/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 167 ms]
    (12) BOUNDS(1, n^1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) 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:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0) -> 0
   round(0) -> s(0)
   round(s(0)) -> s(0)
   round(s(s(x))) -> s(s(round(x)))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0) -> 0
   round(0) -> s(0)
   round(s(0)) -> s(0)
   round(s(s(x))) -> s(s(round(x)))

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

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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


The TRS R consists of the following rules:

   f(s(x), x) -> f(s(x), round(s(x))) [1]
   round(0) -> 0 [1]
   round(0) -> s(0) [1]
   round(s(0)) -> s(0) [1]
   round(s(s(x))) -> s(s(round(x))) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(s(x), x) -> f(s(x), round(s(x))) [1]
   round(0) -> 0 [1]
   round(0) -> s(0) [1]
   round(s(0)) -> s(0) [1]
   round(s(s(x))) -> s(s(round(x))) [1]

The TRS has the following type information:
f :: s:0 -> s:0 -> f
s :: s:0 -> s:0
round :: s:0 -> s:0
0 :: s:0

Rewrite Strategy: INNERMOST
----------------------------------------

(7) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_f

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

(8)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(s(x), x) -> f(s(x), round(s(x))) [1]
   round(0) -> 0 [1]
   round(0) -> s(0) [1]
   round(s(0)) -> s(0) [1]
   round(s(s(x))) -> s(s(round(x))) [1]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: s:0 -> s:0 -> null_f
s :: s:0 -> s:0
round :: s:0 -> s:0
0 :: s:0
null_f :: null_f

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   null_f => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> f(1 + x, round(1 + x)) :|: z' = x, x >= 0, z = 1 + x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   round(z) -{ 1 }-> 0 :|: z = 0
   round(z) -{ 1 }-> 1 + 0 :|: z = 0
   round(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   round(z) -{ 1 }-> 1 + (1 + round(x)) :|: x >= 0, z = 1 + (1 + x)

Only complete derivations are relevant for the runtime complexity.

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

(11) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V),0,[f(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[round(V1, Out)],[V1 >= 0]).
eq(f(V1, V, Out),1,[round(1 + V2, Ret1),f(1 + V2, Ret1, Ret)],[Out = Ret,V = V2,V2 >= 0,V1 = 1 + V2]).
eq(round(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(round(V1, Out),1,[],[Out = 1,V1 = 0]).
eq(round(V1, Out),1,[],[Out = 1,V1 = 1]).
eq(round(V1, Out),1,[round(V3, Ret11)],[Out = 2 + Ret11,V3 >= 0,V1 = 2 + V3]).
eq(f(V1, V, Out),0,[],[Out = 0,V5 >= 0,V4 >= 0,V1 = V5,V = V4]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(round(V1,Out),[V1],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [round/2]
1. recursive  : [f/3]
2. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into round/2
1. SCC is partially evaluated into f/3
2. SCC is partially evaluated into start/2

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations round/2 
* CE 8 is refined into CE [9] 
* CE 7 is refined into CE [10] 
* CE 6 is refined into CE [11] 
* CE 5 is refined into CE [12] 


### Cost equations --> "Loop" of round/2 
* CEs [10] --> Loop 8 
* CEs [11] --> Loop 9 
* CEs [12] --> Loop 10 
* CEs [9] --> Loop 11 

### Ranking functions of CR round(V1,Out) 
* RF of phase [11]: [V1-1]

#### Partial ranking functions of CR round(V1,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V1-1


### Specialization of cost equations f/3 
* CE 4 is refined into CE [13] 
* CE 3 is refined into CE [14,15,16] 


### Cost equations --> "Loop" of f/3 
* CEs [16] --> Loop 12 
* CEs [15] --> Loop 13 
* CEs [14] --> Loop 14 
* CEs [13] --> Loop 15 

### Ranking functions of CR f(V1,V,Out) 

#### Partial ranking functions of CR f(V1,V,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [17,18] 
* CE 2 is refined into CE [19,20,21,22,23] 


### Cost equations --> "Loop" of start/2 
* CEs [22,23] --> Loop 16 
* CEs [17,18] --> Loop 17 
* CEs [21] --> Loop 18 
* CEs [19,20] --> Loop 19 

### Ranking functions of CR start(V1,V) 

#### Partial ranking functions of CR start(V1,V) 


Computing Bounds
=====================================

#### Cost of chains of round(V1,Out):
* Chain [[11],10]: 1*it(11)+1
  Such that:it(11) =< Out

  with precondition: [V1=Out,V1>=2] 

* Chain [[11],9]: 1*it(11)+1
  Such that:it(11) =< Out

  with precondition: [V1+1=Out,V1>=2] 

* Chain [[11],8]: 1*it(11)+1
  Such that:it(11) =< Out

  with precondition: [V1=Out,V1>=3] 

* Chain [10]: 1
  with precondition: [V1=0,Out=0] 

* Chain [9]: 1
  with precondition: [V1=0,Out=1] 

* Chain [8]: 1
  with precondition: [V1=1,Out=1] 


#### Cost of chains of f(V1,V,Out):
* Chain [15]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [14,15]: 2
  with precondition: [V1=1,V=0,Out=0] 

* Chain [13,15]: 1*s(3)+2
  Such that:s(3) =< V1+1

  with precondition: [Out=0,V1=V+1,V1>=2] 

* Chain [12,15]: 2*s(5)+2
  Such that:s(4) =< V1
s(5) =< s(4)

  with precondition: [Out=0,V+1=V1,V>=1] 


#### Cost of chains of start(V1,V):
* Chain [19]: 1
  with precondition: [V1=0] 

* Chain [18]: 1
  with precondition: [V1=1] 

* Chain [17]: 1*s(10)+2*s(11)+2
  Such that:s(9) =< V+1
s(10) =< V+2
s(11) =< s(9)

  with precondition: [V1>=0,V>=0] 

* Chain [16]: 1*s(12)+2*s(14)+1
  Such that:s(13) =< V1
s(12) =< V1+1
s(14) =< s(13)

  with precondition: [V1>=2] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [19] with precondition: [V1=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [18] with precondition: [V1=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [17] with precondition: [V1>=0,V>=0] 
    - Upper bound: 3*V+6 
    - Complexity: n 
* Chain [16] with precondition: [V1>=2] 
    - Upper bound: 3*V1+2 
    - Complexity: n 

### Maximum cost of start(V1,V): max([3*V1+1,nat(V+1)*2+1+nat(V+2)])+1 
Asymptotic class: n 
* Total analysis performed in 101 ms.


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

(12)
BOUNDS(1, n^1)

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

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

(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:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0) -> 0
   round(0) -> s(0)
   round(s(0)) -> s(0)
   round(s(s(x))) -> s(s(round(x)))

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

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

The rewrite sequence

round(s(s(x))) ->^+ s(s(round(x)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].

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

The result substitution is [ ].




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

(16)
Complex Obligation (BEST)

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

(17)
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:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0) -> 0
   round(0) -> s(0)
   round(s(0)) -> s(0)
   round(s(s(x))) -> s(s(round(x)))

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

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

(19)
BOUNDS(n^1, INF)

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

(20)
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:

   f(s(x), x) -> f(s(x), round(s(x)))
   round(0) -> 0
   round(0) -> s(0)
   round(s(0)) -> s(0)
   round(s(s(x))) -> s(s(round(x)))

S is empty.
Rewrite Strategy: FULL