YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ) Problem 1: Dependency Pairs Processor: -> Pairs: A(a(d(d(x1:S)))) -> B(b(x1:S)) A(a(d(d(x1:S)))) -> B(x1:S) A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) B(b(d(d(b(b(x1:S)))))) -> C(c(x1:S)) B(b(d(d(b(b(x1:S)))))) -> C(x1:S) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: SCC Processor: -> Pairs: A(a(d(d(x1:S)))) -> B(b(x1:S)) A(a(d(d(x1:S)))) -> B(x1:S) A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) B(b(d(d(b(b(x1:S)))))) -> C(c(x1:S)) B(b(d(d(b(b(x1:S)))))) -> C(x1:S) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(d(d(x1:S)))) -> B(b(x1:S)) A(a(d(d(x1:S)))) -> B(x1:S) A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(d(d(x1:S)))) -> B(b(x1:S)) A(a(d(d(x1:S)))) -> B(x1:S) A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X [b](X) = X [c](X) = X + 1 [d](X) = X + 1 [A](X) = 2.X + 2 [B](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A(a(d(d(x1:S)))) -> B(x1:S) A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(d(d(x1:S)))) -> B(x1:S) A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(d(d(x1:S)))) -> B(x1:S) A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X [b](X) = X [c](X) = 2.X + 2 [d](X) = 2.X + 2 [A](X) = 2.X + 2 [B](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) A(a(x1:S)) -> B(x1:S) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 1 [b](X) = 2 [c](X) = 0 [d](X) = 0 [A](X) = 2.X + 2 [B](X) = X + 2 Problem 1: SCC Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) B(b(d(d(b(b(x1:S)))))) -> A(c(c(x1:S))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 1 [b](X) = 2 [c](X) = 0 [d](X) = 0 [A](X) = 2.X + 2 [B](X) = 2.X Problem 1: SCC Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(b(b(x1:S)))))) A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 2 [b](X) = X + 1/2 [c](X) = 2.X + 2 [d](X) = 2.X + 2 [A](X) = X + 1 [B](X) = X Problem 1: SCC Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(b(x1:S))))) A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 2 [b](X) = X + 1/2 [c](X) = 2.X + 2 [d](X) = 2.X + 2 [A](X) = 2.X + 1 [B](X) = 2.X Problem 1: SCC Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(x1:S)) -> B(b(b(b(x1:S)))) A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 2 [b](X) = X + 1/2 [c](X) = 2.X [d](X) = 2.X [A](X) = 1/2.X + 1 [B](X) = 1/2.X Problem 1: SCC Processor: -> Pairs: A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(x1:S)) -> B(b(b(x1:S))) A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 2 [b](X) = X + 1/2 [c](X) = 2.X + 2 [d](X) = 2.X + 2 [A](X) = 2.X + 1 [B](X) = 2.X Problem 1: SCC Processor: -> Pairs: A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) ->->-> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) Problem 1: Reduction Pair Processor: -> Pairs: A(a(x1:S)) -> B(b(x1:S)) B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) -> Usable rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [a](X) = X + 2 [b](X) = X + 1/2 [c](X) = 2.X + 1 [d](X) = 2.X + 1 [A](X) = 2.X + 2 [B](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: B(b(d(d(b(b(x1:S)))))) -> A(a(c(c(x1:S)))) -> Rules: a(a(d(d(x1:S)))) -> d(d(b(b(x1:S)))) a(a(x1:S)) -> b(b(b(b(b(b(x1:S)))))) b(b(d(d(b(b(x1:S)))))) -> a(a(c(c(x1:S)))) c(c(x1:S)) -> d(d(x1:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.