YES Problem 1: (VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S) (RULES a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__F(X:S) -> MARK(X:S) A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue Problem 1: SCC Processor: -> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__F(X:S) -> MARK(X:S) A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__F(X:S) -> MARK(X:S) A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) ->->-> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue Problem 1: Reduction Pairs Processor: -> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__F(X:S) -> MARK(X:S) A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue -> Usable rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__f](X) = 2.X + 2 [a__if](X1,X2,X3) = X1 + 2.X2 + 2.X3 [mark](X) = 2.X [c] = 0 [f](X) = 2.X + 1 [fSNonEmpty] = 0 [false] = 1 [if](X1,X2,X3) = X1 + 2.X2 + 2.X3 [true] = 0 [A__F](X) = 2.X + 2 [A__IF](X1,X2,X3) = 2.X2 + 2.X3 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) ->->-> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue Problem 1: Reduction Pairs Processor: -> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue -> Usable rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__f](X) = 2.X + 2 [a__if](X1,X2,X3) = X1 + 2.X2 + 2.X3 [mark](X) = 2.X [c] = 0 [f](X) = 2.X + 1 [fSNonEmpty] = 0 [false] = 2 [if](X1,X2,X3) = X1 + 2.X2 + X3 [true] = 0 [A__F](X) = 2.X + 2 [A__IF](X1,X2,X3) = X1 + 2.X2 + 2.X3 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) ->->-> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue Problem 1: Reduction Pairs Processor: -> Pairs: A__F(X:S) -> A__IF(mark(X:S),c,f(ttrue)) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue -> Usable rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__f](X) = 2.X + 2 [a__if](X1,X2,X3) = X1 + 2.X2 + 2.X3 [mark](X) = 2.X [c] = 0 [f](X) = 2.X + 1 [fSNonEmpty] = 0 [false] = 1 [if](X1,X2,X3) = X1 + 2.X2 + X3 [true] = 0 [A__F](X) = 2.X + 2 [A__IF](X1,X2,X3) = 2.X2 + X3 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> A__F(mark(X:S)) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) ->->-> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue Problem 1: Reduction Pairs Processor: -> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(f(X:S)) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue -> Usable rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__f](X) = 2.X + 2 [a__if](X1,X2,X3) = X1 + 2.X2 + 2.X3 [mark](X) = 2.X [c] = 0 [f](X) = 2.X + 1 [fSNonEmpty] = 0 [false] = 2 [if](X1,X2,X3) = X1 + 2.X2 + 2.X3 [true] = 0 [A__F](X) = 0 [A__IF](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) ->->-> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue Problem 1: Reduction Pairs Processor: -> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue -> Usable rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__f](X) = 2.X.X + 2.X + 1 [a__if](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X2.X3 + X1 + 2.X2 + 1 [mark](X) = X [c] = 0 [f](X) = 2.X.X + 2.X + 1 [fSNonEmpty] = 0 [false] = 1 [if](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X1.X3 + 2.X2.X3 + X1 + 2.X2 + 1 [true] = 0 [A__F](X) = 0 [A__IF](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 2.X2 + 2 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),mark(X2:S),X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) ->->-> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue Problem 1: Subterm Processor: -> Pairs: MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X2:S) -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Projection: pi(MARK) = 1 Problem 1: SCC Processor: -> Pairs: Empty -> Rules: a__f(X:S) -> a__if(mark(X:S),c,f(ttrue)) a__f(X:S) -> f(X:S) a__if(ffalse,X:S,Y:S) -> mark(Y:S) a__if(ttrue,X:S,Y:S) -> mark(X:S) a__if(X1:S,X2:S,X3:S) -> if(X1:S,X2:S,X3:S) mark(c) -> c mark(f(X:S)) -> a__f(mark(X:S)) mark(ffalse) -> ffalse mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),mark(X2:S),X3:S) mark(ttrue) -> ttrue ->Strongly Connected Components: There is no strongly connected component The problem is finite.