(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:
not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)
Rewrite Strategy: FULL
(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS is a non-duplicating overlay system, 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:
not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)
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 3.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
true0() → 0
false0() → 0
00() → 0
s0(0) → 0
not0(0) → 1
evenodd0(0, 0) → 2
false1() → 1
true1() → 1
01() → 5
s1(5) → 4
evenodd1(0, 4) → 3
not1(3) → 2
false1() → 2
01() → 6
evenodd1(0, 6) → 2
02() → 9
s2(9) → 8
evenodd2(0, 8) → 7
not2(7) → 2
false1() → 3
evenodd1(0, 6) → 3
true2() → 2
not2(7) → 3
false1() → 7
evenodd1(0, 6) → 7
not2(7) → 7
true2() → 3
true2() → 7
false2() → 2
false3() → 2
false3() → 3
false3() → 7
true3() → 2
true3() → 3
true3() → 7
(4) BOUNDS(1, n^1)