### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

TERMS(z0) → c(SQR(z0))
TERMS(z0) → c1
SQR(0) → c2
SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
DBL(0) → c4
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ADD(0, z0) → c6
FIRST(0, z0) → c8
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))
FIRST(z0, z1) → c10
S(z0) → c11
ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c15
S tuples:

TERMS(z0) → c(SQR(z0))
TERMS(z0) → c1
SQR(0) → c2
SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
DBL(0) → c4
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
ADD(0, z0) → c6
FIRST(0, z0) → c8
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))
FIRST(z0, z1) → c10
S(z0) → c11
ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c15
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

TERMS, SQR, DBL, ADD, FIRST, S, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))
Removed 12 trailing nodes:

S(z0) → c11
FIRST(z0, z1) → c10
TERMS(z0) → c(SQR(z0))
ACTIVATE(z0) → c15
TERMS(z0) → c1
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
SQR(0) → c2
DBL(0) → c4
ADD(0, z0) → c6
FIRST(0, z0) → c8

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__terms(z0)) → c12(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c12, c13, c14

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

Removed 3 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c12, c13, c14

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0)))
terms(z0) → n__terms(z0)
sqr(0) → 0
sqr(s(z0)) → s(add(sqr(z0), dbl(z0)))
dbl(0) → 0
dbl(s(z0)) → s(s(dbl(z0)))
add(0, z0) → z0
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
s(z0) → n__s(z0)
activate(n__terms(z0)) → terms(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c12, c13, c14

### (9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [5] + [4]x1
POL(c12(x1)) = x1
POL(c13(x1)) = x1
POL(c14(x1, x2)) = x1 + x2
POL(n__first(x1, x2)) = [4] + x1 + x2
POL(n__s(x1)) = [2] + x1
POL(n__terms(x1)) = [5] + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:

ACTIVATE(n__terms(z0)) → c12(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c14(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c12, c13, c14

### (11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty