/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^3))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, n^3).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 82 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, 339 ms]
    (18) BOUNDS(1, n^3)
(19) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(20) TRS for Loop Detection
    (21) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(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(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(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(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

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^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(c_w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   w(x0) -> c_w(x0)

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^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(c_w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   w(x0) -> c_w(x0)

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^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x)) [1]
   b(r(x)) -> r(b(x)) [1]
   b(c_w(x)) -> w(b(x)) [1]
   encArg(r(x_1)) -> r(encArg(x_1)) [0]
   encArg(cons_w(x_1)) -> w(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_w(x_1) -> w(encArg(x_1)) [0]
   encode_r(x_1) -> r(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   w(x0) -> c_w(x0) [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:

   w(r(x)) -> r(w(x)) [1]
   b(r(x)) -> r(b(x)) [1]
   b(c_w(x)) -> w(b(x)) [1]
   encArg(r(x_1)) -> r(encArg(x_1)) [0]
   encArg(cons_w(x_1)) -> w(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_w(x_1) -> w(encArg(x_1)) [0]
   encode_r(x_1) -> r(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   w(x0) -> c_w(x0) [0]

The TRS has the following type information:
w :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
r :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
b :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
c_w :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
encArg :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
cons_w :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
cons_b :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
encode_w :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
encode_r :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b
encode_b :: r:c_w:cons_w:cons_b -> r:c_w:cons_w:cons_b

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_w(v0) -> null_encode_w [0]
   encode_r(v0) -> null_encode_r [0]
   encode_b(v0) -> null_encode_b [0]
   w(v0) -> null_w [0]
   b(v0) -> null_b [0]

And the following fresh constants:    null_encArg, null_encode_w, null_encode_r, null_encode_b, null_w, null_b

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

(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:

   w(r(x)) -> r(w(x)) [1]
   b(r(x)) -> r(b(x)) [1]
   b(c_w(x)) -> w(b(x)) [1]
   encArg(r(x_1)) -> r(encArg(x_1)) [0]
   encArg(cons_w(x_1)) -> w(encArg(x_1)) [0]
   encArg(cons_b(x_1)) -> b(encArg(x_1)) [0]
   encode_w(x_1) -> w(encArg(x_1)) [0]
   encode_r(x_1) -> r(encArg(x_1)) [0]
   encode_b(x_1) -> b(encArg(x_1)) [0]
   w(x0) -> c_w(x0) [0]
   encArg(v0) -> null_encArg [0]
   encode_w(v0) -> null_encode_w [0]
   encode_r(v0) -> null_encode_r [0]
   encode_b(v0) -> null_encode_b [0]
   w(v0) -> null_w [0]
   b(v0) -> null_b [0]

The TRS has the following type information:
w :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
r :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
b :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
c_w :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
encArg :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
cons_w :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
cons_b :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
encode_w :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
encode_r :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
encode_b :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b -> r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
null_encArg :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
null_encode_w :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
null_encode_r :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
null_encode_b :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
null_w :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b
null_b :: r:c_w:cons_w:cons_b:null_encArg:null_encode_w:null_encode_r:null_encode_b:null_w:null_b

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:

   null_encArg => 0
   null_encode_w => 0
   null_encode_r => 0
   null_encode_b => 0
   null_w => 0
   null_b => 0

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   b(z) -{ 1 }-> w(b(x)) :|: x >= 0, z = 1 + x
   b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   b(z) -{ 1 }-> 1 + b(x) :|: x >= 0, z = 1 + x
   encArg(z) -{ 0 }-> w(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> b(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_b(z) -{ 0 }-> b(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_b(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_r(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_r(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_w(z) -{ 0 }-> w(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_w(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   w(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   w(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   w(z) -{ 1 }-> 1 + w(x) :|: x >= 0, z = 1 + x

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),0,[w(V, Out)],[V >= 0]).
eq(start(V),0,[b(V, Out)],[V >= 0]).
eq(start(V),0,[encArg(V, Out)],[V >= 0]).
eq(start(V),0,[fun(V, Out)],[V >= 0]).
eq(start(V),0,[fun1(V, Out)],[V >= 0]).
eq(start(V),0,[fun2(V, Out)],[V >= 0]).
eq(w(V, Out),1,[w(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 1 + V1]).
eq(b(V, Out),1,[b(V2, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 1 + V2]).
eq(b(V, Out),1,[b(V3, Ret0),w(Ret0, Ret)],[Out = Ret,V3 >= 0,V = 1 + V3]).
eq(encArg(V, Out),0,[encArg(V4, Ret12)],[Out = 1 + Ret12,V = 1 + V4,V4 >= 0]).
eq(encArg(V, Out),0,[encArg(V5, Ret01),w(Ret01, Ret2)],[Out = Ret2,V = 1 + V5,V5 >= 0]).
eq(encArg(V, Out),0,[encArg(V6, Ret02),b(Ret02, Ret3)],[Out = Ret3,V = 1 + V6,V6 >= 0]).
eq(fun(V, Out),0,[encArg(V7, Ret03),w(Ret03, Ret4)],[Out = Ret4,V7 >= 0,V = V7]).
eq(fun1(V, Out),0,[encArg(V8, Ret13)],[Out = 1 + Ret13,V8 >= 0,V = V8]).
eq(fun2(V, Out),0,[encArg(V9, Ret04),b(Ret04, Ret5)],[Out = Ret5,V9 >= 0,V = V9]).
eq(w(V, Out),0,[],[Out = 1 + V10,V = V10,V10 >= 0]).
eq(encArg(V, Out),0,[],[Out = 0,V11 >= 0,V = V11]).
eq(fun(V, Out),0,[],[Out = 0,V12 >= 0,V = V12]).
eq(fun1(V, Out),0,[],[Out = 0,V13 >= 0,V = V13]).
eq(fun2(V, Out),0,[],[Out = 0,V14 >= 0,V = V14]).
eq(w(V, Out),0,[],[Out = 0,V15 >= 0,V = V15]).
eq(b(V, Out),0,[],[Out = 0,V16 >= 0,V = V16]).
input_output_vars(w(V,Out),[V],[Out]).
input_output_vars(b(V,Out),[V],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fun1(V,Out),[V],[Out]).
input_output_vars(fun2(V,Out),[V],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into w/2
1. SCC is partially evaluated into b/2
2. SCC is partially evaluated into encArg/2
3. SCC is partially evaluated into fun/2
4. SCC is partially evaluated into fun1/2
5. SCC is partially evaluated into fun2/2
6. SCC is partially evaluated into start/1

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

### Specialization of cost equations w/2 
* CE 8 is refined into CE [23] 
* CE 9 is refined into CE [24] 
* CE 7 is refined into CE [25] 


### Cost equations --> "Loop" of w/2 
* CEs [25] --> Loop 12 
* CEs [23] --> Loop 13 
* CEs [24] --> Loop 14 

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

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


### Specialization of cost equations b/2 
* CE 12 is refined into CE [26] 
* CE 10 is refined into CE [27] 
* CE 11 is refined into CE [28,29,30] 


### Cost equations --> "Loop" of b/2 
* CEs [30] --> Loop 15 
* CEs [27,29] --> Loop 16 
* CEs [28] --> Loop 17 
* CEs [26] --> Loop 18 

### Ranking functions of CR b(V,Out) 
* RF of phase [15,16,17]: [V]

#### Partial ranking functions of CR b(V,Out) 
* Partial RF of phase [15,16,17]:
  - RF of loop [15:1,16:1,17:1]:
    V


### Specialization of cost equations encArg/2 
* CE 16 is refined into CE [31] 
* CE 13 is refined into CE [32] 
* CE 14 is refined into CE [33,34,35] 
* CE 15 is refined into CE [36,37] 


### Cost equations --> "Loop" of encArg/2 
* CEs [35,37] --> Loop 19 
* CEs [32,34] --> Loop 20 
* CEs [33,36] --> Loop 21 
* CEs [31] --> Loop 22 

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

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


### Specialization of cost equations fun/2 
* CE 17 is refined into CE [38,39,40,41,42] 
* CE 18 is refined into CE [43] 


### Cost equations --> "Loop" of fun/2 
* CEs [41,42] --> Loop 23 
* CEs [39] --> Loop 24 
* CEs [38,40,43] --> Loop 25 

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

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


### Specialization of cost equations fun1/2 
* CE 19 is refined into CE [44,45] 
* CE 20 is refined into CE [46] 


### Cost equations --> "Loop" of fun1/2 
* CEs [45] --> Loop 26 
* CEs [44] --> Loop 27 
* CEs [46] --> Loop 28 

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

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


### Specialization of cost equations fun2/2 
* CE 21 is refined into CE [47,48,49] 
* CE 22 is refined into CE [50] 


### Cost equations --> "Loop" of fun2/2 
* CEs [49] --> Loop 29 
* CEs [47,48,50] --> Loop 30 

### Ranking functions of CR fun2(V,Out) 

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


### Specialization of cost equations start/1 
* CE 1 is refined into CE [51,52,53] 
* CE 2 is refined into CE [54,55] 
* CE 3 is refined into CE [56,57] 
* CE 4 is refined into CE [58,59,60] 
* CE 5 is refined into CE [61,62,63] 
* CE 6 is refined into CE [64,65] 


### Cost equations --> "Loop" of start/1 
* CEs [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65] --> Loop 31 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of w(V,Out):
* Chain [[12],14]: 1*it(12)+0
  Such that:it(12) =< Out

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

* Chain [[12],13]: 1*it(12)+0
  Such that:it(12) =< Out

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

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

* Chain [13]: 0
  with precondition: [V+1=Out,V>=0] 


#### Cost of chains of b(V,Out):
* Chain [[15,16,17],18]: 3*it(15)+1*s(6)+1*s(7)+0
  Such that:aux(5) =< V
it(15) =< aux(5)
aux(2) =< aux(5)+1
s(6) =< it(15)*aux(5)
s(7) =< it(15)*aux(2)

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

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


#### Cost of chains of encArg(V,Out):
* Chain [[19,20,21],22]: 4*s(21)+1*s(22)+1*s(23)+1*s(25)+0
  Such that:aux(11) =< V
aux(8) =< aux(11)+1
aux(7) =< aux(11)
s(25) =< aux(11)*aux(8)
s(24) =< aux(11)*aux(7)
s(21) =< s(24)
s(22) =< s(21)*aux(11)
s(23) =< s(21)*aux(8)

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

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


#### Cost of chains of fun(V,Out):
* Chain [25]: 1*s(29)+4*s(31)+1*s(32)+1*s(33)+0
  Such that:s(26) =< V
s(27) =< s(26)+1
s(28) =< s(26)
s(29) =< s(26)*s(27)
s(30) =< s(26)*s(28)
s(31) =< s(30)
s(32) =< s(31)*s(26)
s(33) =< s(31)*s(27)

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

* Chain [24]: 1*s(34)+0
  Such that:s(34) =< 1

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

* Chain [23]: 2*s(38)+8*s(40)+2*s(41)+2*s(42)+1*s(43)+1*s(52)+0
  Such that:s(43) =< V+1
aux(13) =< V
s(52) =< aux(13)
s(36) =< aux(13)+1
s(37) =< aux(13)
s(38) =< aux(13)*s(36)
s(39) =< aux(13)*s(37)
s(40) =< s(39)
s(41) =< s(40)*aux(13)
s(42) =< s(40)*s(36)

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


#### Cost of chains of fun1(V,Out):
* Chain [28]: 0
  with precondition: [Out=0,V>=0] 

* Chain [27]: 0
  with precondition: [Out=1,V>=0] 

* Chain [26]: 1*s(56)+4*s(58)+1*s(59)+1*s(60)+0
  Such that:s(53) =< V
s(54) =< s(53)+1
s(55) =< s(53)
s(56) =< s(53)*s(54)
s(57) =< s(53)*s(55)
s(58) =< s(57)
s(59) =< s(58)*s(53)
s(60) =< s(58)*s(54)

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


#### Cost of chains of fun2(V,Out):
* Chain [30]: 1*s(64)+4*s(66)+1*s(67)+1*s(68)+0
  Such that:s(61) =< V
s(62) =< s(61)+1
s(63) =< s(61)
s(64) =< s(61)*s(62)
s(65) =< s(61)*s(63)
s(66) =< s(65)
s(67) =< s(66)*s(61)
s(68) =< s(66)*s(62)

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

* Chain [29]: 1*s(72)+4*s(74)+1*s(75)+1*s(76)+3*s(78)+1*s(80)+1*s(81)+0
  Such that:aux(14) =< V
s(78) =< aux(14)
s(70) =< aux(14)+1
s(80) =< s(78)*aux(14)
s(81) =< s(78)*s(70)
s(71) =< aux(14)
s(72) =< aux(14)*s(70)
s(73) =< aux(14)*s(71)
s(74) =< s(73)
s(75) =< s(74)*aux(14)
s(76) =< s(74)*s(70)

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


#### Cost of chains of start(V):
* Chain [31]: 2*s(82)+8*s(83)+2*s(87)+2*s(88)+7*s(92)+28*s(94)+7*s(95)+7*s(96)+1*s(105)+0
  Such that:s(105) =< 1
aux(15) =< V
aux(16) =< V+1
s(83) =< aux(15)
s(82) =< aux(16)
s(86) =< aux(15)+1
s(87) =< s(83)*aux(15)
s(88) =< s(83)*s(86)
s(91) =< aux(15)
s(92) =< aux(15)*s(86)
s(93) =< aux(15)*s(91)
s(94) =< s(93)
s(95) =< s(94)*aux(15)
s(96) =< s(94)*s(86)

  with precondition: [V>=0] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [31] with precondition: [V>=0] 
    - Upper bound: 17*V+1+46*V*V+14*V*V*V+(2*V+2) 
    - Complexity: n^3 

### Maximum cost of start(V): 17*V+1+46*V*V+14*V*V*V+(2*V+2) 
Asymptotic class: n^3 
* Total analysis performed in 256 ms.


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(18)
BOUNDS(1, n^3)

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(19) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(20)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

Rewrite Strategy: FULL
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(21) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

b(r(x)) ->^+ r(b(x))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x / r(x)].

The result substitution is [ ].




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(22)
Complex Obligation (BEST)

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(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

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

(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

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

   encArg(r(x_1)) -> r(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_w(x_1) -> w(encArg(x_1))
   encode_r(x_1) -> r(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))

Rewrite Strategy: FULL