### (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(nil) → nil

f(.(nil, y)) → .(nil, f(y))

f(.(.(x, y), z)) → f(.(x, .(y, z)))

g(nil) → nil

g(.(x, nil)) → .(g(x), nil)

g(.(x, .(y, z))) → g(.(.(x, y), z))

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(nil) → nil

f(.(nil, y)) → .(nil, f(y))

f(.(.(x, y), z)) → f(.(x, .(y, z)))

g(nil) → nil

g(.(x, nil)) → .(g(x), nil)

g(.(x, .(y, z))) → g(.(.(x, y), z))

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 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:

final states : [1, 2]

transitions:

nil0() → 0

.0(0, 0) → 0

f0(0) → 1

g0(0) → 2

nil1() → 1

nil1() → 3

f1(0) → 4

.1(3, 4) → 1

.1(0, 0) → 6

.1(0, 6) → 5

f1(5) → 1

nil1() → 2

g1(0) → 7

nil1() → 8

.1(7, 8) → 2

.1(0, 0) → 10

.1(10, 0) → 9

g1(9) → 2

nil1() → 4

.1(3, 4) → 4

f1(6) → 4

f1(5) → 4

.1(0, 6) → 6

nil1() → 7

.1(7, 8) → 7

g1(10) → 7

g1(9) → 7

.1(10, 0) → 10

### (4) BOUNDS(1, n^1)