/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 199 ms]
    (10) BOUNDS(1, n^1)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 47 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:

   nonZero(0) -> false
   nonZero(s(x)) -> true
   p(0) -> 0
   p(s(x)) -> x
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0)
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(y))

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:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(y)) [1]

Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

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

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(y)) [1]

The TRS has the following type information:
nonZero :: 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
p :: 0:s -> 0:s
id_inc :: 0:s -> 0:s
random :: 0:s -> 0:s
rand :: 0:s -> 0:s -> 0:s
if :: false:true -> 0:s -> 0:s -> 0:s

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:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(y)) [1]

The TRS has the following type information:
nonZero :: 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
p :: 0:s -> 0:s
id_inc :: 0:s -> 0:s
random :: 0:s -> 0:s
rand :: 0:s -> 0:s -> 0:s
if :: false:true -> 0:s -> 0:s -> 0:s

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
   false => 0
   true => 1

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

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

   id_inc(z) -{ 1 }-> x :|: x >= 0, z = x
   id_inc(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if(z, z', z'') -{ 1 }-> rand(p(x), id_inc(y)) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   nonZero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   rand(z, z') -{ 1 }-> if(nonZero(x), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   random(z) -{ 1 }-> rand(x, 0) :|: x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

eq(start(V, V6, V9),0,[nonZero(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[p(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[random(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[rand(V, V6, Out)],[V >= 0,V6 >= 0]).
eq(start(V, V6, V9),0,[if(V, V6, V9, Out)],[V >= 0,V6 >= 0,V9 >= 0]).
eq(nonZero(V, Out),1,[],[Out = 0,V = 0]).
eq(nonZero(V, Out),1,[],[Out = 1,V1 >= 0,V = 1 + V1]).
eq(p(V, Out),1,[],[Out = 0,V = 0]).
eq(p(V, Out),1,[],[Out = V2,V2 >= 0,V = 1 + V2]).
eq(fun(V, Out),1,[],[Out = V3,V3 >= 0,V = V3]).
eq(fun(V, Out),1,[],[Out = 1 + V4,V4 >= 0,V = V4]).
eq(random(V, Out),1,[rand(V5, 0, Ret)],[Out = Ret,V5 >= 0,V = V5]).
eq(rand(V, V6, Out),1,[nonZero(V7, Ret0),if(Ret0, V7, V8, Ret1)],[Out = Ret1,V7 >= 0,V8 >= 0,V = V7,V6 = V8]).
eq(if(V, V6, V9, Out),1,[],[Out = V11,V6 = V10,V9 = V11,V10 >= 0,V11 >= 0,V = 0]).
eq(if(V, V6, V9, Out),1,[p(V13, Ret01),fun(V12, Ret11),rand(Ret01, Ret11, Ret2)],[Out = Ret2,V6 = V13,V9 = V12,V = 1,V13 >= 0,V12 >= 0]).
input_output_vars(nonZero(V,Out),[V],[Out]).
input_output_vars(p(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(random(V,Out),[V],[Out]).
input_output_vars(rand(V,V6,Out),[V,V6],[Out]).
input_output_vars(if(V,V6,V9,Out),[V,V6,V9],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [fun/2]
1. non_recursive  : [p/2]
2. non_recursive  : [nonZero/2]
3. recursive  : [if/4,rand/3]
4. non_recursive  : [random/2]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/2
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into nonZero/2
3. SCC is partially evaluated into rand/3
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations fun/2 
* CE 10 is refined into CE [16] 
* CE 11 is refined into CE [17] 


### Cost equations --> "Loop" of fun/2 
* CEs [16] --> Loop 11 
* CEs [17] --> Loop 12 

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

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


### Specialization of cost equations p/2 
* CE 9 is refined into CE [18] 
* CE 8 is refined into CE [19] 


### Cost equations --> "Loop" of p/2 
* CEs [18] --> Loop 13 
* CEs [19] --> Loop 14 

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

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


### Specialization of cost equations nonZero/2 
* CE 15 is refined into CE [20] 
* CE 14 is refined into CE [21] 


### Cost equations --> "Loop" of nonZero/2 
* CEs [20] --> Loop 15 
* CEs [21] --> Loop 16 

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

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


### Specialization of cost equations rand/3 
* CE 13 is refined into CE [22] 
* CE 12 is refined into CE [23,24] 


### Cost equations --> "Loop" of rand/3 
* CEs [24] --> Loop 17 
* CEs [23] --> Loop 18 
* CEs [22] --> Loop 19 

### Ranking functions of CR rand(V,V6,Out) 
* RF of phase [17,18]: [V]

#### Partial ranking functions of CR rand(V,V6,Out) 
* Partial RF of phase [17,18]:
  - RF of loop [17:1,18:1]:
    V


### Specialization of cost equations start/3 
* CE 1 is refined into CE [25,26,27,28,29,30] 
* CE 2 is refined into CE [31] 
* CE 3 is refined into CE [32,33] 
* CE 4 is refined into CE [34,35] 
* CE 5 is refined into CE [36,37] 
* CE 6 is refined into CE [38,39] 
* CE 7 is refined into CE [40,41] 


### Cost equations --> "Loop" of start/3 
* CEs [28,30] --> Loop 20 
* CEs [27,29] --> Loop 21 
* CEs [25,26] --> Loop 22 
* CEs [31,32,33,34,35,36,37,38,39,40,41] --> Loop 23 

### Ranking functions of CR start(V,V6,V9) 

#### Partial ranking functions of CR start(V,V6,V9) 


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

#### Cost of chains of fun(V,Out):
* Chain [12]: 1
  with precondition: [V+1=Out,V>=0] 

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


#### Cost of chains of p(V,Out):
* Chain [14]: 1
  with precondition: [V=0,Out=0] 

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


#### Cost of chains of nonZero(V,Out):
* Chain [16]: 1
  with precondition: [V=0,Out=0] 

* Chain [15]: 1
  with precondition: [Out=1,V>=1] 


#### Cost of chains of rand(V,V6,Out):
* Chain [[17,18],19]: 10*it(17)+3
  Such that:aux(3) =< V
it(17) =< aux(3)

  with precondition: [V>=1,V6>=0,Out>=V6,V+V6>=Out] 

* Chain [19]: 3
  with precondition: [V=0,V6=Out,V6>=0] 


#### Cost of chains of start(V,V6,V9):
* Chain [23]: 20*s(2)+4
  Such that:aux(4) =< V
s(2) =< aux(4)

  with precondition: [V>=0] 

* Chain [22]: 6
  with precondition: [V=1,V6=0,V9>=0] 

* Chain [21]: 6
  with precondition: [V=1,V6=1,V9>=0] 

* Chain [20]: 20*s(6)+6
  Such that:aux(5) =< V6
s(6) =< aux(5)

  with precondition: [V=1,V6>=2,V9>=0] 


Closed-form bounds of start(V,V6,V9): 
-------------------------------------
* Chain [23] with precondition: [V>=0] 
    - Upper bound: 20*V+4 
    - Complexity: n 
* Chain [22] with precondition: [V=1,V6=0,V9>=0] 
    - Upper bound: 6 
    - Complexity: constant 
* Chain [21] with precondition: [V=1,V6=1,V9>=0] 
    - Upper bound: 6 
    - Complexity: constant 
* Chain [20] with precondition: [V=1,V6>=2,V9>=0] 
    - Upper bound: 20*V6+6 
    - Complexity: n 

### Maximum cost of start(V,V6,V9): max([20*V,2,nat(V6)*20+2])+4 
Asymptotic class: n 
* Total analysis performed in 135 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.
----------------------------------------

(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:

   nonZero(0) -> false
   nonZero(s(x)) -> true
   p(0) -> 0
   p(s(x)) -> x
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0)
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(y))

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

rand(s(x1_0), y) ->^+ rand(x1_0, id_inc(y))

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

The pumping substitution is [x1_0 / s(x1_0)].

The result substitution is [y / id_inc(y)].




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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:

   nonZero(0) -> false
   nonZero(s(x)) -> true
   p(0) -> 0
   p(s(x)) -> x
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0)
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(y))

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:

   nonZero(0) -> false
   nonZero(s(x)) -> true
   p(0) -> 0
   p(s(x)) -> x
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0)
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(y))

S is empty.
Rewrite Strategy: INNERMOST