(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) → mark(0)
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(X)))) → mark(s(half(X)))
active(half(dbl(X))) → mark(X)
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(s(X)) → s(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(half(X)) → half(active(X))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
s(mark(X)) → mark(s(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
half(mark(X)) → mark(half(X))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(half(X)) → half(proper(X))
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
half(ok(X)) → ok(half(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(0)) → c1
ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ACTIVE(dbl(0)) → c3
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(add(0, z0)) → c5
ACTIVE(add(s(z0), z1)) → c6(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(first(0, z0)) → c7
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(half(0)) → c9
ACTIVE(half(s(0))) → c10
ACTIVE(half(s(s(z0)))) → c11(S(half(z0)), HALF(z0))
ACTIVE(half(dbl(z0))) → c12
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(0) → c49
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c53
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(0)) → c1
ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ACTIVE(dbl(0)) → c3
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(add(0, z0)) → c5
ACTIVE(add(s(z0), z1)) → c6(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(first(0, z0)) → c7
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(half(0)) → c9
ACTIVE(half(s(0))) → c10
ACTIVE(half(s(s(z0)))) → c11(S(half(z0)), HALF(z0))
ACTIVE(half(dbl(z0))) → c12
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(0) → c49
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c53
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, terms, cons, recip, sqr, s, add, dbl, first, half, proper, top
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 9 trailing nodes:
ACTIVE(half(s(0))) → c10
PROPER(0) → c49
ACTIVE(sqr(0)) → c1
ACTIVE(add(0, z0)) → c5
PROPER(nil) → c53
ACTIVE(dbl(0)) → c3
ACTIVE(half(0)) → c9
ACTIVE(half(dbl(z0))) → c12
ACTIVE(first(0, z0)) → c7
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(half(s(s(z0)))) → c11(S(half(z0)), HALF(z0))
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
S tuples:
ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(S(add(z0, z1)), ADD(z0, z1))
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(half(s(s(z0)))) → c11(S(half(z0)), HALF(z0))
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:
active, terms, cons, recip, sqr, s, add, dbl, first, half, proper, top
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c, c2, c4, c6, c8, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 9 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(z0))) → c2(SQR(z0), DBL(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
S tuples:
ACTIVE(first(s(z0), cons(z1, z2))) → c8(CONS(z1, first(z0, z2)), FIRST(z0, z2))
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(z0))) → c2(SQR(z0), DBL(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
K tuples:none
Defined Rule Symbols:
active, terms, cons, recip, sqr, s, add, dbl, first, half, proper, top
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c8, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c, c2, c4, c6, c11
(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)
Split RHS of tuples not part of any SCC
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
first(mark(z0), z1) → mark(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
S tuples:
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
K tuples:none
Defined Rule Symbols:
active, terms, cons, recip, sqr, s, add, dbl, first, half, proper, top
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1
(9) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
S tuples:
ACTIVE(terms(z0)) → c13(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1
(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
terms(
z0)) →
c13(
TERMS(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
S tuples:
ACTIVE(cons(z0, z1)) → c14(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
cons(
z0,
z1)) →
c14(
CONS(
active(
z0),
z1),
ACTIVE(
z0)) by
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
S tuples:
ACTIVE(recip(z0)) → c15(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14
(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
recip(
z0)) →
c15(
RECIP(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
S tuples:
ACTIVE(sqr(z0)) → c16(SQR(active(z0)), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
sqr(
z0)) →
c16(
SQR(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
S tuples:
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)), ACTIVE(sqr(0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(sqr(0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)), ACTIVE(sqr(0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)), ACTIVE(sqr(0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16
(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
S tuples:
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16
(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
s(
z0)) →
c17(
S(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)), ACTIVE(sqr(0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
(22) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)), ACTIVE(sqr(0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)), ACTIVE(sqr(0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17
(23) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(24) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c20(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17
(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
dbl(
z0)) →
c20(
DBL(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)), ACTIVE(sqr(0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)), ACTIVE(half(0)))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
(26) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)), ACTIVE(sqr(0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)), ACTIVE(half(0)))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)), ACTIVE(dbl(0)))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(dbl(0)))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)), ACTIVE(dbl(0)))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)), ACTIVE(dbl(0)))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(0))) → c17(S(mark(0)), ACTIVE(dbl(0)))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)), ACTIVE(sqr(0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)), ACTIVE(dbl(0)))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)), ACTIVE(half(0)))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20
(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 7 trailing tuple parts
(28) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)), ACTIVE(half(0)))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(half(z0)) → c23(HALF(active(z0)), ACTIVE(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)), ACTIVE(half(0)))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20
(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
ACTIVE(
half(
z0)) →
c23(
HALF(
active(
z0)),
ACTIVE(
z0)) by
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)), ACTIVE(sqr(0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)), ACTIVE(dbl(0)))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(0))) → c23(HALF(mark(0)), ACTIVE(half(0)))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
(30) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)), ACTIVE(half(0)))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)), ACTIVE(sqr(0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)), ACTIVE(dbl(0)))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(0))) → c23(HALF(mark(0)), ACTIVE(half(0)))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)), ACTIVE(half(0)))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(0)))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)), ACTIVE(half(0)))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)), ACTIVE(half(0)))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(0))) → c17(S(mark(0)), ACTIVE(half(0)))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)), ACTIVE(half(0)))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)), ACTIVE(sqr(0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)), ACTIVE(dbl(0)))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(0))) → c23(HALF(mark(0)), ACTIVE(half(0)))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23
(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 9 trailing tuple parts
(32) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(terms(z0)) → c44(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23
(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
terms(
z0)) →
c44(
TERMS(
proper(
z0)),
PROPER(
z0)) by
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)), PROPER(0))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)), PROPER(nil))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
(34) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)), PROPER(0))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)), PROPER(nil))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)), PROPER(0))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)), PROPER(nil))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44
(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(36) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(cons(z0, z1)) → c45(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c45, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44
(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
cons(
z0,
z1)) →
c45(
CONS(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
(38) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45
(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(40) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(recip(z0)) → c46(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c46, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45
(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
recip(
z0)) →
c46(
RECIP(
proper(
z0)),
PROPER(
z0)) by
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)), PROPER(0))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)), PROPER(nil))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
(42) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)), PROPER(0))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)), PROPER(nil))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)), PROPER(0))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)), PROPER(nil))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46
(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(44) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(sqr(z0)) → c47(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c47, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46
(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
sqr(
z0)) →
c47(
SQR(
proper(
z0)),
PROPER(
z0)) by
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)), PROPER(0))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)), PROPER(nil))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
(46) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)), PROPER(0))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)), PROPER(nil))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)), PROPER(0))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)), PROPER(nil))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47
(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(48) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(s(z0)) → c48(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c48, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47
(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
s(
z0)) →
c48(
S(
proper(
z0)),
PROPER(
z0)) by
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(0)) → c48(S(ok(0)), PROPER(0))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(nil)) → c48(S(ok(nil)), PROPER(nil))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
(50) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(0)) → c48(S(ok(0)), PROPER(0))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(nil)) → c48(S(ok(nil)), PROPER(nil))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(0)) → c48(S(ok(0)), PROPER(0))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(nil)) → c48(S(ok(nil)), PROPER(nil))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48
(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(52) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(add(z0, z1)) → c50(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c50, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48
(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
add(
z0,
z1)) →
c50(
ADD(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
(54) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50
(55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(56) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(dbl(z0)) → c51(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c51, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50
(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
dbl(
z0)) →
c51(
DBL(
proper(
z0)),
PROPER(
z0)) by
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)), PROPER(0))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)), PROPER(nil))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
(58) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)), PROPER(0))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)), PROPER(nil))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)), PROPER(0))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)), PROPER(nil))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51
(59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(60) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(first(z0, z1)) → c52(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c52, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51
(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
first(
z0,
z1)) →
c52(
FIRST(
proper(
z0),
proper(
z1)),
PROPER(
z0),
PROPER(
z1)) by
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
(62) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51, c52
(63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(64) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
PROPER(half(z0)) → c54(HALF(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c54, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51, c52, c52
(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
PROPER(
half(
z0)) →
c54(
HALF(
proper(
z0)),
PROPER(
z0)) by
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(0)) → c54(HALF(ok(0)), PROPER(0))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(nil)) → c54(HALF(ok(nil)), PROPER(nil))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
(66) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(0)) → c54(HALF(ok(0)), PROPER(0))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(nil)) → c54(HALF(ok(nil)), PROPER(nil))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(0)) → c54(HALF(ok(0)), PROPER(0))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(nil)) → c54(HALF(ok(nil)), PROPER(nil))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, TOP, PROPER
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51, c52, c52, c54
(67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(68) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
TOP(mark(z0)) → c55(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, TOP, PROPER
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c55, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51, c52, c52, c54, c54
(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
mark(
z0)) →
c55(
TOP(
proper(
z0)),
PROPER(
z0)) by
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c55(TOP(ok(0)), PROPER(0))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c55(TOP(ok(nil)), PROPER(nil))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
(70) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c55(TOP(ok(0)), PROPER(0))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c55(TOP(ok(nil)), PROPER(nil))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c55(TOP(ok(0)), PROPER(0))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c55(TOP(ok(nil)), PROPER(nil))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, TOP, PROPER
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51, c52, c52, c54, c54, c55
(71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(72) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
TOP(mark(0)) → c55(TOP(ok(0)))
TOP(mark(nil)) → c55(TOP(ok(nil)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
TOP(ok(z0)) → c56(TOP(active(z0)), ACTIVE(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
TOP(mark(0)) → c55(TOP(ok(0)))
TOP(mark(nil)) → c55(TOP(ok(nil)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, TOP, PROPER
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c56, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51, c52, c52, c54, c54, c55, c55
(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
TOP(
ok(
z0)) →
c56(
TOP(
active(
z0)),
ACTIVE(
z0)) by
TOP(ok(terms(z0))) → c56(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(0))) → c56(TOP(mark(0)), ACTIVE(sqr(0)))
TOP(ok(sqr(s(z0)))) → c56(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(0))) → c56(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c56(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c56(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c56(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c56(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c56(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(half(0))) → c56(TOP(mark(0)), ACTIVE(half(0)))
TOP(ok(half(s(0)))) → c56(TOP(mark(0)), ACTIVE(half(s(0))))
TOP(ok(half(s(s(z0))))) → c56(TOP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
TOP(ok(half(dbl(z0)))) → c56(TOP(mark(z0)), ACTIVE(half(dbl(z0))))
TOP(ok(terms(z0))) → c56(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c56(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c56(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c56(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(s(z0))) → c56(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(add(z0, z1))) → c56(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c56(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c56(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c56(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c56(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(half(z0))) → c56(TOP(half(active(z0))), ACTIVE(half(z0)))
(74) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
TOP(mark(0)) → c55(TOP(ok(0)))
TOP(mark(nil)) → c55(TOP(ok(nil)))
TOP(ok(terms(z0))) → c56(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(0))) → c56(TOP(mark(0)), ACTIVE(sqr(0)))
TOP(ok(sqr(s(z0)))) → c56(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(0))) → c56(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c56(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c56(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c56(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c56(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c56(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(half(0))) → c56(TOP(mark(0)), ACTIVE(half(0)))
TOP(ok(half(s(0)))) → c56(TOP(mark(0)), ACTIVE(half(s(0))))
TOP(ok(half(s(s(z0))))) → c56(TOP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
TOP(ok(half(dbl(z0)))) → c56(TOP(mark(z0)), ACTIVE(half(dbl(z0))))
TOP(ok(terms(z0))) → c56(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c56(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c56(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c56(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(s(z0))) → c56(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(add(z0, z1))) → c56(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c56(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c56(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c56(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c56(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(half(z0))) → c56(TOP(half(active(z0))), ACTIVE(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
TOP(mark(0)) → c55(TOP(ok(0)))
TOP(mark(nil)) → c55(TOP(ok(nil)))
TOP(ok(terms(z0))) → c56(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(0))) → c56(TOP(mark(0)), ACTIVE(sqr(0)))
TOP(ok(sqr(s(z0)))) → c56(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(0))) → c56(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c56(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c56(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c56(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c56(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c56(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(half(0))) → c56(TOP(mark(0)), ACTIVE(half(0)))
TOP(ok(half(s(0)))) → c56(TOP(mark(0)), ACTIVE(half(s(0))))
TOP(ok(half(s(s(z0))))) → c56(TOP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
TOP(ok(half(dbl(z0)))) → c56(TOP(mark(z0)), ACTIVE(half(dbl(z0))))
TOP(ok(terms(z0))) → c56(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c56(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c56(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c56(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(s(z0))) → c56(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(add(z0, z1))) → c56(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c56(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c56(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c56(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c56(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(half(z0))) → c56(TOP(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51, c52, c52, c54, c54, c55, c55, c56
(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
TOP(mark(nil)) → c55(TOP(ok(nil)))
TOP(mark(0)) → c55(TOP(ok(0)))
TOP(ok(dbl(0))) → c56(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(sqr(0))) → c56(TOP(mark(0)), ACTIVE(sqr(0)))
TOP(ok(half(0))) → c56(TOP(mark(0)), ACTIVE(half(0)))
(76) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
TOP(ok(terms(z0))) → c56(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(s(z0)))) → c56(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(s(z0)))) → c56(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c56(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c56(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c56(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c56(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(half(s(0)))) → c56(TOP(mark(0)), ACTIVE(half(s(0))))
TOP(ok(half(s(s(z0))))) → c56(TOP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
TOP(ok(half(dbl(z0)))) → c56(TOP(mark(z0)), ACTIVE(half(dbl(z0))))
TOP(ok(terms(z0))) → c56(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c56(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c56(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c56(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(s(z0))) → c56(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(add(z0, z1))) → c56(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c56(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c56(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c56(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c56(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(half(z0))) → c56(TOP(half(active(z0))), ACTIVE(half(z0)))
S tuples:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
TOP(ok(terms(z0))) → c56(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(s(z0)))) → c56(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(s(z0)))) → c56(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c56(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c56(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c56(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c56(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(half(s(0)))) → c56(TOP(mark(0)), ACTIVE(half(s(0))))
TOP(ok(half(s(s(z0))))) → c56(TOP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
TOP(ok(half(dbl(z0)))) → c56(TOP(mark(z0)), ACTIVE(half(dbl(z0))))
TOP(ok(terms(z0))) → c56(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c56(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c56(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c56(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(s(z0))) → c56(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(add(z0, z1))) → c56(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c56(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c56(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c56(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c56(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(half(z0))) → c56(TOP(half(active(z0))), ACTIVE(half(z0)))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
ACTIVE, TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF, PROPER, TOP
Compound Symbols:
c18, c19, c21, c22, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c4, c6, c11, c1, c13, c14, c15, c16, c13, c14, c15, c16, c17, c17, c20, c20, c23, c23, c44, c44, c45, c45, c46, c46, c47, c47, c48, c48, c50, c50, c51, c51, c52, c52, c54, c54, c55, c56
(77) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(add(z0, z1)) → c18(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c19(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(first(z0, z1)) → c21(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c22(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(s(z0))) → c4(DBL(z0))
ACTIVE(add(s(z0), z1)) → c6(ADD(z0, z1))
ACTIVE(half(s(s(z0)))) → c11(HALF(z0))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(CONS(z1, first(z0, z2)))
ACTIVE(first(s(z0), cons(z1, z2))) → c1(FIRST(z0, z2))
ACTIVE(terms(z0)) → c1(SQR(z0))
ACTIVE(terms(z0)) → c1(S(z0))
ACTIVE(sqr(s(z0))) → c1(SQR(z0))
ACTIVE(sqr(s(z0))) → c1(DBL(z0))
ACTIVE(terms(terms(z0))) → c13(TERMS(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(terms(sqr(s(z0)))) → c13(TERMS(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(terms(dbl(s(z0)))) → c13(TERMS(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(terms(add(0, z0))) → c13(TERMS(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(terms(add(s(z0), z1))) → c13(TERMS(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(terms(first(0, z0))) → c13(TERMS(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(terms(first(s(z0), cons(z1, z2)))) → c13(TERMS(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(terms(half(s(0)))) → c13(TERMS(mark(0)), ACTIVE(half(s(0))))
ACTIVE(terms(half(s(s(z0))))) → c13(TERMS(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(terms(half(dbl(z0)))) → c13(TERMS(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(terms(terms(z0))) → c13(TERMS(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(terms(cons(z0, z1))) → c13(TERMS(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(terms(recip(z0))) → c13(TERMS(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(terms(sqr(z0))) → c13(TERMS(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(terms(s(z0))) → c13(TERMS(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(terms(add(z0, z1))) → c13(TERMS(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(terms(dbl(z0))) → c13(TERMS(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(terms(first(z0, z1))) → c13(TERMS(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(terms(half(z0))) → c13(TERMS(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(mark(cons(recip(sqr(z0)), terms(s(z0)))), x1), ACTIVE(terms(z0)))
ACTIVE(cons(sqr(s(z0)), x1)) → c14(CONS(mark(s(add(sqr(z0), dbl(z0)))), x1), ACTIVE(sqr(s(z0))))
ACTIVE(cons(dbl(s(z0)), x1)) → c14(CONS(mark(s(s(dbl(z0)))), x1), ACTIVE(dbl(s(z0))))
ACTIVE(cons(add(0, z0), x1)) → c14(CONS(mark(z0), x1), ACTIVE(add(0, z0)))
ACTIVE(cons(add(s(z0), z1), x1)) → c14(CONS(mark(s(add(z0, z1))), x1), ACTIVE(add(s(z0), z1)))
ACTIVE(cons(first(0, z0), x1)) → c14(CONS(mark(nil), x1), ACTIVE(first(0, z0)))
ACTIVE(cons(first(s(z0), cons(z1, z2)), x1)) → c14(CONS(mark(cons(z1, first(z0, z2))), x1), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(cons(half(s(0)), x1)) → c14(CONS(mark(0), x1), ACTIVE(half(s(0))))
ACTIVE(cons(half(s(s(z0))), x1)) → c14(CONS(mark(s(half(z0))), x1), ACTIVE(half(s(s(z0)))))
ACTIVE(cons(half(dbl(z0)), x1)) → c14(CONS(mark(z0), x1), ACTIVE(half(dbl(z0))))
ACTIVE(cons(terms(z0), x1)) → c14(CONS(terms(active(z0)), x1), ACTIVE(terms(z0)))
ACTIVE(cons(cons(z0, z1), x1)) → c14(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1)))
ACTIVE(cons(recip(z0), x1)) → c14(CONS(recip(active(z0)), x1), ACTIVE(recip(z0)))
ACTIVE(cons(sqr(z0), x1)) → c14(CONS(sqr(active(z0)), x1), ACTIVE(sqr(z0)))
ACTIVE(cons(s(z0), x1)) → c14(CONS(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(active(z0), z1), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(add(z0, z1), x1)) → c14(CONS(add(z0, active(z1)), x1), ACTIVE(add(z0, z1)))
ACTIVE(cons(dbl(z0), x1)) → c14(CONS(dbl(active(z0)), x1), ACTIVE(dbl(z0)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(active(z0), z1), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(first(z0, z1), x1)) → c14(CONS(first(z0, active(z1)), x1), ACTIVE(first(z0, z1)))
ACTIVE(cons(half(z0), x1)) → c14(CONS(half(active(z0)), x1), ACTIVE(half(z0)))
ACTIVE(recip(terms(z0))) → c15(RECIP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(recip(sqr(s(z0)))) → c15(RECIP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(recip(dbl(s(z0)))) → c15(RECIP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(recip(add(0, z0))) → c15(RECIP(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(recip(add(s(z0), z1))) → c15(RECIP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(recip(first(0, z0))) → c15(RECIP(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(recip(first(s(z0), cons(z1, z2)))) → c15(RECIP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(recip(half(s(0)))) → c15(RECIP(mark(0)), ACTIVE(half(s(0))))
ACTIVE(recip(half(s(s(z0))))) → c15(RECIP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(recip(half(dbl(z0)))) → c15(RECIP(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(recip(terms(z0))) → c15(RECIP(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(recip(cons(z0, z1))) → c15(RECIP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(recip(recip(z0))) → c15(RECIP(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(recip(sqr(z0))) → c15(RECIP(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(recip(s(z0))) → c15(RECIP(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(recip(add(z0, z1))) → c15(RECIP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(recip(dbl(z0))) → c15(RECIP(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(recip(first(z0, z1))) → c15(RECIP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(recip(half(z0))) → c15(RECIP(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(sqr(terms(z0))) → c16(SQR(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(sqr(sqr(s(z0)))) → c16(SQR(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(sqr(dbl(s(z0)))) → c16(SQR(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(sqr(add(0, z0))) → c16(SQR(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(sqr(add(s(z0), z1))) → c16(SQR(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(sqr(first(0, z0))) → c16(SQR(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(sqr(first(s(z0), cons(z1, z2)))) → c16(SQR(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(sqr(half(s(0)))) → c16(SQR(mark(0)), ACTIVE(half(s(0))))
ACTIVE(sqr(half(s(s(z0))))) → c16(SQR(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(sqr(half(dbl(z0)))) → c16(SQR(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(sqr(terms(z0))) → c16(SQR(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(sqr(cons(z0, z1))) → c16(SQR(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(sqr(recip(z0))) → c16(SQR(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(sqr(sqr(z0))) → c16(SQR(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(sqr(s(z0))) → c16(SQR(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(sqr(add(z0, z1))) → c16(SQR(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(sqr(dbl(z0))) → c16(SQR(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(sqr(first(z0, z1))) → c16(SQR(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(sqr(half(z0))) → c16(SQR(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(sqr(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(sqr(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(sqr(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(sqr(0))) → c16(SQR(mark(0)))
ACTIVE(s(terms(z0))) → c17(S(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(s(sqr(s(z0)))) → c17(S(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(s(dbl(s(z0)))) → c17(S(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(s(add(0, z0))) → c17(S(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(s(add(s(z0), z1))) → c17(S(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(s(first(0, z0))) → c17(S(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(s(first(s(z0), cons(z1, z2)))) → c17(S(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(s(half(s(0)))) → c17(S(mark(0)), ACTIVE(half(s(0))))
ACTIVE(s(half(s(s(z0))))) → c17(S(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(s(half(dbl(z0)))) → c17(S(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(s(terms(z0))) → c17(S(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(s(cons(z0, z1))) → c17(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(s(recip(z0))) → c17(S(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(s(sqr(z0))) → c17(S(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(add(z0, z1))) → c17(S(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(s(add(z0, z1))) → c17(S(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(s(dbl(z0))) → c17(S(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(s(first(z0, z1))) → c17(S(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(s(first(z0, z1))) → c17(S(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(s(half(z0))) → c17(S(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(s(sqr(0))) → c17(S(mark(0)))
ACTIVE(dbl(terms(z0))) → c20(DBL(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(dbl(sqr(s(z0)))) → c20(DBL(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(dbl(dbl(s(z0)))) → c20(DBL(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(dbl(add(0, z0))) → c20(DBL(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(dbl(add(s(z0), z1))) → c20(DBL(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(dbl(first(0, z0))) → c20(DBL(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(dbl(first(s(z0), cons(z1, z2)))) → c20(DBL(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(dbl(half(s(0)))) → c20(DBL(mark(0)), ACTIVE(half(s(0))))
ACTIVE(dbl(half(s(s(z0))))) → c20(DBL(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(dbl(half(dbl(z0)))) → c20(DBL(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(dbl(terms(z0))) → c20(DBL(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(dbl(cons(z0, z1))) → c20(DBL(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(dbl(recip(z0))) → c20(DBL(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(dbl(sqr(z0))) → c20(DBL(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(dbl(s(z0))) → c20(DBL(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(dbl(add(z0, z1))) → c20(DBL(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(dbl(dbl(z0))) → c20(DBL(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(dbl(first(z0, z1))) → c20(DBL(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(dbl(half(z0))) → c20(DBL(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(dbl(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(dbl(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(dbl(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(dbl(0))) → c16(SQR(mark(0)))
ACTIVE(s(dbl(0))) → c17(S(mark(0)))
ACTIVE(dbl(sqr(0))) → c20(DBL(mark(0)))
ACTIVE(dbl(dbl(0))) → c20(DBL(mark(0)))
ACTIVE(half(terms(z0))) → c23(HALF(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
ACTIVE(half(sqr(s(z0)))) → c23(HALF(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
ACTIVE(half(dbl(s(z0)))) → c23(HALF(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
ACTIVE(half(add(0, z0))) → c23(HALF(mark(z0)), ACTIVE(add(0, z0)))
ACTIVE(half(add(s(z0), z1))) → c23(HALF(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
ACTIVE(half(first(0, z0))) → c23(HALF(mark(nil)), ACTIVE(first(0, z0)))
ACTIVE(half(first(s(z0), cons(z1, z2)))) → c23(HALF(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
ACTIVE(half(half(s(0)))) → c23(HALF(mark(0)), ACTIVE(half(s(0))))
ACTIVE(half(half(s(s(z0))))) → c23(HALF(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
ACTIVE(half(half(dbl(z0)))) → c23(HALF(mark(z0)), ACTIVE(half(dbl(z0))))
ACTIVE(half(terms(z0))) → c23(HALF(terms(active(z0))), ACTIVE(terms(z0)))
ACTIVE(half(cons(z0, z1))) → c23(HALF(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
ACTIVE(half(recip(z0))) → c23(HALF(recip(active(z0))), ACTIVE(recip(z0)))
ACTIVE(half(sqr(z0))) → c23(HALF(sqr(active(z0))), ACTIVE(sqr(z0)))
ACTIVE(half(s(z0))) → c23(HALF(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(active(z0), z1)), ACTIVE(add(z0, z1)))
ACTIVE(half(add(z0, z1))) → c23(HALF(add(z0, active(z1))), ACTIVE(add(z0, z1)))
ACTIVE(half(dbl(z0))) → c23(HALF(dbl(active(z0))), ACTIVE(dbl(z0)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(active(z0), z1)), ACTIVE(first(z0, z1)))
ACTIVE(half(first(z0, z1))) → c23(HALF(first(z0, active(z1))), ACTIVE(first(z0, z1)))
ACTIVE(half(half(z0))) → c23(HALF(half(active(z0))), ACTIVE(half(z0)))
ACTIVE(terms(half(0))) → c13(TERMS(mark(0)))
ACTIVE(cons(half(0), x1)) → c14(CONS(mark(0), x1))
ACTIVE(recip(half(0))) → c15(RECIP(mark(0)))
ACTIVE(sqr(half(0))) → c16(SQR(mark(0)))
ACTIVE(s(half(0))) → c17(S(mark(0)))
ACTIVE(dbl(half(0))) → c20(DBL(mark(0)))
ACTIVE(half(sqr(0))) → c23(HALF(mark(0)))
ACTIVE(half(dbl(0))) → c23(HALF(mark(0)))
ACTIVE(half(half(0))) → c23(HALF(mark(0)))
PROPER(terms(terms(z0))) → c44(TERMS(terms(proper(z0))), PROPER(terms(z0)))
PROPER(terms(cons(z0, z1))) → c44(TERMS(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(terms(recip(z0))) → c44(TERMS(recip(proper(z0))), PROPER(recip(z0)))
PROPER(terms(sqr(z0))) → c44(TERMS(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(terms(s(z0))) → c44(TERMS(s(proper(z0))), PROPER(s(z0)))
PROPER(terms(add(z0, z1))) → c44(TERMS(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(terms(dbl(z0))) → c44(TERMS(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(terms(first(z0, z1))) → c44(TERMS(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(terms(half(z0))) → c44(TERMS(half(proper(z0))), PROPER(half(z0)))
PROPER(terms(0)) → c44(TERMS(ok(0)))
PROPER(terms(nil)) → c44(TERMS(ok(nil)))
PROPER(cons(x0, terms(z0))) → c45(CONS(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(cons(x0, cons(z0, z1))) → c45(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(cons(x0, recip(z0))) → c45(CONS(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(cons(x0, sqr(z0))) → c45(CONS(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(cons(x0, s(z0))) → c45(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(cons(x0, add(z0, z1))) → c45(CONS(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(cons(x0, dbl(z0))) → c45(CONS(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(cons(x0, first(z0, z1))) → c45(CONS(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(cons(x0, half(z0))) → c45(CONS(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(cons(terms(z0), x1)) → c45(CONS(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(cons(cons(z0, z1), x1)) → c45(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(cons(recip(z0), x1)) → c45(CONS(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(cons(sqr(z0), x1)) → c45(CONS(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(cons(s(z0), x1)) → c45(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(cons(add(z0, z1), x1)) → c45(CONS(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(cons(dbl(z0), x1)) → c45(CONS(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(cons(first(z0, z1), x1)) → c45(CONS(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(cons(half(z0), x1)) → c45(CONS(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(cons(x0, 0)) → c45(CONS(proper(x0), ok(0)), PROPER(x0))
PROPER(cons(x0, nil)) → c45(CONS(proper(x0), ok(nil)), PROPER(x0))
PROPER(cons(0, x1)) → c45(CONS(ok(0), proper(x1)), PROPER(x1))
PROPER(cons(nil, x1)) → c45(CONS(ok(nil), proper(x1)), PROPER(x1))
PROPER(recip(terms(z0))) → c46(RECIP(terms(proper(z0))), PROPER(terms(z0)))
PROPER(recip(cons(z0, z1))) → c46(RECIP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(recip(recip(z0))) → c46(RECIP(recip(proper(z0))), PROPER(recip(z0)))
PROPER(recip(sqr(z0))) → c46(RECIP(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(recip(s(z0))) → c46(RECIP(s(proper(z0))), PROPER(s(z0)))
PROPER(recip(add(z0, z1))) → c46(RECIP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(recip(dbl(z0))) → c46(RECIP(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(recip(first(z0, z1))) → c46(RECIP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(recip(half(z0))) → c46(RECIP(half(proper(z0))), PROPER(half(z0)))
PROPER(recip(0)) → c46(RECIP(ok(0)))
PROPER(recip(nil)) → c46(RECIP(ok(nil)))
PROPER(sqr(terms(z0))) → c47(SQR(terms(proper(z0))), PROPER(terms(z0)))
PROPER(sqr(cons(z0, z1))) → c47(SQR(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(sqr(recip(z0))) → c47(SQR(recip(proper(z0))), PROPER(recip(z0)))
PROPER(sqr(sqr(z0))) → c47(SQR(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(sqr(s(z0))) → c47(SQR(s(proper(z0))), PROPER(s(z0)))
PROPER(sqr(add(z0, z1))) → c47(SQR(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(sqr(dbl(z0))) → c47(SQR(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(sqr(first(z0, z1))) → c47(SQR(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(sqr(half(z0))) → c47(SQR(half(proper(z0))), PROPER(half(z0)))
PROPER(sqr(0)) → c47(SQR(ok(0)))
PROPER(sqr(nil)) → c47(SQR(ok(nil)))
PROPER(s(terms(z0))) → c48(S(terms(proper(z0))), PROPER(terms(z0)))
PROPER(s(cons(z0, z1))) → c48(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(s(recip(z0))) → c48(S(recip(proper(z0))), PROPER(recip(z0)))
PROPER(s(sqr(z0))) → c48(S(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(s(s(z0))) → c48(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(add(z0, z1))) → c48(S(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(s(dbl(z0))) → c48(S(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(s(first(z0, z1))) → c48(S(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(s(half(z0))) → c48(S(half(proper(z0))), PROPER(half(z0)))
PROPER(s(0)) → c48(S(ok(0)))
PROPER(s(nil)) → c48(S(ok(nil)))
PROPER(add(x0, terms(z0))) → c50(ADD(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(add(x0, cons(z0, z1))) → c50(ADD(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(add(x0, recip(z0))) → c50(ADD(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(add(x0, sqr(z0))) → c50(ADD(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(add(x0, s(z0))) → c50(ADD(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(add(x0, add(z0, z1))) → c50(ADD(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(add(x0, dbl(z0))) → c50(ADD(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(add(x0, first(z0, z1))) → c50(ADD(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(add(x0, half(z0))) → c50(ADD(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(add(terms(z0), x1)) → c50(ADD(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(add(cons(z0, z1), x1)) → c50(ADD(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(add(recip(z0), x1)) → c50(ADD(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(add(sqr(z0), x1)) → c50(ADD(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(add(s(z0), x1)) → c50(ADD(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(add(add(z0, z1), x1)) → c50(ADD(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(add(dbl(z0), x1)) → c50(ADD(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(add(first(z0, z1), x1)) → c50(ADD(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(add(half(z0), x1)) → c50(ADD(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(add(x0, 0)) → c50(ADD(proper(x0), ok(0)), PROPER(x0))
PROPER(add(x0, nil)) → c50(ADD(proper(x0), ok(nil)), PROPER(x0))
PROPER(add(0, x1)) → c50(ADD(ok(0), proper(x1)), PROPER(x1))
PROPER(add(nil, x1)) → c50(ADD(ok(nil), proper(x1)), PROPER(x1))
PROPER(dbl(terms(z0))) → c51(DBL(terms(proper(z0))), PROPER(terms(z0)))
PROPER(dbl(cons(z0, z1))) → c51(DBL(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(dbl(recip(z0))) → c51(DBL(recip(proper(z0))), PROPER(recip(z0)))
PROPER(dbl(sqr(z0))) → c51(DBL(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(dbl(s(z0))) → c51(DBL(s(proper(z0))), PROPER(s(z0)))
PROPER(dbl(add(z0, z1))) → c51(DBL(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(dbl(dbl(z0))) → c51(DBL(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(dbl(first(z0, z1))) → c51(DBL(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(dbl(half(z0))) → c51(DBL(half(proper(z0))), PROPER(half(z0)))
PROPER(dbl(0)) → c51(DBL(ok(0)))
PROPER(dbl(nil)) → c51(DBL(ok(nil)))
PROPER(first(x0, terms(z0))) → c52(FIRST(proper(x0), terms(proper(z0))), PROPER(x0), PROPER(terms(z0)))
PROPER(first(x0, cons(z0, z1))) → c52(FIRST(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1)))
PROPER(first(x0, recip(z0))) → c52(FIRST(proper(x0), recip(proper(z0))), PROPER(x0), PROPER(recip(z0)))
PROPER(first(x0, sqr(z0))) → c52(FIRST(proper(x0), sqr(proper(z0))), PROPER(x0), PROPER(sqr(z0)))
PROPER(first(x0, s(z0))) → c52(FIRST(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(first(x0, add(z0, z1))) → c52(FIRST(proper(x0), add(proper(z0), proper(z1))), PROPER(x0), PROPER(add(z0, z1)))
PROPER(first(x0, dbl(z0))) → c52(FIRST(proper(x0), dbl(proper(z0))), PROPER(x0), PROPER(dbl(z0)))
PROPER(first(x0, first(z0, z1))) → c52(FIRST(proper(x0), first(proper(z0), proper(z1))), PROPER(x0), PROPER(first(z0, z1)))
PROPER(first(x0, half(z0))) → c52(FIRST(proper(x0), half(proper(z0))), PROPER(x0), PROPER(half(z0)))
PROPER(first(terms(z0), x1)) → c52(FIRST(terms(proper(z0)), proper(x1)), PROPER(terms(z0)), PROPER(x1))
PROPER(first(cons(z0, z1), x1)) → c52(FIRST(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1))
PROPER(first(recip(z0), x1)) → c52(FIRST(recip(proper(z0)), proper(x1)), PROPER(recip(z0)), PROPER(x1))
PROPER(first(sqr(z0), x1)) → c52(FIRST(sqr(proper(z0)), proper(x1)), PROPER(sqr(z0)), PROPER(x1))
PROPER(first(s(z0), x1)) → c52(FIRST(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(first(add(z0, z1), x1)) → c52(FIRST(add(proper(z0), proper(z1)), proper(x1)), PROPER(add(z0, z1)), PROPER(x1))
PROPER(first(dbl(z0), x1)) → c52(FIRST(dbl(proper(z0)), proper(x1)), PROPER(dbl(z0)), PROPER(x1))
PROPER(first(first(z0, z1), x1)) → c52(FIRST(first(proper(z0), proper(z1)), proper(x1)), PROPER(first(z0, z1)), PROPER(x1))
PROPER(first(half(z0), x1)) → c52(FIRST(half(proper(z0)), proper(x1)), PROPER(half(z0)), PROPER(x1))
PROPER(first(x0, 0)) → c52(FIRST(proper(x0), ok(0)), PROPER(x0))
PROPER(first(x0, nil)) → c52(FIRST(proper(x0), ok(nil)), PROPER(x0))
PROPER(first(0, x1)) → c52(FIRST(ok(0), proper(x1)), PROPER(x1))
PROPER(first(nil, x1)) → c52(FIRST(ok(nil), proper(x1)), PROPER(x1))
PROPER(half(terms(z0))) → c54(HALF(terms(proper(z0))), PROPER(terms(z0)))
PROPER(half(cons(z0, z1))) → c54(HALF(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
PROPER(half(recip(z0))) → c54(HALF(recip(proper(z0))), PROPER(recip(z0)))
PROPER(half(sqr(z0))) → c54(HALF(sqr(proper(z0))), PROPER(sqr(z0)))
PROPER(half(s(z0))) → c54(HALF(s(proper(z0))), PROPER(s(z0)))
PROPER(half(add(z0, z1))) → c54(HALF(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
PROPER(half(dbl(z0))) → c54(HALF(dbl(proper(z0))), PROPER(dbl(z0)))
PROPER(half(first(z0, z1))) → c54(HALF(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
PROPER(half(half(z0))) → c54(HALF(half(proper(z0))), PROPER(half(z0)))
PROPER(half(0)) → c54(HALF(ok(0)))
PROPER(half(nil)) → c54(HALF(ok(nil)))
TOP(mark(terms(z0))) → c55(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c55(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c55(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c55(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c55(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c55(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c55(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c55(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(half(z0))) → c55(TOP(half(proper(z0))), PROPER(half(z0)))
TOP(ok(terms(z0))) → c56(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(s(z0)))) → c56(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(s(z0)))) → c56(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c56(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c56(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c56(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c56(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(half(s(0)))) → c56(TOP(mark(0)), ACTIVE(half(s(0))))
TOP(ok(half(s(s(z0))))) → c56(TOP(mark(s(half(z0)))), ACTIVE(half(s(s(z0)))))
TOP(ok(half(dbl(z0)))) → c56(TOP(mark(z0)), ACTIVE(half(dbl(z0))))
TOP(ok(terms(z0))) → c56(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c56(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c56(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c56(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(s(z0))) → c56(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(add(z0, z1))) → c56(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c56(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c56(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c56(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c56(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(half(z0))) → c56(TOP(half(active(z0))), ACTIVE(half(z0)))
(78) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:none
Defined Rule Symbols:
active, cons, first, terms, recip, sqr, s, add, dbl, half, proper
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(79) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
active(terms(z0)) → mark(cons(recip(sqr(z0)), terms(s(z0))))
active(sqr(0)) → mark(0)
active(sqr(s(z0))) → mark(s(add(sqr(z0), dbl(z0))))
active(dbl(0)) → mark(0)
active(dbl(s(z0))) → mark(s(s(dbl(z0))))
active(add(0, z0)) → mark(z0)
active(add(s(z0), z1)) → mark(s(add(z0, z1)))
active(first(0, z0)) → mark(nil)
active(first(s(z0), cons(z1, z2))) → mark(cons(z1, first(z0, z2)))
active(half(0)) → mark(0)
active(half(s(0))) → mark(0)
active(half(s(s(z0)))) → mark(s(half(z0)))
active(half(dbl(z0))) → mark(z0)
active(terms(z0)) → terms(active(z0))
active(cons(z0, z1)) → cons(active(z0), z1)
active(recip(z0)) → recip(active(z0))
active(sqr(z0)) → sqr(active(z0))
active(s(z0)) → s(active(z0))
active(add(z0, z1)) → add(active(z0), z1)
active(add(z0, z1)) → add(z0, active(z1))
active(dbl(z0)) → dbl(active(z0))
active(first(z0, z1)) → first(active(z0), z1)
active(first(z0, z1)) → first(z0, active(z1))
active(half(z0)) → half(active(z0))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
add(mark(z0), z1) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
half(mark(z0)) → mark(half(z0))
half(ok(z0)) → ok(half(z0))
proper(terms(z0)) → terms(proper(z0))
proper(cons(z0, z1)) → cons(proper(z0), proper(z1))
proper(recip(z0)) → recip(proper(z0))
proper(sqr(z0)) → sqr(proper(z0))
proper(s(z0)) → s(proper(z0))
proper(0) → ok(0)
proper(add(z0, z1)) → add(proper(z0), proper(z1))
proper(dbl(z0)) → dbl(proper(z0))
proper(first(z0, z1)) → first(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(half(z0)) → half(proper(z0))
(80) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(81) 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.
ADD(mark(z0), z1) → c34(ADD(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = x1
POL(CONS(x1, x2)) = 0
POL(DBL(x1)) = 0
POL(FIRST(x1, x2)) = 0
POL(HALF(x1)) = 0
POL(RECIP(x1)) = 0
POL(S(x1)) = 0
POL(SQR(x1)) = 0
POL(TERMS(x1)) = 0
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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = [2] + x1
POL(ok(x1)) = x1
(82) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(83) 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.
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = 0
POL(CONS(x1, x2)) = [4]x2
POL(DBL(x1)) = 0
POL(FIRST(x1, x2)) = 0
POL(HALF(x1)) = 0
POL(RECIP(x1)) = 0
POL(S(x1)) = 0
POL(SQR(x1)) = 0
POL(TERMS(x1)) = 0
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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = 0
POL(ok(x1)) = [2] + x1
(84) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(85) 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.
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = x2
POL(CONS(x1, x2)) = [2]x2
POL(DBL(x1)) = [2]x1
POL(FIRST(x1, x2)) = 0
POL(HALF(x1)) = 0
POL(RECIP(x1)) = 0
POL(S(x1)) = x1
POL(SQR(x1)) = [2]x1
POL(TERMS(x1)) = 0
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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(86) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(87) 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.
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = 0
POL(CONS(x1, x2)) = 0
POL(DBL(x1)) = 0
POL(FIRST(x1, x2)) = x1
POL(HALF(x1)) = 0
POL(RECIP(x1)) = x1
POL(S(x1)) = 0
POL(SQR(x1)) = 0
POL(TERMS(x1)) = 0
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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = [2] + x1
POL(ok(x1)) = [1] + x1
(88) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(89) 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.
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
DBL(ok(z0)) → c38(DBL(z0))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = 0
POL(CONS(x1, x2)) = x1 + [3]x2
POL(DBL(x1)) = [2]x1
POL(FIRST(x1, x2)) = 0
POL(HALF(x1)) = 0
POL(RECIP(x1)) = [3]x1
POL(S(x1)) = [4]x1
POL(SQR(x1)) = [4]x1
POL(TERMS(x1)) = 0
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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = [2] + x1
POL(ok(x1)) = [4] + x1
(90) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
DBL(ok(z0)) → c38(DBL(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(91) 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.
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = [4]x2
POL(CONS(x1, x2)) = [2]x1 + x2
POL(DBL(x1)) = x1
POL(FIRST(x1, x2)) = 0
POL(HALF(x1)) = 0
POL(RECIP(x1)) = 0
POL(S(x1)) = 0
POL(SQR(x1)) = [3]x1
POL(TERMS(x1)) = 0
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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = [1] + x1
(92) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
DBL(ok(z0)) → c38(DBL(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(93) 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.
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = [4]x1
POL(CONS(x1, x2)) = [4]x2
POL(DBL(x1)) = [4]x1
POL(FIRST(x1, x2)) = x2
POL(HALF(x1)) = 0
POL(RECIP(x1)) = [4]x1
POL(S(x1)) = 0
POL(SQR(x1)) = 0
POL(TERMS(x1)) = 0
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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = [1] + x1
POL(ok(x1)) = x1
(94) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
DBL(ok(z0)) → c38(DBL(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(95) 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.
TERMS(ok(z0)) → c25(TERMS(z0))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = 0
POL(CONS(x1, x2)) = [2]x1
POL(DBL(x1)) = [4]x1
POL(FIRST(x1, x2)) = 0
POL(HALF(x1)) = 0
POL(RECIP(x1)) = [2]x1
POL(S(x1)) = 0
POL(SQR(x1)) = x1
POL(TERMS(x1)) = [4]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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = [1] + x1
(96) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
DBL(ok(z0)) → c38(DBL(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
TERMS(ok(z0)) → c25(TERMS(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(97) 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.
TERMS(mark(z0)) → c24(TERMS(z0))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = [2]x1 + [4]x2
POL(CONS(x1, x2)) = x2
POL(DBL(x1)) = [2]x1
POL(FIRST(x1, x2)) = 0
POL(HALF(x1)) = 0
POL(RECIP(x1)) = 0
POL(S(x1)) = [2]x1
POL(SQR(x1)) = 0
POL(TERMS(x1)) = [4]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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = [4] + x1
POL(ok(x1)) = x1
(98) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
DBL(ok(z0)) → c38(DBL(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
TERMS(ok(z0)) → c25(TERMS(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(99) 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.
HALF(mark(z0)) → c42(HALF(z0))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = [4]x1 + [2]x2
POL(CONS(x1, x2)) = x1 + [4]x2
POL(DBL(x1)) = [2]x1
POL(FIRST(x1, x2)) = [3]x1 + [2]x2
POL(HALF(x1)) = x1
POL(RECIP(x1)) = 0
POL(S(x1)) = [3]x1
POL(SQR(x1)) = [4]x1
POL(TERMS(x1)) = [3]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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = [2] + x1
POL(ok(x1)) = x1
(100) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:
HALF(ok(z0)) → c43(HALF(z0))
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
DBL(ok(z0)) → c38(DBL(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
TERMS(ok(z0)) → c25(TERMS(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
HALF(mark(z0)) → c42(HALF(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(101) 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.
HALF(ok(z0)) → c43(HALF(z0))
We considered the (Usable) Rules:none
And the Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ADD(x1, x2)) = [2]x2
POL(CONS(x1, x2)) = [2]x1 + [3]x2
POL(DBL(x1)) = [4]x1
POL(FIRST(x1, x2)) = [2]x2
POL(HALF(x1)) = x1
POL(RECIP(x1)) = 0
POL(S(x1)) = x1
POL(SQR(x1)) = 0
POL(TERMS(x1)) = 0
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)) = x1
POL(c35(x1)) = x1
POL(c36(x1)) = x1
POL(c37(x1)) = x1
POL(c38(x1)) = x1
POL(c39(x1)) = x1
POL(c40(x1)) = x1
POL(c41(x1)) = x1
POL(c42(x1)) = x1
POL(c43(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = [1] + x1
(102) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TERMS(mark(z0)) → c24(TERMS(z0))
TERMS(ok(z0)) → c25(TERMS(z0))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
SQR(mark(z0)) → c30(SQR(z0))
SQR(ok(z0)) → c31(SQR(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
ADD(mark(z0), z1) → c34(ADD(z0, z1))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
DBL(ok(z0)) → c38(DBL(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
S tuples:none
K tuples:
ADD(mark(z0), z1) → c34(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c27(CONS(z0, z1))
SQR(mark(z0)) → c30(SQR(z0))
S(mark(z0)) → c32(S(z0))
ADD(z0, mark(z1)) → c35(ADD(z0, z1))
DBL(mark(z0)) → c37(DBL(z0))
RECIP(mark(z0)) → c28(RECIP(z0))
RECIP(ok(z0)) → c29(RECIP(z0))
FIRST(mark(z0), z1) → c39(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c41(FIRST(z0, z1))
CONS(mark(z0), z1) → c26(CONS(z0, z1))
SQR(ok(z0)) → c31(SQR(z0))
S(ok(z0)) → c33(S(z0))
DBL(ok(z0)) → c38(DBL(z0))
ADD(ok(z0), ok(z1)) → c36(ADD(z0, z1))
FIRST(z0, mark(z1)) → c40(FIRST(z0, z1))
TERMS(ok(z0)) → c25(TERMS(z0))
TERMS(mark(z0)) → c24(TERMS(z0))
HALF(mark(z0)) → c42(HALF(z0))
HALF(ok(z0)) → c43(HALF(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
TERMS, CONS, RECIP, SQR, S, ADD, DBL, FIRST, HALF
Compound Symbols:
c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43
(103) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(104) BOUNDS(O(1), O(1))