WORST_CASE(Omega(n^1),O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 228 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 160 ms]
    (14) BOUNDS(1, n^1)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) TRS for Loop Detection


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


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))

The (relative) TRS S consists of the following rules:

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))

The (relative) TRS S consists of the following rules:

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))

Rewrite Strategy: INNERMOST
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(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(4)
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:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1)) [1]
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2)) [1]
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2)) [0]

Rewrite Strategy: INNERMOST
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(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

* => times

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(6)
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:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1)) [1]
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2)) [1]
   times(z0, +(z1, z2)) -> +(times(z0, z1), times(z0, z2)) [0]

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1)) [1]
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2)) [1]
   times(z0, +(z1, z2)) -> +(times(z0, z1), times(z0, z2)) [0]

The TRS has the following type information:
*' :: a -> + -> c:c1
+ :: + -> + -> +
c :: c:c1 -> c:c1
c1 :: c:c1 -> c:c1
times :: b -> + -> +

Rewrite Strategy: INNERMOST
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(9) 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:

   times(v0, v1) -> null_times [0]
   *'(v0, v1) -> null_*' [0]

And the following fresh constants:    null_times, null_*', const, const1

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(10)
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:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1)) [1]
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2)) [1]
   times(z0, +(z1, z2)) -> +(times(z0, z1), times(z0, z2)) [0]
   times(v0, v1) -> null_times [0]
   *'(v0, v1) -> null_*' [0]

The TRS has the following type information:
*' :: a -> +:null_times -> c:c1:null_*'
+ :: +:null_times -> +:null_times -> +:null_times
c :: c:c1:null_*' -> c:c1:null_*'
c1 :: c:c1:null_*' -> c:c1:null_*'
times :: b -> +:null_times -> +:null_times
null_times :: +:null_times
null_*' :: c:c1:null_*'
const :: a
const1 :: b

Rewrite Strategy: INNERMOST
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(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   null_times => 0
   null_*' => 0
   const => 0
   const1 => 0

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(12)
Obligation:
Complexity RNTS consisting of the following rules:

   *'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   *'(z, z') -{ 1 }-> 1 + *'(z0, z1) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   *'(z, z') -{ 1 }-> 1 + *'(z0, z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   times(z, z') -{ 0 }-> 1 + times(z0, z1) + times(z0, z2) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0

Only complete derivations are relevant for the runtime complexity.

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

eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[times(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V1 = V3,V2 >= 0,V = 1 + V2 + V4,V3 >= 0,V4 >= 0]).
eq(fun(V1, V, Out),1,[fun(V7, V5, Ret11)],[Out = 1 + Ret11,V1 = V7,V6 >= 0,V = 1 + V5 + V6,V7 >= 0,V5 >= 0]).
eq(times(V1, V, Out),0,[times(V10, V9, Ret01),times(V10, V8, Ret12)],[Out = 1 + Ret01 + Ret12,V1 = V10,V9 >= 0,V = 1 + V8 + V9,V10 >= 0,V8 >= 0]).
eq(times(V1, V, Out),0,[],[Out = 0,V12 >= 0,V11 >= 0,V1 = V12,V = V11]).
eq(fun(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(times(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [fun/3]
1. recursive [multiple] : [times/3]
2. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into times/3
2. SCC is partially evaluated into start/2

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

### Specialization of cost equations fun/3 
* CE 4 is refined into CE [7] 
* CE 3 is refined into CE [8] 


### Cost equations --> "Loop" of fun/3 
* CEs [8] --> Loop 6 
* CEs [7] --> Loop 7 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [6]: [V]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [6]:
  - RF of loop [6:1]:
    V


### Specialization of cost equations times/3 
* CE 6 is refined into CE [9] 
* CE 5 is refined into CE [10] 


### Cost equations --> "Loop" of times/3 
* CEs [10] --> Loop 8 
* CEs [9] --> Loop 9 

### Ranking functions of CR times(V1,V,Out) 
* RF of phase [8]: [V]

#### Partial ranking functions of CR times(V1,V,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1,8:2]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [11,12] 
* CE 2 is refined into CE [13,14] 


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

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

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


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

#### Cost of chains of fun(V1,V,Out):
* Chain [[6],7]: 1*it(6)+0
  Such that:it(6) =< V

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

* Chain [7]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of times(V1,V,Out):
* Chain [9]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [multiple([8],[[9]])]: 0
  with precondition: [V1>=0,Out>=1,V>=Out] 


#### Cost of chains of start(V1,V):
* Chain [10]: 1*s(1)+0
  Such that:s(1) =< V

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [10] with precondition: [V1>=0,V>=0] 
    - Upper bound: V 
    - Complexity: n 

### Maximum cost of start(V1,V): V 
Asymptotic class: n 
* Total analysis performed in 93 ms.


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(14)
BOUNDS(1, n^1)

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(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(16)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))

The (relative) TRS S consists of the following rules:

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))

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

The rewrite sequence

*'(z0, +(z1, z2)) ->^+ c(*'(z0, z1))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [z1 / +(z1, z2)].

The result substitution is [ ].




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(18)
Complex Obligation (BEST)

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(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))

The (relative) TRS S consists of the following rules:

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))

Rewrite Strategy: INNERMOST
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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))

The (relative) TRS S consists of the following rules:

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))

Rewrite Strategy: INNERMOST