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = X + 2 [a#](X) = X + 2 [b#](X) = 2.X + 2 [c#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(A(A(A(x1:S)))) -> a#(x1:S) A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X [B#](X) = 2.X + 2 [C#](X) = 2.X [a#](X) = X + 1 [b#](X) = 2.X [c#](X) = 2.X Problem 1: SCC Processor: -> Pairs: A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(A(A(b(x1:S)))) -> C#(x1:S) A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [b#](X) = X + 1 [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(A(A(b(x1:S)))) -> a#(C(x1:S)) A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X [a#](X) = X [b#](X) = 2.X + 2 [c#](X) = 2.X Problem 1: SCC Processor: -> Pairs: A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(A(A(b(x1:S)))) -> a#(a(C(x1:S))) A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X [b#](X) = 2.X [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(A(A(b(x1:S)))) -> a#(a(a(C(x1:S)))) A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 1 [b#](X) = 2.X + 2 [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(C(x1:S)) -> b#(x1:S) A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [b#](X) = 2.X + 2 [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 1 [b#](X) = 2.X + 2 [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(c(c(c(x1:S)))) -> C#(C(x1:S)) A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = X + 2 [B](X) = X + 2 [C](X) = X + 2 [a](X) = X + 2 [b](X) = X + 2 [c](X) = X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [b#](X) = 2.X + 1 [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(c(c(c(x1:S)))) -> C#(x1:S) A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = X + 2 [b#](X) = 2.X + 2 [c#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: A#(c(c(c(x1:S)))) -> b#(C(C(C(x1:S)))) B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [A#](X) = 2.X + 1 [B#](X) = 2.X [C#](X) = 2.X + 2 [a#](X) = X [b#](X) = 2.X [c#](X) = 2.X Problem 1: SCC Processor: -> Pairs: B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> A#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) B#(a(a(a(x1:S)))) -> A#(A(x1:S)) B#(a(a(a(x1:S)))) -> A#(x1:S) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(a(a(a(x1:S)))) -> A#(A(A(x1:S))) a#(a(a(a(x1:S)))) -> A#(A(x1:S)) a#(a(a(a(x1:S)))) -> A#(x1:S) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) b#(c(x1:S)) -> A#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: B#(A(x1:S)) -> c#(x1:S) B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = X + 2 [B](X) = X + 2 [C](X) = X + 2 [a](X) = X + 2 [b](X) = X + 2 [c](X) = X + 2 [B#](X) = X + 1 [C#](X) = X + 2 [a#](X) = X + 2 [b#](X) = X [c#](X) = X + 1 Problem 1: SCC Processor: -> Pairs: B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: B#(B(B(B(x1:S)))) -> b#(b(b(x1:S))) B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [b#](X) = X [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: B#(B(B(B(x1:S)))) -> b#(b(x1:S)) B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [b#](X) = X [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: B#(B(B(B(x1:S)))) -> b#(x1:S) B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [B#](X) = 2.X + 1 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [b#](X) = 2.X + 1 [c#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: B#(B(B(c(x1:S)))) -> b#(A(x1:S)) B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [B#](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [b#](X) = 2.X [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: B#(B(B(c(x1:S)))) -> b#(b(A(x1:S))) B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [B#](X) = X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [b#](X) = X + 1 [c#](X) = X + 1 Problem 1: SCC Processor: -> Pairs: B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: B#(B(B(c(x1:S)))) -> b#(b(b(A(x1:S)))) B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [B#](X) = 2.X [C#](X) = 2.X + 1 [a#](X) = 2.X + 1 [b#](X) = X + 2 [c#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) b#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) b#(b(b(b(x1:S)))) -> B#(B(x1:S)) b#(b(b(b(x1:S)))) -> B#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: B#(a(a(a(x1:S)))) -> c#(A(A(A(x1:S)))) C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [B#](X) = 2.X [C#](X) = 2.X + 2 [a#](X) = 2.X + 1 [c#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> B#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> B#(B(B(x1:S))) C#(b(b(b(x1:S)))) -> B#(B(x1:S)) C#(b(b(b(x1:S)))) -> B#(x1:S) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(a(x1:S)) -> B#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: C#(B(x1:S)) -> a#(x1:S) C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = X + 1 [B](X) = X + 1 [C](X) = X + 1 [a](X) = X + 1 [b](X) = X + 1 [c](X) = X + 1 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [c#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: C#(C(C(C(x1:S)))) -> c#(c(c(x1:S))) C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: C#(C(C(C(x1:S)))) -> c#(c(x1:S)) C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = X + 1 [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: C#(C(C(C(x1:S)))) -> c#(x1:S) C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [C#](X) = X + 2 [a#](X) = X + 2 [c#](X) = X + 2 Problem 1: SCC Processor: -> Pairs: C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: C#(C(C(a(x1:S)))) -> c#(B(x1:S)) C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = X + 2 [B](X) = X + 2 [C](X) = X + 2 [a](X) = X + 2 [b](X) = X + 2 [c](X) = X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X + 2 [c#](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: C#(C(C(a(x1:S)))) -> c#(c(B(x1:S))) C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = 2.X [c#](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: C#(C(C(a(x1:S)))) -> c#(c(c(B(x1:S)))) C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 2 [B](X) = 2.X + 2 [C](X) = 2.X + 2 [a](X) = 2.X + 2 [b](X) = 2.X + 2 [c](X) = 2.X + 2 [C#](X) = 2.X + 2 [a#](X) = X [c#](X) = X Problem 1: SCC Processor: -> Pairs: C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) c#(c(c(c(x1:S)))) -> C#(C(C(x1:S))) c#(c(c(c(x1:S)))) -> C#(C(x1:S)) c#(c(c(c(x1:S)))) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) ->->-> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) Problem 1: Reduction Pair Processor: -> Pairs: C#(b(b(b(x1:S)))) -> a#(B(B(B(x1:S)))) a#(b(x1:S)) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) -> Usable rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [A](X) = 2.X + 1 [B](X) = 2.X + 1 [C](X) = 2.X + 1 [a](X) = 2.X + 1 [b](X) = 2.X + 1 [c](X) = 2.X + 1 [C#](X) = 2.X + 1 [a#](X) = 2.X Problem 1: SCC Processor: -> Pairs: a#(b(x1:S)) -> C#(x1:S) -> Rules: A(A(A(A(x1:S)))) -> a(a(a(x1:S))) A(A(A(b(x1:S)))) -> a(a(a(C(x1:S)))) A(C(x1:S)) -> b(x1:S) A(a(x1:S)) -> x1:S A(c(c(c(x1:S)))) -> b(C(C(C(x1:S)))) B(A(x1:S)) -> c(x1:S) B(B(B(B(x1:S)))) -> b(b(b(x1:S))) B(B(B(c(x1:S)))) -> b(b(b(A(x1:S)))) B(a(a(a(x1:S)))) -> c(A(A(A(x1:S)))) B(b(x1:S)) -> x1:S C(B(x1:S)) -> a(x1:S) C(C(C(C(x1:S)))) -> c(c(c(x1:S))) C(C(C(a(x1:S)))) -> c(c(c(B(x1:S)))) C(b(b(b(x1:S)))) -> a(B(B(B(x1:S)))) C(c(x1:S)) -> x1:S a(A(x1:S)) -> x1:S a(a(a(a(x1:S)))) -> A(A(A(x1:S))) a(b(x1:S)) -> C(x1:S) b(B(x1:S)) -> x1:S b(b(b(b(x1:S)))) -> B(B(B(x1:S))) b(c(x1:S)) -> A(x1:S) c(C(x1:S)) -> x1:S c(a(x1:S)) -> B(x1:S) c(c(c(c(x1:S)))) -> C(C(C(x1:S))) ->Strongly Connected Components: There is no strongly connected component The problem is finite.