/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 155 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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxWeightedTrs
(13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTypedWeightedTrs
(15) CompletionProof [UPPER BOUND(ID), 0 ms]
(16) CpxTypedWeightedCompleteTrs
(17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) CompleteCoflocoProof [FINISHED, 755 ms]
(20) 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:

   +(x, +(y, z)) -> +(+(x, y), z)
   +(*(x, y), +(x, z)) -> *(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

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(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(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:

   +(x, +(y, z)) -> +(+(x, y), z)
   +(*(x, y), +(x, z)) -> *(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

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

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(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:

   +(x, +(y, z)) -> +(+(x, y), z)
   +(*(x, y), +(x, z)) -> *(x, +(y, z))
   +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

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

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(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:

   +(x, c_+(y, z)) -> +(+(x, y), z)
   +(*(x, y), c_+(x, z)) -> *(x, +(y, z))
   +(*(x, y), c_+(*(x, z), u)) -> +(*(x, +(y, z)), u)

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

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   +(x0, x1) -> c_+(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:

   +(x, c_+(y, z)) -> +(+(x, y), z)
   +(*(x, y), c_+(x, z)) -> *(x, +(y, z))
   +(*(x, y), c_+(*(x, z), u)) -> +(*(x, +(y, z)), u)

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

   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   +(x0, x1) -> c_+(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:

   +(x, c_+(y, z)) -> +(+(x, y), z) [1]
   +(*(x, y), c_+(x, z)) -> *(x, +(y, z)) [1]
   +(*(x, y), c_+(*(x, z), u)) -> +(*(x, +(y, z)), u) [1]
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) [0]
   +(x0, x1) -> c_+(x0, x1) [0]

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

(11) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

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

(12)
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:

   plus(x, c_+(y, z)) -> plus(plus(x, y), z) [1]
   plus(*(x, y), c_+(x, z)) -> *(x, plus(y, z)) [1]
   plus(*(x, y), c_+(*(x, z), u)) -> plus(*(x, plus(y, z)), u) [1]
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   plus(x, c_+(y, z)) -> plus(plus(x, y), z) [1]
   plus(*(x, y), c_+(x, z)) -> *(x, plus(y, z)) [1]
   plus(*(x, y), c_+(*(x, z), u)) -> plus(*(x, plus(y, z)), u) [1]
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]

The TRS has the following type information:
plus :: c_+:*:cons_+ -> c_+:*:cons_+ -> c_+:*:cons_+
c_+ :: c_+:*:cons_+ -> c_+:*:cons_+ -> c_+:*:cons_+
* :: c_+:*:cons_+ -> c_+:*:cons_+ -> c_+:*:cons_+
encArg :: c_+:*:cons_+ -> c_+:*:cons_+
cons_+ :: c_+:*:cons_+ -> c_+:*:cons_+ -> c_+:*:cons_+
encode_+ :: c_+:*:cons_+ -> c_+:*:cons_+ -> c_+:*:cons_+
encode_* :: c_+:*:cons_+ -> c_+:*:cons_+ -> c_+:*:cons_+

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

(15) 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_+(v0, v1) -> null_encode_+ [0]
   encode_*(v0, v1) -> null_encode_* [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_encArg, null_encode_+, null_encode_*, null_plus

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

(16)
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:

   plus(x, c_+(y, z)) -> plus(plus(x, y), z) [1]
   plus(*(x, y), c_+(x, z)) -> *(x, plus(y, z)) [1]
   plus(*(x, y), c_+(*(x, z), u)) -> plus(*(x, plus(y, z)), u) [1]
   encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) [0]
   plus(x0, x1) -> c_+(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_+(v0, v1) -> null_encode_+ [0]
   encode_*(v0, v1) -> null_encode_* [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
plus :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
c_+ :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
* :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
encArg :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
cons_+ :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
encode_+ :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
encode_* :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus -> c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
null_encArg :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
null_encode_+ :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
null_encode_* :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus
null_plus :: c_+:*:cons_+:null_encArg:null_encode_+:null_encode_*:null_plus

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

(17) 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_+ => 0
   null_encode_* => 0
   null_plus => 0

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V2),0,[plus(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V2),0,[fun(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[fun1(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(plus(V, V2, Out),1,[plus(V1, V3, Ret0),plus(Ret0, V4, Ret)],[Out = Ret,V4 >= 0,V = V1,V1 >= 0,V3 >= 0,V2 = 1 + V3 + V4]).
eq(plus(V, V2, Out),1,[plus(V7, V6, Ret1)],[Out = 1 + Ret1 + V5,V6 >= 0,V = 1 + V5 + V7,V2 = 1 + V5 + V6,V5 >= 0,V7 >= 0]).
eq(plus(V, V2, Out),1,[plus(V10, V11, Ret01),plus(1 + V8 + Ret01, V9, Ret2)],[Out = Ret2,V11 >= 0,V = 1 + V10 + V8,V8 >= 0,V10 >= 0,V2 = 2 + V11 + V8 + V9,V9 >= 0]).
eq(encArg(V, Out),0,[encArg(V13, Ret011),encArg(V12, Ret11)],[Out = 1 + Ret011 + Ret11,V13 >= 0,V12 >= 0,V = 1 + V12 + V13]).
eq(encArg(V, Out),0,[encArg(V14, Ret02),encArg(V15, Ret12),plus(Ret02, Ret12, Ret3)],[Out = Ret3,V14 >= 0,V15 >= 0,V = 1 + V14 + V15]).
eq(fun(V, V2, Out),0,[encArg(V17, Ret03),encArg(V16, Ret13),plus(Ret03, Ret13, Ret4)],[Out = Ret4,V17 >= 0,V = V17,V16 >= 0,V2 = V16]).
eq(fun1(V, V2, Out),0,[encArg(V19, Ret012),encArg(V18, Ret14)],[Out = 1 + Ret012 + Ret14,V19 >= 0,V = V19,V18 >= 0,V2 = V18]).
eq(plus(V, V2, Out),0,[],[Out = 1 + V20 + V21,V2 = V20,V21 >= 0,V20 >= 0,V = V21]).
eq(encArg(V, Out),0,[],[Out = 0,V22 >= 0,V = V22]).
eq(fun(V, V2, Out),0,[],[Out = 0,V24 >= 0,V23 >= 0,V2 = V23,V = V24]).
eq(fun1(V, V2, Out),0,[],[Out = 0,V26 >= 0,V25 >= 0,V2 = V25,V = V26]).
eq(plus(V, V2, Out),0,[],[Out = 0,V27 >= 0,V28 >= 0,V2 = V28,V = V27]).
input_output_vars(plus(V,V2,Out),[V,V2],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,V2,Out),[V,V2],[Out]).
input_output_vars(fun1(V,V2,Out),[V,V2],[Out]).


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

#### Computed strongly connected components 
0. recursive [multiple] : [plus/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 plus/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 plus/3 
* CE 8 is refined into CE [17] 
* CE 9 is refined into CE [18] 
* CE 7 is refined into CE [19] 
* CE 5 is refined into CE [20] 
* CE 6 is refined into CE [21] 


### Cost equations --> "Loop" of plus/3 
* CEs [21] --> Loop 11 
* CEs [19] --> Loop 12 
* CEs [20] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [18] --> Loop 15 

### Ranking functions of CR plus(V,V2,Out) 
* RF of phase [11,12,13]: [V2]

#### Partial ranking functions of CR plus(V,V2,Out) 
* Partial RF of phase [11,12,13]:
  - RF of loop [11:1,12:1]:
    V depends on loops [12:2,13:2] 
  - RF of loop [11:1,13:1,13:2]:
    V2
  - RF of loop [12:1,12:2]:
    V2/2-1/2


### Specialization of cost equations encArg/2 
* CE 12 is refined into CE [22] 
* CE 10 is refined into CE [23] 
* CE 11 is refined into CE [24,25,26] 


### Cost equations --> "Loop" of encArg/2 
* CEs [26] --> Loop 16 
* CEs [23,25] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [22] --> Loop 19 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [16,17,18]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [16,17,18]:
  - RF of loop [16:1,16:2,17:1,17:2,18:1,18:2]:
    V


### Specialization of cost equations fun/3 
* CE 13 is refined into CE [27,28,29,30,31,32,33,34,35,36] 
* CE 14 is refined into CE [37] 


### Cost equations --> "Loop" of fun/3 
* CEs [35,36] --> Loop 20 
* CEs [33] --> Loop 21 
* CEs [30,31] --> Loop 22 
* CEs [28] --> Loop 23 
* CEs [27,29,32,34,37] --> Loop 24 

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

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


### Specialization of cost equations fun1/3 
* CE 15 is refined into CE [38,39,40,41] 
* CE 16 is refined into CE [42] 


### Cost equations --> "Loop" of fun1/3 
* CEs [41] --> Loop 25 
* CEs [40] --> Loop 26 
* CEs [39] --> Loop 27 
* CEs [38] --> Loop 28 
* CEs [42] --> Loop 29 

### Ranking functions of CR fun1(V,V2,Out) 

#### Partial ranking functions of CR fun1(V,V2,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [43,44,45] 
* CE 2 is refined into CE [46,47] 
* CE 3 is refined into CE [48,49,50,51,52] 
* CE 4 is refined into CE [53,54,55,56,57] 


### Cost equations --> "Loop" of start/2 
* CEs [43,44,45,46,47,48,49,50,51,52,53,54,55,56,57] --> Loop 30 

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

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


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

#### Cost of chains of plus(V,V2,Out):
* Chain [15]: 0
  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [14]: 0
  with precondition: [V+V2+1=Out,V>=0,V2>=0] 

* Chain [multiple([11,12,13],[[15],[14]])]: 1*it(11)+1*it(12)+1*it(13)+0
  Such that:aux(8) =< V+V2
it(12) =< V2/2
aux(17) =< V
aux(18) =< V2
it(11) =< aux(18)
it(12) =< aux(18)
it(13) =< aux(18)
aux(10) =< aux(8)+2
aux(1) =< it(12)*aux(8)
aux(2) =< it(13)*aux(10)
it(11) =< aux(2)+aux(1)+aux(17)

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


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

* Chain [multiple([16,17,18],[[19]])]: 1*s(19)+1*s(20)+1*s(21)+0
  Such that:aux(22) =< V
aux(23) =< 2*V+1
it(16) =< aux(23)
aux(19) =< aux(22)-1
aux(21) =< it(16)*aux(19)
s(22) =< it(16)*aux(19)
s(19) =< aux(21)*(1/2)
s(20) =< aux(21)
s(19) =< aux(21)
s(21) =< aux(21)
s(16) =< aux(22)+2
s(23) =< s(19)*aux(22)
s(24) =< s(21)*s(16)
s(20) =< s(24)+s(23)+s(22)

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


#### Cost of chains of fun(V,V2,Out):
* Chain [24]: 2*s(32)+2*s(33)+2*s(34)+2*s(44)+2*s(45)+2*s(46)+0
  Such that:aux(24) =< V
aux(25) =< 2*V+1
aux(26) =< V2
aux(27) =< 2*V2+1
s(41) =< aux(24)-1
s(42) =< aux(25)*s(41)
s(44) =< s(42)*(1/2)
s(45) =< s(42)
s(44) =< s(42)
s(46) =< s(42)
s(47) =< aux(24)+2
s(48) =< s(44)*aux(24)
s(49) =< s(46)*s(47)
s(45) =< s(49)+s(48)+s(42)
s(29) =< aux(26)-1
s(30) =< aux(27)*s(29)
s(32) =< s(30)*(1/2)
s(33) =< s(30)
s(32) =< s(30)
s(34) =< s(30)
s(35) =< aux(26)+2
s(36) =< s(32)*aux(26)
s(37) =< s(34)*s(35)
s(33) =< s(37)+s(36)+s(30)

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

* Chain [23]: 0
  with precondition: [Out=1,V>=0,V2>=0] 

* Chain [22]: 2*s(80)+2*s(81)+2*s(82)+1*s(99)+1*s(102)+1*s(103)+0
  Such that:s(99) =< V2/2
aux(29) =< V2
aux(30) =< 2*V2+1
s(77) =< aux(29)-1
s(78) =< aux(30)*s(77)
s(80) =< s(78)*(1/2)
s(81) =< s(78)
s(80) =< s(78)
s(82) =< s(78)
s(83) =< aux(29)+2
s(84) =< s(80)*aux(29)
s(85) =< s(82)*s(83)
s(81) =< s(85)+s(84)+s(78)
s(102) =< aux(29)
s(99) =< aux(29)
s(103) =< aux(29)
s(105) =< s(99)*aux(29)
s(106) =< s(103)*s(83)
s(102) =< s(106)+s(105)

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

* Chain [21]: 1*s(113)+1*s(114)+1*s(115)+0
  Such that:s(107) =< V
s(108) =< 2*V+1
s(110) =< s(107)-1
s(111) =< s(108)*s(110)
s(113) =< s(111)*(1/2)
s(114) =< s(111)
s(113) =< s(111)
s(115) =< s(111)
s(116) =< s(107)+2
s(117) =< s(113)*s(107)
s(118) =< s(115)*s(116)
s(114) =< s(118)+s(117)+s(111)

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

* Chain [20]: 2*s(125)+2*s(126)+2*s(127)+2*s(137)+2*s(138)+2*s(139)+1*s(168)+1*s(171)+1*s(172)+0
  Such that:s(167) =< V+V2
s(168) =< V2/2
aux(33) =< V
aux(34) =< 2*V+1
aux(35) =< V2
aux(36) =< 2*V2+1
s(134) =< aux(35)-1
s(135) =< aux(36)*s(134)
s(137) =< s(135)*(1/2)
s(138) =< s(135)
s(137) =< s(135)
s(139) =< s(135)
s(140) =< aux(35)+2
s(141) =< s(137)*aux(35)
s(142) =< s(139)*s(140)
s(138) =< s(142)+s(141)+s(135)
s(122) =< aux(33)-1
s(123) =< aux(34)*s(122)
s(125) =< s(123)*(1/2)
s(126) =< s(123)
s(125) =< s(123)
s(127) =< s(123)
s(128) =< aux(33)+2
s(129) =< s(125)*aux(33)
s(130) =< s(127)*s(128)
s(126) =< s(130)+s(129)+s(123)
s(171) =< aux(35)
s(168) =< aux(35)
s(172) =< aux(35)
s(173) =< s(167)+2
s(174) =< s(168)*s(167)
s(175) =< s(172)*s(173)
s(171) =< s(175)+s(174)+aux(33)

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


#### Cost of chains of fun1(V,V2,Out):
* Chain [29]: 0
  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [28]: 0
  with precondition: [Out=1,V>=0,V2>=0] 

* Chain [27]: 1*s(182)+1*s(183)+1*s(184)+0
  Such that:s(176) =< V2
s(177) =< 2*V2+1
s(179) =< s(176)-1
s(180) =< s(177)*s(179)
s(182) =< s(180)*(1/2)
s(183) =< s(180)
s(182) =< s(180)
s(184) =< s(180)
s(185) =< s(176)+2
s(186) =< s(182)*s(176)
s(187) =< s(184)*s(185)
s(183) =< s(187)+s(186)+s(180)

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

* Chain [26]: 1*s(194)+1*s(195)+1*s(196)+0
  Such that:s(188) =< V
s(189) =< 2*V+1
s(191) =< s(188)-1
s(192) =< s(189)*s(191)
s(194) =< s(192)*(1/2)
s(195) =< s(192)
s(194) =< s(192)
s(196) =< s(192)
s(197) =< s(188)+2
s(198) =< s(194)*s(188)
s(199) =< s(196)*s(197)
s(195) =< s(199)+s(198)+s(192)

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

* Chain [25]: 1*s(206)+1*s(207)+1*s(208)+1*s(218)+1*s(219)+1*s(220)+0
  Such that:s(200) =< V
s(201) =< 2*V+1
s(212) =< V2
s(213) =< 2*V2+1
s(215) =< s(212)-1
s(216) =< s(213)*s(215)
s(218) =< s(216)*(1/2)
s(219) =< s(216)
s(218) =< s(216)
s(220) =< s(216)
s(221) =< s(212)+2
s(222) =< s(218)*s(212)
s(223) =< s(220)*s(221)
s(219) =< s(223)+s(222)+s(216)
s(203) =< s(200)-1
s(204) =< s(201)*s(203)
s(206) =< s(204)*(1/2)
s(207) =< s(204)
s(206) =< s(204)
s(208) =< s(204)
s(209) =< s(200)+2
s(210) =< s(206)*s(200)
s(211) =< s(208)*s(209)
s(207) =< s(211)+s(210)+s(204)

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


#### Cost of chains of start(V,V2):
* Chain [30]: 3*s(225)+2*s(228)+3*s(229)+8*s(239)+8*s(240)+8*s(241)+8*s(259)+8*s(260)+8*s(261)+1*s(276)+0
  Such that:aux(37) =< V
aux(38) =< V+V2
aux(39) =< 2*V+1
aux(40) =< V2
aux(41) =< 2*V2+1
aux(42) =< V2/2
s(225) =< aux(42)
s(228) =< aux(40)
s(225) =< aux(40)
s(229) =< aux(40)
s(230) =< aux(38)+2
s(231) =< s(225)*aux(38)
s(232) =< s(229)*s(230)
s(228) =< s(232)+s(231)+aux(37)
s(236) =< aux(37)-1
s(237) =< aux(39)*s(236)
s(239) =< s(237)*(1/2)
s(240) =< s(237)
s(239) =< s(237)
s(241) =< s(237)
s(242) =< aux(37)+2
s(243) =< s(239)*aux(37)
s(244) =< s(241)*s(242)
s(240) =< s(244)+s(243)+s(237)
s(257) =< aux(40)-1
s(258) =< aux(41)*s(257)
s(259) =< s(258)*(1/2)
s(260) =< s(258)
s(259) =< s(258)
s(261) =< s(258)
s(262) =< aux(40)+2
s(263) =< s(259)*aux(40)
s(264) =< s(261)*s(262)
s(260) =< s(264)+s(263)+s(258)
s(276) =< aux(40)
s(278) =< s(225)*aux(40)
s(279) =< s(229)*s(262)
s(276) =< s(279)+s(278)

  with precondition: [V>=0] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [30] with precondition: [V>=0] 
    - Upper bound: (2*V+1)*(nat(V-1)*20)+nat(V2)*6+nat(nat(V2)+ -1)*20*nat(2*V2+1)+nat(V2/2)*3 
    - Complexity: n^2 

### Maximum cost of start(V,V2): (2*V+1)*(nat(V-1)*20)+nat(V2)*6+nat(nat(V2)+ -1)*20*nat(2*V2+1)+nat(V2/2)*3 
Asymptotic class: n^2 
* Total analysis performed in 612 ms.


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

(20)
BOUNDS(1, n^2)