WORST_CASE(?,O(n^1))
proof of input_HkVn7xOqY5.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 62 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:

   concat(leaf, Y) -> Y
   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
   lessleaves(X, leaf) -> false
   lessleaves(leaf, cons(W, Z)) -> true
   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))

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:

   concat(leaf, Y) -> Y
   concat(cons(U, V), Y) -> cons(U, concat(V, Y))
   lessleaves(X, leaf) -> false
   lessleaves(leaf, cons(W, Z)) -> true
   lessleaves(cons(U, V), cons(W, Z)) -> lessleaves(concat(U, V), concat(W, Z))

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]
transitions: 
leaf0() -> 0
cons0(0, 0) -> 0
false0() -> 0
true0() -> 0
concat0(0, 0) -> 1
lessleaves0(0, 0) -> 2
concat1(0, 0) -> 3
cons1(0, 3) -> 1
false1() -> 2
true1() -> 2
concat1(0, 0) -> 4
concat1(0, 0) -> 5
lessleaves1(4, 5) -> 2
cons1(0, 3) -> 3
cons1(0, 3) -> 4
cons1(0, 3) -> 5
concat1(0, 3) -> 5
concat1(0, 3) -> 4
concat2(0, 3) -> 6
concat2(0, 3) -> 7
lessleaves2(6, 7) -> 2
concat1(0, 3) -> 3
0 -> 1
0 -> 3
0 -> 4
0 -> 5
3 -> 4
3 -> 5
3 -> 6
3 -> 7

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

(4)
BOUNDS(1, n^1)