Runtime Complexity (innermost) proof of /tmp/tmpOFDMCR/Ex26_Luc03b

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 88 ms)

↳2 CdtProblem

↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CdtProblem

↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 3460 ms)

↳6 CdtProblem

↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 28 ms)

↳8 CdtProblem

↳9 CdtUnreachableProof (⇔, 0 ms)

↳10 CdtProblem

↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CdtProblem

↳15 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳16 CdtProblem

↳17 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳18 CdtProblem

↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CdtProblem

↳21 CdtUnreachableProof (⇔, 0 ms)

↳22 CdtProblem

↳23 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳24 CdtProblem

↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 3 ms)

↳26 CdtProblem

↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))), 36 ms)

↳28 CdtProblem

↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))), 0 ms)

↳30 CdtProblem

↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))), 0 ms)

↳32 CdtProblem

↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 11 ms)

↳34 CdtProblem

↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 21 ms)

↳36 CdtProblem

↳37 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳38 CdtProblem

↳39 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 14 ms)

↳40 CdtProblem

↳41 SIsEmptyProof (⇔, 0 ms)

↳42 BOUNDS(O(1), O(1))

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS 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(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(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))
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))
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))
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)
s(ok(z0)) → ok(s(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(sqr(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(terms(z0)) → c(CONS(recip(sqr(z0)), terms(s(z0))), RECIP(sqr(z0)), SQR(z0), TERMS(s(z0)), S(z0))
ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c, c2, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c44, c45, c46

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

Removed 3 trailing tuple parts

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

Tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))

S tuples:

ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c2, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c44, c45, c46, c

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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(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(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))
first(z0, mark(z1)) → mark(first(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
add(z0, mark(z1)) → mark(add(z0, z1))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
sqr(mark(z0)) → mark(sqr(z0))
sqr(ok(z0)) → ok(sqr(z0))
recip(mark(z0)) → mark(recip(z0))
recip(ok(z0)) → ok(recip(z0))
cons(mark(z0), z1) → mark(cons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
terms(mark(z0)) → mark(terms(z0))
terms(ok(z0)) → ok(terms(z0))
s(ok(z0)) → ok(s(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)

And the Tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]
POL(ACTIVE(x₁)) = 0
POL(ADD(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(DBL(x₁)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(PROPER(x₁)) = 0
POL(RECIP(x₁)) = 0
POL(S(x₁)) = 0
POL(SQR(x₁)) = 0
POL(TERMS(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(add(x₁, x₂)) = [2]x₁ + x₂
POL(c(x₁, x₂)) = x₁ + x₂
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁, x₂)) = x₁ + x₂
POL(c12(x₁, x₂)) = x₁ + x₂
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂)) = x₁ + x₂
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁, x₂)) = x₁ + x₂
POL(c17(x₁, x₂)) = x₁ + x₂
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁, x₂)) = x₁ + x₂
POL(c35(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c36(x₁, x₂)) = x₁ + x₂
POL(c37(x₁, x₂)) = x₁ + x₂
POL(c38(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c40(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c41(x₁, x₂)) = x₁ + x₂
POL(c42(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c44(x₁)) = x₁
POL(c45(x₁, x₂)) = x₁ + x₂
POL(c46(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁
POL(dbl(x₁)) = [4]x₁
POL(first(x₁, x₂)) = [2]x₁ + [2]x₂
POL(mark(x₁)) = [1] + x₁
POL(nil) = [1]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(recip(x₁)) = [1] + [2]x₁
POL(s(x₁)) = [2]
POL(sqr(x₁)) = [2]x₁
POL(terms(x₁)) = [4] + [4]x₁

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

Tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))

S tuples:

ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))

K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c2, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c44, c45, c46, c

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

Use narrowing to replace ACTIVE(sqr(s(z0))) → c2(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0)) by

ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(ok(z0)))) → c2(S(add(sqr(ok(z0)), ok(dbl(z0)))), ADD(sqr(ok(z0)), dbl(ok(z0))), SQR(ok(z0)), DBL(ok(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(ok(z0)))) → c2(S(add(ok(sqr(z0)), dbl(ok(z0)))), ADD(sqr(ok(z0)), dbl(ok(z0))), SQR(ok(z0)), DBL(ok(z0)))
ACTIVE(sqr(s(x0))) → c2

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

Tuples:

ACTIVE(dbl(s(z0))) → c4(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(ok(z0)))) → c2(S(add(sqr(ok(z0)), ok(dbl(z0)))), ADD(sqr(ok(z0)), dbl(ok(z0))), SQR(ok(z0)), DBL(ok(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(ok(z0)))) → c2(S(add(ok(sqr(z0)), dbl(ok(z0)))), ADD(sqr(ok(z0)), dbl(ok(z0))), SQR(ok(z0)), DBL(ok(z0)))
ACTIVE(sqr(s(x0))) → c2

S tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(ok(z0)))) → c2(S(add(sqr(ok(z0)), ok(dbl(z0)))), ADD(sqr(ok(z0)), dbl(ok(z0))), SQR(ok(z0)), DBL(ok(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(ok(z0)))) → c2(S(add(ok(sqr(z0)), dbl(ok(z0)))), ADD(sqr(ok(z0)), dbl(ok(z0))), SQR(ok(z0)), DBL(ok(z0)))
ACTIVE(sqr(s(x0))) → c2

K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c40, c41, c42, c44, c45, c46, c, c2, c2

(9) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(sqr(s(ok(z0)))) → c2(S(add(sqr(ok(z0)), ok(dbl(z0)))), ADD(sqr(ok(z0)), dbl(ok(z0))), SQR(ok(z0)), DBL(ok(z0)))
ACTIVE(sqr(s(ok(z0)))) → c2(S(add(ok(sqr(z0)), dbl(ok(z0)))), ADD(sqr(ok(z0)), dbl(ok(z0))), SQR(ok(z0)), DBL(ok(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(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(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))
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))
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))
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)
s(ok(z0)) → ok(s(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(x0))) → c2
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

S tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(x0))) → c2

K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44, c45, c46, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c, c2, c2, c34, c35, c36, c37, c38, c40, c41, c42

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

Removed 1 trailing nodes:

ACTIVE(sqr(s(x0))) → c2

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

S tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))

K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44, c45, c46, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c, c2, c34, c35, c36, c37, c38, c40, c41, c42

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

Use narrowing to replace TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(terms(z0))) → c45(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c45(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c45(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c45(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c45(TOP(ok(0)), PROPER(0))
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c45(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c45(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c45(TOP(ok(nil)), PROPER(nil))
TOP(mark(x0)) → c45

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(terms(z0))) → c45(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c45(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c45(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c45(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c45(TOP(ok(0)), PROPER(0))
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c45(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c45(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c45(TOP(ok(nil)), PROPER(nil))
TOP(mark(x0)) → c45

S tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))

K tuples:

TOP(mark(z0)) → c45(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44, c46, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c, c2, c34, c35, c36, c37, c38, c40, c41, c42, c45, c45

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c45

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(terms(z0))) → c45(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c45(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c45(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c45(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(0)) → c45(TOP(ok(0)), PROPER(0))
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c45(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c45(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(nil)) → c45(TOP(ok(nil)), PROPER(nil))

S tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44, c46, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c, c2, c34, c35, c36, c37, c38, c40, c41, c42, c45

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

Removed 2 trailing tuple parts

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(terms(z0))) → c45(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c45(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c45(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c45(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c45(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c45(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(0)) → c45(TOP(ok(0)))
TOP(mark(nil)) → c45(TOP(ok(nil)))

S tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44, c46, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c, c2, c34, c35, c36, c37, c38, c40, c41, c42, c45, c45

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

Use narrowing to replace TOP(ok(z0)) → c46(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(terms(z0))) → c46(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(0))) → c46(TOP(mark(0)), ACTIVE(sqr(0)))
TOP(ok(sqr(s(z0)))) → c46(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(0))) → c46(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c46(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c46(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c46(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c46(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c46(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(terms(z0))) → c46(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c46(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c46(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c46(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(add(z0, z1))) → c46(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c46(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c46(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c46(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c46(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(x0)) → c46

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(terms(z0))) → c45(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c45(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c45(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c45(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c45(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c45(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(mark(0)) → c45(TOP(ok(0)))
TOP(mark(nil)) → c45(TOP(ok(nil)))
TOP(ok(terms(z0))) → c46(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(0))) → c46(TOP(mark(0)), ACTIVE(sqr(0)))
TOP(ok(sqr(s(z0)))) → c46(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(0))) → c46(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c46(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c46(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c46(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c46(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c46(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(terms(z0))) → c46(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c46(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c46(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c46(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(add(z0, z1))) → c46(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c46(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c46(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c46(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c46(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(x0)) → c46

S tuples:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
S(ok(z0)) → c44(S(z0))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
TOP(ok(terms(z0))) → c46(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(0))) → c46(TOP(mark(0)), ACTIVE(sqr(0)))
TOP(ok(sqr(s(z0)))) → c46(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(0))) → c46(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c46(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c46(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c46(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c46(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c46(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(terms(z0))) → c46(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c46(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c46(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c46(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(add(z0, z1))) → c46(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c46(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c46(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c46(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c46(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))
TOP(ok(x0)) → c46

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44, c4, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c, c2, c34, c35, c36, c37, c38, c40, c41, c42, c45, c45, c46, c46

(21) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

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(terms(z0)) → c9(TERMS(active(z0)), ACTIVE(z0))
ACTIVE(cons(z0, z1)) → c10(CONS(active(z0), z1), ACTIVE(z0))
ACTIVE(recip(z0)) → c11(RECIP(active(z0)), ACTIVE(z0))
ACTIVE(sqr(z0)) → c12(SQR(active(z0)), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c13(ADD(active(z0), z1), ACTIVE(z0))
ACTIVE(add(z0, z1)) → c14(ADD(z0, active(z1)), ACTIVE(z1))
ACTIVE(dbl(z0)) → c15(DBL(active(z0)), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c16(FIRST(active(z0), z1), ACTIVE(z0))
ACTIVE(first(z0, z1)) → c17(FIRST(z0, active(z1)), ACTIVE(z1))
ACTIVE(terms(z0)) → c(SQR(z0), S(z0))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(sqr(mark(z0)), mark(dbl(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
ACTIVE(sqr(s(mark(z0)))) → c2(S(add(mark(sqr(z0)), dbl(mark(z0)))), ADD(sqr(mark(z0)), dbl(mark(z0))), SQR(mark(z0)), DBL(mark(z0)))
PROPER(terms(z0)) → c34(TERMS(proper(z0)), PROPER(z0))
PROPER(cons(z0, z1)) → c35(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(recip(z0)) → c36(RECIP(proper(z0)), PROPER(z0))
PROPER(sqr(z0)) → c37(SQR(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c38(S(proper(z0)), PROPER(z0))
PROPER(add(z0, z1)) → c40(ADD(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(dbl(z0)) → c41(DBL(proper(z0)), PROPER(z0))
PROPER(first(z0, z1)) → c42(FIRST(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(terms(z0))) → c45(TOP(terms(proper(z0))), PROPER(terms(z0)))
TOP(mark(cons(z0, z1))) → c45(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1)))
TOP(mark(recip(z0))) → c45(TOP(recip(proper(z0))), PROPER(recip(z0)))
TOP(mark(sqr(z0))) → c45(TOP(sqr(proper(z0))), PROPER(sqr(z0)))
TOP(mark(s(z0))) → c45(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(add(z0, z1))) → c45(TOP(add(proper(z0), proper(z1))), PROPER(add(z0, z1)))
TOP(mark(dbl(z0))) → c45(TOP(dbl(proper(z0))), PROPER(dbl(z0)))
TOP(mark(first(z0, z1))) → c45(TOP(first(proper(z0), proper(z1))), PROPER(first(z0, z1)))
TOP(ok(terms(z0))) → c46(TOP(mark(cons(recip(sqr(z0)), terms(s(z0))))), ACTIVE(terms(z0)))
TOP(ok(sqr(0))) → c46(TOP(mark(0)), ACTIVE(sqr(0)))
TOP(ok(sqr(s(z0)))) → c46(TOP(mark(s(add(sqr(z0), dbl(z0))))), ACTIVE(sqr(s(z0))))
TOP(ok(dbl(0))) → c46(TOP(mark(0)), ACTIVE(dbl(0)))
TOP(ok(dbl(s(z0)))) → c46(TOP(mark(s(s(dbl(z0))))), ACTIVE(dbl(s(z0))))
TOP(ok(add(0, z0))) → c46(TOP(mark(z0)), ACTIVE(add(0, z0)))
TOP(ok(add(s(z0), z1))) → c46(TOP(mark(s(add(z0, z1)))), ACTIVE(add(s(z0), z1)))
TOP(ok(first(0, z0))) → c46(TOP(mark(nil)), ACTIVE(first(0, z0)))
TOP(ok(first(s(z0), cons(z1, z2)))) → c46(TOP(mark(cons(z1, first(z0, z2)))), ACTIVE(first(s(z0), cons(z1, z2))))
TOP(ok(terms(z0))) → c46(TOP(terms(active(z0))), ACTIVE(terms(z0)))
TOP(ok(cons(z0, z1))) → c46(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1)))
TOP(ok(recip(z0))) → c46(TOP(recip(active(z0))), ACTIVE(recip(z0)))
TOP(ok(sqr(z0))) → c46(TOP(sqr(active(z0))), ACTIVE(sqr(z0)))
TOP(ok(add(z0, z1))) → c46(TOP(add(active(z0), z1)), ACTIVE(add(z0, z1)))
TOP(ok(add(z0, z1))) → c46(TOP(add(z0, active(z1))), ACTIVE(add(z0, z1)))
TOP(ok(dbl(z0))) → c46(TOP(dbl(active(z0))), ACTIVE(dbl(z0)))
TOP(ok(first(z0, z1))) → c46(TOP(first(active(z0), z1)), ACTIVE(first(z0, z1)))
TOP(ok(first(z0, z1))) → c46(TOP(first(z0, active(z1))), ACTIVE(first(z0, z1)))

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))
TOP(mark(0)) → c45(TOP(ok(0)))
TOP(mark(nil)) → c45(TOP(ok(nil)))
TOP(ok(x0)) → c46

S tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))
TOP(ok(x0)) → c46

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44, c45, c46

(23) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 3 leading nodes:

TOP(mark(0)) → c45(TOP(ok(0)))
TOP(mark(nil)) → c45(TOP(ok(nil)))
TOP(ok(x0)) → c46

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(25) CdtPolyRedPairProof (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(z0, mark(z1)) → c27(ADD(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ADD(x₁, x₂)) = x₂
POL(CONS(x₁, x₂)) = 0
POL(DBL(x₁)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(RECIP(x₁)) = 0
POL(S(x₁)) = 0
POL(SQR(x₁)) = 0
POL(TERMS(x₁)) = 0
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

K tuples:

ADD(z0, mark(z1)) → c27(ADD(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
S(ok(z0)) → c44(S(z0))

We considered the (Usable) Rules:none
And the Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ADD(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = [2]x₂
POL(DBL(x₁)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(RECIP(x₁)) = 0
POL(S(x₁)) = [3]x₁ + [3]x₁²
POL(SQR(x₁)) = 0
POL(TERMS(x₁)) = 0
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [3] + x₁

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))

K tuples:

ADD(z0, mark(z1)) → c27(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
S(ok(z0)) → c44(S(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(29) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

RECIP(ok(z0)) → c23(RECIP(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ADD(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = x₂³
POL(DBL(x₁)) = x₁³
POL(FIRST(x₁, x₂)) = x₁ + x₁² + x₁³ + x₁²·x₂
POL(RECIP(x₁)) = x₁²
POL(S(x₁)) = 0
POL(SQR(x₁)) = x₁³
POL(TERMS(x₁)) = 0
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))

K tuples:

ADD(z0, mark(z1)) → c27(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
S(ok(z0)) → c44(S(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(31) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
SQR(mark(z0)) → c24(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ADD(x₁, x₂)) = x₁
POL(CONS(x₁, x₂)) = x₂² + x₁²
POL(DBL(x₁)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(RECIP(x₁)) = 0
POL(S(x₁)) = x₁²
POL(SQR(x₁)) = x₁
POL(TERMS(x₁)) = x₁³
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:

RECIP(mark(z0)) → c22(RECIP(z0))
DBL(mark(z0)) → c29(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))

K tuples:

ADD(z0, mark(z1)) → c27(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
S(ok(z0)) → c44(S(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
SQR(mark(z0)) → c24(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(33) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

DBL(mark(z0)) → c29(DBL(z0))

We considered the (Usable) Rules:none
And the Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(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)) = x₁
POL(FIRST(x₁, x₂)) = 0
POL(RECIP(x₁)) = 0
POL(S(x₁)) = [4]x₁
POL(SQR(x₁)) = 0
POL(TERMS(x₁)) = 0
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:

RECIP(mark(z0)) → c22(RECIP(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))

K tuples:

ADD(z0, mark(z1)) → c27(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
S(ok(z0)) → c44(S(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
SQR(mark(z0)) → c24(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(35) CdtPolyRedPairProof (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)) → c32(FIRST(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(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)) = [4]x₁ + [2]x₂
POL(DBL(x₁)) = [2]x₁
POL(FIRST(x₁, x₂)) = x₂
POL(RECIP(x₁)) = 0
POL(S(x₁)) = [3]x₁
POL(SQR(x₁)) = 0
POL(TERMS(x₁)) = 0
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:

RECIP(mark(z0)) → c22(RECIP(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))

K tuples:

ADD(z0, mark(z1)) → c27(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
S(ok(z0)) → c44(S(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
SQR(mark(z0)) → c24(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(37) CdtPolyRedPairProof (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)) → c22(RECIP(z0))

We considered the (Usable) Rules:none
And the Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ADD(x₁, x₂)) = [3]x₂
POL(CONS(x₁, x₂)) = [2]x₂
POL(DBL(x₁)) = [2]x₁
POL(FIRST(x₁, x₂)) = 0
POL(RECIP(x₁)) = [4]x₁
POL(S(x₁)) = [2]x₁
POL(SQR(x₁)) = 0
POL(TERMS(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = 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(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(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))
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))
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))
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)
s(ok(z0)) → ok(s(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:

FIRST(mark(z0), z1) → c31(FIRST(z0, z1))

K tuples:

ADD(z0, mark(z1)) → c27(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
S(ok(z0)) → c44(S(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
SQR(mark(z0)) → c24(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(39) CdtPolyRedPairProof (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(mark(z0), z1) → c31(FIRST(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(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)) = [4]x₁
POL(RECIP(x₁)) = [4]x₁
POL(S(x₁)) = 0
POL(SQR(x₁)) = 0
POL(TERMS(x₁)) = [2]x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

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

Tuples:

TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(mark(z0)) → c24(SQR(z0))
SQR(ok(z0)) → c25(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(z0, mark(z1)) → c27(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
S(ok(z0)) → c44(S(z0))

S tuples:none
K tuples:

ADD(z0, mark(z1)) → c27(ADD(z0, z1))
CONS(ok(z0), ok(z1)) → c21(CONS(z0, z1))
S(ok(z0)) → c44(S(z0))
RECIP(ok(z0)) → c23(RECIP(z0))
SQR(ok(z0)) → c25(SQR(z0))
DBL(ok(z0)) → c30(DBL(z0))
FIRST(ok(z0), ok(z1)) → c33(FIRST(z0, z1))
TERMS(mark(z0)) → c18(TERMS(z0))
TERMS(ok(z0)) → c19(TERMS(z0))
CONS(mark(z0), z1) → c20(CONS(z0, z1))
SQR(mark(z0)) → c24(SQR(z0))
ADD(mark(z0), z1) → c26(ADD(z0, z1))
ADD(ok(z0), ok(z1)) → c28(ADD(z0, z1))
DBL(mark(z0)) → c29(DBL(z0))
FIRST(z0, mark(z1)) → c32(FIRST(z0, z1))
RECIP(mark(z0)) → c22(RECIP(z0))
FIRST(mark(z0), z1) → c31(FIRST(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c44

(41) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(6) Obligation:

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

(8) Obligation:

(9) CdtUnreachableProof (EQUIVALENT transformation)

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

(21) CdtUnreachableProof (EQUIVALENT transformation)

(22) Obligation:

(23) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(24) Obligation:

(25) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(26) Obligation:

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(28) Obligation:

(29) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

(30) Obligation:

(31) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

(32) Obligation:

(33) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(34) Obligation:

(35) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(36) Obligation:

(37) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(38) Obligation:

(39) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(40) Obligation:

(41) SIsEmptyProof (EQUIVALENT transformation)

(42) BOUNDS(O(1), O(1))