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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 612 ms]
    (10) BOUNDS(1, n^2)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
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:

   length(nil) -> 0 [1]
   length(cons(x, l)) -> s(length(l)) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   head(cons(x, l)) -> x [1]
   head(nil) -> undefined [1]
   tail(nil) -> nil [1]
   tail(cons(x, l)) -> l [1]
   reverse(l) -> rev(0, l, nil, l) [1]
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig) [1]
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig) [1]
   if(false, x, l, accu, orig) -> accu [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   length(nil) -> 0 [1]
   length(cons(x, l)) -> s(length(l)) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   head(cons(x, l)) -> x [1]
   head(nil) -> undefined [1]
   tail(nil) -> nil [1]
   tail(cons(x, l)) -> l [1]
   reverse(l) -> rev(0, l, nil, l) [1]
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig) [1]
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig) [1]
   if(false, x, l, accu, orig) -> accu [1]

The TRS has the following type information:
length :: nil:cons -> 0:s
nil :: nil:cons
0 :: 0:s
cons :: undefined -> nil:cons -> nil:cons
s :: 0:s -> 0:s
lt :: 0:s -> 0:s -> false:true
false :: false:true
true :: false:true
head :: nil:cons -> undefined
undefined :: undefined
tail :: nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: 0:s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
if :: false:true -> 0:s -> nil:cons -> nil:cons -> nil:cons -> nil:cons

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

(5) 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:
none

And the following fresh constants: none

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

(6)
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:

   length(nil) -> 0 [1]
   length(cons(x, l)) -> s(length(l)) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   head(cons(x, l)) -> x [1]
   head(nil) -> undefined [1]
   tail(nil) -> nil [1]
   tail(cons(x, l)) -> l [1]
   reverse(l) -> rev(0, l, nil, l) [1]
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig) [1]
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig) [1]
   if(false, x, l, accu, orig) -> accu [1]

The TRS has the following type information:
length :: nil:cons -> 0:s
nil :: nil:cons
0 :: 0:s
cons :: undefined -> nil:cons -> nil:cons
s :: 0:s -> 0:s
lt :: 0:s -> 0:s -> false:true
false :: false:true
true :: false:true
head :: nil:cons -> undefined
undefined :: undefined
tail :: nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: 0:s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
if :: false:true -> 0:s -> nil:cons -> nil:cons -> nil:cons -> nil:cons

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

(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0
   0 => 0
   false => 0
   true => 1
   undefined => 0

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

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   head(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   head(z) -{ 1 }-> 0 :|: z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> accu :|: z2 = orig, z' = x, z1 = accu, orig >= 0, x >= 0, l >= 0, z = 0, accu >= 0, z'' = l
   if(z, z', z'', z1, z2) -{ 1 }-> rev(1 + x, tail(l), 1 + head(l) + accu, orig) :|: z2 = orig, z' = x, z1 = accu, z = 1, orig >= 0, x >= 0, l >= 0, accu >= 0, z'' = l
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 1 }-> 1 + length(l) :|: x >= 0, l >= 0, z = 1 + x + l
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   rev(z, z', z'', z1) -{ 1 }-> if(lt(x, length(orig)), x, l, accu, orig) :|: z' = l, orig >= 0, x >= 0, l >= 0, z'' = accu, z = x, accu >= 0, z1 = orig
   reverse(z) -{ 1 }-> rev(0, l, 0, l) :|: z = l, l >= 0
   tail(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tail(z) -{ 1 }-> 0 :|: z = 0

Only complete derivations are relevant for the runtime complexity.

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

(9) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V, V3, V14, V18, V23),0,[length(V, Out)],[V >= 0]).
eq(start(V, V3, V14, V18, V23),0,[lt(V, V3, Out)],[V >= 0,V3 >= 0]).
eq(start(V, V3, V14, V18, V23),0,[head(V, Out)],[V >= 0]).
eq(start(V, V3, V14, V18, V23),0,[tail(V, Out)],[V >= 0]).
eq(start(V, V3, V14, V18, V23),0,[reverse(V, Out)],[V >= 0]).
eq(start(V, V3, V14, V18, V23),0,[rev(V, V3, V14, V18, Out)],[V >= 0,V3 >= 0,V14 >= 0,V18 >= 0]).
eq(start(V, V3, V14, V18, V23),0,[if(V, V3, V14, V18, V23, Out)],[V >= 0,V3 >= 0,V14 >= 0,V18 >= 0,V23 >= 0]).
eq(length(V, Out),1,[],[Out = 0,V = 0]).
eq(length(V, Out),1,[length(V1, Ret1)],[Out = 1 + Ret1,V2 >= 0,V1 >= 0,V = 1 + V1 + V2]).
eq(lt(V, V3, Out),1,[],[Out = 0,V4 >= 0,V = V4,V3 = 0]).
eq(lt(V, V3, Out),1,[],[Out = 1,V3 = 1 + V5,V5 >= 0,V = 0]).
eq(lt(V, V3, Out),1,[lt(V6, V7, Ret)],[Out = Ret,V3 = 1 + V7,V6 >= 0,V7 >= 0,V = 1 + V6]).
eq(head(V, Out),1,[],[Out = V8,V8 >= 0,V9 >= 0,V = 1 + V8 + V9]).
eq(head(V, Out),1,[],[Out = 0,V = 0]).
eq(tail(V, Out),1,[],[Out = 0,V = 0]).
eq(tail(V, Out),1,[],[Out = V11,V10 >= 0,V11 >= 0,V = 1 + V10 + V11]).
eq(reverse(V, Out),1,[rev(0, V12, 0, V12, Ret2)],[Out = Ret2,V = V12,V12 >= 0]).
eq(rev(V, V3, V14, V18, Out),1,[length(V13, Ret01),lt(V16, Ret01, Ret0),if(Ret0, V16, V17, V15, V13, Ret3)],[Out = Ret3,V3 = V17,V13 >= 0,V16 >= 0,V17 >= 0,V14 = V15,V = V16,V15 >= 0,V18 = V13]).
eq(if(V, V3, V14, V18, V23, Out),1,[tail(V19, Ret11),head(V19, Ret201),rev(1 + V22, Ret11, 1 + Ret201 + V20, V21, Ret4)],[Out = Ret4,V23 = V21,V3 = V22,V18 = V20,V = 1,V21 >= 0,V22 >= 0,V19 >= 0,V20 >= 0,V14 = V19]).
eq(if(V, V3, V14, V18, V23, Out),1,[],[Out = V24,V23 = V27,V3 = V26,V18 = V24,V27 >= 0,V26 >= 0,V25 >= 0,V = 0,V24 >= 0,V14 = V25]).
input_output_vars(length(V,Out),[V],[Out]).
input_output_vars(lt(V,V3,Out),[V,V3],[Out]).
input_output_vars(head(V,Out),[V],[Out]).
input_output_vars(tail(V,Out),[V],[Out]).
input_output_vars(reverse(V,Out),[V],[Out]).
input_output_vars(rev(V,V3,V14,V18,Out),[V,V3,V14,V18],[Out]).
input_output_vars(if(V,V3,V14,V18,V23,Out),[V,V3,V14,V18,V23],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [head/2]
1. recursive  : [length/2]
2. recursive  : [lt/3]
3. non_recursive  : [tail/2]
4. recursive  : [if/6,rev/5]
5. non_recursive  : [reverse/2]
6. non_recursive  : [start/5]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into head/2
1. SCC is partially evaluated into length/2
2. SCC is partially evaluated into lt/3
3. SCC is partially evaluated into tail/2
4. SCC is partially evaluated into rev/5
5. SCC is completely evaluated into other SCCs
6. SCC is partially evaluated into start/5

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

### Specialization of cost equations head/2 
* CE 11 is refined into CE [20] 
* CE 12 is refined into CE [21] 


### Cost equations --> "Loop" of head/2 
* CEs [20] --> Loop 14 
* CEs [21] --> Loop 15 

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

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


### Specialization of cost equations length/2 
* CE 16 is refined into CE [22] 
* CE 15 is refined into CE [23] 


### Cost equations --> "Loop" of length/2 
* CEs [23] --> Loop 16 
* CEs [22] --> Loop 17 

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

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


### Specialization of cost equations lt/3 
* CE 19 is refined into CE [24] 
* CE 17 is refined into CE [25] 
* CE 18 is refined into CE [26] 


### Cost equations --> "Loop" of lt/3 
* CEs [25] --> Loop 18 
* CEs [26] --> Loop 19 
* CEs [24] --> Loop 20 

### Ranking functions of CR lt(V,V3,Out) 
* RF of phase [20]: [V,V3]

#### Partial ranking functions of CR lt(V,V3,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V
    V3


### Specialization of cost equations tail/2 
* CE 10 is refined into CE [27] 
* CE 9 is refined into CE [28] 


### Cost equations --> "Loop" of tail/2 
* CEs [27] --> Loop 21 
* CEs [28] --> Loop 22 

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

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


### Specialization of cost equations rev/5 
* CE 14 is refined into CE [29,30,31,32] 
* CE 13 is refined into CE [33,34] 


### Cost equations --> "Loop" of rev/5 
* CEs [34] --> Loop 23 
* CEs [33] --> Loop 24 
* CEs [32] --> Loop 25 
* CEs [31] --> Loop 26 
* CEs [30] --> Loop 27 
* CEs [29] --> Loop 28 

### Ranking functions of CR rev(V,V3,V14,V18,Out) 
* RF of phase [25]: [-V+V18,V3]
* RF of phase [26]: [-V+V18]

#### Partial ranking functions of CR rev(V,V3,V14,V18,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    -V+V18
    V3
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    -V+V18


### Specialization of cost equations start/5 
* CE 2 is refined into CE [35,36,37,38,39,40,41] 
* CE 1 is refined into CE [42] 
* CE 3 is refined into CE [43,44] 
* CE 4 is refined into CE [45,46,47,48] 
* CE 5 is refined into CE [49,50] 
* CE 6 is refined into CE [51,52] 
* CE 7 is refined into CE [53,54,55,56] 
* CE 8 is refined into CE [57,58,59,60,61,62,63,64,65] 


### Cost equations --> "Loop" of start/5 
* CEs [63] --> Loop 29 
* CEs [46,62] --> Loop 30 
* CEs [38,40,41] --> Loop 31 
* CEs [39] --> Loop 32 
* CEs [35,37] --> Loop 33 
* CEs [36,44,47,48,50,52,53,54,55,64,65] --> Loop 34 
* CEs [42,43,45,49,51,56,57,58,59,60,61] --> Loop 35 

### Ranking functions of CR start(V,V3,V14,V18,V23) 

#### Partial ranking functions of CR start(V,V3,V14,V18,V23) 


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

#### Cost of chains of head(V,Out):
* Chain [15]: 1
  with precondition: [V=0,Out=0] 

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


#### Cost of chains of length(V,Out):
* Chain [[17],16]: 1*it(17)+1
  Such that:it(17) =< V

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

* Chain [16]: 1
  with precondition: [V=0,Out=0] 


#### Cost of chains of lt(V,V3,Out):
* Chain [[20],19]: 1*it(20)+1
  Such that:it(20) =< V

  with precondition: [Out=1,V>=1,V3>=V+1] 

* Chain [[20],18]: 1*it(20)+1
  Such that:it(20) =< V3

  with precondition: [Out=0,V3>=1,V>=V3] 

* Chain [19]: 1
  with precondition: [V=0,Out=1,V3>=1] 

* Chain [18]: 1
  with precondition: [V3=0,Out=0,V>=0] 


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

* Chain [21]: 1
  with precondition: [Out>=0,V>=Out+1] 


#### Cost of chains of rev(V,V3,V14,V18,Out):
* Chain [[26],23]: 6*it(26)+2*s(1)+2*s(7)+4
  Such that:it(26) =< -V14+Out
aux(5) =< V18
s(1) =< aux(5)
s(7) =< it(26)*aux(5)

  with precondition: [V3=0,V>=1,V14>=0,Out>=V14+1,V14+V18>=Out+V] 

* Chain [[25],[26],23]: 12*it(25)+2*s(1)+4*s(7)+4
  Such that:aux(9) =< -V+V18
aux(10) =< V18
it(25) =< aux(9)
s(1) =< aux(10)
s(7) =< it(25)*aux(10)

  with precondition: [V>=1,V3>=1,V14>=0,V18>=V+2,Out>=V14+2] 

* Chain [[25],23]: 6*it(25)+2*s(1)+2*s(13)+4
  Such that:it(25) =< -V+V18
aux(11) =< V18
s(1) =< aux(11)
s(13) =< it(25)*aux(11)

  with precondition: [V>=1,V3>=1,V14>=0,V18>=V+1,Out>=V14+1] 

* Chain [28,[26],23]: 6*it(26)+3*s(1)+2*s(7)+10
  Such that:it(26) =< -V14+Out
aux(12) =< V18
s(1) =< aux(12)
s(7) =< it(26)*aux(12)

  with precondition: [V=0,V3=0,V14>=0,Out>=V14+2,V14+V18>=Out] 

* Chain [28,23]: 3*s(1)+10
  Such that:aux(13) =< V18
s(1) =< aux(13)

  with precondition: [V=0,V3=0,Out=V14+1,V18>=1,Out>=1] 

* Chain [27,[26],23]: 9*it(26)+2*s(7)+10
  Such that:aux(14) =< V18
it(26) =< aux(14)
s(7) =< it(26)*aux(14)

  with precondition: [V=0,V3>=1,V14>=0,V18>=2,Out>=V14+2,V3+V14+V18>=Out+1] 

* Chain [27,[25],[26],23]: 15*it(25)+4*s(7)+10
  Such that:aux(15) =< V18
it(25) =< aux(15)
s(7) =< it(25)*aux(15)

  with precondition: [V=0,V3>=2,V14>=0,V18>=3,Out>=V14+3] 

* Chain [27,[25],23]: 9*it(25)+2*s(13)+10
  Such that:aux(16) =< V18
it(25) =< aux(16)
s(13) =< it(25)*aux(16)

  with precondition: [V=0,V3>=2,V14>=0,V18>=2,Out>=V14+2] 

* Chain [27,23]: 3*s(1)+10
  Such that:aux(17) =< V18
s(1) =< aux(17)

  with precondition: [V=0,V14>=0,V18>=1,Out>=V14+1,V3+V14>=Out] 

* Chain [24]: 4
  with precondition: [V18=0,V14=Out,V>=0,V3>=0,V14>=0] 

* Chain [23]: 2*s(1)+4
  Such that:aux(1) =< V18
s(1) =< aux(1)

  with precondition: [V14=Out,V>=1,V3>=0,V14>=0,V18>=1] 


#### Cost of chains of start(V,V3,V14,V18,V23):
* Chain [35]: 48*s(33)+10*s(37)+10
  Such that:aux(22) =< V18
s(33) =< aux(22)
s(37) =< s(33)*aux(22)

  with precondition: [V=0] 

* Chain [34]: 38*s(46)+1*s(47)+8*s(51)+6*s(58)+18*s(61)+6*s(63)+11
  Such that:s(59) =< -V+V18
s(47) =< V3
aux(23) =< V
aux(24) =< V18
s(46) =< aux(23)
s(61) =< s(59)
s(58) =< aux(24)
s(63) =< s(61)*aux(24)
s(51) =< s(46)*aux(23)

  with precondition: [V>=1] 

* Chain [33]: 6*s(64)+4*s(66)+2*s(67)+7
  Such that:s(64) =< -V3+V23
aux(25) =< V23
s(66) =< aux(25)
s(67) =< s(64)*aux(25)

  with precondition: [V=1,V14=0,V3>=0,V18>=0,V23>=1] 

* Chain [32]: 7
  with precondition: [V=1,V23=0,V3>=0,V14>=1,V18>=0] 

* Chain [31]: 24*s(70)+8*s(72)+8*s(73)+7
  Such that:aux(26) =< -V3+V23
aux(27) =< V23
s(70) =< aux(26)
s(72) =< aux(27)
s(73) =< s(70)*aux(27)

  with precondition: [V=1,V3>=0,V14>=1,V18>=0,V23>=1] 

* Chain [30]: 6*s(81)+2*s(83)+2*s(84)+4
  Such that:s(81) =< -V+V18
s(82) =< V18
s(83) =< s(82)
s(84) =< s(81)*s(82)

  with precondition: [V3=0,V>=0] 

* Chain [29]: 4
  with precondition: [V18=0,V>=0,V3>=0,V14>=0] 


Closed-form bounds of start(V,V3,V14,V18,V23): 
-------------------------------------
* Chain [35] with precondition: [V=0] 
    - Upper bound: nat(V18)*48+10+nat(V18)*10*nat(V18) 
    - Complexity: n^2 
* Chain [34] with precondition: [V>=1] 
    - Upper bound: 38*V+11+8*V*V+nat(V3)+nat(V18)*6+nat(V18)*6*nat(-V+V18)+nat(-V+V18)*18 
    - Complexity: n^2 
* Chain [33] with precondition: [V=1,V14=0,V3>=0,V18>=0,V23>=1] 
    - Upper bound: 4*V23+7+2*V23*nat(-V3+V23)+nat(-V3+V23)*6 
    - Complexity: n^2 
* Chain [32] with precondition: [V=1,V23=0,V3>=0,V14>=1,V18>=0] 
    - Upper bound: 7 
    - Complexity: constant 
* Chain [31] with precondition: [V=1,V3>=0,V14>=1,V18>=0,V23>=1] 
    - Upper bound: 8*V23+7+8*V23*nat(-V3+V23)+nat(-V3+V23)*24 
    - Complexity: n^2 
* Chain [30] with precondition: [V3=0,V>=0] 
    - Upper bound: nat(V18)*2+4+nat(V18)*2*nat(-V+V18)+nat(-V+V18)*6 
    - Complexity: n^2 
* Chain [29] with precondition: [V18=0,V>=0,V3>=0,V14>=0] 
    - Upper bound: 4 
    - Complexity: constant 

### Maximum cost of start(V,V3,V14,V18,V23): max([nat(V18)*2+max([nat(V18)*2*nat(-V+V18)+nat(-V+V18)*6,nat(V18)*4+6+max([nat(V18)*10*nat(V18)+nat(V18)*42,38*V+1+8*V*V+nat(V3)+nat(V18)*6*nat(-V+V18)+nat(-V+V18)*18])]),nat(V23)*6*nat(-V3+V23)+nat(V23)*4+nat(-V3+V23)*18+(nat(V23)*2*nat(-V3+V23)+nat(V23)*4+nat(-V3+V23)*6)+3])+4 
Asymptotic class: n^2 
* Total analysis performed in 511 ms.


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(10)
BOUNDS(1, n^2)

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(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

S is empty.
Rewrite Strategy: INNERMOST
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(13) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

lt(s(x), s(y)) ->^+ lt(x, y)

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

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

The result substitution is [ ].




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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

S is empty.
Rewrite Strategy: INNERMOST
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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

S is empty.
Rewrite Strategy: INNERMOST