YES Problem 1: (VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S Z:S) (RULES a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__ADD(0,X:S) -> MARK(X:S) A__AND(ttrue,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(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),X2:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> A__FROM(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue Problem 1: SCC Processor: -> Pairs: A__ADD(0,X:S) -> MARK(X:S) A__AND(ttrue,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(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),X2:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> A__FROM(X:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__ADD(0,X:S) -> MARK(X:S) A__AND(ttrue,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(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),X2:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue Problem 1: Reduction Pairs Processor: -> Pairs: A__ADD(0,X:S) -> MARK(X:S) A__AND(ttrue,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(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),X2:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = 2.X1 + 2.X2 + 1 [a__and](X1,X2) = 2.X1 + 2.X2 + 2 [a__first](X1,X2) = 2.X1 + 2.X2 + 2 [a__from](X) = 2 [a__if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X [0] = 2 [add](X1,X2) = 2.X1 + X2 + 1 [and](X1,X2) = 2.X1 + 2.X2 + 2 [cons](X1,X2) = 2.X2 [fSNonEmpty] = 0 [false] = 1 [first](X1,X2) = 2.X1 + 2.X2 + 2 [from](X) = 1 [if](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 1 [nil] = 0 [s](X) = 2.X + 2 [true] = 2 [A__ADD](X1,X2) = X1 + 2.X2 + 2 [A__AND](X1,X2) = X1 + 2.X2 [A__FIRST](X1,X2) = 0 [A__FROM](X) = 0 [A__IF](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__AND(ttrue,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(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),X2:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__AND(ttrue,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(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue Problem 1: Reduction Pairs Processor: -> Pairs: A__AND(ttrue,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(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = 2.X1 + 2.X2 + 2 [a__and](X1,X2) = 2.X1 + 2.X2 + 2 [a__first](X1,X2) = X1 + X2 + 2 [a__from](X) = 2.X + 2 [a__if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X + 2 [0] = 2 [add](X1,X2) = 2.X1 + X2 + 2 [and](X1,X2) = 2.X1 + X2 + 2 [cons](X1,X2) = X2 [fSNonEmpty] = 0 [false] = 1 [first](X1,X2) = X1 + X2 + 2 [from](X) = X [if](X1,X2,X3) = 2.X1 + X2 + X3 + 2 [nil] = 2 [s](X) = 2.X + 2 [true] = 2 [A__ADD](X1,X2) = 0 [A__AND](X1,X2) = 2.X1 + 2.X2 + 1 [A__FIRST](X1,X2) = 0 [A__FROM](X) = 0 [A__IF](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue Problem 1: Reduction Pairs Processor: -> Pairs: A__IF(ffalse,X:S,Y:S) -> MARK(Y:S) A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = 2.X1 + 2.X2 + 1 [a__and](X1,X2) = X1 + 2.X2 [a__first](X1,X2) = 2.X1 + X2 + 2 [a__from](X) = 2.X + 1 [a__if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X + 1 [0] = 0 [add](X1,X2) = 2.X1 + 2.X2 + 1 [and](X1,X2) = X1 + X2 [cons](X1,X2) = 1 [fSNonEmpty] = 0 [false] = 1 [first](X1,X2) = 2.X1 + X2 + 2 [from](X) = 2.X + 1 [if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [nil] = 2 [s](X) = X [true] = 1 [A__ADD](X1,X2) = 0 [A__AND](X1,X2) = 0 [A__FIRST](X1,X2) = 0 [A__FROM](X) = 0 [A__IF](X1,X2,X3) = 2.X2 + 2.X3 + 1 [MARK](X) = 2.X Problem 1: SCC Processor: -> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue Problem 1: Subterm Processor: -> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(if(X1:S,X2:S,X3:S)) -> A__IF(mark(X1:S),X2:S,X3:S) MARK(if(X1:S,X2:S,X3:S)) -> MARK(X1:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Projection: pi(A__IF) = 2 pi(MARK) = 1 Problem 1: SCC Processor: -> Pairs: A__IF(ttrue,X:S,Y:S) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__and(ffalse,Y:S) -> ffalse a__and(ttrue,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__from(X:S) -> cons(X:S,from(s(X:S))) a__from(X:S) -> from(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(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S) mark(ffalse) -> ffalse mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(from(X:S)) -> a__from(X:S) mark(if(X1:S,X2:S,X3:S)) -> a__if(mark(X1:S),X2:S,X3:S) mark(nil) -> nil mark(s(X:S)) -> s(X:S) mark(ttrue) -> ttrue ->Strongly Connected Components: There is no strongly connected component The problem is finite.