(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:
decrease(Cons(x, xs)) → decrease(xs)
decrease(Nil) → number42(Nil)
number42(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(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 2.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3]
transitions:
Cons0(0, 0) → 0
Nil0() → 0
decrease0(0) → 1
number420(0) → 2
goal0(0) → 3
decrease1(0) → 1
Nil1() → 4
number421(4) → 1
Nil1() → 5
Nil1() → 8
Cons1(5, 8) → 7
Cons1(5, 7) → 6
Cons1(5, 6) → 6
Cons1(5, 6) → 2
decrease1(0) → 3
number421(4) → 3
Nil2() → 9
Nil2() → 12
Cons2(9, 12) → 11
Cons2(9, 11) → 10
Cons2(9, 10) → 10
Cons2(9, 10) → 1
Cons2(9, 10) → 3
(2) BOUNDS(1, n^1)