(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(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
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(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
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))
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))
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(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
s(ok(z0)) → ok(s(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(dbl(0)) → c3
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(first(0, z0)) → c7
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(0) → c39
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c43
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(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(dbl(0)) → c3
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(first(0, z0)) → c7
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(0) → c39
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c43
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

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

Removed 6 trailing nodes:

PROPER(0) → c39
PROPER(nil) → c43
ACTIVE(sqr(0)) → c1
ACTIVE(dbl(0)) → c3
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(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
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(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
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))
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))
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(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
s(ok(z0)) → ok(s(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(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(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(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c, c2, c4, c6, 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, c40, c41, c42, c44, c45, c46

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

Removed 3 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(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
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(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
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))
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))
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(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
s(ok(z0)) → ok(s(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
S tuples:

ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c2, c4, c6, 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, c40, c41, c42, c44, c45, c46, c

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

ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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))
S tuples:

ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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, c40, c41, c42, c44, c45, c46, 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(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
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(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
s(ok(z0)) → ok(s(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
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))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(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(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
Tuples:

ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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))
S tuples:

ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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, c40, c41, c42, c44, c45, c46, c1

(11) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

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

ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]
POL(ACTIVE(x1)) = 0
POL(CONS(x1, x2)) = 0
POL(DBL(x1)) = 0
POL(FIRST(x1, x2)) = 0
POL(PROPER(x1)) = 0
POL(RECIP(x1)) = 0
POL(S(x1)) = 0
POL(SQR(x1)) = 0
POL(TERMS(x1)) = 0
POL(TOP(x1)) = [2]x1
POL(active(x1)) = x1
POL(add(x1, x2)) = [5]x1 + x2
POL(c1(x1)) = x1
POL(c10(x1, x2)) = x1 + x2
POL(c11(x1, x2)) = x1 + x2
POL(c12(x1, x2)) = x1 + x2
POL(c13(x1, x2)) = x1 + x2
POL(c14(x1, x2)) = x1 + x2
POL(c15(x1, x2)) = x1 + x2
POL(c16(x1, x2)) = x1 + x2
POL(c17(x1, x2)) = x1 + x2
POL(c18(x1)) = x1
POL(c19(x1)) = x1
POL(c20(x1)) = x1
POL(c21(x1)) = x1
POL(c22(x1)) = x1
POL(c23(x1)) = x1
POL(c24(x1)) = x1
POL(c25(x1)) = x1
POL(c26(x1)) = x1
POL(c27(x1)) = x1
POL(c28(x1)) = x1
POL(c29(x1)) = x1
POL(c30(x1)) = x1
POL(c31(x1)) = x1
POL(c32(x1)) = x1
POL(c33(x1)) = x1
POL(c34(x1, x2)) = x1 + x2
POL(c35(x1, x2, x3)) = x1 + x2 + x3
POL(c36(x1, x2)) = x1 + x2
POL(c37(x1, x2)) = x1 + x2
POL(c38(x1, x2)) = x1 + x2
POL(c40(x1, x2, x3)) = x1 + x2 + x3
POL(c41(x1, x2)) = x1 + x2
POL(c42(x1, x2, x3)) = x1 + x2 + x3
POL(c44(x1)) = x1
POL(c45(x1, x2)) = x1 + x2
POL(c46(x1, x2)) = x1 + x2
POL(c9(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = [2] + x1
POL(dbl(x1)) = [3]x1
POL(first(x1, x2)) = [2]x1 + [4]x2
POL(mark(x1)) = [2] + x1
POL(nil) = 0
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(recip(x1)) = x1
POL(s(x1)) = [2]
POL(sqr(x1)) = [3]x1
POL(terms(x1)) = [4] + [4]x1

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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))
S tuples:

ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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))
K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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, c40, c41, c42, c44, c45, c46, c1

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

Use narrowing to replace ACTIVE(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0)) by

ACTIVE(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
S tuples:

ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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, c40, c41, c42, c44, c45, c46, c1, c9

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
S tuples:

ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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, c40, c41, c42, c44, c45, c46, c1, c9, c10

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

Use narrowing to replace ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0)) by

ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
S tuples:

ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

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, c40, c41, c42, c44, c45, c46, c1, c9, c10, c11

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

Use narrowing to replace ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0)) by

ACTIVE(sqr(terms(z0))) → c12(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c12(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(dbl(0))) → c12(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c12(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(first(0, z0))) → c12(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c12(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(terms(z0))) → c12(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c12(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c12(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c12(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(dbl(z0))) → c12(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(terms(z0))) → c12(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c12(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(dbl(0))) → c12(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c12(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(first(0, z0))) → c12(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c12(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(terms(z0))) → c12(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c12(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c12(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c12(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(dbl(z0))) → c12(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
S tuples:

ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(terms(z0))) → c12(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c12(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(dbl(0))) → c12(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c12(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(first(0, z0))) → c12(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c12(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(terms(z0))) → c12(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c12(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c12(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c12(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(dbl(z0))) → c12(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, ADD, DBL, FIRST, PROPER, S, 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, c40, c41, c42, c44, c45, c46, c1, c9, c10, c11, c12

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

Removed 4 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(terms(z0))) → c12(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(dbl(0))) → c12(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c12(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(first(0, z0))) → c12(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c12(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(terms(z0))) → c12(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c12(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c12(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c12(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(dbl(z0))) → c12(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c12(SQR(mark(0)))
S tuples:

ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c10(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(terms(z0))) → c12(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(dbl(0))) → c12(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c12(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(first(0, z0))) → c12(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c12(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(terms(z0))) → c12(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c12(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c12(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c12(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(dbl(z0))) → c12(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c12(SQR(mark(0)))
K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, ADD, DBL, FIRST, PROPER, S, 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, c40, c41, c42, c44, c45, c46, c1, c9, c10, c11, c12, c9, c10, c11, c12

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

Split RHS of tuples not part of any SCC

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(terms(z0))) → c12(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(dbl(0))) → c12(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c12(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(first(0, z0))) → c12(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c12(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(terms(z0))) → c12(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c12(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c12(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c12(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(dbl(z0))) → c12(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c12(SQR(mark(0)))
ACTIVE(terms(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
ACTIVE(cons(sqr(s(z0)), x1)) → c(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1))
ACTIVE(cons(sqr(s(z0)), x1)) → c(ACTIVE(sqr(s(z0))))
ACTIVE(recip(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
ACTIVE(sqr(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
S tuples:

ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(terms(z0))) → c12(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(dbl(0))) → c12(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c12(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(first(0, z0))) → c12(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c12(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(terms(z0))) → c12(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c12(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c12(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c12(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(dbl(z0))) → c12(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c12(SQR(mark(0)))
ACTIVE(terms(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
ACTIVE(cons(sqr(s(z0)), x1)) → c(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1))
ACTIVE(cons(sqr(s(z0)), x1)) → c(ACTIVE(sqr(s(z0))))
ACTIVE(recip(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
ACTIVE(sqr(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVE, TERMS, CONS, RECIP, SQR, ADD, DBL, FIRST, PROPER, S, 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, c40, c41, c42, c44, c45, c46, c1, c9, c10, c11, c12, c9, c10, c11, c12, c

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

Use narrowing to replace ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0)) by

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

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(dbl(s(z0))) → c1(S(s(dbl(z0))))
ACTIVE(dbl(s(z0))) → c1(S(dbl(z0)))
ACTIVE(dbl(s(z0))) → c1(DBL(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(terms(terms(z0))) → c9(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(dbl(0))) → c9(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c9(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(first(0, z0))) → c9(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c9(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(terms(z0))) → c9(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c9(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c9(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c9(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(dbl(z0))) → c9(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c9(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(dbl(0), x1)) → c10(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c10(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(first(0, z0), x1)) → c10(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c10(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(terms(z0), x1)) → c10(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c10(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c10(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c10(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(dbl(z0), x1)) → c10(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c10(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(recip(terms(z0))) → c11(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(dbl(0))) → c11(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c11(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(first(0, z0))) → c11(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c11(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(terms(z0))) → c11(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c11(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c11(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c11(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(dbl(z0))) → c11(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c11(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(terms(z0))) → c12(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(dbl(0))) → c12(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c12(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(first(0, z0))) → c12(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c12(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(terms(z0))) → c12(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c12(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c12(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c12(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(dbl(z0))) → c12(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c12(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(sqr(0))) → c9(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c10(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c11(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c12(SQR(mark(0)))
ACTIVE(terms(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
ACTIVE(cons(sqr(s(z0)), x1)) → c(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1))
ACTIVE(cons(sqr(s(z0)), x1)) → c(ACTIVE(sqr(s(z0))))
ACTIVE(recip(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
ACTIVE(sqr(sqr(s(z0)))) → c(ACTIVE(sqr(s(z0))))
ACTIVE(dbl(terms(z0))) → c15(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(0))) → c15(DBL(mark(0)), ACTIVE(sqr(0)))
ACTIVE(dbl(dbl(0))) → c15(DBL(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbl(dbl(s(z0)))) → c15(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(first(0, z0))) → c15(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c15(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(terms(z0))) → c15(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c15(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c15(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c15(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))