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

(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
add(0, X) → X
add(s, Y) → 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
add(0, X) → X
add(s, Y) → 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
add(0, z0) → z0
add(s, z0) → 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
ADD(0, z0) → c5
ADD(s, z0) → c6
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
ADD(0, z0) → c5
ADD(s, z0) → c6
FIRST(0, z0) → c7
FIRST(s, cons(z0)) → c8

K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first

Defined Pair Symbols:

TERMS, SQR, DBL, ADD, FIRST

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))
ADD(0, z0) → c5
ADD(s, z0) → c6
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
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

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

The set S is empty

(8) BOUNDS(1, 1)