(0) 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:
odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(x)
Rewrite Strategy: INNERMOST
(1) 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:
Cons0(0, 0) → 0
Nil0() → 0
False0() → 0
True0() → 0
odd0(0) → 1
even0(0) → 2
notEmpty0(0) → 3
evenodd0(0) → 4
even1(0) → 1
False1() → 1
odd1(0) → 2
True1() → 3
False1() → 3
True1() → 2
even1(0) → 4
even1(0) → 2
False1() → 2
odd1(0) → 1
odd1(0) → 4
True1() → 1
True1() → 4
False1() → 4
(2) BOUNDS(1, n^1)