/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) 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), 144 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
(9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxWeightedTrs
(11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedTrs
(13) CompletionProof [UPPER BOUND(ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) CompleteCoflocoProof [FINISHED, 452 ms]
(18) BOUNDS(1, n^2)


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

(0)
Obligation:
The Derivational Complexity (full) 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), f(y, y)) -> f(g(y, x), y)

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


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

(2)
Obligation:
The Runtime Complexity (full) 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), f(y, y)) -> f(g(y, x), y)

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

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

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^2).


The TRS R consists of the following rules:

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

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

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

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

(5) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) 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), c_f(y, y)) -> f(g(y, x), y)

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

   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   f(x0, x1) -> c_f(x0, x1)

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

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

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

(8)
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), c_f(y, y)) -> f(g(y, x), y)

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

   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   f(x0, x1) -> c_f(x0, x1)

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

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

(10)
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), c_f(y, y)) -> f(g(y, x), y) [1]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(g(x, y), c_f(y, y)) -> f(g(y, x), y) [1]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]

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

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

(13) 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, v1) -> null_encode_g [0]
   f(v0, v1) -> null_f [0]

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

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

(14)
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), c_f(y, y)) -> f(g(y, x), y) [1]
   encArg(g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_g(v0, v1) -> null_encode_g [0]
   f(v0, v1) -> null_f [0]

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

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

(15) 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

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   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) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 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, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z, z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   f(z, z') -{ 1 }-> f(1 + y + x, y) :|: z = 1 + x + y, z' = 1 + y + y, x >= 0, y >= 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1

Only complete derivations are relevant for the runtime complexity.

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

(17) 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,[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, V, Out)],[V1 >= 0,V >= 0]).
eq(f(V1, V, Out),1,[f(1 + V3 + V2, V3, Ret)],[Out = Ret,V1 = 1 + V2 + V3,V = 1 + 2*V3,V2 >= 0,V3 >= 0]).
eq(encArg(V1, Out),0,[encArg(V5, Ret01),encArg(V4, Ret1)],[Out = 1 + Ret01 + Ret1,V5 >= 0,V1 = 1 + V4 + V5,V4 >= 0]).
eq(encArg(V1, Out),0,[encArg(V6, Ret0),encArg(V7, Ret11),f(Ret0, Ret11, Ret2)],[Out = Ret2,V6 >= 0,V1 = 1 + V6 + V7,V7 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V9, Ret02),encArg(V8, Ret12),f(Ret02, Ret12, Ret3)],[Out = Ret3,V9 >= 0,V8 >= 0,V1 = V9,V = V8]).
eq(fun1(V1, V, Out),0,[encArg(V11, Ret011),encArg(V10, Ret13)],[Out = 1 + Ret011 + Ret13,V11 >= 0,V10 >= 0,V1 = V11,V = V10]).
eq(f(V1, V, Out),0,[],[Out = 1 + V12 + V13,V1 = V13,V13 >= 0,V12 >= 0,V = V12]).
eq(encArg(V1, Out),0,[],[Out = 0,V14 >= 0,V1 = V14]).
eq(fun(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(fun1(V1, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V1 = V18,V = V17]).
eq(f(V1, V, Out),0,[],[Out = 0,V19 >= 0,V20 >= 0,V1 = V19,V = V20]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).


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

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

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

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

### Specialization of cost equations f/3 
* CE 6 is refined into CE [15] 
* CE 7 is refined into CE [16] 
* CE 5 is refined into CE [17] 


### Cost equations --> "Loop" of f/3 
* CEs [17] --> Loop 9 
* CEs [15] --> Loop 10 
* CEs [16] --> Loop 11 

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

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


### Specialization of cost equations encArg/2 
* CE 10 is refined into CE [18] 
* CE 8 is refined into CE [19] 
* CE 9 is refined into CE [20,21,22] 


### Cost equations --> "Loop" of encArg/2 
* CEs [22] --> Loop 12 
* CEs [19,21] --> Loop 13 
* CEs [20] --> Loop 14 
* CEs [18] --> Loop 15 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [12,13,14]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [12,13,14]:
  - RF of loop [12:1,12:2,13:1,13:2,14:1,14:2]:
    V1


### Specialization of cost equations fun/3 
* CE 11 is refined into CE [23,24,25,26,27,28,29,30,31] 
* CE 12 is refined into CE [32] 


### Cost equations --> "Loop" of fun/3 
* CEs [30,31] --> Loop 16 
* CEs [28] --> Loop 17 
* CEs [26] --> Loop 18 
* CEs [24] --> Loop 19 
* CEs [23,25,27,29,32] --> Loop 20 

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

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


### Specialization of cost equations fun1/3 
* CE 13 is refined into CE [33,34,35,36] 
* CE 14 is refined into CE [37] 


### Cost equations --> "Loop" of fun1/3 
* CEs [36] --> Loop 21 
* CEs [35] --> Loop 22 
* CEs [34] --> Loop 23 
* CEs [33] --> Loop 24 
* CEs [37] --> Loop 25 

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

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


### Specialization of cost equations start/2 
* CE 1 is refined into CE [38,39,40] 
* CE 2 is refined into CE [41,42] 
* CE 3 is refined into CE [43,44,45,46,47] 
* CE 4 is refined into CE [48,49,50,51,52] 


### Cost equations --> "Loop" of start/2 
* CEs [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52] --> Loop 26 

### 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 [[9],11]: 1*it(9)+0
  Such that:it(9) =< V

  with precondition: [Out=0,V>=1,2*V1>=V+1] 

* Chain [[9],10]: 1*it(9)+0
  Such that:it(9) =< V1+V-Out+1

  with precondition: [Out>=V1+1,2*V1>=V+1,V+2*V1+1>=2*Out] 

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

* Chain [10]: 0
  with precondition: [V+V1+1=Out,V1>=0,V>=0] 


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

* Chain [multiple([12,13,14],[[15]])]: 1*s(6)+1*s(7)+0
  Such that:it([15]) =< V1+1
aux(4) =< V1
aux(5) =< V1/2
it(13) =< aux(4)
it(12) =< aux(5)
aux(3) =< aux(4)+1
it(12) =< it([15])*(1/2)+aux(5)
it(13) =< it([15])*(1/2)+aux(5)
s(7) =< it(13)*aux(3)
s(6) =< it(12)*aux(4)

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


#### Cost of chains of fun(V1,V,Out):
* Chain [20]: 2*s(15)+2*s(16)+2*s(17)+2*s(24)+2*s(25)+0
  Such that:aux(8) =< V1
aux(9) =< V1+1
aux(10) =< V1/2
aux(11) =< V
aux(12) =< V+1
aux(13) =< V/2
s(21) =< aux(8)
s(22) =< aux(10)
s(23) =< aux(8)+1
s(22) =< aux(9)*(1/2)+aux(10)
s(21) =< aux(9)*(1/2)+aux(10)
s(24) =< s(21)*s(23)
s(25) =< s(22)*aux(8)
s(17) =< aux(11)
s(12) =< aux(11)
s(13) =< aux(13)
s(14) =< aux(11)+1
s(13) =< aux(12)*(1/2)+aux(13)
s(12) =< aux(12)*(1/2)+aux(13)
s(15) =< s(12)*s(14)
s(16) =< s(13)*aux(11)

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

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

* Chain [18]: 1*s(50)+1*s(51)+0
  Such that:s(45) =< V
s(44) =< V+1
s(46) =< V/2
s(47) =< s(45)
s(48) =< s(46)
s(49) =< s(45)+1
s(48) =< s(44)*(1/2)+s(46)
s(47) =< s(44)*(1/2)+s(46)
s(50) =< s(47)*s(49)
s(51) =< s(48)*s(45)

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

* Chain [17]: 1*s(58)+1*s(59)+0
  Such that:s(53) =< V1
s(52) =< V1+1
s(54) =< V1/2
s(55) =< s(53)
s(56) =< s(54)
s(57) =< s(53)+1
s(56) =< s(52)*(1/2)+s(54)
s(55) =< s(52)*(1/2)+s(54)
s(58) =< s(55)*s(57)
s(59) =< s(56)*s(53)

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

* Chain [16]: 2*s(66)+2*s(67)+2*s(74)+2*s(75)+1*s(92)+0
  Such that:aux(15) =< V1
aux(16) =< V1+1
aux(17) =< V1/2
aux(18) =< V
aux(19) =< V+1
aux(20) =< V/2
s(71) =< aux(18)
s(72) =< aux(20)
s(73) =< aux(18)+1
s(72) =< aux(19)*(1/2)+aux(20)
s(71) =< aux(19)*(1/2)+aux(20)
s(74) =< s(71)*s(73)
s(75) =< s(72)*aux(18)
s(63) =< aux(15)
s(64) =< aux(17)
s(65) =< aux(15)+1
s(64) =< aux(16)*(1/2)+aux(17)
s(63) =< aux(16)*(1/2)+aux(17)
s(66) =< s(63)*s(65)
s(67) =< s(64)*aux(15)
s(92) =< aux(18)

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


#### Cost of chains of fun1(V1,V,Out):
* Chain [25]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [24]: 0
  with precondition: [Out=1,V1>=0,V>=0] 

* Chain [23]: 1*s(99)+1*s(100)+0
  Such that:s(94) =< V
s(93) =< V+1
s(95) =< V/2
s(96) =< s(94)
s(97) =< s(95)
s(98) =< s(94)+1
s(97) =< s(93)*(1/2)+s(95)
s(96) =< s(93)*(1/2)+s(95)
s(99) =< s(96)*s(98)
s(100) =< s(97)*s(94)

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

* Chain [22]: 1*s(107)+1*s(108)+0
  Such that:s(102) =< V1
s(101) =< V1+1
s(103) =< V1/2
s(104) =< s(102)
s(105) =< s(103)
s(106) =< s(102)+1
s(105) =< s(101)*(1/2)+s(103)
s(104) =< s(101)*(1/2)+s(103)
s(107) =< s(104)*s(106)
s(108) =< s(105)*s(102)

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

* Chain [21]: 1*s(115)+1*s(116)+1*s(123)+1*s(124)+0
  Such that:s(110) =< V1
s(109) =< V1+1
s(111) =< V1/2
s(118) =< V
s(117) =< V+1
s(119) =< V/2
s(120) =< s(118)
s(121) =< s(119)
s(122) =< s(118)+1
s(121) =< s(117)*(1/2)+s(119)
s(120) =< s(117)*(1/2)+s(119)
s(123) =< s(120)*s(122)
s(124) =< s(121)*s(118)
s(112) =< s(110)
s(113) =< s(111)
s(114) =< s(110)+1
s(113) =< s(109)*(1/2)+s(111)
s(112) =< s(109)*(1/2)+s(111)
s(115) =< s(112)*s(114)
s(116) =< s(113)*s(110)

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


#### Cost of chains of start(V1,V):
* Chain [26]: 5*s(125)+8*s(133)+8*s(134)+7*s(150)+7*s(151)+0
  Such that:aux(21) =< V1
aux(22) =< V1+1
aux(23) =< V1/2
aux(24) =< V
aux(25) =< V+1
aux(26) =< V/2
s(125) =< aux(24)
s(130) =< aux(21)
s(131) =< aux(23)
s(132) =< aux(21)+1
s(131) =< aux(22)*(1/2)+aux(23)
s(130) =< aux(22)*(1/2)+aux(23)
s(133) =< s(130)*s(132)
s(134) =< s(131)*aux(21)
s(147) =< aux(24)
s(148) =< aux(26)
s(149) =< aux(24)+1
s(148) =< aux(25)*(1/2)+aux(26)
s(147) =< aux(25)*(1/2)+aux(26)
s(150) =< s(147)*s(149)
s(151) =< s(148)*aux(24)

  with precondition: [V1>=0] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [26] with precondition: [V1>=0] 
    - Upper bound: 8*V1*V1+8*V1+V1/2*(8*V1)+nat(V)*12+nat(V)*7*nat(V)+nat(V)*7*nat(V/2) 
    - Complexity: n^2 

### Maximum cost of start(V1,V): 8*V1*V1+8*V1+V1/2*(8*V1)+nat(V)*12+nat(V)*7*nat(V)+nat(V)*7*nat(V/2) 
Asymptotic class: n^2 
* Total analysis performed in 365 ms.


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

(18)
BOUNDS(1, n^2)