(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:
@(Cons(x, xs), ys) → Cons(x, @(xs, ys))
@(Nil, ys) → ys
game(p1, Cons(x', xs'), Cons(Capture, xs)) → game(Cons(x', p1), xs', xs)
game(p1, p2, Cons(Swap, xs)) → game(p2, p1, xs)
equal(Capture, Capture) → True
equal(Capture, Swap) → False
equal(Swap, Capture) → False
equal(Swap, Swap) → True
game(p1, p2, Nil) → @(p1, p2)
goal(p1, p2, moves) → game(p1, p2, moves)
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]
transitions:
Cons0(0, 0) → 0
Nil0() → 0
Capture0() → 0
Swap0() → 0
True0() → 0
False0() → 0
@0(0, 0) → 1
game0(0, 0, 0) → 2
equal0(0, 0) → 3
goal0(0, 0, 0) → 4
@1(0, 0) → 5
Cons1(0, 5) → 1
Cons1(0, 0) → 6
game1(6, 0, 0) → 2
game1(0, 0, 0) → 2
True1() → 3
False1() → 3
@1(0, 0) → 2
game1(0, 0, 0) → 4
Cons1(0, 5) → 2
Cons1(0, 5) → 5
Cons1(0, 6) → 6
game1(6, 0, 0) → 4
game1(0, 6, 0) → 2
@1(6, 0) → 2
@1(0, 0) → 4
Cons1(0, 5) → 4
@2(0, 0) → 7
Cons2(0, 7) → 2
@2(6, 0) → 7
game1(6, 6, 0) → 2
game1(0, 6, 0) → 4
@1(0, 6) → 2
@1(6, 0) → 4
@1(0, 6) → 5
Cons2(0, 7) → 4
game1(6, 6, 0) → 4
@1(6, 6) → 2
@1(0, 6) → 4
Cons1(0, 5) → 7
Cons2(0, 7) → 7
@2(0, 6) → 7
@2(6, 6) → 7
@1(6, 6) → 4
0 → 1
0 → 2
0 → 5
0 → 4
0 → 7
6 → 2
6 → 4
6 → 5
6 → 7
(2) BOUNDS(1, n^1)