/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 68 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, 387 ms]
    (14) BOUNDS(1, n^2)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 328 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 1340 ms]
            (28) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) 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(0)) -> 0
   half(s(s(x))) -> s(half(x))
   bits(0) -> 0
   bits(s(x)) -> s(bits(half(s(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(s(x_1)) -> s(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_bits(x_1)) -> bits(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_bits(x_1) -> bits(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (innermost) 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(0)) -> 0
   half(s(s(x))) -> s(half(x))
   bits(0) -> 0
   bits(s(x)) -> s(bits(half(s(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_bits(x_1)) -> bits(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_bits(x_1) -> bits(encArg(x_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(n^1, n^2).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   bits(0) -> 0
   bits(s(x)) -> s(bits(half(s(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_bits(x_1)) -> bits(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_bits(x_1) -> bits(encArg(x_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^2).


The TRS R consists of the following rules:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   bits(0) -> 0 [1]
   bits(s(x)) -> s(bits(half(s(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_bits(x_1)) -> bits(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_bits(x_1) -> bits(encArg(x_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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   bits(0) -> 0 [1]
   bits(s(x)) -> s(bits(half(s(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_bits(x_1)) -> bits(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_bits(x_1) -> bits(encArg(x_1)) [0]

The TRS has the following type information:
half :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits
0 :: 0:s:cons_half:cons_bits
s :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits
bits :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits
encArg :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits
cons_half :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits
cons_bits :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits
encode_half :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits
encode_0 :: 0:s:cons_half:cons_bits
encode_s :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits
encode_bits :: 0:s:cons_half:cons_bits -> 0:s:cons_half:cons_bits

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_half(v0) -> null_encode_half [0]
   encode_0 -> null_encode_0 [0]
   encode_s(v0) -> null_encode_s [0]
   encode_bits(v0) -> null_encode_bits [0]
   half(v0) -> null_half [0]
   bits(v0) -> null_bits [0]

And the following fresh constants:    null_encArg, null_encode_half, null_encode_0, null_encode_s, null_encode_bits, null_half, null_bits

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

(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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   bits(0) -> 0 [1]
   bits(s(x)) -> s(bits(half(s(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_bits(x_1)) -> bits(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_bits(x_1) -> bits(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_bits(v0) -> null_encode_bits [0]
   half(v0) -> null_half [0]
   bits(v0) -> null_bits [0]

The TRS has the following type information:
half :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
0 :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
s :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
bits :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
encArg :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
cons_half :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
cons_bits :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
encode_half :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
encode_0 :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
encode_s :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
encode_bits :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits -> 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
null_encArg :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
null_encode_half :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
null_encode_0 :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
null_encode_s :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
null_encode_bits :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
null_half :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits
null_bits :: 0:s:cons_half:cons_bits:null_encArg:null_encode_half:null_encode_0:null_encode_s:null_encode_bits:null_half:null_bits

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
   null_encArg => 0
   null_encode_half => 0
   null_encode_0 => 0
   null_encode_s => 0
   null_encode_bits => 0
   null_half => 0
   null_bits => 0

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

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

   bits(z) -{ 1 }-> 0 :|: z = 0
   bits(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   bits(z) -{ 1 }-> 1 + bits(half(1 + x)) :|: x >= 0, z = 1 + x
   encArg(z) -{ 0 }-> half(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> bits(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_bits(z) -{ 0 }-> bits(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_bits(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_half(z) -{ 0 }-> half(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_half(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) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)

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),0,[half(V, Out)],[V >= 0]).
eq(start(V),0,[bits(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,[],[Out = 0,V = 1]).
eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
eq(bits(V, Out),1,[],[Out = 0,V = 0]).
eq(bits(V, Out),1,[half(1 + V2, Ret10),bits(Ret10, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 1 + 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),bits(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),bits(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(bits(V, Out),0,[],[Out = 0,V14 >= 0,V = V14]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(bits(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  : [bits/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 bits/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 9 is refined into CE [25] 
* CE 8 is refined into CE [26] 
* CE 11 is refined into CE [27] 
* CE 10 is refined into CE [28] 


### Cost equations --> "Loop" of half/2 
* CEs [28] --> Loop 12 
* CEs [25] --> Loop 13 
* CEs [26,27] --> Loop 14 

### Ranking functions of CR half(V,Out) 
* RF of phase [12]: [V-1]

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


### Specialization of cost equations bits/2 
* CE 12 is refined into CE [29] 
* CE 14 is refined into CE [30] 
* CE 13 is refined into CE [31,32] 


### Cost equations --> "Loop" of bits/2 
* CEs [32] --> Loop 15 
* CEs [31] --> Loop 16 
* CEs [29,30] --> Loop 17 

### Ranking functions of CR bits(V,Out) 
* RF of phase [15]: [V-1]

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


### Specialization of cost equations encArg/2 
* CE 15 is refined into CE [33] 
* CE 16 is refined into CE [34] 
* CE 17 is refined into CE [35,36] 
* CE 18 is refined into CE [37,38,39,40] 


### Cost equations --> "Loop" of encArg/2 
* CEs [40] --> Loop 18 
* CEs [36,39] --> Loop 19 
* CEs [34] --> Loop 20 
* CEs [38] --> Loop 21 
* CEs [35,37] --> Loop 22 
* CEs [33] --> Loop 23 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [18,19,20,21,22]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [18,19,20,21,22]:
  - RF of loop [18:1,19:1,20:1,21:1,22:1]:
    V


### Specialization of cost equations fun/2 
* CE 19 is refined into CE [41,42,43] 
* CE 20 is refined into CE [44] 


### Cost equations --> "Loop" of fun/2 
* CEs [43] --> Loop 24 
* CEs [41,42,44] --> Loop 25 

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

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


### Specialization of cost equations fun2/2 
* CE 21 is refined into CE [45,46] 
* CE 22 is refined into CE [47] 


### Cost equations --> "Loop" of fun2/2 
* CEs [46] --> Loop 26 
* CEs [45] --> Loop 27 
* CEs [47] --> Loop 28 

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

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


### Specialization of cost equations fun3/2 
* CE 23 is refined into CE [48,49,50,51,52] 
* CE 24 is refined into CE [53] 


### Cost equations --> "Loop" of fun3/2 
* CEs [52] --> Loop 29 
* CEs [51] --> Loop 30 
* CEs [50] --> Loop 31 
* CEs [48,49,53] --> Loop 32 

### 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 [54,55] 
* CE 2 is refined into CE [56,57,58,59] 
* CE 3 is refined into CE [60,61] 
* CE 4 is refined into CE [62,63] 
* CE 5 is refined into CE [64] 
* CE 6 is refined into CE [65,66,67] 
* CE 7 is refined into CE [68,69,70,71] 


### Cost equations --> "Loop" of start/1 
* CEs [54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71] --> Loop 33 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of half(V,Out):
* Chain [[12],14]: 1*it(12)+1
  Such that:it(12) =< 2*Out

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

* Chain [[12],13]: 1*it(12)+1
  Such that:it(12) =< 2*Out

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

* Chain [14]: 1
  with precondition: [Out=0,V>=0] 

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


#### Cost of chains of bits(V,Out):
* Chain [[15],17]: 2*it(15)+2*s(7)+1
  Such that:it(15) =< V
s(8) =< 2*V
s(7) =< s(8)

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

* Chain [[15],16,17]: 2*it(15)+2*s(7)+3
  Such that:it(15) =< V
s(8) =< 2*V
s(7) =< s(8)

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

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

* Chain [16,17]: 3
  with precondition: [Out=1,V>=1] 


#### Cost of chains of encArg(V,Out):
* Chain [[18,19,20,21,22],23]: 8*it(18)+2*s(24)+2*s(25)+4*s(27)+2*s(28)+0
  Such that:aux(9) =< V
it(18) =< aux(9)
aux(5) =< aux(9)
aux(4) =< it(18)*aux(9)
s(24) =< it(18)*aux(9)
aux(6) =< it(18)*aux(5)
s(26) =< aux(4)*2
s(29) =< aux(6)*2
s(27) =< aux(6)
s(28) =< s(29)
s(25) =< s(26)

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

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


#### Cost of chains of fun(V,Out):
* Chain [25]: 8*s(32)+2*s(35)+4*s(39)+2*s(40)+2*s(41)+1
  Such that:s(31) =< V
s(32) =< s(31)
s(33) =< s(31)
s(34) =< s(32)*s(31)
s(35) =< s(32)*s(31)
s(36) =< s(32)*s(33)
s(37) =< s(34)*2
s(38) =< s(36)*2
s(39) =< s(36)
s(40) =< s(38)
s(41) =< s(37)

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

* Chain [24]: 10*s(43)+2*s(46)+4*s(50)+2*s(51)+2*s(52)+1
  Such that:aux(10) =< V
s(43) =< aux(10)
s(44) =< aux(10)
s(45) =< s(43)*aux(10)
s(46) =< s(43)*aux(10)
s(47) =< s(43)*s(44)
s(48) =< s(45)*2
s(49) =< s(47)*2
s(50) =< s(47)
s(51) =< s(49)
s(52) =< s(48)

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


#### Cost of chains of fun2(V,Out):
* Chain [28]: 0
  with precondition: [Out=0,V>=0] 

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

* Chain [26]: 8*s(56)+2*s(59)+4*s(63)+2*s(64)+2*s(65)+0
  Such that:s(55) =< V
s(56) =< s(55)
s(57) =< s(55)
s(58) =< s(56)*s(55)
s(59) =< s(56)*s(55)
s(60) =< s(56)*s(57)
s(61) =< s(58)*2
s(62) =< s(60)*2
s(63) =< s(60)
s(64) =< s(62)
s(65) =< s(61)

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


#### Cost of chains of fun3(V,Out):
* Chain [32]: 8*s(67)+2*s(70)+4*s(74)+2*s(75)+2*s(76)+1
  Such that:s(66) =< V
s(67) =< s(66)
s(68) =< s(66)
s(69) =< s(67)*s(66)
s(70) =< s(67)*s(66)
s(71) =< s(67)*s(68)
s(72) =< s(69)*2
s(73) =< s(71)*2
s(74) =< s(71)
s(75) =< s(73)
s(76) =< s(72)

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

* Chain [31]: 8*s(78)+2*s(81)+4*s(85)+2*s(86)+2*s(87)+3
  Such that:s(77) =< V
s(78) =< s(77)
s(79) =< s(77)
s(80) =< s(78)*s(77)
s(81) =< s(78)*s(77)
s(82) =< s(78)*s(79)
s(83) =< s(80)*2
s(84) =< s(82)*2
s(85) =< s(82)
s(86) =< s(84)
s(87) =< s(83)

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

* Chain [30]: 10*s(89)+2*s(92)+4*s(96)+2*s(97)+2*s(98)+2*s(101)+1
  Such that:s(100) =< 2*V
aux(11) =< V
s(89) =< aux(11)
s(101) =< s(100)
s(90) =< aux(11)
s(91) =< s(89)*aux(11)
s(92) =< s(89)*aux(11)
s(93) =< s(89)*s(90)
s(94) =< s(91)*2
s(95) =< s(93)*2
s(96) =< s(93)
s(97) =< s(95)
s(98) =< s(94)

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

* Chain [29]: 10*s(103)+2*s(106)+4*s(110)+2*s(111)+2*s(112)+2*s(115)+3
  Such that:s(114) =< 2*V
aux(12) =< V
s(103) =< aux(12)
s(115) =< s(114)
s(104) =< aux(12)
s(105) =< s(103)*aux(12)
s(106) =< s(103)*aux(12)
s(107) =< s(103)*s(104)
s(108) =< s(105)*2
s(109) =< s(107)*2
s(110) =< s(107)
s(111) =< s(109)
s(112) =< s(108)

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


#### Cost of chains of start(V):
* Chain [33]: 76*s(117)+8*s(120)+16*s(128)+32*s(132)+16*s(133)+16*s(134)+3
  Such that:aux(13) =< V
aux(14) =< 2*V
s(117) =< aux(13)
s(120) =< aux(14)
s(126) =< aux(13)
s(127) =< s(117)*aux(13)
s(128) =< s(117)*aux(13)
s(129) =< s(117)*s(126)
s(130) =< s(127)*2
s(131) =< s(129)*2
s(132) =< s(129)
s(133) =< s(131)
s(134) =< s(130)

  with precondition: [] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [33] with precondition: [] 
    - Upper bound: nat(V)*76+3+nat(V)*112*nat(V)+nat(2*V)*8 
    - Complexity: n^2 

### Maximum cost of start(V): nat(V)*76+3+nat(V)*112*nat(V)+nat(2*V)*8 
Asymptotic class: n^2 
* Total analysis performed in 297 ms.


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

(14)
BOUNDS(1, n^2)

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

(15) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   bits(0') -> 0'
   bits(s(x)) -> s(bits(half(s(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_bits(x_1)) -> bits(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_bits(x_1) -> bits(encArg(x_1))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
bits(0') -> 0'
bits(s(x)) -> s(bits(half(s(x))))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_bits(x_1)) -> bits(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_bits(x_1) -> bits(encArg(x_1))

Types:
half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
0' :: 0':s:cons_half:cons_bits
s :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encArg :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
cons_half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
cons_bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_0 :: 0':s:cons_half:cons_bits
encode_s :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
hole_0':s:cons_half:cons_bits1_2 :: 0':s:cons_half:cons_bits
gen_0':s:cons_half:cons_bits2_2 :: Nat -> 0':s:cons_half:cons_bits

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
half, bits, encArg

They will be analysed ascendingly in the following order:
half < bits
half < encArg
bits < encArg

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

(20)
Obligation:
Innermost TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
bits(0') -> 0'
bits(s(x)) -> s(bits(half(s(x))))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_bits(x_1)) -> bits(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_bits(x_1) -> bits(encArg(x_1))

Types:
half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
0' :: 0':s:cons_half:cons_bits
s :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encArg :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
cons_half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
cons_bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_0 :: 0':s:cons_half:cons_bits
encode_s :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
hole_0':s:cons_half:cons_bits1_2 :: 0':s:cons_half:cons_bits
gen_0':s:cons_half:cons_bits2_2 :: Nat -> 0':s:cons_half:cons_bits


Generator Equations:
gen_0':s:cons_half:cons_bits2_2(0) <=> 0'
gen_0':s:cons_half:cons_bits2_2(+(x, 1)) <=> s(gen_0':s:cons_half:cons_bits2_2(x))


The following defined symbols remain to be analysed:
half, bits, encArg

They will be analysed ascendingly in the following order:
half < bits
half < encArg
bits < encArg

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s:cons_half:cons_bits2_2(*(2, n4_2))) -> gen_0':s:cons_half:cons_bits2_2(n4_2), rt in Omega(1 + n4_2)

Induction Base:
half(gen_0':s:cons_half:cons_bits2_2(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s:cons_half:cons_bits2_2(*(2, +(n4_2, 1)))) ->_R^Omega(1)
s(half(gen_0':s:cons_half:cons_bits2_2(*(2, n4_2)))) ->_IH
s(gen_0':s:cons_half:cons_bits2_2(c5_2))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
bits(0') -> 0'
bits(s(x)) -> s(bits(half(s(x))))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_bits(x_1)) -> bits(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_bits(x_1) -> bits(encArg(x_1))

Types:
half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
0' :: 0':s:cons_half:cons_bits
s :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encArg :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
cons_half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
cons_bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_0 :: 0':s:cons_half:cons_bits
encode_s :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
hole_0':s:cons_half:cons_bits1_2 :: 0':s:cons_half:cons_bits
gen_0':s:cons_half:cons_bits2_2 :: Nat -> 0':s:cons_half:cons_bits


Generator Equations:
gen_0':s:cons_half:cons_bits2_2(0) <=> 0'
gen_0':s:cons_half:cons_bits2_2(+(x, 1)) <=> s(gen_0':s:cons_half:cons_bits2_2(x))


The following defined symbols remain to be analysed:
half, bits, encArg

They will be analysed ascendingly in the following order:
half < bits
half < encArg
bits < encArg

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
bits(0') -> 0'
bits(s(x)) -> s(bits(half(s(x))))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_half(x_1)) -> half(encArg(x_1))
encArg(cons_bits(x_1)) -> bits(encArg(x_1))
encode_half(x_1) -> half(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_bits(x_1) -> bits(encArg(x_1))

Types:
half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
0' :: 0':s:cons_half:cons_bits
s :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encArg :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
cons_half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
cons_bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_half :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_0 :: 0':s:cons_half:cons_bits
encode_s :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
encode_bits :: 0':s:cons_half:cons_bits -> 0':s:cons_half:cons_bits
hole_0':s:cons_half:cons_bits1_2 :: 0':s:cons_half:cons_bits
gen_0':s:cons_half:cons_bits2_2 :: Nat -> 0':s:cons_half:cons_bits


Lemmas:
half(gen_0':s:cons_half:cons_bits2_2(*(2, n4_2))) -> gen_0':s:cons_half:cons_bits2_2(n4_2), rt in Omega(1 + n4_2)


Generator Equations:
gen_0':s:cons_half:cons_bits2_2(0) <=> 0'
gen_0':s:cons_half:cons_bits2_2(+(x, 1)) <=> s(gen_0':s:cons_half:cons_bits2_2(x))


The following defined symbols remain to be analysed:
bits, encArg

They will be analysed ascendingly in the following order:
bits < encArg

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:cons_half:cons_bits2_2(n1774_2)) -> gen_0':s:cons_half:cons_bits2_2(n1774_2), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_half:cons_bits2_2(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_half:cons_bits2_2(+(n1774_2, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_half:cons_bits2_2(n1774_2))) ->_IH
s(gen_0':s:cons_half:cons_bits2_2(c1775_2))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(28)
BOUNDS(1, INF)