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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 257 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), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 129 ms]
    (12) BOUNDS(1, n^1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 4 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


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

(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:

   D'(t) -> c
   D'(constant) -> c1
   D'(+(z0, z1)) -> c2(D'(z0))
   D'(+(z0, z1)) -> c3(D'(z1))
   D'(*(z0, z1)) -> c4(D'(z0))
   D'(*(z0, z1)) -> c5(D'(z1))
   D'(-(z0, z1)) -> c6(D'(z0))
   D'(-(z0, z1)) -> c7(D'(z1))

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

   D(t) -> 1
   D(constant) -> 0
   D(+(z0, z1)) -> +(D(z0), D(z1))
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1)))
   D(-(z0, z1)) -> -(D(z0), D(z1))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(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:

   D'(t) -> c
   D'(constant) -> c1
   D'(+(z0, z1)) -> c2(D'(z0))
   D'(+(z0, z1)) -> c3(D'(z1))
   D'(*(z0, z1)) -> c4(D'(z0))
   D'(*(z0, z1)) -> c5(D'(z1))
   D'(-(z0, z1)) -> c6(D'(z0))
   D'(-(z0, z1)) -> c7(D'(z1))

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

   D(t) -> 1
   D(constant) -> 0
   D(+(z0, z1)) -> +(D(z0), D(z1))
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1)))
   D(-(z0, z1)) -> -(D(z0), D(z1))

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

(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:

   D'(t) -> c [1]
   D'(constant) -> c1 [1]
   D'(+(z0, z1)) -> c2(D'(z0)) [1]
   D'(+(z0, z1)) -> c3(D'(z1)) [1]
   D'(*(z0, z1)) -> c4(D'(z0)) [1]
   D'(*(z0, z1)) -> c5(D'(z1)) [1]
   D'(-(z0, z1)) -> c6(D'(z0)) [1]
   D'(-(z0, z1)) -> c7(D'(z1)) [1]
   D(t) -> 1 [0]
   D(constant) -> 0 [0]
   D(+(z0, z1)) -> +(D(z0), D(z1)) [0]
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1))) [0]
   D(-(z0, z1)) -> -(D(z0), D(z1)) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   D'(t) -> c [1]
   D'(constant) -> c1 [1]
   D'(+(z0, z1)) -> c2(D'(z0)) [1]
   D'(+(z0, z1)) -> c3(D'(z1)) [1]
   D'(*(z0, z1)) -> c4(D'(z0)) [1]
   D'(*(z0, z1)) -> c5(D'(z1)) [1]
   D'(-(z0, z1)) -> c6(D'(z0)) [1]
   D'(-(z0, z1)) -> c7(D'(z1)) [1]
   D(t) -> 1 [0]
   D(constant) -> 0 [0]
   D(+(z0, z1)) -> +(D(z0), D(z1)) [0]
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1))) [0]
   D(-(z0, z1)) -> -(D(z0), D(z1)) [0]

The TRS has the following type information:
D' :: t:constant:+:*:-:1:0 -> c:c1:c2:c3:c4:c5:c6:c7
t :: t:constant:+:*:-:1:0
c :: c:c1:c2:c3:c4:c5:c6:c7
constant :: t:constant:+:*:-:1:0
c1 :: c:c1:c2:c3:c4:c5:c6:c7
+ :: t:constant:+:*:-:1:0 -> t:constant:+:*:-:1:0 -> t:constant:+:*:-:1:0
c2 :: c:c1:c2:c3:c4:c5:c6:c7 -> c:c1:c2:c3:c4:c5:c6:c7
c3 :: c:c1:c2:c3:c4:c5:c6:c7 -> c:c1:c2:c3:c4:c5:c6:c7
* :: t:constant:+:*:-:1:0 -> t:constant:+:*:-:1:0 -> t:constant:+:*:-:1:0
c4 :: c:c1:c2:c3:c4:c5:c6:c7 -> c:c1:c2:c3:c4:c5:c6:c7
c5 :: c:c1:c2:c3:c4:c5:c6:c7 -> c:c1:c2:c3:c4:c5:c6:c7
- :: t:constant:+:*:-:1:0 -> t:constant:+:*:-:1:0 -> t:constant:+:*:-:1:0
c6 :: c:c1:c2:c3:c4:c5:c6:c7 -> c:c1:c2:c3:c4:c5:c6:c7
c7 :: c:c1:c2:c3:c4:c5:c6:c7 -> c:c1:c2:c3:c4:c5:c6:c7
D :: t:constant:+:*:-:1:0 -> t:constant:+:*:-:1:0
1 :: t:constant:+:*:-:1:0
0 :: t:constant:+:*:-:1:0

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:

   D(v0) -> null_D [0]
   D'(v0) -> null_D' [0]

And the following fresh constants:    null_D, null_D'

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

(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:

   D'(t) -> c [1]
   D'(constant) -> c1 [1]
   D'(+(z0, z1)) -> c2(D'(z0)) [1]
   D'(+(z0, z1)) -> c3(D'(z1)) [1]
   D'(*(z0, z1)) -> c4(D'(z0)) [1]
   D'(*(z0, z1)) -> c5(D'(z1)) [1]
   D'(-(z0, z1)) -> c6(D'(z0)) [1]
   D'(-(z0, z1)) -> c7(D'(z1)) [1]
   D(t) -> 1 [0]
   D(constant) -> 0 [0]
   D(+(z0, z1)) -> +(D(z0), D(z1)) [0]
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1))) [0]
   D(-(z0, z1)) -> -(D(z0), D(z1)) [0]
   D(v0) -> null_D [0]
   D'(v0) -> null_D' [0]

The TRS has the following type information:
D' :: t:constant:+:*:-:1:0:null_D -> c:c1:c2:c3:c4:c5:c6:c7:null_D'
t :: t:constant:+:*:-:1:0:null_D
c :: c:c1:c2:c3:c4:c5:c6:c7:null_D'
constant :: t:constant:+:*:-:1:0:null_D
c1 :: c:c1:c2:c3:c4:c5:c6:c7:null_D'
+ :: t:constant:+:*:-:1:0:null_D -> t:constant:+:*:-:1:0:null_D -> t:constant:+:*:-:1:0:null_D
c2 :: c:c1:c2:c3:c4:c5:c6:c7:null_D' -> c:c1:c2:c3:c4:c5:c6:c7:null_D'
c3 :: c:c1:c2:c3:c4:c5:c6:c7:null_D' -> c:c1:c2:c3:c4:c5:c6:c7:null_D'
* :: t:constant:+:*:-:1:0:null_D -> t:constant:+:*:-:1:0:null_D -> t:constant:+:*:-:1:0:null_D
c4 :: c:c1:c2:c3:c4:c5:c6:c7:null_D' -> c:c1:c2:c3:c4:c5:c6:c7:null_D'
c5 :: c:c1:c2:c3:c4:c5:c6:c7:null_D' -> c:c1:c2:c3:c4:c5:c6:c7:null_D'
- :: t:constant:+:*:-:1:0:null_D -> t:constant:+:*:-:1:0:null_D -> t:constant:+:*:-:1:0:null_D
c6 :: c:c1:c2:c3:c4:c5:c6:c7:null_D' -> c:c1:c2:c3:c4:c5:c6:c7:null_D'
c7 :: c:c1:c2:c3:c4:c5:c6:c7:null_D' -> c:c1:c2:c3:c4:c5:c6:c7:null_D'
D :: t:constant:+:*:-:1:0:null_D -> t:constant:+:*:-:1:0:null_D
1 :: t:constant:+:*:-:1:0:null_D
0 :: t:constant:+:*:-:1:0:null_D
null_D :: t:constant:+:*:-:1:0:null_D
null_D' :: c:c1:c2:c3:c4:c5:c6:c7:null_D'

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:

   t => 3
   c => 0
   constant => 2
   c1 => 1
   1 => 1
   0 => 0
   null_D => 0
   null_D' => 0

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

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

   D(z) -{ 0 }-> 1 :|: z = 3
   D(z) -{ 0 }-> 0 :|: z = 2
   D(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   D(z) -{ 0 }-> 1 + D(z0) + D(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   D(z) -{ 0 }-> 1 + (1 + z1 + D(z0)) + (1 + z0 + D(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   D'(z) -{ 1 }-> 1 :|: z = 2
   D'(z) -{ 1 }-> 0 :|: z = 3
   D'(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   D'(z) -{ 1 }-> 1 + D'(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   D'(z) -{ 1 }-> 1 + D'(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Only complete derivations are relevant for the runtime complexity.

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

(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,[fun1(V, Out)],[V >= 0]).
eq(fun(V, Out),1,[],[Out = 0,V = 3]).
eq(fun(V, Out),1,[],[Out = 1,V = 2]).
eq(fun(V, Out),1,[fun(V2, Ret1)],[Out = 1 + Ret1,V1 >= 0,V2 >= 0,V = 1 + V1 + V2]).
eq(fun(V, Out),1,[fun(V3, Ret11)],[Out = 1 + Ret11,V3 >= 0,V4 >= 0,V = 1 + V3 + V4]).
eq(fun1(V, Out),0,[],[Out = 1,V = 3]).
eq(fun1(V, Out),0,[],[Out = 0,V = 2]).
eq(fun1(V, Out),0,[fun1(V6, Ret01),fun1(V5, Ret12)],[Out = 1 + Ret01 + Ret12,V5 >= 0,V6 >= 0,V = 1 + V5 + V6]).
eq(fun1(V, Out),0,[fun1(V7, Ret011),fun1(V8, Ret111)],[Out = 3 + Ret011 + Ret111 + V7 + V8,V8 >= 0,V7 >= 0,V = 1 + V7 + V8]).
eq(fun1(V, Out),0,[],[Out = 0,V9 >= 0,V = V9]).
eq(fun(V, Out),0,[],[Out = 0,V10 >= 0,V = V10]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fun1(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [fun/2]
1. recursive [multiple] : [fun1/2]
2. non_recursive  : [start/1]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/2
1. SCC is partially evaluated into fun1/2
2. SCC is partially evaluated into start/1

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

### Specialization of cost equations fun/2 
* CE 3 is refined into CE [11] 
* CE 6 is refined into CE [12] 
* CE 4 is refined into CE [13] 
* CE 5 is refined into CE [14] 


### Cost equations --> "Loop" of fun/2 
* CEs [14] --> Loop 9 
* CEs [11,12] --> Loop 10 
* CEs [13] --> Loop 11 

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

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


### Specialization of cost equations fun1/2 
* CE 8 is refined into CE [15] 
* CE 7 is refined into CE [16] 
* CE 10 is refined into CE [17] 
* CE 9 is refined into CE [18] 


### Cost equations --> "Loop" of fun1/2 
* CEs [17] --> Loop 12 
* CEs [18] --> Loop 13 
* CEs [15] --> Loop 14 
* CEs [16] --> Loop 15 

### Ranking functions of CR fun1(V,Out) 
* RF of phase [12,13]: [V]

#### Partial ranking functions of CR fun1(V,Out) 
* Partial RF of phase [12,13]:
  - RF of loop [12:1,12:2,13:1,13:2]:
    V


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


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

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of fun(V,Out):
* Chain [[9],11]: 1*it(9)+1
  Such that:it(9) =< V

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

* Chain [[9],10]: 1*it(9)+1
  Such that:it(9) =< V

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

* Chain [11]: 1
  with precondition: [V=2,Out=1] 

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


#### Cost of chains of fun1(V,Out):
* Chain [15]: 0
  with precondition: [V=3,Out=1] 

* Chain [14]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([12,13],[[15],[14]])]: 0
  with precondition: [V>=1,Out>=1] 


#### Cost of chains of start(V):
* Chain [16]: 2*s(4)+1
  Such that:s(3) =< V
s(4) =< s(3)

  with precondition: [V>=0] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [16] with precondition: [V>=0] 
    - Upper bound: 2*V+1 
    - Complexity: n 

### Maximum cost of start(V): 2*V+1 
Asymptotic class: n 
* Total analysis performed in 114 ms.


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

(12)
BOUNDS(1, n^1)

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

(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:

   D'(t) -> c
   D'(constant) -> c1
   D'(+(z0, z1)) -> c2(D'(z0))
   D'(+(z0, z1)) -> c3(D'(z1))
   D'(*(z0, z1)) -> c4(D'(z0))
   D'(*(z0, z1)) -> c5(D'(z1))
   D'(-(z0, z1)) -> c6(D'(z0))
   D'(-(z0, z1)) -> c7(D'(z1))

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

   D(t) -> 1
   D(constant) -> 0
   D(+(z0, z1)) -> +(D(z0), D(z1))
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1)))
   D(-(z0, z1)) -> -(D(z0), D(z1))

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

D'(+(z0, z1)) ->^+ c3(D'(z1))

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

The pumping substitution is [z1 / +(z0, z1)].

The result substitution is [ ].




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

(16)
Complex Obligation (BEST)

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

(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:

   D'(t) -> c
   D'(constant) -> c1
   D'(+(z0, z1)) -> c2(D'(z0))
   D'(+(z0, z1)) -> c3(D'(z1))
   D'(*(z0, z1)) -> c4(D'(z0))
   D'(*(z0, z1)) -> c5(D'(z1))
   D'(-(z0, z1)) -> c6(D'(z0))
   D'(-(z0, z1)) -> c7(D'(z1))

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

   D(t) -> 1
   D(constant) -> 0
   D(+(z0, z1)) -> +(D(z0), D(z1))
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1)))
   D(-(z0, z1)) -> -(D(z0), D(z1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

(19)
BOUNDS(n^1, INF)

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

(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:

   D'(t) -> c
   D'(constant) -> c1
   D'(+(z0, z1)) -> c2(D'(z0))
   D'(+(z0, z1)) -> c3(D'(z1))
   D'(*(z0, z1)) -> c4(D'(z0))
   D'(*(z0, z1)) -> c5(D'(z1))
   D'(-(z0, z1)) -> c6(D'(z0))
   D'(-(z0, z1)) -> c7(D'(z1))

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

   D(t) -> 1
   D(constant) -> 0
   D(+(z0, z1)) -> +(D(z0), D(z1))
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1)))
   D(-(z0, z1)) -> -(D(z0), D(z1))

Rewrite Strategy: INNERMOST