WORST_CASE(?,O(n^1))
proof of input_JszSaIkHXG.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 191 ms]
(4) BOUNDS(1, n^1)


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(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:

   a__f(0) -> cons(0, f(s(0)))
   a__f(s(0)) -> a__f(a__p(s(0)))
   a__p(s(0)) -> 0
   mark(f(X)) -> a__f(mark(X))
   mark(p(X)) -> a__p(mark(X))
   mark(0) -> 0
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   a__f(X) -> f(X)
   a__p(X) -> p(X)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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:

   a__f(0) -> cons(0, f(s(0)))
   a__f(s(0)) -> a__f(a__p(s(0)))
   a__p(s(0)) -> 0
   mark(f(X)) -> a__f(mark(X))
   mark(p(X)) -> a__p(mark(X))
   mark(0) -> 0
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   a__f(X) -> f(X)
   a__p(X) -> p(X)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(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 3. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
00() -> 0
cons0(0, 0) -> 0
f0(0) -> 0
s0(0) -> 0
p0(0) -> 0
a__f0(0) -> 1
a__p0(0) -> 2
mark0(0) -> 3
01() -> 4
01() -> 7
s1(7) -> 6
f1(6) -> 5
cons1(4, 5) -> 1
s1(7) -> 9
a__p1(9) -> 8
a__f1(8) -> 1
01() -> 2
mark1(0) -> 10
a__f1(10) -> 3
mark1(0) -> 11
a__p1(11) -> 3
01() -> 3
mark1(0) -> 12
cons1(12, 0) -> 3
mark1(0) -> 13
s1(13) -> 3
f1(0) -> 1
p1(0) -> 2
02() -> 8
a__f1(10) -> 10
a__f1(10) -> 11
a__f1(10) -> 12
a__f1(10) -> 13
a__p1(11) -> 10
a__p1(11) -> 11
a__p1(11) -> 12
a__p1(11) -> 13
01() -> 10
01() -> 11
01() -> 12
01() -> 13
cons1(12, 0) -> 10
cons1(12, 0) -> 11
cons1(12, 0) -> 12
cons1(12, 0) -> 13
s1(13) -> 10
s1(13) -> 11
s1(13) -> 12
s1(13) -> 13
f2(8) -> 1
f2(10) -> 3
p2(11) -> 3
p2(9) -> 8
02() -> 14
02() -> 17
s2(17) -> 16
f2(16) -> 15
cons2(14, 15) -> 3
cons2(14, 15) -> 10
cons2(14, 15) -> 11
cons2(14, 15) -> 12
cons2(14, 15) -> 13
s2(17) -> 19
a__p2(19) -> 18
a__f2(18) -> 3
a__f2(18) -> 10
a__f2(18) -> 11
a__f2(18) -> 12
a__f2(18) -> 13
02() -> 3
02() -> 10
02() -> 11
02() -> 12
02() -> 13
f2(10) -> 10
f2(10) -> 11
f2(10) -> 12
f2(10) -> 13
p2(11) -> 10
p2(11) -> 11
p2(11) -> 12
p2(11) -> 13
cons2(14, 15) -> 1
03() -> 18
f3(18) -> 3
f3(18) -> 10
f3(18) -> 11
f3(18) -> 12
f3(18) -> 13
p3(19) -> 18
03() -> 20
03() -> 23
s3(23) -> 22
f3(22) -> 21
cons3(20, 21) -> 3
cons3(20, 21) -> 10
cons3(20, 21) -> 11
cons3(20, 21) -> 12
cons3(20, 21) -> 13

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(4)
BOUNDS(1, n^1)