### (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:

double(0) → 0
double(s(x)) → s(s(double(x)))
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
if(0, y, z) → y
if(s(x), y, z) → z
half(double(x)) → x

Rewrite Strategy: FULL

### (1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
half(double(x)) → x

### (2) 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:

double(0) → 0
double(s(x)) → s(s(double(x)))
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(s(x), y, z) → z
half(0) → 0
-(s(x), s(y)) → -(x, y)
-(x, 0) → x
if(0, y, z) → y

Rewrite Strategy: FULL

### (3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.

### (4) 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:

double(0) → 0
double(s(x)) → s(s(double(x)))
half(s(0)) → 0
half(s(s(x))) → s(half(x))
if(s(x), y, z) → z
half(0) → 0
-(s(x), s(y)) → -(x, y)
-(x, 0) → x
if(0, y, z) → y

Rewrite Strategy: INNERMOST

### (5) 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, 3, 4]
transitions:
00() → 0
s0(0) → 0
double0(0) → 1
half0(0) → 2
if0(0, 0, 0) → 3
-0(0, 0) → 4
01() → 1
double1(0) → 6
s1(6) → 5
s1(5) → 1
01() → 2
half1(0) → 7
s1(7) → 2
-1(0, 0) → 4
01() → 6
s1(5) → 6
01() → 7
s1(7) → 7
0 → 3
0 → 4