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:

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

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:

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

first(0, z0) → nil
first(s(z0), cons(z1)) → cons(z1)
from(z0) → cons(z0)

Tuples:

FIRST(0, z0) → c
FIRST(s(z0), cons(z1)) → c1
FROM(z0) → c2

S tuples:

FIRST(0, z0) → c
FIRST(s(z0), cons(z1)) → c1
FROM(z0) → c2

K tuples:none
Defined Rule Symbols:

first, from

Defined Pair Symbols:

FIRST, FROM

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

FIRST(s(z0), cons(z1)) → c1
FROM(z0) → c2
FIRST(0, z0) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

first(0, z0) → nil
first(s(z0), cons(z1)) → cons(z1)
from(z0) → cons(z0)

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

first, from

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(8) BOUNDS(1, 1)