(STRATEGY INNERMOST) (VAR N X X1 X2 Y Z) (DATATYPES A = µX.< mark(X), 0, nil, ok(X) >) (SIGNATURES active :: [A] -> A terms :: [A] -> A cons :: [A x A] -> A recip :: [A] -> A sqr :: [A] -> A add :: [A x A] -> A dbl :: [A] -> A first :: [A x A] -> A proper :: [A] -> A s :: [A] -> A top :: [A] -> A) (RULES active(terms(N)) -> mark(cons(recip(sqr(N)) ,terms(s(N)))) active(sqr(0())) -> mark(0()) active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X)))) active(dbl(0())) -> mark(0()) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(terms(X)) -> terms(active(X)) active(cons(X1,X2)) -> cons(active(X1),X2) active(recip(X)) -> recip(active(X)) active(sqr(X)) -> sqr(active(X)) active(add(X1,X2)) -> add(active(X1),X2) active(add(X1,X2)) -> add(X1 ,active(X2)) active(dbl(X)) -> dbl(active(X)) active(first(X1,X2)) -> first(active(X1),X2) active(first(X1,X2)) -> first(X1 ,active(X2)) terms(mark(X)) -> mark(terms(X)) cons(mark(X1),X2) -> mark(cons(X1,X2)) recip(mark(X)) -> mark(recip(X)) sqr(mark(X)) -> mark(sqr(X)) add(mark(X1),X2) -> mark(add(X1 ,X2)) add(X1,mark(X2)) -> mark(add(X1 ,X2)) dbl(mark(X)) -> mark(dbl(X)) first(mark(X1),X2) -> mark(first(X1,X2)) first(X1,mark(X2)) -> mark(first(X1,X2)) proper(terms(X)) -> terms(proper(X)) proper(cons(X1,X2)) -> cons(proper(X1),proper(X2)) proper(recip(X)) -> recip(proper(X)) proper(sqr(X)) -> sqr(proper(X)) proper(s(X)) -> s(proper(X)) proper(0()) -> ok(0()) proper(add(X1,X2)) -> add(proper(X1),proper(X2)) proper(dbl(X)) -> dbl(proper(X)) proper(first(X1,X2)) -> first(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) terms(ok(X)) -> ok(terms(X)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) recip(ok(X)) -> ok(recip(X)) sqr(ok(X)) -> ok(sqr(X)) s(ok(X)) -> ok(s(X)) add(ok(X1),ok(X2)) -> ok(add(X1 ,X2)) dbl(ok(X)) -> ok(dbl(X)) first(ok(X1),ok(X2)) -> ok(first(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)))