(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following 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(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(X)))) → mark(s(half(X)))
active(half(dbl(X))) → mark(X)
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(s(X)) → s(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))
active(half(X)) → half(active(X))
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))
s(mark(X)) → mark(s(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))
half(mark(X)) → mark(half(X))
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)
proper(half(X)) → half(proper(X))
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))
half(ok(X)) → ok(half(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(0)) → c1
ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ACTIVE(dbl(0)) → c3
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(add(0, z0)) → c5
ACTIVE(add(s(z0), z1)) → c6(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(first(0, z0)) → c7
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(half(0)) → c9
ACTIVE(half(s(0))) → c10
ACTIVE(half(s(s(z0)))) → c11(S(half(z0)), HALF(z0))
ACTIVE(half(dbl(z0))) → c12
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(0) → c49
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c53
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(0)) → c1
ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ACTIVE(dbl(0)) → c3
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(add(0, z0)) → c5
ACTIVE(add(s(z0), z1)) → c6(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(first(0, z0)) → c7
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(half(0)) → c9
ACTIVE(half(s(0))) → c10
ACTIVE(half(s(s(z0)))) → c11(S(half(z0)), HALF(z0))
ACTIVE(half(dbl(z0))) → c12
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(0) → c49
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c53
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, terms, cons, recip, sqr, s, add, dbl, first, half, proper, top

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing nodes:

ACTIVE(half(s(0))) → c10
PROPER(0) → c49
ACTIVE(sqr(0)) → c1
ACTIVE(add(0, z0)) → c5
PROPER(nil) → c53
ACTIVE(dbl(0)) → c3
ACTIVE(half(0)) → c9
ACTIVE(half(dbl(z0))) → c12
ACTIVE(first(0, z0)) → c7

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(half(s(s(z0)))) → c11(S(half(z0)), HALF(z0))
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(half(s(s(z0)))) → c11(S(half(z0)), HALF(z0))
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, terms, cons, recip, sqr, s, add, dbl, first, half, proper, top

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c, c2, c4, c6, c8, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(z0))) → c2(SQR(z0), DBL(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
S tuples:

ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(z0))) → c2(SQR(z0), DBL(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
K tuples:none
Defined Rule Symbols:

active, terms, cons, recip, sqr, s, add, dbl, first, half, proper, top

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c8, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c, c2, c4, c6, c11

(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
S tuples:

ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
K tuples:none
Defined Rule Symbols:

active, terms, cons, recip, sqr, s, add, dbl, first, half, proper, top

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:

ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
S tuples:

ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
K tuples:none
Defined Rule Symbols:

active, cons, first, terms, recip, sqr, s, add, dbl, half, proper

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0)) by

ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:

ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
S tuples:

ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:

active, cons, first, terms, recip, sqr, s, add, dbl, half, proper

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13

(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:

ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
S tuples:

ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:

active, cons, first, terms, recip, sqr, s, add, dbl, half, proper

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0)) by

ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:

ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
S tuples:

ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:

active, cons, first, terms, recip, sqr, s, add, dbl, half, proper

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0)) by

ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
S tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:

active, cons, first, terms, recip, sqr, s, add, dbl, half, proper

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
S tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
K tuples:none
Defined Rule Symbols:

active, cons, first, terms, recip, sqr, s, add, dbl, half, proper

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP

Compound Symbols:

c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)), ACTIVE(sqr(0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:

ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)), ACTIVE(sqr(0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
S tuples:

ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0),