/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^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 41 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, 340 ms]
    (16) BOUNDS(1, n^2)
(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(18) TRS for Loop Detection
    (19) DecreasingLoopProof [LOWER BOUND(ID), 4 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) TRS for Loop Detection


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))


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

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(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^2).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(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^2).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(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^2).


The TRS R consists of the following rules:

   half(0) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(half(x)))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_half(x_1)) -> half(encArg(x_1)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_half(x_1) -> half(encArg(x_1)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_log(x_1) -> log(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:

   half(0) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(half(x)))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_half(x_1)) -> half(encArg(x_1)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_half(x_1) -> half(encArg(x_1)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]

The TRS has the following type information:
half :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log
0 :: 0:s:cons_half:cons_log
s :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log
log :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log
encArg :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log
cons_half :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log
cons_log :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log
encode_half :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log
encode_0 :: 0:s:cons_half:cons_log
encode_s :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log
encode_log :: 0:s:cons_half:cons_log -> 0:s:cons_half:cons_log

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_half(v0) -> null_encode_half [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   encode_log(v0) -> null_encode_log [0]
   half(v0) -> null_half [0]
   log(v0) -> null_log [0]

And the following fresh constants:    null_encArg, null_encode_half, null_encode_0, null_encode_s, null_encode_log, null_half, null_log

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

(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:

   half(0) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(half(x)))) [1]
   encArg(0) -> 0 [0]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(cons_half(x_1)) -> half(encArg(x_1)) [0]
   encArg(cons_log(x_1)) -> log(encArg(x_1)) [0]
   encode_half(x_1) -> half(encArg(x_1)) [0]
   encode_0 -> 0 [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_log(x_1) -> log(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_half(v0) -> null_encode_half [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   encode_log(v0) -> null_encode_log [0]
   half(v0) -> null_half [0]
   log(v0) -> null_log [0]

The TRS has the following type information:
half :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
0 :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
s :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
log :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
encArg :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
cons_half :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
cons_log :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
encode_half :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
encode_0 :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
encode_s :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
encode_log :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log -> 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
null_encArg :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
null_encode_half :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
null_encode_0 :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
null_encode_s :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
null_encode_log :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
null_half :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log
null_log :: 0:s:cons_half:cons_log:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_log:null_half:null_log

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:

   0 => 0
   null_encArg => 0
   null_encode_half => 0
   null_encode_0 => 0
   null_encode_s => 0
   null_encode_log => 0
   null_half => 0
   null_log => 0

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

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

   encArg(z) -{ 0 }-> log(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> half(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_0 -{ 0 }-> 0 :|: 
   encode_half(z) -{ 0 }-> half(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_log(z) -{ 0 }-> log(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   log(z) -{ 1 }-> 1 + log(1 + half(x)) :|: x >= 0, z = 1 + (1 + x)

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(V),0,[half(V, Out)],[V >= 0]).
eq(start(V),0,[log(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(Out)],[]).
eq(start(V),0,[fun2(V, Out)],[V >= 0]).
eq(start(V),0,[fun3(V, Out)],[V >= 0]).
eq(half(V, Out),1,[],[Out = 0,V = 0]).
eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
eq(log(V, Out),1,[],[Out = 0,V = 1]).
eq(log(V, Out),1,[half(V2, Ret101),log(1 + Ret101, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 2 + V2]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[encArg(V3, Ret12)],[Out = 1 + Ret12,V = 1 + V3,V3 >= 0]).
eq(encArg(V, Out),0,[encArg(V4, Ret0),half(Ret0, Ret)],[Out = Ret,V = 1 + V4,V4 >= 0]).
eq(encArg(V, Out),0,[encArg(V5, Ret01),log(Ret01, Ret2)],[Out = Ret2,V = 1 + V5,V5 >= 0]).
eq(fun(V, Out),0,[encArg(V6, Ret02),half(Ret02, Ret3)],[Out = Ret3,V6 >= 0,V = V6]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(V, Out),0,[encArg(V7, Ret13)],[Out = 1 + Ret13,V7 >= 0,V = V7]).
eq(fun3(V, Out),0,[encArg(V8, Ret03),log(Ret03, Ret4)],[Out = Ret4,V8 >= 0,V = V8]).
eq(encArg(V, Out),0,[],[Out = 0,V9 >= 0,V = V9]).
eq(fun(V, Out),0,[],[Out = 0,V10 >= 0,V = V10]).
eq(fun2(V, Out),0,[],[Out = 0,V11 >= 0,V = V11]).
eq(fun3(V, Out),0,[],[Out = 0,V12 >= 0,V = V12]).
eq(half(V, Out),0,[],[Out = 0,V13 >= 0,V = V13]).
eq(log(V, Out),0,[],[Out = 0,V14 >= 0,V = V14]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(log(V,Out),[V],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,Out),[V],[Out]).
input_output_vars(fun1(Out),[],[Out]).
input_output_vars(fun2(V,Out),[V],[Out]).
input_output_vars(fun3(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [half/2]
1. recursive  : [log/2]
2. recursive [non_tail] : [encArg/2]
3. non_recursive  : [fun/2]
4. non_recursive  : [fun1/1]
5. non_recursive  : [fun2/2]
6. non_recursive  : [fun3/2]
7. non_recursive  : [start/1]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into half/2
1. SCC is partially evaluated into log/2
2. SCC is partially evaluated into encArg/2
3. SCC is partially evaluated into fun/2
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into fun2/2
6. SCC is partially evaluated into fun3/2
7. SCC is partially evaluated into start/1

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

### Specialization of cost equations half/2 
* CE 8 is refined into CE [24] 
* CE 10 is refined into CE [25] 
* CE 9 is refined into CE [26] 


### Cost equations --> "Loop" of half/2 
* CEs [26] --> Loop 11 
* CEs [24,25] --> Loop 12 

### Ranking functions of CR half(V,Out) 
* RF of phase [11]: [V-1]

#### Partial ranking functions of CR half(V,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V-1


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


### Cost equations --> "Loop" of log/2 
* CEs [30] --> Loop 13 
* CEs [29] --> Loop 14 
* CEs [27,28] --> Loop 15 

### Ranking functions of CR log(V,Out) 
* RF of phase [13]: [V-3]

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


### Specialization of cost equations encArg/2 
* CE 14 is refined into CE [31] 
* CE 15 is refined into CE [32] 
* CE 16 is refined into CE [33,34] 
* CE 17 is refined into CE [35,36,37,38] 


### Cost equations --> "Loop" of encArg/2 
* CEs [37] --> Loop 16 
* CEs [32] --> Loop 17 
* CEs [34,36,38] --> Loop 18 
* CEs [33,35] --> Loop 19 
* CEs [31] --> Loop 20 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [16,17,18,19]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [16,17,18,19]:
  - RF of loop [16:1,17:1,18:1,19:1]:
    V


### Specialization of cost equations fun/2 
* CE 18 is refined into CE [39,40,41] 
* CE 19 is refined into CE [42] 


### Cost equations --> "Loop" of fun/2 
* CEs [41] --> Loop 21 
* CEs [39,40,42] --> Loop 22 

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

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


### Specialization of cost equations fun2/2 
* CE 20 is refined into CE [43,44] 
* CE 21 is refined into CE [45] 


### Cost equations --> "Loop" of fun2/2 
* CEs [44] --> Loop 23 
* CEs [43] --> Loop 24 
* CEs [45] --> Loop 25 

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

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


### Specialization of cost equations fun3/2 
* CE 22 is refined into CE [46,47,48,49,50] 
* CE 23 is refined into CE [51] 


### Cost equations --> "Loop" of fun3/2 
* CEs [50] --> Loop 26 
* CEs [49] --> Loop 27 
* CEs [48] --> Loop 28 
* CEs [46,47,51] --> Loop 29 

### Ranking functions of CR fun3(V,Out) 

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


### Specialization of cost equations start/1 
* CE 1 is refined into CE [52,53] 
* CE 2 is refined into CE [54,55,56,57] 
* CE 3 is refined into CE [58,59] 
* CE 4 is refined into CE [60,61] 
* CE 5 is refined into CE [62] 
* CE 6 is refined into CE [63,64,65] 
* CE 7 is refined into CE [66,67,68,69] 


### Cost equations --> "Loop" of start/1 
* CEs [52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69] --> Loop 30 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of half(V,Out):
* Chain [[11],12]: 1*it(11)+1
  Such that:it(11) =< 2*Out

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

* Chain [12]: 1
  with precondition: [Out=0,V>=0] 


#### Cost of chains of log(V,Out):
* Chain [[13],15]: 2*it(13)+1*s(3)+1
  Such that:it(13) =< V
s(3) =< 2*V

  with precondition: [Out>=1,V>=4*Out] 

* Chain [[13],14,15]: 2*it(13)+1*s(3)+3
  Such that:it(13) =< V
s(3) =< 2*V

  with precondition: [Out>=2,V+4>=4*Out] 

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

* Chain [14,15]: 3
  with precondition: [Out=1,V>=2] 


#### Cost of chains of encArg(V,Out):
* Chain [[16,17,18,19],20]: 5*it(16)+2*s(14)+1*s(15)+3*s(16)+1*s(17)+0
  Such that:aux(8) =< V
it(16) =< aux(8)
aux(4) =< aux(8)
aux(3) =< it(16)*aux(8)
s(14) =< it(16)*aux(8)
aux(5) =< it(16)*aux(4)
s(15) =< aux(3)*2
s(17) =< aux(5)*2
s(16) =< aux(5)

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

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


#### Cost of chains of fun(V,Out):
* Chain [22]: 5*s(20)+2*s(23)+1*s(25)+1*s(26)+3*s(27)+1
  Such that:s(19) =< V
s(20) =< s(19)
s(21) =< s(19)
s(22) =< s(20)*s(19)
s(23) =< s(20)*s(19)
s(24) =< s(20)*s(21)
s(25) =< s(22)*2
s(26) =< s(24)*2
s(27) =< s(24)

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

* Chain [21]: 6*s(29)+2*s(32)+1*s(34)+1*s(35)+3*s(36)+1
  Such that:aux(9) =< V
s(29) =< aux(9)
s(30) =< aux(9)
s(31) =< s(29)*aux(9)
s(32) =< s(29)*aux(9)
s(33) =< s(29)*s(30)
s(34) =< s(31)*2
s(35) =< s(33)*2
s(36) =< s(33)

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


#### Cost of chains of fun2(V,Out):
* Chain [25]: 0
  with precondition: [Out=0,V>=0] 

* Chain [24]: 0
  with precondition: [Out=1,V>=0] 

* Chain [23]: 5*s(39)+2*s(42)+1*s(44)+1*s(45)+3*s(46)+0
  Such that:s(38) =< V
s(39) =< s(38)
s(40) =< s(38)
s(41) =< s(39)*s(38)
s(42) =< s(39)*s(38)
s(43) =< s(39)*s(40)
s(44) =< s(41)*2
s(45) =< s(43)*2
s(46) =< s(43)

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


#### Cost of chains of fun3(V,Out):
* Chain [29]: 5*s(48)+2*s(51)+1*s(53)+1*s(54)+3*s(55)+1
  Such that:s(47) =< V
s(48) =< s(47)
s(49) =< s(47)
s(50) =< s(48)*s(47)
s(51) =< s(48)*s(47)
s(52) =< s(48)*s(49)
s(53) =< s(50)*2
s(54) =< s(52)*2
s(55) =< s(52)

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

* Chain [28]: 5*s(57)+2*s(60)+1*s(62)+1*s(63)+3*s(64)+3
  Such that:s(56) =< V
s(57) =< s(56)
s(58) =< s(56)
s(59) =< s(57)*s(56)
s(60) =< s(57)*s(56)
s(61) =< s(57)*s(58)
s(62) =< s(59)*2
s(63) =< s(61)*2
s(64) =< s(61)

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

* Chain [27]: 7*s(66)+2*s(69)+1*s(71)+1*s(72)+3*s(73)+1*s(75)+1
  Such that:s(75) =< 2*V
aux(10) =< V
s(66) =< aux(10)
s(67) =< aux(10)
s(68) =< s(66)*aux(10)
s(69) =< s(66)*aux(10)
s(70) =< s(66)*s(67)
s(71) =< s(68)*2
s(72) =< s(70)*2
s(73) =< s(70)

  with precondition: [Out>=1,V>=4*Out] 

* Chain [26]: 7*s(77)+2*s(80)+1*s(82)+1*s(83)+3*s(84)+1*s(86)+3
  Such that:s(86) =< 2*V
aux(11) =< V
s(77) =< aux(11)
s(78) =< aux(11)
s(79) =< s(77)*aux(11)
s(80) =< s(77)*aux(11)
s(81) =< s(77)*s(78)
s(82) =< s(79)*2
s(83) =< s(81)*2
s(84) =< s(81)

  with precondition: [Out>=2,V+4>=4*Out] 


#### Cost of chains of start(V):
* Chain [30]: 50*s(87)+4*s(89)+16*s(96)+8*s(98)+8*s(99)+24*s(100)+3
  Such that:aux(12) =< V
aux(13) =< 2*V
s(87) =< aux(12)
s(89) =< aux(13)
s(94) =< aux(12)
s(95) =< s(87)*aux(12)
s(96) =< s(87)*aux(12)
s(97) =< s(87)*s(94)
s(98) =< s(95)*2
s(99) =< s(97)*2
s(100) =< s(97)

  with precondition: [] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [30] with precondition: [] 
    - Upper bound: nat(V)*50+3+nat(V)*72*nat(V)+nat(2*V)*4 
    - Complexity: n^2 

### Maximum cost of start(V): nat(V)*50+3+nat(V)*72*nat(V)+nat(2*V)*4 
Asymptotic class: n^2 
* Total analysis performed in 259 ms.


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(16)
BOUNDS(1, n^2)

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

(18)
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^2).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))

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

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

The rewrite sequence

half(s(s(x))) ->^+ s(half(x))

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

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

The result substitution is [ ].




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

(20)
Complex Obligation (BEST)

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

(21)
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^2).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
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^2).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_log(x_1)) -> log(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_log(x_1) -> log(encArg(x_1))

Rewrite Strategy: FULL