/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 163 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedTrs
(11) CompletionProof [UPPER BOUND(ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(14) CpxRNTS
(15) CompleteCoflocoProof [FINISHED, 314 ms]
(16) 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(X, X) -> f(a, n__b)
   b -> a
   b -> n__b
   activate(n__b) -> b
   activate(X) -> X

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(n__b) -> n__b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_b) -> b
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_n__b -> n__b
   encode_b -> b
   encode_activate(x_1) -> activate(encArg(x_1))


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

(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(X, X) -> f(a, n__b)
   b -> a
   b -> n__b
   activate(n__b) -> b
   activate(X) -> X

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

   encArg(a) -> a
   encArg(n__b) -> n__b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_b) -> b
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_n__b -> n__b
   encode_b -> b
   encode_activate(x_1) -> activate(encArg(x_1))

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(X, X) -> f(a, n__b)
   b -> a
   b -> n__b
   activate(n__b) -> b
   activate(X) -> X

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

   encArg(a) -> a
   encArg(n__b) -> n__b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_b) -> b
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_n__b -> n__b
   encode_b -> b
   encode_activate(x_1) -> activate(encArg(x_1))

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

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

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

(6)
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, X) -> f(a, n__b)
   b -> a
   b -> n__b
   activate(n__b) -> b
   activate(X) -> X

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

   encArg(a) -> a
   encArg(n__b) -> n__b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_b) -> b
   encArg(cons_activate(x_1)) -> activate(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_n__b -> n__b
   encode_b -> b
   encode_activate(x_1) -> activate(encArg(x_1))

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^1).


The TRS R consists of the following rules:

   f(X, X) -> f(a, n__b) [1]
   b -> a [1]
   b -> n__b [1]
   activate(n__b) -> b [1]
   activate(X) -> X [1]
   encArg(a) -> a [0]
   encArg(n__b) -> n__b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_n__b -> n__b [0]
   encode_b -> b [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X, X) -> f(a, n__b) [1]
   b -> a [1]
   b -> n__b [1]
   activate(n__b) -> b [1]
   activate(X) -> X [1]
   encArg(a) -> a [0]
   encArg(n__b) -> n__b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_n__b -> n__b [0]
   encode_b -> b [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]

The TRS has the following type information:
f :: a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate
a :: a:n__b:cons_f:cons_b:cons_activate
n__b :: a:n__b:cons_f:cons_b:cons_activate
b :: a:n__b:cons_f:cons_b:cons_activate
activate :: a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate
encArg :: a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate
cons_f :: a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate
cons_b :: a:n__b:cons_f:cons_b:cons_activate
cons_activate :: a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate
encode_f :: a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate
encode_a :: a:n__b:cons_f:cons_b:cons_activate
encode_n__b :: a:n__b:cons_f:cons_b:cons_activate
encode_b :: a:n__b:cons_f:cons_b:cons_activate
encode_activate :: a:n__b:cons_f:cons_b:cons_activate -> a:n__b:cons_f:cons_b:cons_activate

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

(11) 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_n__b -> null_encode_n__b [0]
   encode_b -> null_encode_b [0]
   encode_activate(v0) -> null_encode_activate [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_a, null_encode_n__b, null_encode_b, null_encode_activate, null_f

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

(12)
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, X) -> f(a, n__b) [1]
   b -> a [1]
   b -> n__b [1]
   activate(n__b) -> b [1]
   activate(X) -> X [1]
   encArg(a) -> a [0]
   encArg(n__b) -> n__b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encArg(cons_activate(x_1)) -> activate(encArg(x_1)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_n__b -> n__b [0]
   encode_b -> b [0]
   encode_activate(x_1) -> activate(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_n__b -> null_encode_n__b [0]
   encode_b -> null_encode_b [0]
   encode_activate(v0) -> null_encode_activate [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
a :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
n__b :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
b :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
activate :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
encArg :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
cons_f :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
cons_b :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
cons_activate :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
encode_f :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
encode_a :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
encode_n__b :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
encode_b :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
encode_activate :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f -> a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
null_encArg :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
null_encode_f :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
null_encode_a :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
null_encode_n__b :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
null_encode_b :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
null_encode_activate :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f
null_f :: a:n__b:cons_f:cons_b:cons_activate:null_encArg:null_encode_f:null_encode_a:null_encode_n__b:null_encode_b:null_encode_activate:null_f

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

(13) 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
   n__b => 2
   cons_b => 1
   null_encArg => 0
   null_encode_f => 0
   null_encode_a => 0
   null_encode_n__b => 0
   null_encode_b => 0
   null_encode_activate => 0
   null_f => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> b :|: z = 2
   b -{ 1 }-> 2 :|: 
   b -{ 1 }-> 0 :|: 
   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 }-> b :|: z = 1
   encArg(z) -{ 0 }-> activate(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 2 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> 0 :|: 
   encode_activate(z) -{ 0 }-> activate(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_b -{ 0 }-> b :|: 
   encode_b -{ 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
   encode_n__b -{ 0 }-> 2 :|: 
   encode_n__b -{ 0 }-> 0 :|: 
   f(z, z') -{ 1 }-> f(0, 2) :|: z' = X, X >= 0, z = X
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(15) 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,[b(Out)],[]).
eq(start(V1, V),0,[activate(V1, Out)],[V1 >= 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(start(V1, V),0,[fun4(V1, Out)],[V1 >= 0]).
eq(f(V1, V, Out),1,[f(0, 2, Ret)],[Out = Ret,V = X1,X1 >= 0,V1 = X1]).
eq(b(Out),1,[],[Out = 0]).
eq(b(Out),1,[],[Out = 2]).
eq(activate(V1, Out),1,[b(Ret1)],[Out = Ret1,V1 = 2]).
eq(activate(V1, Out),1,[],[Out = X2,X2 >= 0,V1 = X2]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[],[Out = 2,V1 = 2]).
eq(encArg(V1, Out),0,[encArg(V3, Ret0),encArg(V2, Ret11),f(Ret0, Ret11, Ret2)],[Out = Ret2,V3 >= 0,V1 = 1 + V2 + V3,V2 >= 0]).
eq(encArg(V1, Out),0,[b(Ret3)],[Out = Ret3,V1 = 1]).
eq(encArg(V1, Out),0,[encArg(V4, Ret01),activate(Ret01, Ret4)],[Out = Ret4,V1 = 1 + V4,V4 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V5, Ret02),encArg(V6, Ret12),f(Ret02, Ret12, Ret5)],[Out = Ret5,V5 >= 0,V6 >= 0,V1 = V5,V = V6]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(Out),0,[],[Out = 2]).
eq(fun3(Out),0,[b(Ret6)],[Out = Ret6]).
eq(fun4(V1, Out),0,[encArg(V7, Ret03),activate(Ret03, Ret7)],[Out = Ret7,V7 >= 0,V1 = V7]).
eq(encArg(V1, Out),0,[],[Out = 0,V8 >= 0,V1 = V8]).
eq(fun(V1, V, Out),0,[],[Out = 0,V10 >= 0,V9 >= 0,V1 = V10,V = V9]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(Out),0,[],[Out = 0]).
eq(fun4(V1, Out),0,[],[Out = 0,V11 >= 0,V1 = V11]).
eq(f(V1, V, Out),0,[],[Out = 0,V12 >= 0,V13 >= 0,V1 = V12,V = V13]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(b(Out),[],[Out]).
input_output_vars(activate(V1,Out),[V1],[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]).
input_output_vars(fun4(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [b/1]
1. non_recursive  : [activate/2]
2. recursive  : [f/3]
3. recursive [non_tail,multiple] : [encArg/2]
4. non_recursive  : [fun/3]
5. non_recursive  : [fun1/1]
6. non_recursive  : [fun2/1]
7. non_recursive  : [fun3/1]
8. non_recursive  : [fun4/2]
9. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into b/1
1. SCC is partially evaluated into activate/2
2. SCC is partially evaluated into f/3
3. SCC is partially evaluated into encArg/2
4. SCC is partially evaluated into fun/3
5. SCC is completely evaluated into other SCCs
6. SCC is partially evaluated into fun2/1
7. SCC is partially evaluated into fun3/1
8. SCC is partially evaluated into fun4/2
9. SCC is partially evaluated into start/2

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

### Specialization of cost equations b/1 
* CE 13 is refined into CE [29] 
* CE 12 is refined into CE [30] 


### Cost equations --> "Loop" of b/1 
* CEs [29] --> Loop 18 
* CEs [30] --> Loop 19 

### Ranking functions of CR b(Out) 

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


### Specialization of cost equations activate/2 
* CE 15 is refined into CE [31] 
* CE 14 is refined into CE [32,33] 


### Cost equations --> "Loop" of activate/2 
* CEs [31,33] --> Loop 20 
* CEs [32] --> Loop 21 

### Ranking functions of CR activate(V1,Out) 

#### Partial ranking functions of CR activate(V1,Out) 


### Specialization of cost equations f/3 
* CE 11 is refined into CE [34] 
* CE 10 is refined into CE [35] 


### Cost equations --> "Loop" of f/3 
* CEs [35] --> Loop 22 
* CEs [34] --> Loop 23 

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

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


### Specialization of cost equations encArg/2 
* CE 16 is refined into CE [36] 
* CE 17 is refined into CE [37] 
* CE 19 is refined into CE [38,39] 
* CE 20 is refined into CE [40,41] 
* CE 18 is refined into CE [42] 


### Cost equations --> "Loop" of encArg/2 
* CEs [42] --> Loop 24 
* CEs [41] --> Loop 25 
* CEs [40] --> Loop 26 
* CEs [37] --> Loop 27 
* CEs [39] --> Loop 28 
* CEs [36,38] --> Loop 29 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [24,25,26]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [24,25,26]:
  - RF of loop [24:1,24:2,25:1,26:1]:
    V1


### Specialization of cost equations fun/3 
* CE 21 is refined into CE [43,44,45,46,47,48,49,50,51] 
* CE 22 is refined into CE [52] 


### Cost equations --> "Loop" of fun/3 
* CEs [46,49] --> Loop 30 
* CEs [43,44,45,47,48,50,51,52] --> Loop 31 

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

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


### Specialization of cost equations fun2/1 
* CE 23 is refined into CE [53] 
* CE 24 is refined into CE [54] 


### Cost equations --> "Loop" of fun2/1 
* CEs [53] --> Loop 32 
* CEs [54] --> Loop 33 

### Ranking functions of CR fun2(Out) 

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


### Specialization of cost equations fun3/1 
* CE 25 is refined into CE [55,56] 
* CE 26 is refined into CE [57] 


### Cost equations --> "Loop" of fun3/1 
* CEs [56] --> Loop 34 
* CEs [55,57] --> Loop 35 

### Ranking functions of CR fun3(Out) 

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


### Specialization of cost equations fun4/2 
* CE 27 is refined into CE [58,59,60,61,62] 
* CE 28 is refined into CE [63] 


### Cost equations --> "Loop" of fun4/2 
* CEs [61] --> Loop 36 
* CEs [59] --> Loop 37 
* CEs [58,60,62,63] --> Loop 38 

### Ranking functions of CR fun4(V1,Out) 

#### Partial ranking functions of CR fun4(V1,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [64] 
* CE 2 is refined into CE [65,66] 
* CE 3 is refined into CE [67,68] 
* CE 4 is refined into CE [69,70,71] 
* CE 5 is refined into CE [72] 
* CE 6 is refined into CE [73] 
* CE 7 is refined into CE [74,75] 
* CE 8 is refined into CE [76,77] 
* CE 9 is refined into CE [78,79,80] 


### Cost equations --> "Loop" of start/2 
* CEs [64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80] --> Loop 39 

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

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


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

#### Cost of chains of b(Out):
* Chain [19]: 1
  with precondition: [Out=0] 

* Chain [18]: 1
  with precondition: [Out=2] 


#### Cost of chains of activate(V1,Out):
* Chain [21]: 2
  with precondition: [V1=2,Out=0] 

* Chain [20]: 2
  with precondition: [V1=Out,V1>=0] 


#### Cost of chains of f(V1,V,Out):
* Chain [23]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [22,23]: 1
  with precondition: [Out=0,V1=V,V1>=0] 


#### Cost of chains of encArg(V1,Out):
* Chain [29]: 1
  with precondition: [Out=0,V1>=0] 

* Chain [28]: 1
  with precondition: [V1=1,Out=2] 

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

* Chain [multiple([24,25,26],[[29],[28],[27]])]: 5*it(24)+1*it([28])+1*it([29])+0
  Such that:it([29]) =< V1+1
it([28]) =< V1/2+1/2
aux(1) =< V1
it(24) =< aux(1)
it([28]) =< aux(1)

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


#### Cost of chains of fun(V1,V,Out):
* Chain [31]: 3*s(6)+3*s(7)+15*s(8)+2*s(10)+2*s(11)+10*s(12)+3
  Such that:aux(2) =< V1
aux(3) =< V1+1
aux(4) =< V1/2+1/2
aux(5) =< V
aux(6) =< V+1
aux(7) =< V/2+1/2
s(10) =< aux(3)
s(11) =< aux(4)
s(6) =< aux(6)
s(7) =< aux(7)
s(8) =< aux(5)
s(7) =< aux(5)
s(12) =< aux(2)
s(11) =< aux(2)

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

* Chain [30]: 1*s(26)+1*s(27)+5*s(28)+3
  Such that:s(25) =< V1
s(26) =< V1+1
s(27) =< V1/2+1/2
s(28) =< s(25)
s(27) =< s(25)

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


#### Cost of chains of fun2(Out):
* Chain [33]: 0
  with precondition: [Out=0] 

* Chain [32]: 0
  with precondition: [Out=2] 


#### Cost of chains of fun3(Out):
* Chain [35]: 1
  with precondition: [Out=0] 

* Chain [34]: 1
  with precondition: [Out=2] 


#### Cost of chains of fun4(V1,Out):
* Chain [38]: 1*s(46)+1*s(47)+5*s(48)+3
  Such that:s(45) =< V1
s(46) =< V1+1
s(47) =< V1/2+1/2
s(48) =< s(45)
s(47) =< s(45)

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

* Chain [37]: 3
  with precondition: [V1=1,Out=2] 

* Chain [36]: 1*s(50)+1*s(51)+5*s(52)+2
  Such that:s(49) =< V1
s(50) =< V1+1
s(51) =< V1/2+1/2
s(52) =< s(49)
s(51) =< s(49)

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


#### Cost of chains of start(V1,V):
* Chain [39]: 6*s(54)+6*s(55)+30*s(56)+3*s(66)+3*s(67)+15*s(68)+3
  Such that:s(57) =< V
s(58) =< V+1
s(59) =< V/2+1/2
aux(11) =< V1
aux(12) =< V1+1
aux(13) =< V1/2+1/2
s(54) =< aux(12)
s(55) =< aux(13)
s(56) =< aux(11)
s(55) =< aux(11)
s(66) =< s(58)
s(67) =< s(59)
s(68) =< s(57)
s(67) =< s(57)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [39] with precondition: [] 
    - Upper bound: nat(V1)*30+3+nat(V)*15+nat(V1+1)*6+nat(V+1)*3+nat(V1/2+1/2)*6+nat(V/2+1/2)*3 
    - Complexity: n 

### Maximum cost of start(V1,V): nat(V1)*30+3+nat(V)*15+nat(V1+1)*6+nat(V+1)*3+nat(V1/2+1/2)*6+nat(V/2+1/2)*3 
Asymptotic class: n 
* Total analysis performed in 240 ms.


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

(16)
BOUNDS(1, n^1)