(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(Leaf(x2)) → cons_x(x2)
walk#1(Node(x5, x3)) → comp_f_g(walk#1(x5), walk#1(x3))
comp_f_g#1(comp_f_g(x4, x5), comp_f_g(x2, x3), x1) → comp_f_g#1(x4, x5, comp_f_g#1(x2, x3, x1))
comp_f_g#1(comp_f_g(x7, x9), cons_x(x2), x4) → comp_f_g#1(x7, x9, Cons(x2, x4))
comp_f_g#1(cons_x(x2), comp_f_g(x5, x7), x3) → Cons(x2, comp_f_g#1(x5, x7, x3))
comp_f_g#1(cons_x(x5), cons_x(x2), x4) → Cons(x5, Cons(x2, x4))
main(Leaf(x4)) → Cons(x4, Nil)
main(Node(x9, x5)) → comp_f_g#1(walk#1(x9), walk#1(x5), 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:
Leaf0(0) → 0
cons_x0(0) → 0
Node0(0, 0) → 0
comp_f_g0(0, 0) → 0
Cons0(0, 0) → 0
Nil0() → 0
walk#10(0) → 1
comp_f_g#10(0, 0, 0) → 2
main0(0) → 3
cons_x1(0) → 1
walk#11(0) → 4
walk#11(0) → 5
comp_f_g1(4, 5) → 1
comp_f_g#11(0, 0, 0) → 6
comp_f_g#11(0, 0, 6) → 2
Cons1(0, 0) → 7
comp_f_g#11(0, 0, 7) → 2
comp_f_g#11(0, 0, 0) → 8
Cons1(0, 8) → 2
Cons1(0, 0) → 9
Cons1(0, 9) → 2
Nil1() → 10
Cons1(0, 10) → 3
walk#11(0) → 11
walk#11(0) → 12
Nil1() → 13
comp_f_g#11(11, 12, 13) → 3
cons_x1(0) → 4
cons_x1(0) → 5
cons_x1(0) → 11
cons_x1(0) → 12
comp_f_g1(4, 5) → 4
comp_f_g1(4, 5) → 5
comp_f_g1(4, 5) → 11
comp_f_g1(4, 5) → 12
comp_f_g#11(0, 0, 6) → 6
comp_f_g#11(0, 0, 7) → 6
comp_f_g#11(0, 0, 6) → 8
Cons1(0, 6) → 7
Cons1(0, 7) → 7
comp_f_g#11(0, 0, 7) → 8
Cons1(0, 8) → 6
Cons1(0, 8) → 8
Cons1(0, 6) → 9
Cons1(0, 7) → 9
Cons1(0, 9) → 6
Cons1(0, 9) → 8
comp_f_g#12(4, 5, 13) → 14
comp_f_g#12(4, 5, 14) → 3
Cons2(0, 13) → 15
comp_f_g#12(4, 5, 15) → 3
comp_f_g#12(4, 5, 13) → 16
Cons2(0, 16) → 3
Cons2(0, 13) → 17
Cons2(0, 17) → 3
comp_f_g#12(4, 5, 14) → 14
comp_f_g#12(4, 5, 15) → 14
comp_f_g#12(4, 5, 14) → 16
Cons2(0, 14) → 15
Cons2(0, 15) → 15
comp_f_g#12(4, 5, 15) → 16
Cons2(0, 16) → 14
Cons2(0, 16) → 16
Cons2(0, 14) → 17
Cons2(0, 15) → 17
Cons2(0, 17) → 14
Cons2(0, 17) → 16
(2) BOUNDS(1, n^1)