/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 172 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, 443 ms]
(18) 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(a, f(a, f(b, f(x, y)))) -> f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
   f(a, f(c, f(x, y))) -> f(b, f(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(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   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_a -> a
   encode_b -> b
   encode_c -> c


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

(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(a, f(a, f(b, f(x, y)))) -> f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
   f(a, f(c, f(x, y))) -> f(b, f(x, y))

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   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_a -> a
   encode_b -> b
   encode_c -> c

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(a, f(a, f(b, f(x, y)))) -> f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
   f(a, f(c, f(x, y))) -> f(b, f(x, y))

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   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_a -> a
   encode_b -> b
   encode_c -> c

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^1).


The TRS R consists of the following rules:

   f(a, c_f(c, c_f(x, y))) -> f(b, f(x, y))
   f(a, c_f(a, c_f(b, c_f(x, y)))) -> f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   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_a -> a
   encode_b -> b
   encode_c -> c
   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^1).


The TRS R consists of the following rules:

   f(a, c_f(c, c_f(x, y))) -> f(b, f(x, y))
   f(a, c_f(a, c_f(b, c_f(x, y)))) -> f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   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_a -> a
   encode_b -> b
   encode_c -> c
   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^1).


The TRS R consists of the following rules:

   f(a, c_f(c, c_f(x, y))) -> f(b, f(x, y)) [1]
   f(a, c_f(a, c_f(b, c_f(x, y)))) -> f(b, f(c, f(b, f(a, f(a, f(a, f(x, y))))))) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(c) -> c [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_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [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(a, c_f(c, c_f(x, y))) -> f(b, f(x, y)) [1]
   f(a, c_f(a, c_f(b, c_f(x, y)))) -> f(b, f(c, f(b, f(a, f(a, f(a, f(x, y))))))) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(c) -> c [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_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [0]
   f(x0, x1) -> c_f(x0, x1) [0]

The TRS has the following type information:
f :: a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f
a :: a:c:c_f:b:cons_f
c_f :: a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f
c :: a:c:c_f:b:cons_f
b :: a:c:c_f:b:cons_f
encArg :: a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f
cons_f :: a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f
encode_f :: a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f -> a:c:c_f:b:cons_f
encode_a :: a:c:c_f:b:cons_f
encode_b :: a:c:c_f:b:cons_f
encode_c :: a:c:c_f:b: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_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_a, null_encode_b, null_encode_c, 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(a, c_f(c, c_f(x, y))) -> f(b, f(x, y)) [1]
   f(a, c_f(a, c_f(b, c_f(x, y)))) -> f(b, f(c, f(b, f(a, f(a, f(a, f(x, y))))))) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(c) -> c [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_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [0]
   f(x0, x1) -> c_f(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
a :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
c_f :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
c :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
b :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encArg :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
cons_f :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_f :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_a :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_b :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_c :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encArg :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_f :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_a :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_b :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_c :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_f :: a:c:c_f:b:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c: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:

   a => 0
   c => 2
   b => 1
   null_encArg => 0
   null_encode_f => 0
   null_encode_a => 0
   null_encode_b => 0
   null_encode_c => 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 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 0 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 2 :|: 
   encode_c -{ 0 }-> 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
   f(z, z') -{ 1 }-> f(1, f(x, y)) :|: x >= 0, y >= 0, z' = 1 + 2 + (1 + x + y), z = 0
   f(z, z') -{ 1 }-> f(1, f(2, f(1, f(0, f(0, f(0, f(x, y))))))) :|: z' = 1 + 0 + (1 + 1 + (1 + x + y)), x >= 0, y >= 0, z = 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(Out)],[]).
eq(start(V1, V),0,[fun2(Out)],[]).
eq(start(V1, V),0,[fun3(Out)],[]).
eq(f(V1, V, Out),1,[f(V2, V3, Ret1),f(1, Ret1, Ret)],[Out = Ret,V2 >= 0,V3 >= 0,V = 4 + V2 + V3,V1 = 0]).
eq(f(V1, V, Out),1,[f(V4, V5, Ret111111),f(0, Ret111111, Ret11111),f(0, Ret11111, Ret1111),f(0, Ret1111, Ret111),f(1, Ret111, Ret11),f(2, Ret11, Ret12),f(1, Ret12, Ret2)],[Out = Ret2,V = 4 + V4 + V5,V4 >= 0,V5 >= 0,V1 = 0]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[],[Out = 1,V1 = 1]).
eq(encArg(V1, Out),0,[],[Out = 2,V1 = 2]).
eq(encArg(V1, Out),0,[encArg(V7, Ret0),encArg(V6, Ret13),f(Ret0, Ret13, Ret3)],[Out = Ret3,V7 >= 0,V1 = 1 + V6 + V7,V6 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V8, Ret01),encArg(V9, Ret14),f(Ret01, Ret14, Ret4)],[Out = Ret4,V8 >= 0,V9 >= 0,V1 = V8,V = V9]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(Out),0,[],[Out = 1]).
eq(fun3(Out),0,[],[Out = 2]).
eq(f(V1, V, Out),0,[],[Out = 1 + V10 + V11,V1 = V11,V11 >= 0,V10 >= 0,V = V10]).
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(fun2(Out),0,[],[Out = 0]).
eq(fun3(Out),0,[],[Out = 0]).
eq(f(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
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(Out),[],[Out]).
input_output_vars(fun2(Out),[],[Out]).
input_output_vars(fun3(Out),[],[Out]).


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

#### Computed strongly connected components 
0. recursive [multiple] : [f/3]
1. recursive [non_tail,multiple] : [encArg/2]
2. non_recursive  : [fun/3]
3. non_recursive  : [fun1/1]
4. non_recursive  : [fun2/1]
5. non_recursive  : [fun3/1]
6. 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 completely evaluated into other SCCs
4. SCC is partially evaluated into fun2/1
5. SCC is partially evaluated into fun3/1
6. SCC is partially evaluated into start/2

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

### Specialization of cost equations f/3 
* CE 9 is refined into CE [21] 
* CE 10 is refined into CE [22] 
* CE 8 is refined into CE [23] 
* CE 7 is refined into CE [24] 


### Cost equations --> "Loop" of f/3 
* CEs [24] --> Loop 15 
* CEs [23] --> Loop 16 
* CEs [21] --> Loop 17 
* CEs [22] --> Loop 18 

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

#### Partial ranking functions of CR f(V1,V,Out) 
* Partial RF of phase [15,16]:
  - RF of loop [15:1,16:1]:
    V-3 depends on loops [15:2,16:2,16:3,16:4,16:5,16:6,16:7] 
    V1+V-3 depends on loops [15:2,16:2,16:3,16:4,16:5,16:6,16:7] 
  - RF of loop [15:2,16:5,16:6,16:7]:
    -V1+1


### Specialization of cost equations encArg/2 
* CE 11 is refined into CE [25] 
* CE 13 is refined into CE [26] 
* CE 12 is refined into CE [27] 
* CE 14 is refined into CE [28,29,30] 


### Cost equations --> "Loop" of encArg/2 
* CEs [30] --> Loop 19 
* CEs [29] --> Loop 20 
* CEs [28] --> Loop 21 
* CEs [25] --> Loop 22 
* CEs [26] --> Loop 23 
* CEs [27] --> Loop 24 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [19,20,21]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [19,20,21]:
  - RF of loop [19:1,19:2,20:1,20:2,21:1,21:2]:
    V1


### Specialization of cost equations fun/3 
* CE 15 is refined into CE [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50] 
* CE 16 is refined into CE [51] 


### Cost equations --> "Loop" of fun/3 
* CEs [50] --> Loop 25 
* CEs [35] --> Loop 26 
* CEs [48] --> Loop 27 
* CEs [34,47] --> Loop 28 
* CEs [37,43] --> Loop 29 
* CEs [32,39,41,45] --> Loop 30 
* CEs [31,36,38,40,42,44,49,51] --> Loop 31 
* CEs [33,46] --> Loop 32 

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

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


### Specialization of cost equations fun2/1 
* CE 17 is refined into CE [52] 
* CE 18 is refined into CE [53] 


### Cost equations --> "Loop" of fun2/1 
* CEs [52] --> Loop 33 
* CEs [53] --> Loop 34 

### Ranking functions of CR fun2(Out) 

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


### Specialization of cost equations fun3/1 
* CE 19 is refined into CE [54] 
* CE 20 is refined into CE [55] 


### Cost equations --> "Loop" of fun3/1 
* CEs [54] --> Loop 35 
* CEs [55] --> Loop 36 

### Ranking functions of CR fun3(Out) 

#### Partial ranking functions of CR fun3(Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [56,57,58] 
* CE 2 is refined into CE [59,60,61] 
* CE 3 is refined into CE [62,63,64,65,66] 
* CE 4 is refined into CE [67] 
* CE 5 is refined into CE [68,69] 
* CE 6 is refined into CE [70,71] 


### Cost equations --> "Loop" of start/2 
* CEs [56,57,59,60,61,62,63,64,65,67,68,69,70,71] --> Loop 37 
* CEs [58,66] --> Loop 38 

### 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 [multiple([15,16],[[],[18],[17]])]...: 2*it(15)+0
  Such that:aux(2) =< -2*V1+1
it(15) =< aux(2)

  with precondition: [V1=0,V>=4] 

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

* Chain [17]: 0
  with precondition: [V+V1+1=Out,V1>=0,V>=0] 


#### Cost of chains of encArg(V1,Out):
* Chain [24]: 0
  with precondition: [V1=1,Out=1] 

* Chain [23]: 0
  with precondition: [V1=2,Out=2] 

* Chain [22]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([19,20,21],[[24],[23],[22]])]: 2*s(5)+0
  Such that:aux(4) =< V1
s(5) =< aux(4)

  with precondition: [V1>=1] 


#### Cost of chains of fun(V1,V,Out):
* Chain [32]...: 2*s(10)+4*s(12)+4*s(14)+0
  Such that:s(9) =< V1
aux(6) =< 1
aux(7) =< V
s(14) =< aux(6)
s(12) =< aux(7)
s(10) =< s(9)

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

* Chain [31]: 4*s(20)+6*s(22)+0
  Such that:aux(8) =< V1
aux(9) =< V
s(22) =< aux(9)
s(20) =< aux(8)

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

* Chain [30]: 2*s(30)+6*s(32)+0
  Such that:s(29) =< V1
aux(10) =< V
s(32) =< aux(10)
s(30) =< s(29)

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

* Chain [29]: 2*s(38)+0
  Such that:s(37) =< V1
s(38) =< s(37)

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

* Chain [28]: 2*s(40)+0
  Such that:s(39) =< V1
s(40) =< s(39)

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

* Chain [27]: 0
  with precondition: [V=2,Out=3,V1>=0] 

* Chain [26]: 2*s(42)+0
  Such that:s(41) =< V1
s(42) =< s(41)

  with precondition: [V=2,V1>=1,Out>=3] 

* Chain [25]: 0
  with precondition: [Out=1,V1>=0,V>=0] 


#### Cost of chains of fun2(Out):
* Chain [34]: 0
  with precondition: [Out=0] 

* Chain [33]: 0
  with precondition: [Out=1] 


#### Cost of chains of fun3(Out):
* Chain [36]: 0
  with precondition: [Out=0] 

* Chain [35]: 0
  with precondition: [Out=2] 


#### Cost of chains of start(V1,V):
* Chain [38]...: 6*s(58)+4*s(63)+2*s(64)+0
  Such that:s(59) =< V1
s(61) =< V
aux(13) =< 1
s(58) =< aux(13)
s(63) =< s(61)
s(64) =< s(59)

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

* Chain [37]: 14*s(66)+12*s(70)+0
  Such that:aux(14) =< V1
aux(15) =< V
s(66) =< aux(14)
s(70) =< aux(15)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [38]... with precondition: [V1>=0,V>=1] 
    - Upper bound: 2*V1+4*V+6 
    - Complexity: n 
* Chain [37] with precondition: [] 
    - Upper bound: nat(V)*12+nat(V1)*14 
    - Complexity: n 

### Maximum cost of start(V1,V): nat(V)*4+nat(V1)*2+max([6,nat(V)*8+nat(V1)*12]) 
Asymptotic class: n 
* Total analysis performed in 367 ms.


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

(18)
BOUNDS(1, n^1)