WORST_CASE(?,O(n^1))
proof of input_dsuwlmK22i.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 97 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:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)

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:

   a__zeros -> cons(0, zeros)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(zeros) -> a__zeros
   mark(tail(X)) -> a__tail(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   a__zeros -> zeros
   a__tail(X) -> tail(X)

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 4. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
cons0(0, 0) -> 0
00() -> 0
zeros0() -> 0
tail0(0) -> 0
a__zeros0() -> 1
a__tail0(0) -> 2
mark0(0) -> 3
01() -> 4
zeros1() -> 5
cons1(4, 5) -> 1
mark1(0) -> 2
a__zeros1() -> 3
mark1(0) -> 6
a__tail1(6) -> 3
mark1(0) -> 7
cons1(7, 0) -> 3
01() -> 3
zeros1() -> 1
tail1(0) -> 2
02() -> 8
zeros2() -> 9
cons2(8, 9) -> 3
a__zeros1() -> 2
a__zeros1() -> 6
a__zeros1() -> 7
a__tail1(6) -> 2
a__tail1(6) -> 6
a__tail1(6) -> 7
cons1(7, 0) -> 2
cons1(7, 0) -> 6
cons1(7, 0) -> 7
01() -> 2
01() -> 6
01() -> 7
zeros2() -> 3
tail2(6) -> 3
cons2(8, 9) -> 2
cons2(8, 9) -> 6
cons2(8, 9) -> 7
mark2(0) -> 2
mark2(0) -> 3
mark2(0) -> 6
mark2(0) -> 7
zeros2() -> 2
zeros2() -> 6
zeros2() -> 7
tail2(6) -> 2
tail2(6) -> 6
tail2(6) -> 7
mark2(9) -> 2
mark2(9) -> 3
mark2(9) -> 6
mark2(9) -> 7
a__zeros3() -> 2
a__zeros3() -> 3
a__zeros3() -> 6
a__zeros3() -> 7
04() -> 10
zeros4() -> 11
cons4(10, 11) -> 2
cons4(10, 11) -> 3
cons4(10, 11) -> 6
cons4(10, 11) -> 7
zeros4() -> 2
zeros4() -> 3
zeros4() -> 6
zeros4() -> 7
mark2(11) -> 2
mark2(11) -> 3
mark2(11) -> 6
mark2(11) -> 7

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

(4)
BOUNDS(1, n^1)