### (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:

implies(not(x), y) → or(x, y)

implies(not(x), or(y, z)) → implies(y, or(x, z))

implies(x, or(y, z)) → or(y, implies(x, 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:

implies(not(x), y) → or(x, y)

implies(not(x), or(y, z)) → implies(y, or(x, z))

implies(x, or(y, z)) → or(y, implies(x, 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 2.

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

final states : [1]

transitions:

not0(0) → 0

or0(0, 0) → 0

implies0(0, 0) → 1

or1(0, 0) → 1

or1(0, 0) → 2

implies1(0, 2) → 1

implies1(0, 0) → 3

or1(0, 3) → 1

or1(0, 2) → 1

or1(0, 0) → 3

implies1(0, 2) → 3

or1(0, 3) → 3

implies2(0, 0) → 4

or2(0, 4) → 1

or1(0, 2) → 3

or2(0, 4) → 3

or1(0, 0) → 4

implies1(0, 2) → 4

or1(0, 3) → 4

or1(0, 2) → 4

or2(0, 4) → 4

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