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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 318 ms]
    (10) BOUNDS(1, n^1)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(x))) -> lastbit(x)
   conv(0) -> cons(nil, 0)
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(2)
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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   lastbit(0) -> 0 [1]
   lastbit(s(0)) -> s(0) [1]
   lastbit(s(s(x))) -> lastbit(x) [1]
   conv(0) -> cons(nil, 0) [1]
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [1]

Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   lastbit(0) -> 0 [1]
   lastbit(s(0)) -> s(0) [1]
   lastbit(s(s(x))) -> lastbit(x) [1]
   conv(0) -> cons(nil, 0) [1]
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lastbit :: 0:s -> 0:s
conv :: 0:s -> nil:cons
cons :: nil:cons -> 0:s -> nil:cons
nil :: nil:cons

Rewrite Strategy: INNERMOST
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(5) 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:
none

And the following fresh constants: none

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(6)
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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   lastbit(0) -> 0 [1]
   lastbit(s(0)) -> s(0) [1]
   lastbit(s(s(x))) -> lastbit(x) [1]
   conv(0) -> cons(nil, 0) [1]
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lastbit :: 0:s -> 0:s
conv :: 0:s -> nil:cons
cons :: nil:cons -> 0:s -> nil:cons
nil :: nil:cons

Rewrite Strategy: INNERMOST
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(7) 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
   nil => 0

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

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   conv(z) -{ 1 }-> 1 + conv(half(1 + x)) + lastbit(1 + x) :|: x >= 0, z = 1 + x
   conv(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
   lastbit(z) -{ 1 }-> lastbit(x) :|: x >= 0, z = 1 + (1 + x)
   lastbit(z) -{ 1 }-> 0 :|: z = 0
   lastbit(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V),0,[half(V, Out)],[V >= 0]).
eq(start(V),0,[lastbit(V, Out)],[V >= 0]).
eq(start(V),0,[conv(V, Out)],[V >= 0]).
eq(half(V, Out),1,[],[Out = 0,V = 0]).
eq(half(V, Out),1,[],[Out = 0,V = 1]).
eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
eq(lastbit(V, Out),1,[],[Out = 0,V = 0]).
eq(lastbit(V, Out),1,[],[Out = 1,V = 1]).
eq(lastbit(V, Out),1,[lastbit(V2, Ret)],[Out = Ret,V2 >= 0,V = 2 + V2]).
eq(conv(V, Out),1,[],[Out = 1,V = 0]).
eq(conv(V, Out),1,[half(1 + V3, Ret010),conv(Ret010, Ret01),lastbit(1 + V3, Ret11)],[Out = 1 + Ret01 + Ret11,V3 >= 0,V = 1 + V3]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(lastbit(V,Out),[V],[Out]).
input_output_vars(conv(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [half/2]
1. recursive  : [lastbit/2]
2. recursive [non_tail] : [conv/2]
3. non_recursive  : [start/1]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into half/2
1. SCC is partially evaluated into lastbit/2
2. SCC is partially evaluated into conv/2
3. SCC is partially evaluated into start/1

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

### Specialization of cost equations half/2 
* CE 6 is refined into CE [12] 
* CE 5 is refined into CE [13] 
* CE 4 is refined into CE [14] 


### Cost equations --> "Loop" of half/2 
* CEs [13] --> Loop 10 
* CEs [14] --> Loop 11 
* CEs [12] --> Loop 12 

### Ranking functions of CR half(V,Out) 
* RF of phase [12]: [V-1]

#### Partial ranking functions of CR half(V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V-1


### Specialization of cost equations lastbit/2 
* CE 9 is refined into CE [15] 
* CE 8 is refined into CE [16] 
* CE 7 is refined into CE [17] 


### Cost equations --> "Loop" of lastbit/2 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [15] --> Loop 15 

### Ranking functions of CR lastbit(V,Out) 
* RF of phase [15]: [V-1]

#### Partial ranking functions of CR lastbit(V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V-1


### Specialization of cost equations conv/2 
* CE 11 is refined into CE [18,19,20,21,22] 
* CE 10 is refined into CE [23] 


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

### Ranking functions of CR conv(V,Out) 
* RF of phase [17,18,19,20]: [V-1]

#### Partial ranking functions of CR conv(V,Out) 
* Partial RF of phase [17,18,19,20]:
  - RF of loop [17:1,18:1]:
    V/2-1
  - RF of loop [19:1]:
    2*V-5
  - RF of loop [20:1]:
    V-1


### Specialization of cost equations start/1 
* CE 1 is refined into CE [24,25,26,27] 
* CE 2 is refined into CE [28,29,30,31] 
* CE 3 is refined into CE [32,33,34] 


### Cost equations --> "Loop" of start/1 
* CEs [26,27,30,31,34] --> Loop 22 
* CEs [25,29,33] --> Loop 23 
* CEs [24,28,32] --> Loop 24 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of half(V,Out):
* Chain [[12],11]: 1*it(12)+1
  Such that:it(12) =< 2*Out

  with precondition: [V=2*Out,V>=2] 

* Chain [[12],10]: 1*it(12)+1
  Such that:it(12) =< 2*Out

  with precondition: [V=2*Out+1,V>=3] 

* Chain [11]: 1
  with precondition: [V=0,Out=0] 

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


#### Cost of chains of lastbit(V,Out):
* Chain [[15],14]: 1*it(15)+1
  Such that:it(15) =< V

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

* Chain [[15],13]: 1*it(15)+1
  Such that:it(15) =< V

  with precondition: [Out=1,V>=3] 

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

* Chain [13]: 1
  with precondition: [V=1,Out=1] 


#### Cost of chains of conv(V,Out):
* Chain [[17,18,19,20],21,16]: 3*it(17)+3*it(18)+3*it(19)+3*it(20)+2*s(17)+2*s(18)+4*s(21)+4
  Such that:aux(5) =< 2*V+4
aux(13) =< V
aux(14) =< 2*V
aux(15) =< 3*V
aux(16) =< 4*V
aux(17) =< V/2
it(19) =< aux(14)
it(17) =< aux(13)
it(18) =< aux(13)
it(19) =< aux(13)
it(20) =< aux(13)
it(20) =< aux(5)
s(22) =< aux(5)
it(20) =< aux(14)
s(22) =< aux(14)
it(18) =< aux(15)
it(19) =< aux(15)
s(18) =< aux(15)
it(20) =< aux(15)
it(18) =< aux(16)
it(19) =< aux(16)
s(17) =< aux(16)
it(20) =< aux(16)
it(17) =< aux(17)
it(18) =< aux(17)
s(21) =< s(22)

  with precondition: [Out>=4,V+8>=2*Out,V+2>=Out] 

* Chain [21,16]: 4
  with precondition: [V=1,Out=3] 

* Chain [16]: 1
  with precondition: [V=0,Out=1] 


#### Cost of chains of start(V):
* Chain [24]: 1
  with precondition: [V=0] 

* Chain [23]: 4
  with precondition: [V=1] 

* Chain [22]: 4*s(25)+3*s(35)+3*s(36)+3*s(37)+3*s(38)+2*s(40)+2*s(41)+4*s(42)+4
  Such that:s(31) =< 2*V
s(29) =< 2*V+4
s(32) =< 3*V
s(33) =< 4*V
s(34) =< V/2
aux(18) =< V
s(25) =< aux(18)
s(35) =< s(31)
s(36) =< aux(18)
s(37) =< aux(18)
s(35) =< aux(18)
s(38) =< aux(18)
s(38) =< s(29)
s(39) =< s(29)
s(38) =< s(31)
s(39) =< s(31)
s(37) =< s(32)
s(35) =< s(32)
s(40) =< s(32)
s(38) =< s(32)
s(37) =< s(33)
s(35) =< s(33)
s(41) =< s(33)
s(38) =< s(33)
s(36) =< s(34)
s(37) =< s(34)
s(42) =< s(39)

  with precondition: [V>=2] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [24] with precondition: [V=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [23] with precondition: [V=1] 
    - Upper bound: 4 
    - Complexity: constant 
* Chain [22] with precondition: [V>=2] 
    - Upper bound: 41*V+20 
    - Complexity: n 

### Maximum cost of start(V): 41*V+20 
Asymptotic class: n 
* Total analysis performed in 229 ms.


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(10)
BOUNDS(1, n^1)

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(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(x))) -> lastbit(x)
   conv(0) -> cons(nil, 0)
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

S is empty.
Rewrite Strategy: INNERMOST
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(13) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

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

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

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

The result substitution is [ ].




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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(x))) -> lastbit(x)
   conv(0) -> cons(nil, 0)
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

S is empty.
Rewrite Strategy: INNERMOST
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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(x))) -> lastbit(x)
   conv(0) -> cons(nil, 0)
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

S is empty.
Rewrite Strategy: INNERMOST