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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 163 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, 332 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), 1131 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), 18 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:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(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(a) -> a
   encArg(b) -> b
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))


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

(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:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))

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:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))

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:

   rev(a) -> a [1]
   rev(b) -> b [1]
   rev(++(x, y)) -> ++(rev(y), rev(x)) [1]
   rev(++(x, x)) -> rev(x) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) [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:

   rev(a) -> a [1]
   rev(b) -> b [1]
   rev(++(x, y)) -> ++(rev(y), rev(x)) [1]
   rev(++(x, x)) -> rev(x) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) [0]

The TRS has the following type information:
rev :: a:b:++:cons_rev -> a:b:++:cons_rev
a :: a:b:++:cons_rev
b :: a:b:++:cons_rev
++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
encArg :: a:b:++:cons_rev -> a:b:++:cons_rev
cons_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_a :: a:b:++:cons_rev
encode_b :: a:b:++:cons_rev
encode_++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev

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_rev(v0) -> null_encode_rev [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   encode_++(v0, v1) -> null_encode_++ [0]
   rev(v0) -> null_rev [0]

And the following fresh constants:    null_encArg, null_encode_rev, null_encode_a, null_encode_b, null_encode_++, null_rev

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

(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:

   rev(a) -> a [1]
   rev(b) -> b [1]
   rev(++(x, y)) -> ++(rev(y), rev(x)) [1]
   rev(++(x, x)) -> rev(x) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_rev(x_1)) -> rev(encArg(x_1)) [0]
   encode_rev(x_1) -> rev(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> null_encArg [0]
   encode_rev(v0) -> null_encode_rev [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   encode_++(v0, v1) -> null_encode_++ [0]
   rev(v0) -> null_rev [0]

The TRS has the following type information:
rev :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev -> a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
a :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
b :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
++ :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev -> a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev -> a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
encArg :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev -> a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
cons_rev :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev -> a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
encode_rev :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev -> a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
encode_a :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
encode_b :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
encode_++ :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev -> a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev -> a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
null_encArg :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
null_encode_rev :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
null_encode_a :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
null_encode_b :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
null_encode_++ :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev
null_rev :: a:b:++:cons_rev:null_encArg:null_encode_rev:null_encode_a:null_encode_b:null_encode_++:null_rev

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:

   a => 0
   b => 1
   null_encArg => 0
   null_encode_rev => 0
   null_encode_a => 0
   null_encode_b => 0
   null_encode_++ => 0
   null_rev => 0

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

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

   encArg(z) -{ 0 }-> rev(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encode_++(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_++(z, z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 0 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_rev(z) -{ 0 }-> rev(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_rev(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   rev(z) -{ 1 }-> rev(x) :|: z = 1 + x + x, x >= 0
   rev(z) -{ 1 }-> 1 :|: z = 1
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   rev(z) -{ 1 }-> 1 + rev(y) + rev(x) :|: z = 1 + x + y, x >= 0, y >= 0

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, V8),0,[rev(V, Out)],[V >= 0]).
eq(start(V, V8),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V8),0,[fun(V, Out)],[V >= 0]).
eq(start(V, V8),0,[fun1(Out)],[]).
eq(start(V, V8),0,[fun2(Out)],[]).
eq(start(V, V8),0,[fun3(V, V8, Out)],[V >= 0,V8 >= 0]).
eq(rev(V, Out),1,[],[Out = 0,V = 0]).
eq(rev(V, Out),1,[],[Out = 1,V = 1]).
eq(rev(V, Out),1,[rev(V1, Ret01),rev(V2, Ret1)],[Out = 1 + Ret01 + Ret1,V = 1 + V1 + V2,V2 >= 0,V1 >= 0]).
eq(rev(V, Out),1,[rev(V3, Ret)],[Out = Ret,V = 1 + 2*V3,V3 >= 0]).
eq(encArg(V, Out),0,[],[Out = 0,V = 0]).
eq(encArg(V, Out),0,[],[Out = 1,V = 1]).
eq(encArg(V, Out),0,[encArg(V5, Ret011),encArg(V4, Ret11)],[Out = 1 + Ret011 + Ret11,V5 >= 0,V = 1 + V4 + V5,V4 >= 0]).
eq(encArg(V, Out),0,[encArg(V6, Ret0),rev(Ret0, Ret2)],[Out = Ret2,V = 1 + V6,V6 >= 0]).
eq(fun(V, Out),0,[encArg(V7, Ret02),rev(Ret02, Ret3)],[Out = Ret3,V7 >= 0,V = V7]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(Out),0,[],[Out = 1]).
eq(fun3(V, V8, Out),0,[encArg(V10, Ret012),encArg(V9, Ret12)],[Out = 1 + Ret012 + Ret12,V10 >= 0,V9 >= 0,V = V10,V8 = V9]).
eq(encArg(V, Out),0,[],[Out = 0,V11 >= 0,V = V11]).
eq(fun(V, Out),0,[],[Out = 0,V12 >= 0,V = V12]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(V, V8, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V = V14,V8 = V13]).
eq(rev(V, Out),0,[],[Out = 0,V15 >= 0,V = V15]).
input_output_vars(rev(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(Out),[],[Out]).
input_output_vars(fun3(V,V8,Out),[V,V8],[Out]).


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

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

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

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

### Specialization of cost equations rev/2 
* CE 8 is refined into CE [22] 
* CE 7 is refined into CE [23] 
* CE 11 is refined into CE [24] 
* CE 10 is refined into CE [25] 
* CE 9 is refined into CE [26] 


### Cost equations --> "Loop" of rev/2 
* CEs [26] --> Loop 14 
* CEs [25] --> Loop 15 
* CEs [22] --> Loop 16 
* CEs [23,24] --> Loop 17 

### Ranking functions of CR rev(V,Out) 
* RF of phase [14,15]: [V]

#### Partial ranking functions of CR rev(V,Out) 
* Partial RF of phase [14,15]:
  - RF of loop [14:1,14:2,15:1]:
    V


### Specialization of cost equations encArg/2 
* CE 12 is refined into CE [27] 
* CE 13 is refined into CE [28] 
* CE 15 is refined into CE [29,30] 
* CE 14 is refined into CE [31] 


### Cost equations --> "Loop" of encArg/2 
* CEs [31] --> Loop 18 
* CEs [29] --> Loop 19 
* CEs [30] --> Loop 20 
* CEs [27] --> Loop 21 
* CEs [28] --> Loop 22 

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

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [18,19,20]:
  - RF of loop [18:1,18:2,19:1,20:1]:
    V


### Specialization of cost equations fun/2 
* CE 16 is refined into CE [32,33,34] 
* CE 17 is refined into CE [35] 


### Cost equations --> "Loop" of fun/2 
* CEs [32] --> Loop 23 
* CEs [33,34,35] --> Loop 24 

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

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


### Specialization of cost equations fun2/1 
* CE 18 is refined into CE [36] 
* CE 19 is refined into CE [37] 


### Cost equations --> "Loop" of fun2/1 
* CEs [36] --> Loop 25 
* CEs [37] --> Loop 26 

### Ranking functions of CR fun2(Out) 

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


### Specialization of cost equations fun3/3 
* CE 20 is refined into CE [38,39,40,41] 
* CE 21 is refined into CE [42] 


### Cost equations --> "Loop" of fun3/3 
* CEs [38] --> Loop 27 
* CEs [39] --> Loop 28 
* CEs [40] --> Loop 29 
* CEs [41] --> Loop 30 
* CEs [42] --> Loop 31 

### Ranking functions of CR fun3(V,V8,Out) 

#### Partial ranking functions of CR fun3(V,V8,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [43,44] 
* CE 2 is refined into CE [45,46] 
* CE 3 is refined into CE [47,48] 
* CE 4 is refined into CE [49] 
* CE 5 is refined into CE [50,51] 
* CE 6 is refined into CE [52,53,54,55,56] 


### Cost equations --> "Loop" of start/2 
* CEs [43,44,45,46,47,48,49,50,51,52,53,54,55,56] --> Loop 32 

### Ranking functions of CR start(V,V8) 

#### Partial ranking functions of CR start(V,V8) 


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

#### Cost of chains of rev(V,Out):
* Chain [17]: 1
  with precondition: [Out=0,V>=0] 

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

* Chain [multiple([14,15],[[17],[16]])]: 2*it(14)+1*it([16])+1*it([17])+0
  Such that:it([17]) =< V+1
it([16]) =< V/2+1/2
aux(1) =< V
it(14) =< aux(1)
it([16]) =< aux(1)

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


#### Cost of chains of encArg(V,Out):
* Chain [22]: 0
  with precondition: [V=1,Out=1] 

* Chain [21]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([18,19,20],[[22],[21]])]: 2*it(19)+1*s(13)+1*s(14)+2*s(15)+0
  Such that:aux(5) =< V
it(19) =< aux(5)
aux(3) =< aux(5)+1
s(16) =< it(19)*aux(5)
aux(4) =< it(19)*aux(3)
s(13) =< it(19)*aux(3)
s(14) =< aux(4)*(1/2)
s(15) =< s(16)
s(14) =< s(16)

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


#### Cost of chains of fun(V,Out):
* Chain [24]: 2*s(26)+1*s(30)+1*s(31)+2*s(32)+1
  Such that:s(25) =< V
s(26) =< s(25)
s(27) =< s(25)+1
s(28) =< s(26)*s(25)
s(29) =< s(26)*s(27)
s(30) =< s(26)*s(27)
s(31) =< s(29)*(1/2)
s(32) =< s(28)
s(31) =< s(28)

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

* Chain [23]: 4*s(34)+1*s(38)+1*s(39)+2*s(40)+1*s(42)+1*s(43)+1
  Such that:s(42) =< V+1
s(43) =< V/2+1/2
aux(7) =< V
s(34) =< aux(7)
s(43) =< aux(7)
s(35) =< aux(7)+1
s(36) =< s(34)*aux(7)
s(37) =< s(34)*s(35)
s(38) =< s(34)*s(35)
s(39) =< s(37)*(1/2)
s(40) =< s(36)
s(39) =< s(36)

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


#### Cost of chains of fun2(Out):
* Chain [26]: 0
  with precondition: [Out=0] 

* Chain [25]: 0
  with precondition: [Out=1] 


#### Cost of chains of fun3(V,V8,Out):
* Chain [31]: 0
  with precondition: [Out=0,V>=0,V8>=0] 

* Chain [30]: 0
  with precondition: [Out=1,V>=0,V8>=0] 

* Chain [29]: 2*s(46)+1*s(50)+1*s(51)+2*s(52)+0
  Such that:s(45) =< V8
s(46) =< s(45)
s(47) =< s(45)+1
s(48) =< s(46)*s(45)
s(49) =< s(46)*s(47)
s(50) =< s(46)*s(47)
s(51) =< s(49)*(1/2)
s(52) =< s(48)
s(51) =< s(48)

  with precondition: [V>=0,V8>=1,Out>=1,V8+1>=Out] 

* Chain [28]: 2*s(54)+1*s(58)+1*s(59)+2*s(60)+0
  Such that:s(53) =< V
s(54) =< s(53)
s(55) =< s(53)+1
s(56) =< s(54)*s(53)
s(57) =< s(54)*s(55)
s(58) =< s(54)*s(55)
s(59) =< s(57)*(1/2)
s(60) =< s(56)
s(59) =< s(56)

  with precondition: [V>=1,V8>=0,Out>=1,V+1>=Out] 

* Chain [27]: 2*s(62)+1*s(66)+1*s(67)+2*s(68)+2*s(70)+1*s(74)+1*s(75)+2*s(76)+0
  Such that:s(61) =< V
s(69) =< V8
s(70) =< s(69)
s(71) =< s(69)+1
s(72) =< s(70)*s(69)
s(73) =< s(70)*s(71)
s(74) =< s(70)*s(71)
s(75) =< s(73)*(1/2)
s(76) =< s(72)
s(75) =< s(72)
s(62) =< s(61)
s(63) =< s(61)+1
s(64) =< s(62)*s(61)
s(65) =< s(62)*s(63)
s(66) =< s(62)*s(63)
s(67) =< s(65)*(1/2)
s(68) =< s(64)
s(67) =< s(64)

  with precondition: [V>=1,V8>=1,Out>=1,V+V8+1>=Out] 


#### Cost of chains of start(V,V8):
* Chain [32]: 2*s(78)+2*s(79)+14*s(80)+5*s(86)+5*s(87)+10*s(88)+4*s(108)+2*s(112)+2*s(113)+4*s(114)+1
  Such that:aux(8) =< V
aux(9) =< V+1
aux(10) =< V/2+1/2
aux(11) =< V8
s(78) =< aux(9)
s(79) =< aux(10)
s(80) =< aux(8)
s(79) =< aux(8)
s(83) =< aux(8)+1
s(84) =< s(80)*aux(8)
s(85) =< s(80)*s(83)
s(86) =< s(80)*s(83)
s(87) =< s(85)*(1/2)
s(88) =< s(84)
s(87) =< s(84)
s(108) =< aux(11)
s(109) =< aux(11)+1
s(110) =< s(108)*aux(11)
s(111) =< s(108)*s(109)
s(112) =< s(108)*s(109)
s(113) =< s(111)*(1/2)
s(114) =< s(110)
s(113) =< s(110)

  with precondition: [] 


Closed-form bounds of start(V,V8): 
-------------------------------------
* Chain [32] with precondition: [] 
    - Upper bound: 43/2*nat(V)+1+35/2*nat(V)*nat(V)+nat(V8)*7+nat(V8)*7*nat(V8)+nat(V+1)*2+nat(V/2+1/2)*2 
    - Complexity: n^2 

### Maximum cost of start(V,V8): 43/2*nat(V)+1+35/2*nat(V)*nat(V)+nat(V8)*7+nat(V8)*7*nat(V8)+nat(V+1)*2+nat(V/2+1/2)*2 
Asymptotic class: n^2 
* Total analysis performed in 255 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:

   rev(a) -> a
   rev(b) -> b
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(++(x, x)) -> rev(x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_rev(x_1)) -> rev(encArg(x_1))
   encode_rev(x_1) -> rev(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
rev(a) -> a
rev(b) -> b
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(++(x, x)) -> rev(x)
encArg(a) -> a
encArg(b) -> b
encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encode_rev(x_1) -> rev(encArg(x_1))
encode_a -> a
encode_b -> b
encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))

Types:
rev :: a:b:++:cons_rev -> a:b:++:cons_rev
a :: a:b:++:cons_rev
b :: a:b:++:cons_rev
++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
encArg :: a:b:++:cons_rev -> a:b:++:cons_rev
cons_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_a :: a:b:++:cons_rev
encode_b :: a:b:++:cons_rev
encode_++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
hole_a:b:++:cons_rev1_0 :: a:b:++:cons_rev
gen_a:b:++:cons_rev2_0 :: Nat -> a:b:++:cons_rev

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

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

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

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

(20)
Obligation:
Innermost TRS:
Rules:
rev(a) -> a
rev(b) -> b
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(++(x, x)) -> rev(x)
encArg(a) -> a
encArg(b) -> b
encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encode_rev(x_1) -> rev(encArg(x_1))
encode_a -> a
encode_b -> b
encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))

Types:
rev :: a:b:++:cons_rev -> a:b:++:cons_rev
a :: a:b:++:cons_rev
b :: a:b:++:cons_rev
++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
encArg :: a:b:++:cons_rev -> a:b:++:cons_rev
cons_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_a :: a:b:++:cons_rev
encode_b :: a:b:++:cons_rev
encode_++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
hole_a:b:++:cons_rev1_0 :: a:b:++:cons_rev
gen_a:b:++:cons_rev2_0 :: Nat -> a:b:++:cons_rev


Generator Equations:
gen_a:b:++:cons_rev2_0(0) <=> a
gen_a:b:++:cons_rev2_0(+(x, 1)) <=> ++(a, gen_a:b:++:cons_rev2_0(x))


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev(gen_a:b:++:cons_rev2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
rev(gen_a:b:++:cons_rev2_0(+(1, 0)))

Induction Step:
rev(gen_a:b:++:cons_rev2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
++(rev(gen_a:b:++:cons_rev2_0(+(1, n4_0))), rev(a)) ->_IH
++(*3_0, rev(a)) ->_R^Omega(1)
++(*3_0, a)

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:
rev(a) -> a
rev(b) -> b
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(++(x, x)) -> rev(x)
encArg(a) -> a
encArg(b) -> b
encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encode_rev(x_1) -> rev(encArg(x_1))
encode_a -> a
encode_b -> b
encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))

Types:
rev :: a:b:++:cons_rev -> a:b:++:cons_rev
a :: a:b:++:cons_rev
b :: a:b:++:cons_rev
++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
encArg :: a:b:++:cons_rev -> a:b:++:cons_rev
cons_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_a :: a:b:++:cons_rev
encode_b :: a:b:++:cons_rev
encode_++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
hole_a:b:++:cons_rev1_0 :: a:b:++:cons_rev
gen_a:b:++:cons_rev2_0 :: Nat -> a:b:++:cons_rev


Generator Equations:
gen_a:b:++:cons_rev2_0(0) <=> a
gen_a:b:++:cons_rev2_0(+(x, 1)) <=> ++(a, gen_a:b:++:cons_rev2_0(x))


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

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

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
rev(a) -> a
rev(b) -> b
rev(++(x, y)) -> ++(rev(y), rev(x))
rev(++(x, x)) -> rev(x)
encArg(a) -> a
encArg(b) -> b
encArg(++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
encArg(cons_rev(x_1)) -> rev(encArg(x_1))
encode_rev(x_1) -> rev(encArg(x_1))
encode_a -> a
encode_b -> b
encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))

Types:
rev :: a:b:++:cons_rev -> a:b:++:cons_rev
a :: a:b:++:cons_rev
b :: a:b:++:cons_rev
++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
encArg :: a:b:++:cons_rev -> a:b:++:cons_rev
cons_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_rev :: a:b:++:cons_rev -> a:b:++:cons_rev
encode_a :: a:b:++:cons_rev
encode_b :: a:b:++:cons_rev
encode_++ :: a:b:++:cons_rev -> a:b:++:cons_rev -> a:b:++:cons_rev
hole_a:b:++:cons_rev1_0 :: a:b:++:cons_rev
gen_a:b:++:cons_rev2_0 :: Nat -> a:b:++:cons_rev


Lemmas:
rev(gen_a:b:++:cons_rev2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_a:b:++:cons_rev2_0(0) <=> a
gen_a:b:++:cons_rev2_0(+(x, 1)) <=> ++(a, gen_a:b:++:cons_rev2_0(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:b:++:cons_rev2_0(n11844_0)) -> gen_a:b:++:cons_rev2_0(n11844_0), rt in Omega(0)

Induction Base:
encArg(gen_a:b:++:cons_rev2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_a:b:++:cons_rev2_0(+(n11844_0, 1))) ->_R^Omega(0)
++(encArg(a), encArg(gen_a:b:++:cons_rev2_0(n11844_0))) ->_R^Omega(0)
++(a, encArg(gen_a:b:++:cons_rev2_0(n11844_0))) ->_IH
++(a, gen_a:b:++:cons_rev2_0(c11845_0))

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

(28)
BOUNDS(1, INF)