/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^2), 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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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^2, 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, 167 ms]
    (10) BOUNDS(1, n^2)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) SlicingProof [LOWER BOUND(ID), 0 ms]
    (14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 312 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 49 ms]
            (26) proven lower bound
            (27) LowerBoundPropagationProof [FINISHED, 0 ms]
            (28) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).


The TRS R consists of the following rules:

   naiverev(Cons(x, xs)) -> app(naiverev(xs), Cons(x, Nil))
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   naiverev(Nil) -> Nil
   app(Nil, ys) -> ys
   goal(xs) -> naiverev(xs)

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^2).


The TRS R consists of the following rules:

   naiverev(Cons(x, xs)) -> app(naiverev(xs), Cons(x, Nil)) [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   naiverev(Nil) -> Nil [1]
   app(Nil, ys) -> ys [1]
   goal(xs) -> naiverev(xs) [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:

   naiverev(Cons(x, xs)) -> app(naiverev(xs), Cons(x, Nil)) [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   naiverev(Nil) -> Nil [1]
   app(Nil, ys) -> ys [1]
   goal(xs) -> naiverev(xs) [1]

The TRS has the following type information:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil

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:
none

And the following fresh constants:    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:

   naiverev(Cons(x, xs)) -> app(naiverev(xs), Cons(x, Nil)) [1]
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys)) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   naiverev(Nil) -> Nil [1]
   app(Nil, ys) -> ys [1]
   goal(xs) -> naiverev(xs) [1]

The TRS has the following type information:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
const :: a

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:

   Nil => 0
   True => 1
   False => 0
   const => 0

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

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

   app(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0
   app(z, z') -{ 1 }-> 1 + x + app(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
   goal(z) -{ 1 }-> naiverev(xs) :|: xs >= 0, z = xs
   naiverev(z) -{ 1 }-> app(naiverev(xs), 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0
   naiverev(z) -{ 1 }-> 0 :|: z = 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(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(V, V3),0,[naiverev(V, Out)],[V >= 0]).
eq(start(V, V3),0,[app(V, V3, Out)],[V >= 0,V3 >= 0]).
eq(start(V, V3),0,[notEmpty(V, Out)],[V >= 0]).
eq(start(V, V3),0,[goal(V, Out)],[V >= 0]).
eq(naiverev(V, Out),1,[naiverev(V1, Ret0),app(Ret0, 1 + V2 + 0, Ret)],[Out = Ret,V = 1 + V1 + V2,V1 >= 0,V2 >= 0]).
eq(app(V, V3, Out),1,[app(V5, V6, Ret1)],[Out = 1 + Ret1 + V4,V = 1 + V4 + V5,V5 >= 0,V3 = V6,V6 >= 0,V4 >= 0]).
eq(notEmpty(V, Out),1,[],[Out = 1,V = 1 + V7 + V8,V8 >= 0,V7 >= 0]).
eq(notEmpty(V, Out),1,[],[Out = 0,V = 0]).
eq(naiverev(V, Out),1,[],[Out = 0,V = 0]).
eq(app(V, V3, Out),1,[],[Out = V9,V3 = V9,V9 >= 0,V = 0]).
eq(goal(V, Out),1,[naiverev(V10, Ret2)],[Out = Ret2,V10 >= 0,V = V10]).
input_output_vars(naiverev(V,Out),[V],[Out]).
input_output_vars(app(V,V3,Out),[V,V3],[Out]).
input_output_vars(notEmpty(V,Out),[V],[Out]).
input_output_vars(goal(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [app/3]
1. recursive [non_tail] : [naiverev/2]
2. non_recursive  : [goal/2]
3. non_recursive  : [notEmpty/2]
4. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into app/3
1. SCC is partially evaluated into naiverev/2
2. SCC is completely evaluated into other SCCs
3. SCC is partially evaluated into notEmpty/2
4. SCC is partially evaluated into start/2

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

### Specialization of cost equations app/3 
* CE 8 is refined into CE [11] 
* CE 7 is refined into CE [12] 


### Cost equations --> "Loop" of app/3 
* CEs [12] --> Loop 8 
* CEs [11] --> Loop 9 

### Ranking functions of CR app(V,V3,Out) 
* RF of phase [8]: [V]

#### Partial ranking functions of CR app(V,V3,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1]:
    V


### Specialization of cost equations naiverev/2 
* CE 6 is refined into CE [13] 
* CE 5 is refined into CE [14,15] 


### Cost equations --> "Loop" of naiverev/2 
* CEs [15] --> Loop 10 
* CEs [14] --> Loop 11 
* CEs [13] --> Loop 12 

### Ranking functions of CR naiverev(V,Out) 
* RF of phase [10]: [V]

#### Partial ranking functions of CR naiverev(V,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V


### Specialization of cost equations notEmpty/2 
* CE 9 is refined into CE [16] 
* CE 10 is refined into CE [17] 


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

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

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


### Specialization of cost equations start/2 
* CE 1 is refined into CE [18,19] 
* CE 2 is refined into CE [20,21] 
* CE 3 is refined into CE [22,23] 
* CE 4 is refined into CE [24,25] 


### Cost equations --> "Loop" of start/2 
* CEs [19,21,23,25] --> Loop 15 
* CEs [18,20,22,24] --> Loop 16 

### Ranking functions of CR start(V,V3) 

#### Partial ranking functions of CR start(V,V3) 


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

#### Cost of chains of app(V,V3,Out):
* Chain [[8],9]: 1*it(8)+1
  Such that:it(8) =< -V3+Out

  with precondition: [V+V3=Out,V>=1,V3>=0] 

* Chain [9]: 1
  with precondition: [V=0,V3=Out,V3>=0] 


#### Cost of chains of naiverev(V,Out):
* Chain [[10],11,12]: 2*it(10)+1*s(3)+3
  Such that:aux(3) =< Out
it(10) =< aux(3)
s(3) =< it(10)*aux(3)

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

* Chain [12]: 1
  with precondition: [V=0,Out=0] 

* Chain [11,12]: 3
  with precondition: [V=Out,V>=1] 


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

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


#### Cost of chains of start(V,V3):
* Chain [16]: 2
  with precondition: [V=0] 

* Chain [15]: 5*s(8)+2*s(9)+4
  Such that:aux(4) =< V
s(8) =< aux(4)
s(9) =< s(8)*aux(4)

  with precondition: [V>=1] 


Closed-form bounds of start(V,V3): 
-------------------------------------
* Chain [16] with precondition: [V=0] 
    - Upper bound: 2 
    - Complexity: constant 
* Chain [15] with precondition: [V>=1] 
    - Upper bound: 5*V+4+2*V*V 
    - Complexity: n^2 

### Maximum cost of start(V,V3): 5*V+2+2*V*V+2 
Asymptotic class: n^2 
* Total analysis performed in 100 ms.


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

(10)
BOUNDS(1, n^2)

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

(11) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   naiverev(Cons(x, xs)) -> app(naiverev(xs), Cons(x, Nil))
   app(Cons(x, xs), ys) -> Cons(x, app(xs, ys))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   naiverev(Nil) -> Nil
   app(Nil, ys) -> ys
   goal(xs) -> naiverev(xs)

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

(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
   app(Cons(xs), ys) -> Cons(app(xs, ys))
   notEmpty(Cons(xs)) -> True
   notEmpty(Nil) -> False
   naiverev(Nil) -> Nil
   app(Nil, ys) -> ys
   goal(xs) -> naiverev(xs)

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

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

(16)
Obligation:
Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
naiverev, app

They will be analysed ascendingly in the following order:
app < naiverev

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

(18)
Obligation:
Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
app, naiverev

They will be analysed ascendingly in the following order:
app < naiverev

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) -> gen_Cons:Nil3_0(+(n5_0, b)), rt in Omega(1 + n5_0)

Induction Base:
app(gen_Cons:Nil3_0(0), gen_Cons:Nil3_0(b)) ->_R^Omega(1)
gen_Cons:Nil3_0(b)

Induction Step:
app(gen_Cons:Nil3_0(+(n5_0, 1)), gen_Cons:Nil3_0(b)) ->_R^Omega(1)
Cons(app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b))) ->_IH
Cons(gen_Cons:Nil3_0(+(b, c6_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
app, naiverev

They will be analysed ascendingly in the following order:
app < naiverev

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Lemmas:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) -> gen_Cons:Nil3_0(+(n5_0, b)), rt in Omega(1 + n5_0)


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
naiverev
----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
naiverev(gen_Cons:Nil3_0(n489_0)) -> gen_Cons:Nil3_0(n489_0), rt in Omega(1 + n489_0 + n489_0^2)

Induction Base:
naiverev(gen_Cons:Nil3_0(0)) ->_R^Omega(1)
Nil

Induction Step:
naiverev(gen_Cons:Nil3_0(+(n489_0, 1))) ->_R^Omega(1)
app(naiverev(gen_Cons:Nil3_0(n489_0)), Cons(Nil)) ->_IH
app(gen_Cons:Nil3_0(c490_0), Cons(Nil)) ->_L^Omega(1 + n489_0)
gen_Cons:Nil3_0(+(n489_0, +(0, 1)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(26)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
naiverev(Cons(xs)) -> app(naiverev(xs), Cons(Nil))
app(Cons(xs), ys) -> Cons(app(xs, ys))
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
naiverev(Nil) -> Nil
app(Nil, ys) -> ys
goal(xs) -> naiverev(xs)

Types:
naiverev :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
app :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_True:False2_0 :: True:False
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Lemmas:
app(gen_Cons:Nil3_0(n5_0), gen_Cons:Nil3_0(b)) -> gen_Cons:Nil3_0(+(n5_0, b)), rt in Omega(1 + n5_0)


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
naiverev
----------------------------------------

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

(28)
BOUNDS(n^2, INF)