(0) Obligation:

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

The TRS R consists of the following rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__p(s(0)) → 0
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

(2) Obligation:

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

The TRS R consists of the following rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__p(s(0)) → 0
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Rewrite Strategy: INNERMOST

(3) 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 3.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3]
transitions:
00() → 0
cons0(0, 0) → 0
f0(0) → 0
s0(0) → 0
p0(0) → 0
a__f0(0) → 1
a__p0(0) → 2
mark0(0) → 3
01() → 4
01() → 7
s1(7) → 6
f1(6) → 5
cons1(4, 5) → 1
s1(7) → 9
a__p1(9) → 8
a__f1(8) → 1
01() → 2
mark1(0) → 10
a__f1(10) → 3
mark1(0) → 11
a__p1(11) → 3
01() → 3
mark1(0) → 12
cons1(12, 0) → 3
mark1(0) → 13
s1(13) → 3
f1(0) → 1
p1(0) → 2
02() → 8
a__f1(10) → 10
a__f1(10) → 11
a__f1(10) → 12
a__f1(10) → 13
a__p1(11) → 10
a__p1(11) → 11
a__p1(11) → 12
a__p1(11) → 13
01() → 10
01() → 11
01() → 12
01() → 13
cons1(12, 0) → 10
cons1(12, 0) → 11
cons1(12, 0) → 12
cons1(12, 0) → 13
s1(13) → 10
s1(13) → 11
s1(13) → 12
s1(13) → 13
f2(8) → 1
f2(10) → 3
p2(11) → 3
p2(9) → 8
02() → 14
02() → 17
s2(17) → 16
f2(16) → 15
cons2(14, 15) → 3
cons2(14, 15) → 10
cons2(14, 15) → 11
cons2(14, 15) → 12
cons2(14, 15) → 13
s2(17) → 19
a__p2(19) → 18
a__f2(18) → 3
a__f2(18) → 10
a__f2(18) → 11
a__f2(18) → 12
a__f2(18) → 13
02() → 3
02() → 10
02() → 11
02() → 12
02() → 13
f2(10) → 10
f2(10) → 11
f2(10) → 12
f2(10) → 13
p2(11) → 10
p2(11) → 11
p2(11) → 12
p2(11) → 13
cons2(14, 15) → 1
03() → 18
f3(18) → 3
f3(18) → 10
f3(18) → 11
f3(18) → 12
f3(18) → 13
p3(19) → 18
03() → 20
03() → 23
s3(23) → 22
f3(22) → 21
cons3(20, 21) → 3
cons3(20, 21) → 10
cons3(20, 21) → 11
cons3(20, 21) → 12
cons3(20, 21) → 13