(0) Obligation:

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

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
first(0, X) → nil
first(s, cons(Y)) → cons(Y)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

(2) Obligation:

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

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
first(0, X) → nil
first(s, cons(Y)) → cons(Y)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(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
first(0, z0) → nil
first(s, cons(z0)) → cons(z0)
Tuples:

TERMS(z0) → c(SQR(z0))
SQR(0) → c1
SQR(s) → c2
DBL(0) → c3
DBL(s) → c4
FIRST(0, z0) → c7
FIRST(s, cons(z0)) → c8
S tuples:

TERMS(z0) → c(SQR(z0))
SQR(0) → c1
SQR(s) → c2
DBL(0) → c3
DBL(s) → c4
FIRST(0, z0) → c7
FIRST(s, cons(z0)) → c8
K tuples:none
Defined Rule Symbols:

Defined Pair Symbols:

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8

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

Removed 9 trailing nodes:

SQR(0) → c1
FIRST(0, z0) → c7
DBL(0) → c3
SQR(s) → c2
DBL(s) → c4
TERMS(z0) → c(SQR(z0))
FIRST(s, cons(z0)) → c8

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
first(0, z0) → nil
first(s, cons(z0)) → cons(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols: