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


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


WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 150 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedTrs
(11) CompletionProof [UPPER BOUND(ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(14) CpxRNTS
(15) CompleteCoflocoProof [FINISHED, 212 ms]
(16) BOUNDS(1, n^1)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(x, x) -> f(g(x), x)
   g(x) -> s(x)

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(x, x) -> f(g(x), x)
   g(x) -> s(x)

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(x, x) -> f(g(x), x)
   g(x) -> s(x)

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(5) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(x, x) -> f(g(x), x)
   g(x) -> s(x)

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))

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

(7) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(8)
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:

   f(x, x) -> f(g(x), x) [1]
   g(x) -> s(x) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, x) -> f(g(x), x) [1]
   g(x) -> s(x) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]

The TRS has the following type information:
f :: s:cons_f:cons_g -> s:cons_f:cons_g -> s:cons_f:cons_g
g :: s:cons_f:cons_g -> s:cons_f:cons_g
s :: s:cons_f:cons_g -> s:cons_f:cons_g
encArg :: s:cons_f:cons_g -> s:cons_f:cons_g
cons_f :: s:cons_f:cons_g -> s:cons_f:cons_g -> s:cons_f:cons_g
cons_g :: s:cons_f:cons_g -> s:cons_f:cons_g
encode_f :: s:cons_f:cons_g -> s:cons_f:cons_g -> s:cons_f:cons_g
encode_g :: s:cons_f:cons_g -> s:cons_f:cons_g
encode_s :: s:cons_f:cons_g -> s:cons_f:cons_g

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

(11) 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:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_g(v0) -> null_encode_g [0]
   encode_s(v0) -> null_encode_s [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_g, null_encode_s, null_f

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

(12)
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:

   f(x, x) -> f(g(x), x) [1]
   g(x) -> s(x) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_g(v0) -> null_encode_g [0]
   encode_s(v0) -> null_encode_s [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
g :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
s :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
encArg :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
cons_f :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
cons_g :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
encode_f :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
encode_g :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
encode_s :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f -> s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
null_encArg :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
null_encode_f :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
null_encode_g :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
null_encode_s :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f
null_f :: s:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_s:null_f

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

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

   null_encArg => 0
   null_encode_f => 0
   null_encode_g => 0
   null_encode_s => 0
   null_f => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   encArg(z) -{ 0 }-> g(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z) -{ 0 }-> g(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   f(z, z') -{ 1 }-> f(g(x), x) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[f(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[g(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun2(V1, Out)],[V1 >= 0]).
eq(f(V1, V, Out),1,[g(V2, Ret0),f(Ret0, V2, Ret)],[Out = Ret,V = V2,V2 >= 0,V1 = V2]).
eq(g(V1, Out),1,[],[Out = 1 + V3,V3 >= 0,V1 = V3]).
eq(encArg(V1, Out),0,[encArg(V4, Ret1)],[Out = 1 + Ret1,V1 = 1 + V4,V4 >= 0]).
eq(encArg(V1, Out),0,[encArg(V5, Ret01),encArg(V6, Ret11),f(Ret01, Ret11, Ret2)],[Out = Ret2,V5 >= 0,V1 = 1 + V5 + V6,V6 >= 0]).
eq(encArg(V1, Out),0,[encArg(V7, Ret02),g(Ret02, Ret3)],[Out = Ret3,V1 = 1 + V7,V7 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V9, Ret03),encArg(V8, Ret12),f(Ret03, Ret12, Ret4)],[Out = Ret4,V9 >= 0,V8 >= 0,V1 = V9,V = V8]).
eq(fun1(V1, Out),0,[encArg(V10, Ret04),g(Ret04, Ret5)],[Out = Ret5,V10 >= 0,V1 = V10]).
eq(fun2(V1, Out),0,[encArg(V11, Ret13)],[Out = 1 + Ret13,V11 >= 0,V1 = V11]).
eq(encArg(V1, Out),0,[],[Out = 0,V12 >= 0,V1 = V12]).
eq(fun(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(fun1(V1, Out),0,[],[Out = 0,V15 >= 0,V1 = V15]).
eq(fun2(V1, Out),0,[],[Out = 0,V16 >= 0,V1 = V16]).
eq(f(V1, V, Out),0,[],[Out = 0,V17 >= 0,V18 >= 0,V1 = V17,V = V18]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(g(V1,Out),[V1],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,Out),[V1],[Out]).
input_output_vars(fun2(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [g/2]
1. recursive  : [f/3]
2. recursive [non_tail,multiple] : [encArg/2]
3. non_recursive  : [fun/3]
4. non_recursive  : [fun1/2]
5. non_recursive  : [fun2/2]
6. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into f/3
2. SCC is partially evaluated into encArg/2
3. SCC is partially evaluated into fun/3
4. SCC is partially evaluated into fun1/2
5. SCC is partially evaluated into fun2/2
6. SCC is partially evaluated into start/2

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

### Specialization of cost equations f/3 
* CE 8 is refined into CE [19] 
* CE 7 is refined into CE [20] 


### Cost equations --> "Loop" of f/3 
* CEs [20] --> Loop 11 
* CEs [19] --> Loop 12 

### Ranking functions of CR f(V1,V,Out) 

#### Partial ranking functions of CR f(V1,V,Out) 


### Specialization of cost equations encArg/2 
* CE 12 is refined into CE [21] 
* CE 9 is refined into CE [22] 
* CE 11 is refined into CE [23] 
* CE 10 is refined into CE [24] 


### Cost equations --> "Loop" of encArg/2 
* CEs [24] --> Loop 13 
* CEs [22,23] --> Loop 14 
* CEs [21] --> Loop 15 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [13,14]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [13,14]:
  - RF of loop [13:1,13:2,14:1]:
    V1


### Specialization of cost equations fun/3 
* CE 13 is refined into CE [25,26,27,28] 
* CE 14 is refined into CE [29] 


### Cost equations --> "Loop" of fun/3 
* CEs [25,26,27,28,29] --> Loop 16 

### Ranking functions of CR fun(V1,V,Out) 

#### Partial ranking functions of CR fun(V1,V,Out) 


### Specialization of cost equations fun1/2 
* CE 15 is refined into CE [30,31] 
* CE 16 is refined into CE [32] 


### Cost equations --> "Loop" of fun1/2 
* CEs [31] --> Loop 17 
* CEs [30] --> Loop 18 
* CEs [32] --> Loop 19 

### Ranking functions of CR fun1(V1,Out) 

#### Partial ranking functions of CR fun1(V1,Out) 


### Specialization of cost equations fun2/2 
* CE 17 is refined into CE [33,34] 
* CE 18 is refined into CE [35] 


### Cost equations --> "Loop" of fun2/2 
* CEs [34] --> Loop 20 
* CEs [33] --> Loop 21 
* CEs [35] --> Loop 22 

### Ranking functions of CR fun2(V1,Out) 

#### Partial ranking functions of CR fun2(V1,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [36] 
* CE 2 is refined into CE [37] 
* CE 3 is refined into CE [38,39] 
* CE 4 is refined into CE [40] 
* CE 5 is refined into CE [41,42,43] 
* CE 6 is refined into CE [44,45,46] 


### Cost equations --> "Loop" of start/2 
* CEs [36,37,38,39,40,41,42,43,44,45,46] --> Loop 23 

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

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


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

#### Cost of chains of f(V1,V,Out):
* Chain [12]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [11,12]: 2
  with precondition: [Out=0,V1=V,V1>=0] 


#### Cost of chains of encArg(V1,Out):
* Chain [15]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([13,14],[[15]])]: 3*it(13)+0
  Such that:aux(1) =< V1
it(13) =< aux(1)

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


#### Cost of chains of fun(V1,V,Out):
* Chain [16]: 6*s(2)+6*s(4)+2
  Such that:aux(2) =< V1
aux(3) =< V
s(4) =< aux(2)
s(2) =< aux(3)

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


#### Cost of chains of fun1(V1,Out):
* Chain [19]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [18]: 1
  with precondition: [Out=1,V1>=0] 

* Chain [17]: 3*s(10)+1
  Such that:s(9) =< V1
s(10) =< s(9)

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


#### Cost of chains of fun2(V1,Out):
* Chain [22]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [21]: 0
  with precondition: [Out=1,V1>=0] 

* Chain [20]: 3*s(12)+0
  Such that:s(11) =< V1
s(12) =< s(11)

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


#### Cost of chains of start(V1,V):
* Chain [23]: 15*s(14)+6*s(18)+2
  Such that:s(16) =< V
aux(4) =< V1
s(14) =< aux(4)
s(18) =< s(16)

  with precondition: [V1>=0] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [23] with precondition: [V1>=0] 
    - Upper bound: 15*V1+2+nat(V)*6 
    - Complexity: n 

### Maximum cost of start(V1,V): 15*V1+2+nat(V)*6 
Asymptotic class: n 
* Total analysis performed in 151 ms.


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

(16)
BOUNDS(1, n^1)