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


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


WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 160 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, 325 ms]
(14) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   f(x, y, z) -> g(x, y, z)
   g(0, 1, x) -> f(x, x, x)

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(1) -> 1
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2, x_3)) -> g(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, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 1


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

(2)
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(x, y, z) -> g(x, y, z)
   g(0, 1, x) -> f(x, x, x)

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2, x_3)) -> g(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, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 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^1).


The TRS R consists of the following rules:

   f(x, y, z) -> g(x, y, z)
   g(0, 1, x) -> f(x, x, x)

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2, x_3)) -> g(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, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_0 -> 0
   encode_1 -> 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^1).


The TRS R consists of the following rules:

   f(x, y, z) -> g(x, y, z) [1]
   g(0, 1, x) -> f(x, x, x) [1]
   encArg(0) -> 0 [0]
   encArg(1) -> 1 [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1, x_2, x_3)) -> g(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, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_1 -> 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(x, y, z) -> g(x, y, z) [1]
   g(0, 1, x) -> f(x, x, x) [1]
   encArg(0) -> 0 [0]
   encArg(1) -> 1 [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1, x_2, x_3)) -> g(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, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_1 -> 1 [0]

The TRS has the following type information:
f :: 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g
g :: 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g
0 :: 0:1:cons_f:cons_g
1 :: 0:1:cons_f:cons_g
encArg :: 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g
cons_f :: 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g
cons_g :: 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g
encode_f :: 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g
encode_g :: 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g -> 0:1:cons_f:cons_g
encode_0 :: 0:1:cons_f:cons_g
encode_1 :: 0:1:cons_f:cons_g

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, v1, v2) -> null_encode_g [0]
   encode_0 -> null_encode_0 [0]
   encode_1 -> null_encode_1 [0]
   g(v0, v1, v2) -> null_g [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_g, null_encode_0, null_encode_1, null_g

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

(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(x, y, z) -> g(x, y, z) [1]
   g(0, 1, x) -> f(x, x, x) [1]
   encArg(0) -> 0 [0]
   encArg(1) -> 1 [0]
   encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encArg(cons_g(x_1, x_2, x_3)) -> g(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, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_0 -> 0 [0]
   encode_1 -> 1 [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1, v2) -> null_encode_f [0]
   encode_g(v0, v1, v2) -> null_encode_g [0]
   encode_0 -> null_encode_0 [0]
   encode_1 -> null_encode_1 [0]
   g(v0, v1, v2) -> null_g [0]

The TRS has the following type information:
f :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
g :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
0 :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
1 :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
encArg :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
cons_f :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
cons_g :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
encode_f :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
encode_g :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g -> 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
encode_0 :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
encode_1 :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
null_encArg :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
null_encode_f :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
null_encode_g :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
null_encode_0 :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
null_encode_1 :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g
null_g :: 0:1:cons_f:cons_g:null_encArg:null_encode_f:null_encode_g:null_encode_0:null_encode_1:null_g

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:

   0 => 0
   1 => 1
   null_encArg => 0
   null_encode_f => 0
   null_encode_g => 0
   null_encode_0 => 0
   null_encode_1 => 0
   null_g => 0

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

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

   encArg(z') -{ 0 }-> g(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 }-> 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 }-> 1 :|: z' = 1
   encArg(z') -{ 0 }-> 0 :|: z' = 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_0 -{ 0 }-> 0 :|: 
   encode_1 -{ 0 }-> 1 :|: 
   encode_1 -{ 0 }-> 0 :|: 
   encode_f(z', z'', z1) -{ 0 }-> f(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2
   encode_f(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   encode_g(z', z'', z1) -{ 0 }-> g(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2
   encode_g(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   f(z', z'', z1) -{ 1 }-> g(x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   g(z', z'', z1) -{ 1 }-> f(x, x, x) :|: x >= 0, z'' = 1, z1 = x, z' = 0
   g(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0

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(V, V1, V5),0,[f(V, V1, V5, Out)],[V >= 0,V1 >= 0,V5 >= 0]).
eq(start(V, V1, V5),0,[g(V, V1, V5, Out)],[V >= 0,V1 >= 0,V5 >= 0]).
eq(start(V, V1, V5),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V1, V5),0,[fun(V, V1, V5, Out)],[V >= 0,V1 >= 0,V5 >= 0]).
eq(start(V, V1, V5),0,[fun1(V, V1, V5, Out)],[V >= 0,V1 >= 0,V5 >= 0]).
eq(start(V, V1, V5),0,[fun2(Out)],[]).
eq(start(V, V1, V5),0,[fun3(Out)],[]).
eq(f(V, V1, V5, Out),1,[g(V4, V2, V3, Ret)],[Out = Ret,V5 = V3,V3 >= 0,V = V4,V1 = V2,V4 >= 0,V2 >= 0]).
eq(g(V, V1, V5, Out),1,[f(V6, V6, V6, Ret1)],[Out = Ret1,V6 >= 0,V1 = 1,V5 = V6,V = 0]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[],[Out = 1,V = 1]).
eq(encArg(V, Out),0,[encArg(V9, Ret0),encArg(V8, Ret11),encArg(V7, Ret2),f(Ret0, Ret11, Ret2, Ret3)],[Out = Ret3,V9 >= 0,V = 1 + V7 + V8 + V9,V7 >= 0,V8 >= 0]).
eq(encArg(V, Out),0,[encArg(V10, Ret01),encArg(V12, Ret12),encArg(V11, Ret21),g(Ret01, Ret12, Ret21, Ret4)],[Out = Ret4,V10 >= 0,V = 1 + V10 + V11 + V12,V11 >= 0,V12 >= 0]).
eq(fun(V, V1, V5, Out),0,[encArg(V14, Ret02),encArg(V13, Ret13),encArg(V15, Ret22),f(Ret02, Ret13, Ret22, Ret5)],[Out = Ret5,V14 >= 0,V5 = V15,V = V14,V15 >= 0,V13 >= 0,V1 = V13]).
eq(fun1(V, V1, V5, Out),0,[encArg(V18, Ret03),encArg(V17, Ret14),encArg(V16, Ret23),g(Ret03, Ret14, Ret23, Ret6)],[Out = Ret6,V18 >= 0,V5 = V16,V = V18,V16 >= 0,V17 >= 0,V1 = V17]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(Out),0,[],[Out = 1]).
eq(encArg(V, Out),0,[],[Out = 0,V19 >= 0,V = V19]).
eq(fun(V, V1, V5, Out),0,[],[Out = 0,V21 >= 0,V5 = V22,V20 >= 0,V1 = V20,V22 >= 0,V = V21]).
eq(fun1(V, V1, V5, Out),0,[],[Out = 0,V25 >= 0,V5 = V23,V24 >= 0,V1 = V24,V23 >= 0,V = V25]).
eq(fun3(Out),0,[],[Out = 0]).
eq(g(V, V1, V5, Out),0,[],[Out = 0,V26 >= 0,V5 = V27,V28 >= 0,V1 = V28,V27 >= 0,V = V26]).
input_output_vars(f(V,V1,V5,Out),[V,V1,V5],[Out]).
input_output_vars(g(V,V1,V5,Out),[V,V1,V5],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,V1,V5,Out),[V,V1,V5],[Out]).
input_output_vars(fun1(V,V1,V5,Out),[V,V1,V5],[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  : [f/4,g/4]
1. recursive [non_tail,multiple] : [encArg/2]
2. non_recursive  : [fun/4]
3. non_recursive  : [fun1/4]
4. non_recursive  : [fun2/1]
5. non_recursive  : [fun3/1]
6. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into g/4
1. SCC is partially evaluated into encArg/2
2. SCC is partially evaluated into fun/4
3. SCC is partially evaluated into fun1/4
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into fun3/1
6. SCC is partially evaluated into start/3

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

### Specialization of cost equations g/4 
* CE 9 is refined into CE [20] 
* CE 8 is refined into CE [21] 


### Cost equations --> "Loop" of g/4 
* CEs [21] --> Loop 11 
* CEs [20] --> Loop 12 

### Ranking functions of CR g(V,V1,V5,Out) 

#### Partial ranking functions of CR g(V,V1,V5,Out) 


### Specialization of cost equations encArg/2 
* CE 11 is refined into CE [22] 
* CE 12 is refined into CE [23] 
* CE 10 is refined into CE [24] 
* CE 13 is refined into CE [25] 


### Cost equations --> "Loop" of encArg/2 
* CEs [24,25] --> Loop 13 
* CEs [22] --> Loop 14 
* CEs [23] --> Loop 15 

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

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


### Specialization of cost equations fun/4 
* CE 14 is refined into CE [26,27,28,29,30,31,32,33] 
* CE 15 is refined into CE [34] 


### Cost equations --> "Loop" of fun/4 
* CEs [30,31] --> Loop 16 
* CEs [28,32] --> Loop 17 
* CEs [26,27,29,33,34] --> Loop 18 

### Ranking functions of CR fun(V,V1,V5,Out) 

#### Partial ranking functions of CR fun(V,V1,V5,Out) 


### Specialization of cost equations fun1/4 
* CE 16 is refined into CE [35,36,37,38,39,40,41,42] 
* CE 17 is refined into CE [43] 


### Cost equations --> "Loop" of fun1/4 
* CEs [39,40] --> Loop 19 
* CEs [37,41] --> Loop 20 
* CEs [35,36,38,42,43] --> Loop 21 

### Ranking functions of CR fun1(V,V1,V5,Out) 

#### Partial ranking functions of CR fun1(V,V1,V5,Out) 


### Specialization of cost equations fun3/1 
* CE 18 is refined into CE [44] 
* CE 19 is refined into CE [45] 


### Cost equations --> "Loop" of fun3/1 
* CEs [44] --> Loop 22 
* CEs [45] --> Loop 23 

### Ranking functions of CR fun3(Out) 

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [46] 
* CE 2 is refined into CE [47] 
* CE 3 is refined into CE [48,49] 
* CE 4 is refined into CE [50,51] 
* CE 5 is refined into CE [52,53] 
* CE 6 is refined into CE [54] 
* CE 7 is refined into CE [55,56] 


### Cost equations --> "Loop" of start/3 
* CEs [46,47,48,49,50,51,52,53,54,55,56] --> Loop 24 

### Ranking functions of CR start(V,V1,V5) 

#### Partial ranking functions of CR start(V,V1,V5) 


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

#### Cost of chains of g(V,V1,V5,Out):
* Chain [12]: 0
  with precondition: [Out=0,V>=0,V1>=0,V5>=0] 

* Chain [11,12]: 2
  with precondition: [V=0,V1=1,Out=0,V5>=0] 


#### Cost of chains of encArg(V,Out):
* Chain [15]: 0
  with precondition: [V=1,Out=1] 

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

* Chain [multiple([13],[[15],[14]])]: 3*it(13)+0
  Such that:aux(1) =< V
it(13) =< aux(1)

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


#### Cost of chains of fun(V,V1,V5,Out):
* Chain [18]: 9*s(4)+6*s(6)+3*s(10)+3
  Such that:s(9) =< V
aux(3) =< V1
aux(4) =< V5
s(4) =< aux(4)
s(6) =< aux(3)
s(10) =< s(9)

  with precondition: [Out=0,V>=0,V1>=0,V5>=0] 

* Chain [17]: 6*s(16)+3*s(18)+3
  Such that:s(17) =< V
aux(5) =< V1
s(16) =< aux(5)
s(18) =< s(17)

  with precondition: [V5=1,Out=0,V>=0,V1>=0] 

* Chain [16]: 6*s(22)+3*s(26)+3
  Such that:s(25) =< V5
aux(6) =< V
s(22) =< aux(6)
s(26) =< s(25)

  with precondition: [V1=1,Out=0,V>=0,V5>=0] 


#### Cost of chains of fun1(V,V1,V5,Out):
* Chain [21]: 9*s(38)+6*s(40)+3*s(44)+2
  Such that:s(43) =< V
aux(9) =< V1
aux(10) =< V5
s(38) =< aux(10)
s(40) =< aux(9)
s(44) =< s(43)

  with precondition: [Out=0,V>=0,V1>=0,V5>=0] 

* Chain [20]: 6*s(50)+3*s(52)+2
  Such that:s(51) =< V
aux(11) =< V1
s(50) =< aux(11)
s(52) =< s(51)

  with precondition: [V5=1,Out=0,V>=0,V1>=0] 

* Chain [19]: 6*s(56)+3*s(60)+2
  Such that:s(59) =< V5
aux(12) =< V
s(56) =< aux(12)
s(60) =< s(59)

  with precondition: [V1=1,Out=0,V>=0,V5>=0] 


#### Cost of chains of fun3(Out):
* Chain [23]: 0
  with precondition: [Out=0] 

* Chain [22]: 0
  with precondition: [Out=1] 


#### Cost of chains of start(V,V1,V5):
* Chain [24]: 27*s(72)+24*s(76)+24*s(77)+3
  Such that:aux(15) =< V
aux(16) =< V1
aux(17) =< V5
s(72) =< aux(15)
s(76) =< aux(17)
s(77) =< aux(16)

  with precondition: [] 


Closed-form bounds of start(V,V1,V5): 
-------------------------------------
* Chain [24] with precondition: [] 
    - Upper bound: nat(V)*27+3+nat(V1)*24+nat(V5)*24 
    - Complexity: n 

### Maximum cost of start(V,V1,V5): nat(V)*27+3+nat(V1)*24+nat(V5)*24 
Asymptotic class: n 
* Total analysis performed in 251 ms.


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

(14)
BOUNDS(1, n^1)