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


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


WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 156 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [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, 411 ms]
(14) BOUNDS(1, n^2)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

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

(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(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))


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

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


The TRS R consists of the following rules:

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

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1) -> g(encArg(x_1))

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

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

(6)
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:

   f(g(x), y, y) -> g(f(x, x, y)) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(g(x), y, y) -> g(f(x, x, y)) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]

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

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

(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:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_g(v0) -> null_encode_g [0]
   f(v0, v1, v2) -> null_f [0]

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

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

(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:

   f(g(x), y, y) -> g(f(x, x, y)) [1]
   encArg(g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_g(v0) -> null_encode_g [0]
   f(v0, v1, v2) -> null_f [0]

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

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

(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_encArg => 0
   null_encode_f => 0
   null_encode_g => 0
   null_f => 0

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, 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', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_g(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   f(z, z', z'') -{ 1 }-> 1 + f(x, x, y) :|: z'' = y, x >= 0, y >= 0, z = 1 + x, z' = y

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V2),0,[fun(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[fun1(V1, Out)],[V1 >= 0]).
eq(f(V1, V, V2, Out),1,[f(V4, V4, V3, Ret1)],[Out = 1 + Ret1,V2 = V3,V4 >= 0,V3 >= 0,V1 = 1 + V4,V = V3]).
eq(encArg(V1, Out),0,[encArg(V5, Ret11)],[Out = 1 + Ret11,V1 = 1 + V5,V5 >= 0]).
eq(encArg(V1, Out),0,[encArg(V7, Ret0),encArg(V8, Ret12),encArg(V6, Ret2),f(Ret0, Ret12, Ret2, Ret)],[Out = Ret,V7 >= 0,V1 = 1 + V6 + V7 + V8,V6 >= 0,V8 >= 0]).
eq(fun(V1, V, V2, Out),0,[encArg(V9, Ret01),encArg(V11, Ret13),encArg(V10, Ret21),f(Ret01, Ret13, Ret21, Ret3)],[Out = Ret3,V9 >= 0,V10 >= 0,V11 >= 0,V1 = V9,V = V11,V2 = V10]).
eq(fun1(V1, Out),0,[encArg(V12, Ret14)],[Out = 1 + Ret14,V12 >= 0,V1 = V12]).
eq(encArg(V1, Out),0,[],[Out = 0,V13 >= 0,V1 = V13]).
eq(fun(V1, V, V2, Out),0,[],[Out = 0,V15 >= 0,V2 = V16,V14 >= 0,V1 = V15,V = V14,V16 >= 0]).
eq(fun1(V1, Out),0,[],[Out = 0,V17 >= 0,V1 = V17]).
eq(f(V1, V, V2, Out),0,[],[Out = 0,V18 >= 0,V2 = V19,V20 >= 0,V1 = V18,V = V20,V19 >= 0]).
input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(fun1(V1,Out),[V1],[Out]).


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

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

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

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

### Specialization of cost equations f/4 
* CE 6 is refined into CE [14] 
* CE 5 is refined into CE [15] 


### Cost equations --> "Loop" of f/4 
* CEs [15] --> Loop 9 
* CEs [14] --> Loop 10 

### Ranking functions of CR f(V1,V,V2,Out) 
* RF of phase [9]: [V1]

#### Partial ranking functions of CR f(V1,V,V2,Out) 
* Partial RF of phase [9]:
  - RF of loop [9:1]:
    V1


### Specialization of cost equations encArg/2 
* CE 9 is refined into CE [16] 
* CE 8 is refined into CE [17,18] 
* CE 7 is refined into CE [19] 


### Cost equations --> "Loop" of encArg/2 
* CEs [19] --> Loop 11 
* CEs [18] --> Loop 12 
* CEs [17] --> Loop 13 
* CEs [16] --> Loop 14 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [11,12,13]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [11,12,13]:
  - RF of loop [11:1,12:1,12:2,12:3,13:1,13:2,13:3]:
    V1


### Specialization of cost equations fun/4 
* CE 10 is refined into CE [20,21,22,23,24,25,26,27,28,29,30,31] 
* CE 11 is refined into CE [32] 


### Cost equations --> "Loop" of fun/4 
* CEs [31] --> Loop 15 
* CEs [25,27,29] --> Loop 16 
* CEs [20,21,22,23,24,26,28,30,32] --> Loop 17 

### Ranking functions of CR fun(V1,V,V2,Out) 

#### Partial ranking functions of CR fun(V1,V,V2,Out) 


### Specialization of cost equations fun1/2 
* CE 12 is refined into CE [33,34] 
* CE 13 is refined into CE [35] 


### Cost equations --> "Loop" of fun1/2 
* CEs [34] --> Loop 18 
* CEs [33] --> Loop 19 
* CEs [35] --> Loop 20 

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

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


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


### Cost equations --> "Loop" of start/3 
* CEs [36,37,38,39,40,41,42,43,44,45] --> Loop 21 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of f(V1,V,V2,Out):
* Chain [[9],10]: 1*it(9)+0
  Such that:it(9) =< Out

  with precondition: [V=V2,Out>=1,V1>=Out,V+1>=Out] 

* Chain [10]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 


#### Cost of chains of encArg(V1,Out):
* Chain [14]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([11,12,13],[[14]])]: 1*s(3)+0
  Such that:aux(6) =< V1
aux(7) =< 3/5*V1
it(12) =< aux(6)
it(12) =< aux(7)
s(3) =< it(12)*aux(6)

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


#### Cost of chains of fun(V1,V,V2,Out):
* Chain [17]: 4*s(7)+4*s(11)+4*s(23)+0
  Such that:aux(8) =< V1
aux(9) =< 3/5*V1
aux(10) =< V
aux(11) =< 3/5*V
aux(12) =< V2
aux(13) =< 3/5*V2
s(22) =< aux(8)
s(22) =< aux(9)
s(23) =< s(22)*aux(8)
s(6) =< aux(12)
s(6) =< aux(13)
s(7) =< s(6)*aux(12)
s(10) =< aux(10)
s(10) =< aux(11)
s(11) =< s(10)*aux(10)

  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [16]: 3*s(55)+3*s(56)+1*s(64)+1*s(73)+0
  Such that:s(70) =< V
s(71) =< 3/5*V
s(61) =< V2
s(62) =< 3/5*V2
aux(14) =< 1
aux(15) =< V1
aux(16) =< 3/5*V1
s(56) =< aux(14)
s(54) =< aux(15)
s(54) =< aux(16)
s(55) =< s(54)*aux(15)
s(63) =< s(61)
s(63) =< s(62)
s(64) =< s(63)*s(61)
s(72) =< s(70)
s(72) =< s(71)
s(73) =< s(72)*s(70)

  with precondition: [Out=1,V1>=1,V>=0,V2>=0] 

* Chain [15]: 1*s(78)+1*s(82)+1*s(86)+1*s(87)+0
  Such that:s(76) =< 3/5*V1
s(79) =< V
s(80) =< 3/5*V
s(83) =< V2
s(84) =< 3/5*V2
aux(17) =< V1
s(87) =< aux(17)
s(85) =< s(83)
s(85) =< s(84)
s(86) =< s(85)*s(83)
s(81) =< s(79)
s(81) =< s(80)
s(82) =< s(81)*s(79)
s(77) =< aux(17)
s(77) =< s(76)
s(78) =< s(77)*aux(17)

  with precondition: [V>=1,V2>=1,Out>=1,V1>=Out,V+1>=Out,V2+1>=Out] 


#### Cost of chains of fun1(V1,Out):
* Chain [20]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [19]: 0
  with precondition: [Out=1,V1>=0] 

* Chain [18]: 1*s(91)+0
  Such that:s(88) =< V1
s(89) =< 3/5*V1
s(90) =< s(88)
s(90) =< s(89)
s(91) =< s(90)*s(88)

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


#### Cost of chains of start(V1,V,V2):
* Chain [21]: 1*s(92)+10*s(96)+6*s(106)+6*s(108)+3*s(116)+1*s(129)+0
  Such that:s(113) =< 1
s(92) =< V2+1
aux(18) =< V1
aux(19) =< 3/5*V1
aux(20) =< V
aux(21) =< 3/5*V
aux(22) =< V2
aux(23) =< 3/5*V2
s(116) =< s(113)
s(95) =< aux(18)
s(95) =< aux(19)
s(96) =< s(95)*aux(18)
s(105) =< aux(22)
s(105) =< aux(23)
s(106) =< s(105)*aux(22)
s(107) =< aux(20)
s(107) =< aux(21)
s(108) =< s(107)*aux(20)
s(129) =< aux(18)

  with precondition: [V1>=0] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [21] with precondition: [V1>=0] 
    - Upper bound: V1+3+10*V1*V1+nat(V)*6*nat(V)+nat(V2)*6*nat(V2)+nat(V2+1) 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V2): V1+3+10*V1*V1+nat(V)*6*nat(V)+nat(V2)*6*nat(V2)+nat(V2+1) 
Asymptotic class: n^2 
* Total analysis performed in 329 ms.


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

(14)
BOUNDS(1, n^2)