(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:
compS_f#1(compS_f(x2), x1) → compS_f#1(x2, S(x1))
compS_f#1(id, x3) → S(x3)
iter#3(0) → id
iter#3(S(x6)) → compS_f(iter#3(x6))
main(0) → 0
main(S(x9)) → compS_f#1(iter#3(x9), 0)
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:
compS_f0(0) → 0
S0(0) → 0
id0() → 0
00() → 0
compS_f#10(0, 0) → 1
iter#30(0) → 2
main0(0) → 3
S1(0) → 4
compS_f#11(0, 4) → 1
S1(0) → 1
id1() → 2
iter#31(0) → 5
compS_f1(5) → 2
01() → 3
iter#31(0) → 6
01() → 7
compS_f#11(6, 7) → 3
S1(4) → 4
S1(4) → 1
id1() → 5
id1() → 6
compS_f1(5) → 5
compS_f1(5) → 6
S2(7) → 8
compS_f#12(5, 8) → 3
S2(7) → 3
S2(8) → 8
S2(8) → 3
(2) BOUNDS(1, n^1)