### (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:

f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)

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, n^1).

The TRS R consists of the following rules:

f(a, a) → f(a, b)
f(a, b) → f(s(a), c)
f(s(X), c) → f(X, c)
f(c, c) → f(a, a)

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 4.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : 
transitions:
a0() → 0
b0() → 0
s0(0) → 0
c0() → 0
f0(0, 0) → 1
a1() → 2
b1() → 3
f1(2, 3) → 1
a1() → 5
s1(5) → 4
c1() → 6
f1(4, 6) → 1
f1(0, 6) → 1
a1() → 7
f1(2, 7) → 1
a2() → 8
b2() → 9
f2(8, 9) → 1
a2() → 11
s2(11) → 10
c2() → 12
f2(10, 12) → 1
f2(5, 12) → 1
a3() → 14
s3(14) → 13
c3() → 15
f3(13, 15) → 1
f3(11, 15) → 1
c4() → 16
f4(14, 16) → 1