174.81/179.23 YES 174.81/179.23 174.81/179.23 Problem 1: 174.81/179.23 174.81/179.23 (VAR v_NonEmpty:S N:S X:S X1:S X2:S Y:S Z:S) 174.81/179.23 (RULES 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 ) 174.81/179.23 (STRATEGY INNERMOST) 174.81/179.23 174.81/179.23 Problem 1: 174.81/179.23 174.81/179.23 Dependency Pairs Processor: 174.81/179.23 -> Pairs: 174.81/179.23 A__ADD(0,X:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 -> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 174.81/179.23 Problem 1: 174.81/179.23 174.81/179.23 SCC Processor: 174.81/179.23 -> Pairs: 174.81/179.23 A__ADD(0,X:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 -> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 ->Strongly Connected Components: 174.81/179.23 ->->Cycle: 174.81/179.23 ->->-> Pairs: 174.81/179.23 A__ADD(0,X:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 ->->-> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 174.81/179.23 Problem 1: 174.81/179.23 174.81/179.23 Reduction Pairs Processor: 174.81/179.23 -> Pairs: 174.81/179.23 A__ADD(0,X:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 -> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 -> Usable rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 ->Interpretation type: 174.81/179.23 Simple mixed 174.81/179.23 ->Coefficients: 174.81/179.23 Natural Numbers 174.81/179.23 ->Dimension: 174.81/179.23 1 174.81/179.23 ->Bound: 174.81/179.23 2 174.81/179.23 ->Interpretation: 174.81/179.23 174.81/179.23 [a__add](X1,X2) = X1 + X2 + 2 174.81/179.23 [a__dbl](X) = 2.X + 2 174.81/179.23 [a__first](X1,X2) = 2.X1 + 2.X2 + 2 174.81/179.23 [a__sqr](X) = X.X + 2.X 174.81/179.23 [a__terms](X) = 2.X.X + 2.X + 2 174.81/179.23 [mark](X) = X 174.81/179.23 [0] = 2 174.81/179.23 [add](X1,X2) = X1 + X2 + 2 174.81/179.23 [cons](X1,X2) = X1 174.81/179.23 [dbl](X) = 2.X + 2 174.81/179.23 [fSNonEmpty] = 0 174.81/179.23 [first](X1,X2) = 2.X1 + 2.X2 + 2 174.81/179.23 [nil] = 1 174.81/179.23 [recip](X) = X + 2 174.81/179.23 [s](X) = X + 2 174.81/179.23 [sqr](X) = X.X + 2.X 174.81/179.23 [terms](X) = 2.X.X + 2.X + 2 174.81/179.23 [A__ADD](X1,X2) = 2.X1 + 2.X2 + 1 174.81/179.23 [A__DBL](X) = 2.X + 2 174.81/179.23 [A__FIRST](X1,X2) = 2.X1 + 2.X2 + 1 174.81/179.23 [A__SQR](X) = 2.X.X + X + 2 174.81/179.23 [A__TERMS](X) = 2.X.X + 2.X + 2 174.81/179.23 [MARK](X) = 2.X + 2 174.81/179.23 174.81/179.23 Problem 1: 174.81/179.23 174.81/179.23 SCC Processor: 174.81/179.23 -> Pairs: 174.81/179.23 A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 -> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 ->Strongly Connected Components: 174.81/179.23 ->->Cycle: 174.81/179.23 ->->-> Pairs: 174.81/179.23 A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 ->->-> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 174.81/179.23 Problem 1: 174.81/179.23 174.81/179.23 Reduction Pairs Processor: 174.81/179.23 -> Pairs: 174.81/179.23 A__ADD(s(X:S),Y:S) -> A__ADD(mark(X:S),mark(Y:S)) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 -> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 -> Usable rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 ->Interpretation type: 174.81/179.23 Simple mixed 174.81/179.23 ->Coefficients: 174.81/179.23 Natural Numbers 174.81/179.23 ->Dimension: 174.81/179.23 1 174.81/179.23 ->Bound: 174.81/179.23 2 174.81/179.23 ->Interpretation: 174.81/179.23 174.81/179.23 [a__add](X1,X2) = X1 + 2.X2 + 2 174.81/179.23 [a__dbl](X) = 2.X + 1 174.81/179.23 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 174.81/179.23 [a__sqr](X) = X.X + X 174.81/179.23 [a__terms](X) = 2.X.X + 2.X 174.81/179.23 [mark](X) = X 174.81/179.23 [0] = 2 174.81/179.23 [add](X1,X2) = X1 + 2.X2 + 2 174.81/179.23 [cons](X1,X2) = X1 174.81/179.23 [dbl](X) = 2.X + 1 174.81/179.23 [fSNonEmpty] = 0 174.81/179.23 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 174.81/179.23 [nil] = 1 174.81/179.23 [recip](X) = 2.X 174.81/179.23 [s](X) = X + 2 174.81/179.23 [sqr](X) = X.X + X 174.81/179.23 [terms](X) = 2.X.X + 2.X 174.81/179.23 [A__ADD](X1,X2) = 2.X1 + 2.X2 + 2 174.81/179.23 [A__DBL](X) = 2.X + 2 174.81/179.23 [A__FIRST](X1,X2) = 2.X1.X2 + 2.X1 + 2 174.81/179.23 [A__SQR](X) = 2.X.X + X + 1 174.81/179.23 [A__TERMS](X) = 2.X.X + 2.X + 2 174.81/179.23 [MARK](X) = 2.X + 2 174.81/179.23 174.81/179.23 Problem 1: 174.81/179.23 174.81/179.23 SCC Processor: 174.81/179.23 -> Pairs: 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 -> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 ->Strongly Connected Components: 174.81/179.23 ->->Cycle: 174.81/179.23 ->->-> Pairs: 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.81/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.81/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.81/179.23 MARK(s(X:S)) -> MARK(X:S) 174.81/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.81/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.81/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.81/179.23 ->->-> Rules: 174.81/179.23 a__add(0,X:S) -> mark(X:S) 174.81/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.81/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.81/179.23 a__dbl(0) -> 0 174.81/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.81/179.23 a__dbl(X:S) -> dbl(X:S) 174.81/179.23 a__first(0,X:S) -> nil 174.81/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.81/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.81/179.23 a__sqr(0) -> 0 174.81/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.81/179.23 a__sqr(X:S) -> sqr(X:S) 174.81/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.81/179.23 a__terms(X:S) -> terms(X:S) 174.81/179.23 mark(0) -> 0 174.81/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.81/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.81/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.81/179.23 mark(nil) -> nil 174.81/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.81/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.81/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.81/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.81/179.23 174.81/179.23 Problem 1: 174.81/179.23 174.81/179.23 Reduction Pairs Processor: 174.81/179.23 -> Pairs: 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(X:S) 174.81/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.81/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.81/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.81/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.81/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.81/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.81/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.81/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 -> Usable rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Interpretation type: 174.93/179.23 Simple mixed 174.93/179.23 ->Coefficients: 174.93/179.23 Natural Numbers 174.93/179.23 ->Dimension: 174.93/179.23 1 174.93/179.23 ->Bound: 174.93/179.23 2 174.93/179.23 ->Interpretation: 174.93/179.23 174.93/179.23 [a__add](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [a__dbl](X) = 2.X + 2 174.93/179.23 [a__first](X1,X2) = 2.X1 + 2.X2 + 1 174.93/179.23 [a__sqr](X) = X.X + 2.X + 2 174.93/179.23 [a__terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [mark](X) = X 174.93/179.23 [0] = 2 174.93/179.23 [add](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [cons](X1,X2) = X1 174.93/179.23 [dbl](X) = 2.X + 2 174.93/179.23 [fSNonEmpty] = 0 174.93/179.23 [first](X1,X2) = 2.X1 + 2.X2 + 1 174.93/179.23 [nil] = 0 174.93/179.23 [recip](X) = X 174.93/179.23 [s](X) = X + 2 174.93/179.23 [sqr](X) = X.X + 2.X + 2 174.93/179.23 [terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [A__ADD](X1,X2) = 2.X1 + 2.X2 + 1 174.93/179.23 [A__DBL](X) = 2.X + 1 174.93/179.23 [A__FIRST](X1,X2) = 2.X2 + 2 174.93/179.23 [A__SQR](X) = 2.X.X + 2.X + 2 174.93/179.23 [A__TERMS](X) = 2.X.X + 2.X + 2 174.93/179.23 [MARK](X) = 2.X + 2 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 SCC Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.93/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Strongly Connected Components: 174.93/179.23 ->->Cycle: 174.93/179.23 ->->-> Pairs: 174.93/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.93/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 ->->-> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 Reduction Pairs Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__ADD(s(X:S),Y:S) -> MARK(Y:S) 174.93/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 -> Usable rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Interpretation type: 174.93/179.23 Simple mixed 174.93/179.23 ->Coefficients: 174.93/179.23 Natural Numbers 174.93/179.23 ->Dimension: 174.93/179.23 1 174.93/179.23 ->Bound: 174.93/179.23 2 174.93/179.23 ->Interpretation: 174.93/179.23 174.93/179.23 [a__add](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [a__dbl](X) = 2.X + 1 174.93/179.23 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 174.93/179.23 [a__sqr](X) = 2.X.X + 2.X + 2 174.93/179.23 [a__terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [mark](X) = X 174.93/179.23 [0] = 2 174.93/179.23 [add](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [cons](X1,X2) = X1 174.93/179.23 [dbl](X) = 2.X + 1 174.93/179.23 [fSNonEmpty] = 0 174.93/179.23 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 174.93/179.23 [nil] = 2 174.93/179.23 [recip](X) = X 174.93/179.23 [s](X) = X + 2 174.93/179.23 [sqr](X) = 2.X.X + 2.X + 2 174.93/179.23 [terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [A__ADD](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [A__DBL](X) = 2.X + 2 174.93/179.23 [A__FIRST](X1,X2) = 2.X2 + 1 174.93/179.23 [A__SQR](X) = 2.X.X + X + 2 174.93/179.23 [A__TERMS](X) = 2.X.X + 2.X + 2 174.93/179.23 [MARK](X) = 2.X + 1 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 SCC Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__ADD(a__sqr(mark(X:S)),a__dbl(mark(X:S))) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> A__ADD(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Strongly Connected Components: 174.93/179.23 ->->Cycle: 174.93/179.23 ->->-> Pairs: 174.93/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 ->->-> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 Reduction Pairs Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__DBL(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 -> Usable rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Interpretation type: 174.93/179.23 Simple mixed 174.93/179.23 ->Coefficients: 174.93/179.23 Natural Numbers 174.93/179.23 ->Dimension: 174.93/179.23 1 174.93/179.23 ->Bound: 174.93/179.23 2 174.93/179.23 ->Interpretation: 174.93/179.23 174.93/179.23 [a__add](X1,X2) = X1 + 2.X2 + 1 174.93/179.23 [a__dbl](X) = 2.X + 1 174.93/179.23 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 174.93/179.23 [a__sqr](X) = 2.X.X + X 174.93/179.23 [a__terms](X) = 2.X.X + X 174.93/179.23 [mark](X) = X 174.93/179.23 [0] = 2 174.93/179.23 [add](X1,X2) = X1 + 2.X2 + 1 174.93/179.23 [cons](X1,X2) = X1 174.93/179.23 [dbl](X) = 2.X + 1 174.93/179.23 [fSNonEmpty] = 0 174.93/179.23 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 174.93/179.23 [nil] = 0 174.93/179.23 [recip](X) = X 174.93/179.23 [s](X) = X + 2 174.93/179.23 [sqr](X) = 2.X.X + X 174.93/179.23 [terms](X) = 2.X.X + X 174.93/179.23 [A__ADD](X1,X2) = 0 174.93/179.23 [A__DBL](X) = 2.X + 2 174.93/179.23 [A__FIRST](X1,X2) = 2.X1.X2 + 2.X2 + 2 174.93/179.23 [A__SQR](X) = 2.X 174.93/179.23 [A__TERMS](X) = 2.X.X + 2.X + 1 174.93/179.23 [MARK](X) = 2.X + 1 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 SCC Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Strongly Connected Components: 174.93/179.23 ->->Cycle: 174.93/179.23 ->->-> Pairs: 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 ->->-> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 Reduction Pairs Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__DBL(s(X:S)) -> MARK(X:S) 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 -> Usable rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Interpretation type: 174.93/179.23 Simple mixed 174.93/179.23 ->Coefficients: 174.93/179.23 Natural Numbers 174.93/179.23 ->Dimension: 174.93/179.23 1 174.93/179.23 ->Bound: 174.93/179.23 2 174.93/179.23 ->Interpretation: 174.93/179.23 174.93/179.23 [a__add](X1,X2) = X1 + 2.X2 174.93/179.23 [a__dbl](X) = 2.X + 2 174.93/179.23 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 174.93/179.23 [a__sqr](X) = 2.X.X + 2.X 174.93/179.23 [a__terms](X) = 2.X.X + 2.X + 1 174.93/179.23 [mark](X) = X 174.93/179.23 [0] = 1 174.93/179.23 [add](X1,X2) = X1 + 2.X2 174.93/179.23 [cons](X1,X2) = X1 174.93/179.23 [dbl](X) = 2.X + 2 174.93/179.23 [fSNonEmpty] = 0 174.93/179.23 [first](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 2 174.93/179.23 [nil] = 2 174.93/179.23 [recip](X) = X + 1 174.93/179.23 [s](X) = X + 2 174.93/179.23 [sqr](X) = 2.X.X + 2.X 174.93/179.23 [terms](X) = 2.X.X + 2.X + 1 174.93/179.23 [A__ADD](X1,X2) = 0 174.93/179.23 [A__DBL](X) = 2.X 174.93/179.23 [A__FIRST](X1,X2) = 2.X1.X2 + X1 + 1 174.93/179.23 [A__SQR](X) = 2.X.X + X 174.93/179.23 [A__TERMS](X) = 2.X.X + 2.X + 2 174.93/179.23 [MARK](X) = 2.X + 2 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 SCC Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> A__DBL(mark(X:S)) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Strongly Connected Components: 174.93/179.23 ->->Cycle: 174.93/179.23 ->->-> Pairs: 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 ->->-> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 Reduction Pairs Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__FIRST(s(X:S),cons(Y:S,Z:S)) -> MARK(Y:S) 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 -> Usable rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Interpretation type: 174.93/179.23 Simple mixed 174.93/179.23 ->Coefficients: 174.93/179.23 Natural Numbers 174.93/179.23 ->Dimension: 174.93/179.23 1 174.93/179.23 ->Bound: 174.93/179.23 2 174.93/179.23 ->Interpretation: 174.93/179.23 174.93/179.23 [a__add](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [a__dbl](X) = 2.X + 1 174.93/179.23 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 174.93/179.23 [a__sqr](X) = 2.X.X + 2.X + 1 174.93/179.23 [a__terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [mark](X) = X 174.93/179.23 [0] = 1 174.93/179.23 [add](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [cons](X1,X2) = X1 + 1 174.93/179.23 [dbl](X) = 2.X + 1 174.93/179.23 [fSNonEmpty] = 0 174.93/179.23 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 174.93/179.23 [nil] = 2 174.93/179.23 [recip](X) = X 174.93/179.23 [s](X) = X + 2 174.93/179.23 [sqr](X) = 2.X.X + 2.X + 1 174.93/179.23 [terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [A__ADD](X1,X2) = 0 174.93/179.23 [A__DBL](X) = 0 174.93/179.23 [A__FIRST](X1,X2) = 2.X1.X2 + 2.X2 + 2 174.93/179.23 [A__SQR](X) = 2.X + 2 174.93/179.23 [A__TERMS](X) = X.X + 2.X + 2 174.93/179.23 [MARK](X) = 2.X + 2 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 SCC Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> A__FIRST(mark(X1:S),mark(X2:S)) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Strongly Connected Components: 174.93/179.23 ->->Cycle: 174.93/179.23 ->->-> Pairs: 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 ->->-> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 Reduction Pairs Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__SQR(s(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 -> Usable rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Interpretation type: 174.93/179.23 Simple mixed 174.93/179.23 ->Coefficients: 174.93/179.23 Natural Numbers 174.93/179.23 ->Dimension: 174.93/179.23 1 174.93/179.23 ->Bound: 174.93/179.23 2 174.93/179.23 ->Interpretation: 174.93/179.23 174.93/179.23 [a__add](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [a__dbl](X) = 2.X + 2 174.93/179.23 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 174.93/179.23 [a__sqr](X) = 2.X.X + 2.X + 2 174.93/179.23 [a__terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [mark](X) = X 174.93/179.23 [0] = 0 174.93/179.23 [add](X1,X2) = X1 + 2.X2 + 2 174.93/179.23 [cons](X1,X2) = X1 174.93/179.23 [dbl](X) = 2.X + 2 174.93/179.23 [fSNonEmpty] = 0 174.93/179.23 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 174.93/179.23 [nil] = 1 174.93/179.23 [recip](X) = X 174.93/179.23 [s](X) = X + 2 174.93/179.23 [sqr](X) = 2.X.X + 2.X + 2 174.93/179.23 [terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [A__ADD](X1,X2) = 0 174.93/179.23 [A__DBL](X) = 0 174.93/179.23 [A__FIRST](X1,X2) = 0 174.93/179.23 [A__SQR](X) = 2.X.X + 2 174.93/179.23 [A__TERMS](X) = 2.X.X + 2.X + 2 174.93/179.23 [MARK](X) = 2.X.X + 2.X + 2 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 SCC Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Strongly Connected Components: 174.93/179.23 ->->Cycle: 174.93/179.23 ->->-> Pairs: 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 ->->-> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 Reduction Pairs Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__SQR(s(X:S)) -> MARK(X:S) 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 -> Usable rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Interpretation type: 174.93/179.23 Simple mixed 174.93/179.23 ->Coefficients: 174.93/179.23 Natural Numbers 174.93/179.23 ->Dimension: 174.93/179.23 1 174.93/179.23 ->Bound: 174.93/179.23 2 174.93/179.23 ->Interpretation: 174.93/179.23 174.93/179.23 [a__add](X1,X2) = X1 + 2.X2 174.93/179.23 [a__dbl](X) = 2.X + 2 174.93/179.23 [a__first](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 174.93/179.23 [a__sqr](X) = X.X + X + 1 174.93/179.23 [a__terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [mark](X) = X 174.93/179.23 [0] = 2 174.93/179.23 [add](X1,X2) = X1 + 2.X2 174.93/179.23 [cons](X1,X2) = X1 174.93/179.23 [dbl](X) = 2.X + 2 174.93/179.23 [fSNonEmpty] = 0 174.93/179.23 [first](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 174.93/179.23 [nil] = 0 174.93/179.23 [recip](X) = 2.X 174.93/179.23 [s](X) = X + 2 174.93/179.23 [sqr](X) = X.X + X + 1 174.93/179.23 [terms](X) = 2.X.X + 2.X + 2 174.93/179.23 [A__ADD](X1,X2) = 0 174.93/179.23 [A__DBL](X) = 0 174.93/179.23 [A__FIRST](X1,X2) = 0 174.93/179.23 [A__SQR](X) = 2.X.X + 2.X + 2 174.93/179.23 [A__TERMS](X) = 2.X.X + 2.X + 2 174.93/179.23 [MARK](X) = 2.X + 2 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 SCC Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__TERMS(N:S) -> A__SQR(mark(N:S)) 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> A__SQR(mark(X:S)) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 -> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 ->Strongly Connected Components: 174.93/179.23 ->->Cycle: 174.93/179.23 ->->-> Pairs: 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.23 ->->-> Rules: 174.93/179.23 a__add(0,X:S) -> mark(X:S) 174.93/179.23 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.23 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.23 a__dbl(0) -> 0 174.93/179.23 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.23 a__dbl(X:S) -> dbl(X:S) 174.93/179.23 a__first(0,X:S) -> nil 174.93/179.23 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.23 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.23 a__sqr(0) -> 0 174.93/179.23 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.23 a__sqr(X:S) -> sqr(X:S) 174.93/179.23 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.23 a__terms(X:S) -> terms(X:S) 174.93/179.23 mark(0) -> 0 174.93/179.23 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.23 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.23 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.23 mark(nil) -> nil 174.93/179.23 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.23 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.23 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.23 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.23 174.93/179.23 Problem 1: 174.93/179.23 174.93/179.23 Reduction Pairs Processor: 174.93/179.23 -> Pairs: 174.93/179.23 A__TERMS(N:S) -> MARK(N:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.23 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.23 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.23 MARK(s(X:S)) -> MARK(X:S) 174.93/179.23 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.23 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.23 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.24 -> Rules: 174.93/179.24 a__add(0,X:S) -> mark(X:S) 174.93/179.24 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.24 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.24 a__dbl(0) -> 0 174.93/179.24 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.24 a__dbl(X:S) -> dbl(X:S) 174.93/179.24 a__first(0,X:S) -> nil 174.93/179.24 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.24 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.24 a__sqr(0) -> 0 174.93/179.24 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.24 a__sqr(X:S) -> sqr(X:S) 174.93/179.24 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.24 a__terms(X:S) -> terms(X:S) 174.93/179.24 mark(0) -> 0 174.93/179.24 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.24 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.24 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(nil) -> nil 174.93/179.24 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.24 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.24 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.24 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.24 -> Usable rules: 174.93/179.24 a__add(0,X:S) -> mark(X:S) 174.93/179.24 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.24 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.24 a__dbl(0) -> 0 174.93/179.24 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.24 a__dbl(X:S) -> dbl(X:S) 174.93/179.24 a__first(0,X:S) -> nil 174.93/179.24 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.24 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.24 a__sqr(0) -> 0 174.93/179.24 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.24 a__sqr(X:S) -> sqr(X:S) 174.93/179.24 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.24 a__terms(X:S) -> terms(X:S) 174.93/179.24 mark(0) -> 0 174.93/179.24 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.24 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.24 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(nil) -> nil 174.93/179.24 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.24 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.24 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.24 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.24 ->Interpretation type: 174.93/179.24 Simple mixed 174.93/179.24 ->Coefficients: 174.93/179.24 Natural Numbers 174.93/179.24 ->Dimension: 174.93/179.24 1 174.93/179.24 ->Bound: 174.93/179.24 2 174.93/179.24 ->Interpretation: 174.93/179.24 174.93/179.24 [a__add](X1,X2) = X1 + 2.X2 + 2 174.93/179.24 [a__dbl](X) = 2.X + 2 174.93/179.24 [a__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 174.93/179.24 [a__sqr](X) = 2.X.X + 2.X 174.93/179.24 [a__terms](X) = 2.X.X + 2.X + 2 174.93/179.24 [mark](X) = X 174.93/179.24 [0] = 1 174.93/179.24 [add](X1,X2) = X1 + 2.X2 + 2 174.93/179.24 [cons](X1,X2) = X1 174.93/179.24 [dbl](X) = 2.X + 2 174.93/179.24 [fSNonEmpty] = 0 174.93/179.24 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 174.93/179.24 [nil] = 2 174.93/179.24 [recip](X) = X + 2 174.93/179.24 [s](X) = X + 2 174.93/179.24 [sqr](X) = 2.X.X + 2.X 174.93/179.24 [terms](X) = 2.X.X + 2.X + 2 174.93/179.24 [A__ADD](X1,X2) = 0 174.93/179.24 [A__DBL](X) = 0 174.93/179.24 [A__FIRST](X1,X2) = 0 174.93/179.24 [A__SQR](X) = 0 174.93/179.24 [A__TERMS](X) = 2.X.X + 2.X + 2 174.93/179.24 [MARK](X) = 2.X.X + 2.X + 1 174.93/179.24 174.93/179.24 Problem 1: 174.93/179.24 174.93/179.24 SCC Processor: 174.93/179.24 -> Pairs: 174.93/179.24 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.24 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.24 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.24 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.24 MARK(s(X:S)) -> MARK(X:S) 174.93/179.24 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.24 MARK(terms(X:S)) -> A__TERMS(mark(X:S)) 174.93/179.24 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.24 -> Rules: 174.93/179.24 a__add(0,X:S) -> mark(X:S) 174.93/179.24 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.24 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.24 a__dbl(0) -> 0 174.93/179.24 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.24 a__dbl(X:S) -> dbl(X:S) 174.93/179.24 a__first(0,X:S) -> nil 174.93/179.24 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.24 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.24 a__sqr(0) -> 0 174.93/179.24 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.24 a__sqr(X:S) -> sqr(X:S) 174.93/179.24 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.24 a__terms(X:S) -> terms(X:S) 174.93/179.24 mark(0) -> 0 174.93/179.24 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.24 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.24 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(nil) -> nil 174.93/179.24 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.24 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.24 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.24 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.24 ->Strongly Connected Components: 174.93/179.24 ->->Cycle: 174.93/179.24 ->->-> Pairs: 174.93/179.24 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.24 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.24 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.24 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.24 MARK(s(X:S)) -> MARK(X:S) 174.93/179.24 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.24 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.24 ->->-> Rules: 174.93/179.24 a__add(0,X:S) -> mark(X:S) 174.93/179.24 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.24 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.24 a__dbl(0) -> 0 174.93/179.24 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.24 a__dbl(X:S) -> dbl(X:S) 174.93/179.24 a__first(0,X:S) -> nil 174.93/179.24 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.24 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.24 a__sqr(0) -> 0 174.93/179.24 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.24 a__sqr(X:S) -> sqr(X:S) 174.93/179.24 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.24 a__terms(X:S) -> terms(X:S) 174.93/179.24 mark(0) -> 0 174.93/179.24 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.24 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.24 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(nil) -> nil 174.93/179.24 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.24 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.24 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.24 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.24 174.93/179.24 Problem 1: 174.93/179.24 174.93/179.24 Subterm Processor: 174.93/179.24 -> Pairs: 174.93/179.24 MARK(add(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(add(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.24 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(dbl(X:S)) -> MARK(X:S) 174.93/179.24 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 174.93/179.24 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 174.93/179.24 MARK(recip(X:S)) -> MARK(X:S) 174.93/179.24 MARK(s(X:S)) -> MARK(X:S) 174.93/179.24 MARK(sqr(X:S)) -> MARK(X:S) 174.93/179.24 MARK(terms(X:S)) -> MARK(X:S) 174.93/179.24 -> Rules: 174.93/179.24 a__add(0,X:S) -> mark(X:S) 174.93/179.24 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.24 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.24 a__dbl(0) -> 0 174.93/179.24 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.24 a__dbl(X:S) -> dbl(X:S) 174.93/179.24 a__first(0,X:S) -> nil 174.93/179.24 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.24 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.24 a__sqr(0) -> 0 174.93/179.24 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.24 a__sqr(X:S) -> sqr(X:S) 174.93/179.24 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.24 a__terms(X:S) -> terms(X:S) 174.93/179.24 mark(0) -> 0 174.93/179.24 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.24 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.24 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(nil) -> nil 174.93/179.24 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.24 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.24 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.24 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.24 ->Projection: 174.93/179.24 pi(MARK) = 1 174.93/179.24 174.93/179.24 Problem 1: 174.93/179.24 174.93/179.24 SCC Processor: 174.93/179.24 -> Pairs: 174.93/179.24 Empty 174.93/179.24 -> Rules: 174.93/179.24 a__add(0,X:S) -> mark(X:S) 174.93/179.24 a__add(s(X:S),Y:S) -> s(a__add(mark(X:S),mark(Y:S))) 174.93/179.24 a__add(X1:S,X2:S) -> add(X1:S,X2:S) 174.93/179.24 a__dbl(0) -> 0 174.93/179.24 a__dbl(s(X:S)) -> s(s(a__dbl(mark(X:S)))) 174.93/179.24 a__dbl(X:S) -> dbl(X:S) 174.93/179.24 a__first(0,X:S) -> nil 174.93/179.24 a__first(s(X:S),cons(Y:S,Z:S)) -> cons(mark(Y:S),first(X:S,Z:S)) 174.93/179.24 a__first(X1:S,X2:S) -> first(X1:S,X2:S) 174.93/179.24 a__sqr(0) -> 0 174.93/179.24 a__sqr(s(X:S)) -> s(a__add(a__sqr(mark(X:S)),a__dbl(mark(X:S)))) 174.93/179.24 a__sqr(X:S) -> sqr(X:S) 174.93/179.24 a__terms(N:S) -> cons(recip(a__sqr(mark(N:S))),terms(s(N:S))) 174.93/179.24 a__terms(X:S) -> terms(X:S) 174.93/179.24 mark(0) -> 0 174.93/179.24 mark(add(X1:S,X2:S)) -> a__add(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) 174.93/179.24 mark(dbl(X:S)) -> a__dbl(mark(X:S)) 174.93/179.24 mark(first(X1:S,X2:S)) -> a__first(mark(X1:S),mark(X2:S)) 174.93/179.24 mark(nil) -> nil 174.93/179.24 mark(recip(X:S)) -> recip(mark(X:S)) 174.93/179.24 mark(s(X:S)) -> s(mark(X:S)) 174.93/179.24 mark(sqr(X:S)) -> a__sqr(mark(X:S)) 174.93/179.24 mark(terms(X:S)) -> a__terms(mark(X:S)) 174.93/179.24 ->Strongly Connected Components: 174.93/179.24 There is no strongly connected component 174.93/179.24 174.93/179.24 The problem is finite. 174.93/179.24 EOF