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

(0) Obligation:

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

terms(N) → cons(recip(sqr(N)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
add(0, X) → X
add(s, Y) → s
first(0, X) → nil
first(s, cons(Y)) → cons(Y)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
add(0, z0) → z0
add(s, z0) → s
first(0, z0) → nil
first(s, cons(z0)) → cons(z0)

Tuples:

TERMS(z0) → c(SQR(z0))

S tuples:

TERMS(z0) → c(SQR(z0))

K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first

Defined Pair Symbols:

TERMS

Compound Symbols:

c

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

Removed 1 trailing nodes:

TERMS(z0) → c(SQR(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
add(0, z0) → z0
add(s, z0) → s
first(0, z0) → nil
first(s, cons(z0)) → cons(z0)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))