(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:
walk#1(Nil) → walk_xs
walk#1(Cons(x4, x3)) → comp_f_g(walk#1(x3), walk_xs_3(x4))
comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) → comp_f_g#1(x7, x9, Cons(x8, x12))
comp_f_g#1(walk_xs, walk_xs_3(x8), x12) → Cons(x8, x12)
main(Nil) → Nil
main(Cons(x4, x5)) → comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)
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:
Nil0() → 0
walk_xs0() → 0
Cons0(0, 0) → 0
comp_f_g0(0, 0) → 0
walk_xs_30(0) → 0
walk#10(0) → 1
comp_f_g#10(0, 0, 0) → 2
main0(0) → 3
walk_xs1() → 1
walk#11(0) → 4
walk_xs_31(0) → 5
comp_f_g1(4, 5) → 1
Cons1(0, 0) → 6
comp_f_g#11(0, 0, 6) → 2
Cons1(0, 0) → 2
Nil1() → 3
walk#11(0) → 7
walk_xs_31(0) → 8
Nil1() → 9
comp_f_g#11(7, 8, 9) → 3
walk_xs1() → 4
walk_xs1() → 7
comp_f_g1(4, 5) → 4
comp_f_g1(4, 5) → 7
Cons1(0, 6) → 6
Cons1(0, 6) → 2
Cons2(0, 9) → 10
comp_f_g#12(4, 5, 10) → 3
Cons2(0, 9) → 3
Cons2(0, 10) → 10
Cons2(0, 10) → 3
(2) BOUNDS(1, n^1)