/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 152 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) CpxWeightedTrsRenamingProof [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, 434 ms]
(18) 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(+(x, 0)) -> f(x)
   +(x, +(y, z)) -> +(+(x, y), z)

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(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0


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

(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(+(x, 0)) -> f(x)
   +(x, +(y, z)) -> +(+(x, y), z)

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

   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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(+(x, 0)) -> f(x)
   +(x, +(y, z)) -> +(+(x, y), z)

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

   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0

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

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

(6)
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(c_+(x, 0)) -> f(x)
   +(x, c_+(y, z)) -> +(+(x, y), z)

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

   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   +(x0, x1) -> c_+(x0, x1)

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^2).


The TRS R consists of the following rules:

   f(c_+(x, 0)) -> f(x) [1]
   +(x, c_+(y, z)) -> +(+(x, y), z) [1]
   encArg(0) -> 0 [0]
   encArg(cons_f(x_1)) -> f(encArg(x_1)) [0]
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   +(x0, x1) -> c_+(x0, x1) [0]

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

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

+ => plus

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

(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(c_+(x, 0)) -> f(x) [1]
   plus(x, c_+(y, z)) -> plus(plus(x, y), z) [1]
   encArg(0) -> 0 [0]
   encArg(cons_f(x_1)) -> f(encArg(x_1)) [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   plus(x0, x1) -> c_+(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(c_+(x, 0)) -> f(x) [1]
   plus(x, c_+(y, z)) -> plus(plus(x, y), z) [1]
   encArg(0) -> 0 [0]
   encArg(cons_f(x_1)) -> f(encArg(x_1)) [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   plus(x0, x1) -> c_+(x0, x1) [0]

The TRS has the following type information:
f :: 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+
c_+ :: 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+
0 :: 0:c_+:cons_f:cons_+
plus :: 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+
encArg :: 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+
cons_f :: 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+
cons_+ :: 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+
encode_f :: 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+
encode_+ :: 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+ -> 0:c_+:cons_f:cons_+
encode_0 :: 0:c_+:cons_f:cons_+

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) -> null_encode_f [0]
   encode_+(v0, v1) -> null_encode_+ [0]
   encode_0 -> null_encode_0 [0]
   plus(v0, v1) -> null_plus [0]
   f(v0) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_+, null_encode_0, null_plus, 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(c_+(x, 0)) -> f(x) [1]
   plus(x, c_+(y, z)) -> plus(plus(x, y), z) [1]
   encArg(0) -> 0 [0]
   encArg(cons_f(x_1)) -> f(encArg(x_1)) [0]
   encArg(cons_+(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1) -> f(encArg(x_1)) [0]
   encode_+(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) [0]
   encode_0 -> 0 [0]
   plus(x0, x1) -> c_+(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0) -> null_encode_f [0]
   encode_+(v0, v1) -> null_encode_+ [0]
   encode_0 -> null_encode_0 [0]
   plus(v0, v1) -> null_plus [0]
   f(v0) -> null_f [0]

The TRS has the following type information:
f :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
c_+ :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
0 :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
plus :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
encArg :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
cons_f :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
cons_+ :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
encode_f :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
encode_+ :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f -> 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
encode_0 :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
null_encArg :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
null_encode_f :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
null_encode_+ :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
null_encode_0 :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
null_plus :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus:null_f
null_f :: 0:c_+:cons_f:cons_+:null_encArg:null_encode_f:null_encode_+:null_encode_0:null_plus: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:

   0 => 0
   null_encArg => 0
   null_encode_f => 0
   null_encode_+ => 0
   null_encode_0 => 0
   null_plus => 0
   null_f => 0

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

(16)
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 }-> f(encArg(x_1)) :|: x_1 >= 0, z' = 1 + x_1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   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
   encode_0 -{ 0 }-> 0 :|: 
   encode_f(z') -{ 0 }-> f(encArg(x_1)) :|: x_1 >= 0, z' = x_1
   encode_f(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   f(z') -{ 1 }-> f(x) :|: x >= 0, z' = 1 + x + 0
   f(z') -{ 0 }-> 0 :|: v0 >= 0, 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'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0

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(V, V3),0,[f(V, Out)],[V >= 0]).
eq(start(V, V3),0,[plus(V, V3, Out)],[V >= 0,V3 >= 0]).
eq(start(V, V3),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V3),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V3),0,[fun1(V, V3, Out)],[V >= 0,V3 >= 0]).
eq(start(V, V3),0,[fun2(Out)],[]).
eq(f(V, Out),1,[f(V1, Ret)],[Out = Ret,V1 >= 0,V = 1 + V1]).
eq(plus(V, V3, Out),1,[plus(V2, V4, Ret0),plus(Ret0, V5, Ret1)],[Out = Ret1,V5 >= 0,V = V2,V2 >= 0,V4 >= 0,V3 = 1 + V4 + V5]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[encArg(V6, Ret01),f(Ret01, Ret2)],[Out = Ret2,V6 >= 0,V = 1 + V6]).
eq(encArg(V, Out),0,[encArg(V7, Ret02),encArg(V8, Ret11),plus(Ret02, Ret11, Ret3)],[Out = Ret3,V7 >= 0,V8 >= 0,V = 1 + V7 + V8]).
eq(fun(V, Out),0,[encArg(V9, Ret03),f(Ret03, Ret4)],[Out = Ret4,V9 >= 0,V = V9]).
eq(fun1(V, V3, Out),0,[encArg(V11, Ret04),encArg(V10, Ret12),plus(Ret04, Ret12, Ret5)],[Out = Ret5,V11 >= 0,V = V11,V10 >= 0,V3 = V10]).
eq(fun2(Out),0,[],[Out = 0]).
eq(plus(V, V3, Out),0,[],[Out = 1 + V12 + V13,V3 = V12,V13 >= 0,V12 >= 0,V = V13]).
eq(encArg(V, Out),0,[],[Out = 0,V14 >= 0,V = V14]).
eq(fun(V, Out),0,[],[Out = 0,V15 >= 0,V = V15]).
eq(fun1(V, V3, Out),0,[],[Out = 0,V17 >= 0,V16 >= 0,V3 = V16,V = V17]).
eq(plus(V, V3, Out),0,[],[Out = 0,V18 >= 0,V19 >= 0,V3 = V19,V = V18]).
eq(f(V, Out),0,[],[Out = 0,V20 >= 0,V = V20]).
input_output_vars(f(V,Out),[V],[Out]).
input_output_vars(plus(V,V3,Out),[V,V3],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fun1(V,V3,Out),[V,V3],[Out]).
input_output_vars(fun2(Out),[],[Out]).


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

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

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

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

### Specialization of cost equations f/2 
* CE 8 is refined into CE [19] 
* CE 7 is refined into CE [20] 


### Cost equations --> "Loop" of f/2 
* CEs [20] --> Loop 12 
* CEs [19] --> Loop 13 

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

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


### Specialization of cost equations plus/3 
* CE 10 is refined into CE [21] 
* CE 11 is refined into CE [22] 
* CE 9 is refined into CE [23] 


### Cost equations --> "Loop" of plus/3 
* CEs [23] --> Loop 14 
* CEs [21] --> Loop 15 
* CEs [22] --> Loop 16 

### Ranking functions of CR plus(V,V3,Out) 
* RF of phase [14]: [V3]

#### Partial ranking functions of CR plus(V,V3,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1,14:2]:
    V3


### Specialization of cost equations encArg/2 
* CE 12 is refined into CE [24] 
* CE 14 is refined into CE [25,26,27] 
* CE 13 is refined into CE [28] 


### Cost equations --> "Loop" of encArg/2 
* CEs [28] --> Loop 17 
* CEs [27] --> Loop 18 
* CEs [26] --> Loop 19 
* CEs [25] --> Loop 20 
* CEs [24] --> Loop 21 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [17,18,19,20]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [17,18,19,20]:
  - RF of loop [17:1,18:1,18:2,19:1,19:2,20:1,20:2]:
    V


### Specialization of cost equations fun/2 
* CE 15 is refined into CE [29,30] 
* CE 16 is refined into CE [31] 


### Cost equations --> "Loop" of fun/2 
* CEs [29,30,31] --> Loop 22 

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

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


### Specialization of cost equations fun1/3 
* CE 17 is refined into CE [32,33,34,35,36,37,38,39,40,41] 
* CE 18 is refined into CE [42] 


### Cost equations --> "Loop" of fun1/3 
* CEs [40,41] --> Loop 23 
* CEs [38] --> Loop 24 
* CEs [35,36] --> Loop 25 
* CEs [33] --> Loop 26 
* CEs [32,34,37,39,42] --> Loop 27 

### Ranking functions of CR fun1(V,V3,Out) 

#### Partial ranking functions of CR fun1(V,V3,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [43] 
* CE 2 is refined into CE [44,45,46] 
* CE 3 is refined into CE [47,48] 
* CE 4 is refined into CE [49] 
* CE 5 is refined into CE [50,51,52,53,54] 
* CE 6 is refined into CE [55] 


### Cost equations --> "Loop" of start/2 
* CEs [43,44,45,46,47,48,49,50,51,52,53,54,55] --> Loop 28 

### Ranking functions of CR start(V,V3) 

#### Partial ranking functions of CR start(V,V3) 


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

#### Cost of chains of f(V,Out):
* Chain [[12],13]: 1*it(12)+0
  Such that:it(12) =< V

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

* Chain [13]: 0
  with precondition: [Out=0,V>=0] 


#### Cost of chains of plus(V,V3,Out):
* Chain [16]: 0
  with precondition: [Out=0,V>=0,V3>=0] 

* Chain [15]: 0
  with precondition: [V+V3+1=Out,V>=0,V3>=0] 

* Chain [multiple([14],[[16],[15]])]: 1*it(14)+0
  Such that:it(14) =< V3

  with precondition: [V>=0,V3>=1,Out>=0,V+V3+1>=Out] 


#### Cost of chains of encArg(V,Out):
* Chain [21]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([17,18,19,20],[[21]])]: 1*s(6)+1*s(7)+0
  Such that:aux(7) =< V
aux(8) =< 2*V+1
it(17) =< aux(7)
it(18) =< aux(7)
it(18) =< aux(8)
aux(3) =< aux(7)-1
s(6) =< it(17)*aux(7)
s(7) =< it(18)*aux(3)

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


#### Cost of chains of fun(V,Out):
* Chain [22]: 1*s(14)+1*s(15)+1*s(16)+0
  Such that:aux(9) =< V
s(10) =< 2*V+1
s(16) =< aux(9)
s(12) =< aux(9)
s(12) =< s(10)
s(13) =< aux(9)-1
s(14) =< aux(9)*aux(9)
s(15) =< s(12)*s(13)

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


#### Cost of chains of fun1(V,V3,Out):
* Chain [27]: 2*s(22)+2*s(23)+2*s(29)+2*s(30)+0
  Such that:aux(10) =< V
aux(11) =< 2*V+1
aux(12) =< V3
aux(13) =< 2*V3+1
s(27) =< aux(10)
s(27) =< aux(11)
s(28) =< aux(10)-1
s(29) =< aux(10)*aux(10)
s(30) =< s(27)*s(28)
s(20) =< aux(12)
s(20) =< aux(13)
s(21) =< aux(12)-1
s(22) =< aux(12)*aux(12)
s(23) =< s(20)*s(21)

  with precondition: [Out=0,V>=0,V3>=0] 

* Chain [26]: 0
  with precondition: [Out=1,V>=0,V3>=0] 

* Chain [25]: 2*s(50)+2*s(51)+1*s(59)+0
  Such that:aux(15) =< V3
aux(16) =< 2*V3+1
s(48) =< aux(15)
s(48) =< aux(16)
s(49) =< aux(15)-1
s(50) =< aux(15)*aux(15)
s(51) =< s(48)*s(49)
s(59) =< aux(15)

  with precondition: [V>=0,V3>=1,Out>=0,V3+1>=Out] 

* Chain [24]: 1*s(65)+1*s(66)+0
  Such that:s(60) =< V
s(61) =< 2*V+1
s(63) =< s(60)
s(63) =< s(61)
s(64) =< s(60)-1
s(65) =< s(60)*s(60)
s(66) =< s(63)*s(64)

  with precondition: [V>=1,V3>=0,Out>=1,V+1>=Out] 

* Chain [23]: 2*s(72)+2*s(73)+2*s(79)+2*s(80)+1*s(95)+0
  Such that:aux(18) =< V
aux(19) =< 2*V+1
aux(20) =< V3
aux(21) =< 2*V3+1
s(77) =< aux(20)
s(77) =< aux(21)
s(78) =< aux(20)-1
s(79) =< aux(20)*aux(20)
s(80) =< s(77)*s(78)
s(70) =< aux(18)
s(70) =< aux(19)
s(71) =< aux(18)-1
s(72) =< aux(18)*aux(18)
s(73) =< s(70)*s(71)
s(95) =< aux(20)

  with precondition: [V>=1,V3>=1,Out>=0,V+V3+1>=Out] 


#### Cost of chains of start(V,V3):
* Chain [28]: 2*s(96)+3*s(97)+7*s(103)+7*s(104)+6*s(122)+6*s(123)+0
  Such that:aux(22) =< V
aux(23) =< 2*V+1
aux(24) =< V3
aux(25) =< 2*V3+1
s(96) =< aux(22)
s(97) =< aux(24)
s(101) =< aux(22)
s(101) =< aux(23)
s(102) =< aux(22)-1
s(103) =< aux(22)*aux(22)
s(104) =< s(101)*s(102)
s(120) =< aux(24)
s(120) =< aux(25)
s(121) =< aux(24)-1
s(122) =< aux(24)*aux(24)
s(123) =< s(120)*s(121)

  with precondition: [] 


Closed-form bounds of start(V,V3): 
-------------------------------------
* Chain [28] with precondition: [] 
    - Upper bound: nat(V)*7*nat(V)+nat(V)*2+nat(V)*7*nat(nat(V)+ -1)+nat(V3)*3+nat(V3)*6*nat(V3)+nat(V3)*6*nat(nat(V3)+ -1) 
    - Complexity: n^2 

### Maximum cost of start(V,V3): nat(V)*7*nat(V)+nat(V)*2+nat(V)*7*nat(nat(V)+ -1)+nat(V3)*3+nat(V3)*6*nat(V3)+nat(V3)*6*nat(nat(V3)+ -1) 
Asymptotic class: n^2 
* Total analysis performed in 348 ms.


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(18)
BOUNDS(1, n^2)