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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(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, 190 ms]
    (10) BOUNDS(1, n^2)
(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^2).


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(x))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(x))) -> odd(x)
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST
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(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^2).


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(x)) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(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:

   cond(true, x) -> cond(odd(x), p(x)) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond :: true:false -> 0:s -> cond
true :: true:false
odd :: 0:s -> true:false
p :: 0:s -> 0:s
0 :: 0:s
false :: true:false
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:

   cond(v0, v1) -> null_cond [0]

And the following fresh constants:    null_cond

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

(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:

   cond(true, x) -> cond(odd(x), p(x)) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   cond(v0, v1) -> null_cond [0]

The TRS has the following type information:
cond :: true:false -> 0:s -> null_cond
true :: true:false
odd :: 0:s -> true:false
p :: 0:s -> 0:s
0 :: 0:s
false :: true:false
s :: 0:s -> 0:s
null_cond :: null_cond

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:

   true => 1
   0 => 0
   false => 0
   null_cond => 0

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

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

   cond(z, z') -{ 1 }-> cond(odd(x), p(x)) :|: z' = x, z = 1, x >= 0
   cond(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x)
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 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(V1, V),0,[cond(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[odd(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
eq(cond(V1, V, Out),1,[odd(V2, Ret0),p(V2, Ret1),cond(Ret0, Ret1, Ret)],[Out = Ret,V = V2,V1 = 1,V2 >= 0]).
eq(odd(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(odd(V1, Out),1,[],[Out = 1,V1 = 1]).
eq(odd(V1, Out),1,[odd(V3, Ret2)],[Out = Ret2,V3 >= 0,V1 = 2 + V3]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V4,V4 >= 0,V1 = 1 + V4]).
eq(cond(V1, V, Out),0,[],[Out = 0,V6 >= 0,V5 >= 0,V1 = V6,V = V5]).
input_output_vars(cond(V1,V,Out),[V1,V],[Out]).
input_output_vars(odd(V1,Out),[V1],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [odd/2]
1. non_recursive  : [p/2]
2. recursive  : [cond/3]
3. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into odd/2
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into cond/3
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations odd/2 
* CE 8 is refined into CE [11] 
* CE 7 is refined into CE [12] 
* CE 6 is refined into CE [13] 


### Cost equations --> "Loop" of odd/2 
* CEs [12] --> Loop 9 
* CEs [13] --> Loop 10 
* CEs [11] --> Loop 11 

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

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


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


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

### Ranking functions of CR p(V1,Out) 

#### Partial ranking functions of CR p(V1,Out) 


### Specialization of cost equations cond/3 
* CE 5 is refined into CE [16] 
* CE 4 is refined into CE [17,18,19,20] 


### Cost equations --> "Loop" of cond/3 
* CEs [20] --> Loop 14 
* CEs [19] --> Loop 15 
* CEs [18] --> Loop 16 
* CEs [17] --> Loop 17 
* CEs [16] --> Loop 18 

### Ranking functions of CR cond(V1,V,Out) 
* RF of phase [14]: [V-2]

#### Partial ranking functions of CR cond(V1,V,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V-2


### Specialization of cost equations start/2 
* CE 1 is refined into CE [21,22,23] 
* CE 2 is refined into CE [24,25,26,27] 
* CE 3 is refined into CE [28,29] 


### Cost equations --> "Loop" of start/2 
* CEs [21] --> Loop 19 
* CEs [22,23,25,26,27,29] --> Loop 20 
* CEs [24,28] --> Loop 21 

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

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


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

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

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

* Chain [[11],9]: 1*it(11)+1
  Such that:it(11) =< V1

  with precondition: [Out=1,V1>=3] 

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

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


#### Cost of chains of p(V1,Out):
* Chain [13]: 1
  with precondition: [V1=0,Out=0] 

* Chain [12]: 1
  with precondition: [V1=Out+1,V1>=1] 


#### Cost of chains of cond(V1,V,Out):
* Chain [[14],18]: 3*it(14)+1*s(3)+0
  Such that:aux(3) =< V
it(14) =< aux(3)
s(3) =< it(14)*aux(3)

  with precondition: [V1=1,Out=0,V>=3] 

* Chain [[14],15,18]: 4*it(14)+1*s(3)+3
  Such that:aux(4) =< V
it(14) =< aux(4)
s(3) =< it(14)*aux(4)

  with precondition: [V1=1,Out=0,V>=3] 

* Chain [18]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [17,18]: 3
  with precondition: [V1=1,V=0,Out=0] 

* Chain [16,18]: 3
  with precondition: [V1=1,V=1,Out=0] 

* Chain [16,17,18]: 6
  with precondition: [V1=1,V=1,Out=0] 

* Chain [15,18]: 1*s(4)+3
  Such that:s(4) =< V

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


#### Cost of chains of start(V1,V):
* Chain [21]: 1
  with precondition: [V1=0] 

* Chain [20]: 8*s(13)+2*s(14)+2*s(15)+6
  Such that:s(12) =< V
aux(6) =< V1
s(15) =< aux(6)
s(13) =< s(12)
s(14) =< s(13)*s(12)

  with precondition: [V1>=1] 

* Chain [19]: 3
  with precondition: [V1>=0,V>=0] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [21] with precondition: [V1=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [20] with precondition: [V1>=1] 
    - Upper bound: 2*V1+6+nat(V)*8+nat(V)*2*nat(V) 
    - Complexity: n^2 
* Chain [19] with precondition: [V1>=0,V>=0] 
    - Upper bound: 3 
    - Complexity: constant 

### Maximum cost of start(V1,V): max([2,2*V1+5+nat(V)*8+nat(V)*2*nat(V)])+1 
Asymptotic class: n^2 
* Total analysis performed in 137 ms.


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(10)
BOUNDS(1, n^2)

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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^2).


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(x))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(x))) -> odd(x)
   p(0) -> 0
   p(s(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

odd(s(s(x))) ->^+ odd(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^2).


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(x))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(x))) -> odd(x)
   p(0) -> 0
   p(s(x)) -> x

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

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

(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^2).


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(x))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(x))) -> odd(x)
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST