YES Problem 1: (VAR v_NonEmpty:S N:S X:S X1:S X2:S Y:S Z:S) (RULES a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__ADD(0,X:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: SCC Processor: -> Pairs: A__ADD(0,X:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__ADD(0,X:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__ADD(0,X:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + X2 + 2 [a__dbl](X) = 2.X + 2 [a__first](X1,X2) = 2.X1 + 2.X2 + 2 [a__sqr](X) = X.X + 2.X [a__terms](X) = 2.X.X + 2.X + 2 [mark](X) = X [0] = 2 [add](X1,X2) = X1 + X2 + 2 [cons](X1,X2) = X1 [dbl](X) = 2.X + 2 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1 + 2.X2 + 2 [nil] = 1 [recip](X) = X + 2 [s](X) = X + 2 [sqr](X) = X.X + 2.X [terms](X) = 2.X.X + 2.X + 2 [A__ADD](X1,X2) = 2.X1 + 2.X2 + 1 [A__DBL](X) = 2.X + 2 [A__FIRST](X1,X2) = 2.X1 + 2.X2 + 1 [A__SQR](X) = 2.X.X + X + 2 [A__TERMS](X) = 2.X.X + 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 + 2 [a__dbl](X) = 2.X + 1 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [a__sqr](X) = X.X + X [a__terms](X) = 2.X.X + 2.X [mark](X) = X [0] = 2 [add](X1,X2) = X1 + 2.X2 + 2 [cons](X1,X2) = X1 [dbl](X) = 2.X + 1 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 [nil] = 1 [recip](X) = 2.X [s](X) = X + 2 [sqr](X) = X.X + X [terms](X) = 2.X.X + 2.X [A__ADD](X1,X2) = 2.X1 + 2.X2 + 2 [A__DBL](X) = 2.X + 2 [A__FIRST](X1,X2) = 2.X1.X2 + 2.X1 + 2 [A__SQR](X) = 2.X.X + X + 1 [A__TERMS](X) = 2.X.X + 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__ADD(s(X:S),Y:S) -> MARK(X:S) A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 + 2 [a__dbl](X) = 2.X + 2 [a__first](X1,X2) = 2.X1 + 2.X2 + 1 [a__sqr](X) = X.X + 2.X + 2 [a__terms](X) = 2.X.X + 2.X + 2 [mark](X) = X [0] = 2 [add](X1,X2) = X1 + 2.X2 + 2 [cons](X1,X2) = X1 [dbl](X) = 2.X + 2 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1 + 2.X2 + 1 [nil] = 0 [recip](X) = X [s](X) = X + 2 [sqr](X) = X.X + 2.X + 2 [terms](X) = 2.X.X + 2.X + 2 [A__ADD](X1,X2) = 2.X1 + 2.X2 + 1 [A__DBL](X) = 2.X + 1 [A__FIRST](X1,X2) = 2.X2 + 2 [A__SQR](X) = 2.X.X + 2.X + 2 [A__TERMS](X) = 2.X.X + 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__ADD(s(X:S),Y:S) -> MARK(Y:S) A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 + 2 [a__dbl](X) = 2.X + 1 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [a__sqr](X) = 2.X.X + 2.X + 2 [a__terms](X) = 2.X.X + 2.X + 2 [mark](X) = X [0] = 2 [add](X1,X2) = X1 + 2.X2 + 2 [cons](X1,X2) = X1 [dbl](X) = 2.X + 1 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [nil] = 2 [recip](X) = X [s](X) = X + 2 [sqr](X) = 2.X.X + 2.X + 2 [terms](X) = 2.X.X + 2.X + 2 [A__ADD](X1,X2) = X1 + 2.X2 + 2 [A__DBL](X) = 2.X + 2 [A__FIRST](X1,X2) = 2.X2 + 1 [A__SQR](X) = 2.X.X + X + 2 [A__TERMS](X) = 2.X.X + 2.X + 2 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__DBL(s(X:S)) -> A__DBL(mark(X:S)) A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 + 1 [a__dbl](X) = 2.X + 1 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [a__sqr](X) = 2.X.X + X [a__terms](X) = 2.X.X + X [mark](X) = X [0] = 2 [add](X1,X2) = X1 + 2.X2 + 1 [cons](X1,X2) = X1 [dbl](X) = 2.X + 1 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [nil] = 0 [recip](X) = X [s](X) = X + 2 [sqr](X) = 2.X.X + X [terms](X) = 2.X.X + X [A__ADD](X1,X2) = 0 [A__DBL](X) = 2.X + 2 [A__FIRST](X1,X2) = 2.X1.X2 + 2.X2 + 2 [A__SQR](X) = 2.X [A__TERMS](X) = 2.X.X + 2.X + 1 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__DBL(s(X:S)) -> MARK(X:S) A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 [a__dbl](X) = 2.X + 2 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 [a__sqr](X) = 2.X.X + 2.X [a__terms](X) = 2.X.X + 2.X + 1 [mark](X) = X [0] = 1 [add](X1,X2) = X1 + 2.X2 [cons](X1,X2) = X1 [dbl](X) = 2.X + 2 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 [nil] = 2 [recip](X) = X + 1 [s](X) = X + 2 [sqr](X) = 2.X.X + 2.X [terms](X) = 2.X.X + 2.X + 1 [A__ADD](X1,X2) = 0 [A__DBL](X) = 2.X [A__FIRST](X1,X2) = 2.X1.X2 + X1 + 1 [A__SQR](X) = 2.X.X + X [A__TERMS](X) = 2.X.X + 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__DBL(mark(X:S)) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> A__DBL(mark(X:S)) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 + 2 [a__dbl](X) = 2.X + 1 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [a__sqr](X) = 2.X.X + 2.X + 1 [a__terms](X) = 2.X.X + 2.X + 2 [mark](X) = X [0] = 1 [add](X1,X2) = X1 + 2.X2 + 2 [cons](X1,X2) = X1 + 1 [dbl](X) = 2.X + 1 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [nil] = 2 [recip](X) = X [s](X) = X + 2 [sqr](X) = 2.X.X + 2.X + 1 [terms](X) = 2.X.X + 2.X + 2 [A__ADD](X1,X2) = 0 [A__DBL](X) = 0 [A__FIRST](X1,X2) = 2.X1.X2 + 2.X2 + 2 [A__SQR](X) = 2.X + 2 [A__TERMS](X) = X.X + 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X: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(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__SQR(s(X:S)) -> A__SQR(mark(X:S)) A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 + 2 [a__dbl](X) = 2.X + 2 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [a__sqr](X) = 2.X.X + 2.X + 2 [a__terms](X) = 2.X.X + 2.X + 2 [mark](X) = X [0] = 0 [add](X1,X2) = X1 + 2.X2 + 2 [cons](X1,X2) = X1 [dbl](X) = 2.X + 2 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [nil] = 1 [recip](X) = X [s](X) = X + 2 [sqr](X) = 2.X.X + 2.X + 2 [terms](X) = 2.X.X + 2.X + 2 [A__ADD](X1,X2) = 0 [A__DBL](X) = 0 [A__FIRST](X1,X2) = 0 [A__SQR](X) = 2.X.X + 2 [A__TERMS](X) = 2.X.X + 2.X + 2 [MARK](X) = 2.X.X + 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__SQR(s(X:S)) -> MARK(X:S) A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 [a__dbl](X) = 2.X + 2 [a__first](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 [a__sqr](X) = X.X + X + 1 [a__terms](X) = 2.X.X + 2.X + 2 [mark](X) = X [0] = 2 [add](X1,X2) = X1 + 2.X2 [cons](X1,X2) = X1 [dbl](X) = 2.X + 2 [fSNonEmpty] = 0 [first](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 [nil] = 0 [recip](X) = 2.X [s](X) = X + 2 [sqr](X) = X.X + X + 1 [terms](X) = 2.X.X + 2.X + 2 [A__ADD](X1,X2) = 0 [A__DBL](X) = 0 [A__FIRST](X1,X2) = 0 [A__SQR](X) = 2.X.X + 2.X + 2 [A__TERMS](X) = 2.X.X + 2.X + 2 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__TERMS(N:S) -> A__SQR(mark(N:S)) A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> A__SQR(mark(X:S)) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Reduction Pairs Processor: -> Pairs: A__TERMS(N:S) -> MARK(N:S) MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) -> Usable rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__add](X1,X2) = X1 + 2.X2 + 2 [a__dbl](X) = 2.X + 2 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [a__sqr](X) = 2.X.X + 2.X [a__terms](X) = 2.X.X + 2.X + 2 [mark](X) = X [0] = 1 [add](X1,X2) = X1 + 2.X2 + 2 [cons](X1,X2) = X1 [dbl](X) = 2.X + 2 [fSNonEmpty] = 0 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [nil] = 2 [recip](X) = X + 2 [s](X) = X + 2 [sqr](X) = 2.X.X + 2.X [terms](X) = 2.X.X + 2.X + 2 [A__ADD](X1,X2) = 0 [A__DBL](X) = 0 [A__FIRST](X1,X2) = 0 [A__SQR](X) = 0 [A__TERMS](X) = 2.X.X + 2.X + 2 [MARK](X) = 2.X.X + 2.X + 1 Problem 1: SCC Processor: -> Pairs: MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> A__TERMS(mark(X:S)) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> MARK(X:S) ->->-> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) Problem 1: Subterm Processor: -> Pairs: MARK(add(X1:S,X2:S)) -> MARK(X1:S) MARK(add(X1:S,X2:S)) -> MARK(X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(dbl(X:S)) -> MARK(X:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(recip(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(sqr(X:S)) -> MARK(X:S) MARK(terms(X:S)) -> MARK(X:S) -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Projection: pi(MARK) = 1 Problem 1: SCC Processor: -> Pairs: Empty -> Rules: a__add(0,X:S) -> mark(X:S) a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) a__add(X1:S,X2:S) -> add(X1:S,X2:S) a__dbl(0) -> 0 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) a__dbl(X:S) -> dbl(X:S) a__first(0,X:S) -> nil a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) a__first(X1:S,X2:S) -> first(X1:S,X2:S) a__sqr(0) -> 0 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) a__sqr(X:S) -> sqr(X:S) a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) a__terms(X:S) -> terms(X:S) mark(0) -> 0 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(dbl(X:S)) -> a__dbl(mark(X:S)) mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) mark(nil) -> nil mark(recip(X:S)) -> recip(mark(X:S)) mark(s(X:S)) -> s(mark(X:S)) mark(sqr(X:S)) -> a__sqr(mark(X:S)) mark(terms(X:S)) -> a__terms(mark(X:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.