/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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, 187 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


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

(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:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

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

(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:

   from(X) -> cons(X, n__from(s(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
from :: s:0 -> n__from:cons
cons :: s:0 -> n__from:cons -> n__from:cons
n__from :: s:0 -> n__from:cons
s :: s:0 -> s:0
sel :: s:0 -> n__from:cons -> s:0
0 :: s:0
activate :: n__from:cons -> n__from:cons

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

(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:

   sel(v0, v1) -> null_sel [0]

And the following fresh constants:    null_sel, const

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

(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:

   from(X) -> cons(X, n__from(s(X))) [1]
   sel(0, cons(X, Y)) -> X [1]
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]
   sel(v0, v1) -> null_sel [0]

The TRS has the following type information:
from :: s:0:null_sel -> n__from:cons
cons :: s:0:null_sel -> n__from:cons -> n__from:cons
n__from :: s:0:null_sel -> n__from:cons
s :: s:0:null_sel -> s:0:null_sel
sel :: s:0:null_sel -> n__from:cons -> s:0:null_sel
0 :: s:0:null_sel
activate :: n__from:cons -> n__from:cons
null_sel :: s:0:null_sel
const :: n__from:cons

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

(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
   null_sel => 0
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
   sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
   sel(z, z') -{ 1 }-> sel(X, activate(Z)) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
   sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

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, V1),0,[from(V, Out)],[V >= 0]).
eq(start(V, V1),0,[sel(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[activate(V, Out)],[V >= 0]).
eq(from(V, Out),1,[],[Out = 3 + 2*X1,X1 >= 0,V = X1]).
eq(sel(V, V1, Out),1,[],[Out = X2,Y1 >= 0,X2 >= 0,V1 = 1 + X2 + Y1,V = 0]).
eq(sel(V, V1, Out),1,[activate(Z1, Ret1),sel(X3, Ret1, Ret)],[Out = Ret,Z1 >= 0,V = 1 + X3,Y2 >= 0,X3 >= 0,V1 = 1 + Y2 + Z1]).
eq(from(V, Out),1,[],[Out = 1 + X4,X4 >= 0,V = X4]).
eq(activate(V, Out),1,[from(X5, Ret2)],[Out = Ret2,V = 1 + X5,X5 >= 0]).
eq(activate(V, Out),1,[],[Out = X6,X6 >= 0,V = X6]).
eq(sel(V, V1, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V = V3,V1 = V2]).
input_output_vars(from(V,Out),[V],[Out]).
input_output_vars(sel(V,V1,Out),[V,V1],[Out]).
input_output_vars(activate(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [from/2]
1. non_recursive  : [activate/2]
2. recursive  : [sel/3]
3. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into from/2
1. SCC is partially evaluated into activate/2
2. SCC is partially evaluated into sel/3
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations from/2 
* CE 4 is refined into CE [11] 
* CE 5 is refined into CE [12] 


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

### Ranking functions of CR from(V,Out) 

#### Partial ranking functions of CR from(V,Out) 


### Specialization of cost equations activate/2 
* CE 9 is refined into CE [13,14] 
* CE 10 is refined into CE [15] 


### Cost equations --> "Loop" of activate/2 
* CEs [14] --> Loop 11 
* CEs [13,15] --> Loop 12 

### Ranking functions of CR activate(V,Out) 

#### Partial ranking functions of CR activate(V,Out) 


### Specialization of cost equations sel/3 
* CE 8 is refined into CE [16] 
* CE 6 is refined into CE [17] 
* CE 7 is refined into CE [18,19] 


### Cost equations --> "Loop" of sel/3 
* CEs [19] --> Loop 13 
* CEs [18] --> Loop 14 
* CEs [16] --> Loop 15 
* CEs [17] --> Loop 16 

### Ranking functions of CR sel(V,V1,Out) 
* RF of phase [13,14]: [V]

#### Partial ranking functions of CR sel(V,V1,Out) 
* Partial RF of phase [13,14]:
  - RF of loop [13:1,14:1]:
    V
  - RF of loop [14:1]:
    V1 depends on loops [13:1] 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [20,21] 
* CE 2 is refined into CE [22,23,24] 
* CE 3 is refined into CE [25,26] 


### Cost equations --> "Loop" of start/2 
* CEs [20,21,22,23,24,25,26] --> Loop 17 

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

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


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

#### Cost of chains of from(V,Out):
* Chain [10]: 1
  with precondition: [V+1=Out,V>=0] 

* Chain [9]: 1
  with precondition: [2*V+3=Out,V>=0] 


#### Cost of chains of activate(V,Out):
* Chain [12]: 2
  with precondition: [V=Out,V>=0] 

* Chain [11]: 2
  with precondition: [2*V+1=Out,V>=1] 


#### Cost of chains of sel(V,V1,Out):
* Chain [[13,14],16]: 6*it(13)+1
  Such that:aux(9) =< V
it(13) =< aux(9)

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

* Chain [[13,14],15]: 6*it(13)+0
  Such that:aux(10) =< V
it(13) =< aux(10)

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

* Chain [16]: 1
  with precondition: [V=0,Out>=0,V1>=Out+1] 

* Chain [15]: 0
  with precondition: [Out=0,V>=0,V1>=0] 


#### Cost of chains of start(V,V1):
* Chain [17]: 12*s(4)+2
  Such that:aux(11) =< V
s(4) =< aux(11)

  with precondition: [V>=0] 


Closed-form bounds of start(V,V1): 
-------------------------------------
* Chain [17] with precondition: [V>=0] 
    - Upper bound: 12*V+2 
    - Complexity: n 

### Maximum cost of start(V,V1): 12*V+2 
Asymptotic class: n 
* Total analysis performed in 106 ms.


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

(10)
BOUNDS(1, n^1)

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

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

(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:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

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

The rewrite sequence

sel(s(X), cons(Y, Z)) ->^+ sel(X, Z)

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

The pumping substitution is [X / s(X), Z / cons(Y, Z)].

The result substitution is [ ].




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

(14)
Complex Obligation (BEST)

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

(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:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

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

(17)
BOUNDS(n^1, INF)

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

(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:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST