WORST_CASE(?,O(n^1))
proof of input_DkX67lU7to.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) CpxTrsMatchBoundsTAProof [FINISHED, 50 ms]
(4) BOUNDS(1, n^1)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-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)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-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)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) CpxTrsMatchBoundsTAProof (FINISHED)
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

----------------------------------------

(4)
BOUNDS(1, n^1)