0.00/0.37 YES 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 (VAR v_NonEmpty:S N:S X:S X1:S X2:S Y:S Z:S) 0.00/0.37 (RULES 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 ) 0.00/0.37 (STRATEGY INNERMOST) 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 Dependency Pairs Processor: 0.00/0.37 -> Pairs: 0.00/0.37 A__ADD(0,X:S) -> MARK(X:S) 0.00/0.37 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.37 A__TERMS(N:S) -> A__SQR(mark(N:S)) 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 -> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 SCC Processor: 0.00/0.37 -> Pairs: 0.00/0.37 A__ADD(0,X:S) -> MARK(X:S) 0.00/0.37 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.37 A__TERMS(N:S) -> A__SQR(mark(N:S)) 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 -> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 ->Strongly Connected Components: 0.00/0.37 ->->Cycle: 0.00/0.37 ->->-> Pairs: 0.00/0.37 A__ADD(0,X:S) -> MARK(X:S) 0.00/0.37 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 ->->-> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 Reduction Pairs Processor: 0.00/0.37 -> Pairs: 0.00/0.37 A__ADD(0,X:S) -> MARK(X:S) 0.00/0.37 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 -> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 -> Usable rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 ->Interpretation type: 0.00/0.37 Linear 0.00/0.37 ->Coefficients: 0.00/0.37 Natural Numbers 0.00/0.37 ->Dimension: 0.00/0.37 1 0.00/0.37 ->Bound: 0.00/0.37 2 0.00/0.37 ->Interpretation: 0.00/0.37 0.00/0.37 [a__add](X1,X2) = 2.X1 + 2.X2 0.00/0.37 [a__dbl](X) = 2.X + 2 0.00/0.37 [a__first](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.37 [a__sqr](X) = X + 2 0.00/0.37 [a__terms](X) = 2.X + 2 0.00/0.37 [mark](X) = 2.X 0.00/0.37 [0] = 2 0.00/0.37 [add](X1,X2) = 2.X1 + 2.X2 0.00/0.37 [cons](X1,X2) = X1 0.00/0.37 [dbl](X) = 2.X + 2 0.00/0.37 [fSNonEmpty] = 0 0.00/0.37 [first](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.37 [nil] = 2 0.00/0.37 [recip](X) = X 0.00/0.37 [s](X) = 1 0.00/0.37 [sqr](X) = X + 2 0.00/0.37 [terms](X) = 2.X + 2 0.00/0.37 [A__ADD](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.37 [A__DBL](X) = 0 0.00/0.37 [A__FIRST](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.37 [A__SQR](X) = 0 0.00/0.37 [A__TERMS](X) = 2.X + 2 0.00/0.37 [MARK](X) = 2.X + 2 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 SCC Processor: 0.00/0.37 -> Pairs: 0.00/0.37 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 -> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 ->Strongly Connected Components: 0.00/0.37 ->->Cycle: 0.00/0.37 ->->-> Pairs: 0.00/0.37 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 ->->-> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 Reduction Pairs Processor: 0.00/0.37 -> Pairs: 0.00/0.37 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 -> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 -> Usable rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 ->Interpretation type: 0.00/0.37 Linear 0.00/0.37 ->Coefficients: 0.00/0.37 Natural Numbers 0.00/0.37 ->Dimension: 0.00/0.37 1 0.00/0.37 ->Bound: 0.00/0.37 2 0.00/0.37 ->Interpretation: 0.00/0.37 0.00/0.37 [a__add](X1,X2) = 2.X1 + 2.X2 0.00/0.37 [a__dbl](X) = X + 2 0.00/0.37 [a__first](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.37 [a__sqr](X) = X + 2 0.00/0.37 [a__terms](X) = 2.X + 2 0.00/0.37 [mark](X) = 2.X 0.00/0.37 [0] = 1 0.00/0.37 [add](X1,X2) = 2.X1 + 2.X2 0.00/0.37 [cons](X1,X2) = X1 0.00/0.37 [dbl](X) = X + 2 0.00/0.37 [fSNonEmpty] = 0 0.00/0.37 [first](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.37 [nil] = 2 0.00/0.37 [recip](X) = X 0.00/0.37 [s](X) = 2 0.00/0.37 [sqr](X) = X + 2 0.00/0.37 [terms](X) = 2.X + 2 0.00/0.37 [A__ADD](X1,X2) = 0 0.00/0.37 [A__DBL](X) = 0 0.00/0.37 [A__FIRST](X1,X2) = 2.X1 + 2.X2 0.00/0.37 [A__SQR](X) = 0 0.00/0.37 [A__TERMS](X) = 2.X + 2 0.00/0.37 [MARK](X) = 2.X + 2 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 SCC Processor: 0.00/0.37 -> Pairs: 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 -> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 ->Strongly Connected Components: 0.00/0.37 ->->Cycle: 0.00/0.37 ->->-> Pairs: 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 ->->-> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 Reduction Pairs Processor: 0.00/0.37 -> Pairs: 0.00/0.37 A__TERMS(N:S) -> MARK(N:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 -> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 -> Usable rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 ->Interpretation type: 0.00/0.37 Linear 0.00/0.37 ->Coefficients: 0.00/0.37 Natural Numbers 0.00/0.37 ->Dimension: 0.00/0.37 1 0.00/0.37 ->Bound: 0.00/0.37 2 0.00/0.37 ->Interpretation: 0.00/0.37 0.00/0.37 [a__add](X1,X2) = X1 + 2.X2 + 2 0.00/0.37 [a__dbl](X) = 2.X + 2 0.00/0.37 [a__first](X1,X2) = X1 + 2.X2 + 2 0.00/0.37 [a__sqr](X) = X + 1 0.00/0.37 [a__terms](X) = 2.X + 2 0.00/0.37 [mark](X) = 2.X + 1 0.00/0.37 [0] = 2 0.00/0.37 [add](X1,X2) = X1 + 2.X2 + 2 0.00/0.37 [cons](X1,X2) = X1 0.00/0.37 [dbl](X) = 2.X + 2 0.00/0.37 [fSNonEmpty] = 0 0.00/0.37 [first](X1,X2) = X1 + 2.X2 + 2 0.00/0.37 [nil] = 2 0.00/0.37 [recip](X) = X 0.00/0.37 [s](X) = 2 0.00/0.37 [sqr](X) = X + 1 0.00/0.37 [terms](X) = 2.X + 2 0.00/0.37 [A__ADD](X1,X2) = 0 0.00/0.37 [A__DBL](X) = 0 0.00/0.37 [A__FIRST](X1,X2) = 0 0.00/0.37 [A__SQR](X) = 0 0.00/0.37 [A__TERMS](X) = X + 2 0.00/0.37 [MARK](X) = X + 1 0.00/0.37 0.00/0.37 Problem 1: 0.00/0.37 0.00/0.37 SCC Processor: 0.00/0.37 -> Pairs: 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 -> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.37 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.37 a__sqr(0) -> 0 0.00/0.37 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.37 a__sqr(X:S) -> sqr(X:S) 0.00/0.37 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.37 a__terms(X:S) -> terms(X:S) 0.00/0.37 mark(0) -> 0 0.00/0.37 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.37 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.37 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.37 mark(nil) -> nil 0.00/0.37 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.37 mark(s(X:S)) -> s(X:S) 0.00/0.37 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.37 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.37 ->Strongly Connected Components: 0.00/0.37 ->->Cycle: 0.00/0.37 ->->-> Pairs: 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.37 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.37 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.37 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.37 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.37 ->->-> Rules: 0.00/0.37 a__add(0,X:S) -> mark(X:S) 0.00/0.37 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.37 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.37 a__dbl(0) -> 0 0.00/0.37 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.37 a__dbl(X:S) -> dbl(X:S) 0.00/0.37 a__first(0,X:S) -> nil 0.00/0.37 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.38 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.38 a__sqr(0) -> 0 0.00/0.38 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.38 a__sqr(X:S) -> sqr(X:S) 0.00/0.38 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.38 a__terms(X:S) -> terms(X:S) 0.00/0.38 mark(0) -> 0 0.00/0.38 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.38 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.38 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.38 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.38 mark(nil) -> nil 0.00/0.38 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.38 mark(s(X:S)) -> s(X:S) 0.00/0.38 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.38 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.38 0.00/0.38 Problem 1: 0.00/0.38 0.00/0.38 Subterm Processor: 0.00/0.38 -> Pairs: 0.00/0.38 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.38 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.38 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.38 MARK(dbl(X:S)) -> MARK(X:S) 0.00/0.38 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.38 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.38 MARK(recip(X:S)) -> MARK(X:S) 0.00/0.38 MARK(sqr(X:S)) -> MARK(X:S) 0.00/0.38 MARK(terms(X:S)) -> MARK(X:S) 0.00/0.38 -> Rules: 0.00/0.38 a__add(0,X:S) -> mark(X:S) 0.00/0.38 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.38 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.38 a__dbl(0) -> 0 0.00/0.38 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.38 a__dbl(X:S) -> dbl(X:S) 0.00/0.38 a__first(0,X:S) -> nil 0.00/0.38 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.38 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.38 a__sqr(0) -> 0 0.00/0.38 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.38 a__sqr(X:S) -> sqr(X:S) 0.00/0.38 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.38 a__terms(X:S) -> terms(X:S) 0.00/0.38 mark(0) -> 0 0.00/0.38 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.38 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.38 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.38 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.38 mark(nil) -> nil 0.00/0.38 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.38 mark(s(X:S)) -> s(X:S) 0.00/0.38 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.38 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.38 ->Projection: 0.00/0.38 pi(MARK) = 1 0.00/0.38 0.00/0.38 Problem 1: 0.00/0.38 0.00/0.38 SCC Processor: 0.00/0.38 -> Pairs: 0.00/0.38 Empty 0.00/0.38 -> Rules: 0.00/0.38 a__add(0,X:S) -> mark(X:S) 0.00/0.38 a__add(s(X:S),Y:S) -> s(add(X:S,Y:S)) 0.00/0.38 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 0.00/0.38 a__dbl(0) -> 0 0.00/0.38 a__dbl(s(X:S)) -> s(s(dbl(X:S))) 0.00/0.38 a__dbl(X:S) -> dbl(X:S) 0.00/0.38 a__first(0,X:S) -> nil 0.00/0.38 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 0.00/0.38 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 0.00/0.38 a__sqr(0) -> 0 0.00/0.38 a__sqr(s(X:S)) -> s(add(sqr(X:S),dbl(X:S))) 0.00/0.38 a__sqr(X:S) -> sqr(X:S) 0.00/0.38 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 0.00/0.38 a__terms(X:S) -> terms(X:S) 0.00/0.38 mark(0) -> 0 0.00/0.38 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 0.00/0.38 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 0.00/0.38 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 0.00/0.38 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 0.00/0.38 mark(nil) -> nil 0.00/0.38 mark(recip(X:S)) -> recip(mark(X:S)) 0.00/0.38 mark(s(X:S)) -> s(X:S) 0.00/0.38 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 0.00/0.38 mark(terms(X:S)) -> a__terms(mark(X:S)) 0.00/0.38 ->Strongly Connected Components: 0.00/0.38 There is no strongly connected component 0.00/0.38 0.00/0.38 The problem is finite. 0.00/0.38 EOF