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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 14 ms)
↳10 CdtProblem
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 BOUNDS(O(1), O(1))

(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
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
half(0) → 0
half(s(0)) → 0
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
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
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(dbl(z0)) → z0
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
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1))
FIRST(0, z0) → c8
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))
FIRST(z0, z1) → c10
HALF(0) → c11
HALF(s(0)) → c12
HALF(s(s(z0))) → c13(S(half(z0)), HALF(z0))
HALF(dbl(z0)) → c14
S(z0) → c15
ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c19
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
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1))
FIRST(0, z0) → c8
FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))
FIRST(z0, z1) → c10
HALF(0) → c11
HALF(s(0)) → c12
HALF(s(s(z0))) → c13(S(half(z0)), HALF(z0))
HALF(dbl(z0)) → c14
S(z0) → c15
ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c19
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

FIRST(s(z0), cons(z1, z2)) → c9(ACTIVATE(z2))
Removed 16 trailing nodes:

HALF(0) → c11
S(z0) → c15
DBL(s(z0)) → c5(S(s(dbl(z0))), S(dbl(z0)), DBL(z0))
HALF(s(0)) → c12
ACTIVATE(z0) → c19
HALF(s(s(z0))) → c13(S(half(z0)), HALF(z0))
ADD(0, z0) → c6
ADD(s(z0), z1) → c7(S(add(z0, z1)), ADD(z0, z1))
FIRST(z0, z1) → c10
TERMS(z0) → c(SQR(z0))
HALF(dbl(z0)) → c14
TERMS(z0) → c1
SQR(s(z0)) → c3(S(add(sqr(z0), dbl(z0))), ADD(sqr(z0), dbl(z0)), SQR(z0), DBL(z0))
DBL(0) → c4
SQR(0) → c2
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
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(dbl(z0)) → z0
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)) → c16(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__terms(z0)) → c16(TERMS(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c16, c17, c18

(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
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(dbl(z0)) → z0
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)) → c16(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__terms(z0)) → c16(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c16, c17, c18

(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
add(s(z0), z1) → s(add(z0, z1))
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
half(0) → 0
half(s(0)) → 0
half(s(s(z0))) → s(half(z0))
half(dbl(z0)) → z0
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)) → c16(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__terms(z0)) → c16(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c16, c17, c18

(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)) → c16(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__terms(z0)) → c16(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [5] + [4]x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(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)) → c16(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:

ACTIVATE(n__terms(z0)) → c16(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c17(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c18(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c16, c17, c18

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

The set S is empty

(12) BOUNDS(O(1), O(1))