(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:
dec(Cons(Nil, Nil)) → Nil
dec(Cons(Nil, Cons(x, xs))) → dec(Cons(x, xs))
dec(Cons(Cons(x, xs), Nil)) → dec(Nil)
dec(Cons(Cons(x', xs'), Cons(x, xs))) → dec(Cons(x, xs))
isNilNil(Cons(Nil, Nil)) → True
isNilNil(Cons(Nil, Cons(x, xs))) → False
isNilNil(Cons(Cons(x, xs), Nil)) → False
isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) → False
nestdec(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)))))))))))))))))
nestdec(Cons(x, xs)) → nestdec(dec(Cons(x, xs)))
number17(n) → 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) → nestdec(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, 4, 5]
transitions:
Cons0(0, 0) → 0
Nil0() → 0
True0() → 0
False0() → 0
dec0(0) → 1
isNilNil0(0) → 2
nestdec0(0) → 3
number170(0) → 4
goal0(0) → 5
Nil1() → 1
Cons1(0, 0) → 6
dec1(6) → 1
Nil1() → 7
dec1(7) → 1
True1() → 2
False1() → 2
Nil1() → 8
Nil1() → 11
Cons1(8, 11) → 10
Cons1(8, 10) → 9
Cons1(8, 9) → 9
Cons1(8, 9) → 3
dec1(6) → 12
nestdec1(12) → 3
Cons1(8, 9) → 4
nestdec1(0) → 5
Nil1() → 12
dec1(7) → 12
Cons1(8, 9) → 5
nestdec1(12) → 5
Nil2() → 13
Nil2() → 16
Cons2(13, 16) → 15
Cons2(13, 15) → 14
Cons2(13, 14) → 14
Cons2(13, 14) → 3
Cons2(13, 14) → 5
(2) BOUNDS(1, n^1)