WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 29 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
half(double(x)) -> x

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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z

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

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z

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

(5) 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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
00() -> 0
s0(0) -> 0
double0(0) -> 1
half0(0) -> 2
-0(0, 0) -> 3
if0(0, 0, 0) -> 4
01() -> 1
double1(0) -> 6
s1(6) -> 5
s1(5) -> 1
01() -> 2
half1(0) -> 7
s1(7) -> 2
-1(0, 0) -> 3
01() -> 6
s1(5) -> 6
01() -> 7
s1(7) -> 7
0 -> 3
0 -> 4

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

(6)
BOUNDS(1, n^1)

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

(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(8)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

(9) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

-(s(x), s(y)) ->^+ -(x, y)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x / s(x), y / s(y)].

The result substitution is [ ].




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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

S is empty.
Rewrite Strategy: FULL