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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 209 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 7 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 97 ms]
    (12) BOUNDS(1, n^1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) 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:

   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))

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

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))

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:

   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))

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

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))

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:

   FIB(0) -> c [1]
   FIB(s(0)) -> c1 [1]
   FIB(s(s(z0))) -> c2(FIB(s(z0))) [1]
   FIB(s(s(z0))) -> c3(FIB(z0)) [1]
   fib(0) -> 0 [0]
   fib(s(0)) -> s(0) [0]
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) [0]

Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   FIB(0) -> c [1]
   FIB(s(0)) -> c1 [1]
   FIB(s(s(z0))) -> c2(FIB(s(z0))) [1]
   FIB(s(s(z0))) -> c3(FIB(z0)) [1]
   fib(0) -> 0 [0]
   fib(s(0)) -> s(0) [0]
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) [0]

The TRS has the following type information:
FIB :: 0:s:+ -> c:c1:c2:c3
0 :: 0:s:+
c :: c:c1:c2:c3
s :: 0:s:+ -> 0:s:+
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
fib :: 0:s:+ -> 0:s:+
+ :: 0:s:+ -> 0:s:+ -> 0:s:+

Rewrite Strategy: INNERMOST
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(7) 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:

   fib(v0) -> null_fib [0]
   FIB(v0) -> null_FIB [0]

And the following fresh constants:    null_fib, null_FIB

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(8)
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:

   FIB(0) -> c [1]
   FIB(s(0)) -> c1 [1]
   FIB(s(s(z0))) -> c2(FIB(s(z0))) [1]
   FIB(s(s(z0))) -> c3(FIB(z0)) [1]
   fib(0) -> 0 [0]
   fib(s(0)) -> s(0) [0]
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0)) [0]
   fib(v0) -> null_fib [0]
   FIB(v0) -> null_FIB [0]

The TRS has the following type information:
FIB :: 0:s:+:null_fib -> c:c1:c2:c3:null_FIB
0 :: 0:s:+:null_fib
c :: c:c1:c2:c3:null_FIB
s :: 0:s:+:null_fib -> 0:s:+:null_fib
c1 :: c:c1:c2:c3:null_FIB
c2 :: c:c1:c2:c3:null_FIB -> c:c1:c2:c3:null_FIB
c3 :: c:c1:c2:c3:null_FIB -> c:c1:c2:c3:null_FIB
fib :: 0:s:+:null_fib -> 0:s:+:null_fib
+ :: 0:s:+:null_fib -> 0:s:+:null_fib -> 0:s:+:null_fib
null_fib :: 0:s:+:null_fib
null_FIB :: c:c1:c2:c3:null_FIB

Rewrite Strategy: INNERMOST
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(9) 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
   c => 0
   c1 => 1
   null_fib => 0
   null_FIB => 0

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(10)
Obligation:
Complexity RNTS consisting of the following rules:

   FIB(z) -{ 1 }-> 1 :|: z = 1 + 0
   FIB(z) -{ 1 }-> 0 :|: z = 0
   FIB(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   FIB(z) -{ 1 }-> 1 + FIB(z0) :|: z0 >= 0, z = 1 + (1 + z0)
   FIB(z) -{ 1 }-> 1 + FIB(1 + z0) :|: z0 >= 0, z = 1 + (1 + z0)
   fib(z) -{ 0 }-> 0 :|: z = 0
   fib(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   fib(z) -{ 0 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 0 }-> 1 + fib(1 + z0) + fib(z0) :|: z0 >= 0, z = 1 + (1 + z0)

Only complete derivations are relevant for the runtime complexity.

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

eq(start(V),0,[fun(V, Out)],[V >= 0]).
eq(start(V),0,[fib(V, Out)],[V >= 0]).
eq(fun(V, Out),1,[],[Out = 0,V = 0]).
eq(fun(V, Out),1,[],[Out = 1,V = 1]).
eq(fun(V, Out),1,[fun(1 + V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
eq(fun(V, Out),1,[fun(V2, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 2 + V2]).
eq(fib(V, Out),0,[],[Out = 0,V = 0]).
eq(fib(V, Out),0,[],[Out = 1,V = 1]).
eq(fib(V, Out),0,[fib(1 + V3, Ret01),fib(V3, Ret12)],[Out = 1 + Ret01 + Ret12,V3 >= 0,V = 2 + V3]).
eq(fib(V, Out),0,[],[Out = 0,V4 >= 0,V = V4]).
eq(fun(V, Out),0,[],[Out = 0,V5 >= 0,V = V5]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fib(V,Out),[V],[Out]).


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

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

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

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

### Specialization of cost equations fib/2 
* CE 8 is refined into CE [11] 
* CE 9 is refined into CE [12] 
* CE 10 is refined into CE [13] 


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

### Ranking functions of CR fib(V,Out) 
* RF of phase [9]: [V-1]

#### Partial ranking functions of CR fib(V,Out) 
* Partial RF of phase [9]:
  - RF of loop [9:1,9:2]:
    V-1


### Specialization of cost equations fun/2 
* CE 4 is refined into CE [14] 
* CE 3 is refined into CE [15] 
* CE 7 is refined into CE [16] 
* CE 5 is refined into CE [17] 
* CE 6 is refined into CE [18] 


### Cost equations --> "Loop" of fun/2 
* CEs [17] --> Loop 12 
* CEs [18] --> Loop 13 
* CEs [14] --> Loop 14 
* CEs [15,16] --> Loop 15 

### Ranking functions of CR fun(V,Out) 
* RF of phase [12,13]: [V-1]

#### Partial ranking functions of CR fun(V,Out) 
* Partial RF of phase [12,13]:
  - RF of loop [12:1,13:1]:
    V-1


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


### Cost equations --> "Loop" of start/1 
* CEs [19,20,21,22,23,24,25] --> Loop 16 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of fib(V,Out):
* Chain [11]: 0
  with precondition: [V=1,Out=1] 

* Chain [10]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([9],[[11],[10]])]: 0
  with precondition: [V>=2,Out>=1] 


#### Cost of chains of fun(V,Out):
* Chain [[12,13],15]: 2*it(12)+1
  Such that:aux(8) =< V
it(12) =< aux(8)

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

* Chain [[12,13],14]: 2*it(12)+1
  Such that:aux(9) =< V
it(12) =< aux(9)

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

* Chain [15]: 1
  with precondition: [Out=0,V>=0] 

* Chain [14]: 1
  with precondition: [V=1,Out=1] 


#### Cost of chains of start(V):
* Chain [16]: 4*s(2)+1
  Such that:aux(10) =< V
s(2) =< aux(10)

  with precondition: [V>=0] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [16] with precondition: [V>=0] 
    - Upper bound: 4*V+1 
    - Complexity: n 

### Maximum cost of start(V): 4*V+1 
Asymptotic class: n 
* Total analysis performed in 107 ms.


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(12)
BOUNDS(1, n^1)

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(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(14)
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:

   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))

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

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))

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

The rewrite sequence

FIB(s(s(z0))) ->^+ c3(FIB(z0))

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

The pumping substitution is [z0 / s(s(z0))].

The result substitution is [ ].




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(16)
Complex Obligation (BEST)

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(17)
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:

   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))

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

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))

Rewrite Strategy: INNERMOST
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(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(19)
BOUNDS(n^1, INF)

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(20)
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:

   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))

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

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))

Rewrite Strategy: INNERMOST