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

(0) 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

(1) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
00() → 0
nil0() → 0
s0(0) → 0
cons0(0) → 0
first0(0, 0) → 1
from0(0) → 2
nil1() → 1
cons1(0) → 1
cons1(0) → 2

(2) BOUNDS(1, n^1)

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

(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)