WORST_CASE(Omega(n^1),O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 127 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 708 ms]
    (12) BOUNDS(1, n^1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y))
   a(Z, y, z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y))
   a(Z, y, z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) [1]
   a(Z, y, z) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) [1]
   a(Z, y, z) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]

The TRS has the following type information:
a :: C:Z -> a -> a -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True
and :: False:True -> False:True -> False:True
False :: False:True
True :: False:True
second :: C:Z -> C:Z
first :: C:Z -> C:Z

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

(7) 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:

   and(v0, v1) -> null_and [0]
   second(v0) -> null_second [0]
   first(v0) -> null_first [0]
   a(v0, v1, v2) -> null_a [0]
   eqZList(v0, v1) -> null_eqZList [0]

And the following fresh constants:    null_and, null_second, null_first, null_a, null_eqZList, const

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

(8)
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:

   a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y)) [1]
   a(Z, y, z) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]
   and(v0, v1) -> null_and [0]
   second(v0) -> null_second [0]
   first(v0) -> null_first [0]
   a(v0, v1, v2) -> null_a [0]
   eqZList(v0, v1) -> null_eqZList [0]

The TRS has the following type information:
a :: C:Z:null_second:null_first:null_a -> a -> a -> C:Z:null_second:null_first:null_a
C :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a
Z :: C:Z:null_second:null_first:null_a
eqZList :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a -> False:True:null_and:null_eqZList
and :: False:True:null_and:null_eqZList -> False:True:null_and:null_eqZList -> False:True:null_and:null_eqZList
False :: False:True:null_and:null_eqZList
True :: False:True:null_and:null_eqZList
second :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a
first :: C:Z:null_second:null_first:null_a -> C:Z:null_second:null_first:null_a
null_and :: False:True:null_and:null_eqZList
null_second :: C:Z:null_second:null_first:null_a
null_first :: C:Z:null_second:null_first:null_a
null_a :: C:Z:null_second:null_first:null_a
null_eqZList :: False:True:null_and:null_eqZList
const :: a

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

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

   Z => 0
   False => 1
   True => 2
   null_and => 0
   null_second => 0
   null_first => 0
   null_a => 0
   null_eqZList => 0
   const => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   a(z', z'', z1) -{ 1 }-> 0 :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
   a(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   a(z', z'', z1) -{ 1 }-> 1 + a(x1, y, z) + a(x2, y, y) :|: z' = 1 + x1 + x2, z1 = z, x1 >= 0, z >= 0, z'' = y, y >= 0, x2 >= 0
   and(z', z'') -{ 0 }-> 2 :|: z' = 2, z'' = 2
   and(z', z'') -{ 0 }-> 1 :|: z' = 1, z'' = 1
   and(z', z'') -{ 0 }-> 1 :|: z' = 2, z'' = 1
   and(z', z'') -{ 0 }-> 1 :|: z' = 1, z'' = 2
   and(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   eqZList(z', z'') -{ 1 }-> and(eqZList(x1, y1), eqZList(x2, y2)) :|: y1 >= 0, z' = 1 + x1 + x2, x1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, x2 >= 0
   eqZList(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eqZList(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
   eqZList(z', z'') -{ 1 }-> 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
   eqZList(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   first(z') -{ 1 }-> x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
   first(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   second(z') -{ 1 }-> x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
   second(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V2, V6),0,[a(V, V2, V6, Out)],[V >= 0,V2 >= 0,V6 >= 0]).
eq(start(V, V2, V6),0,[eqZList(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2, V6),0,[second(V, Out)],[V >= 0]).
eq(start(V, V2, V6),0,[first(V, Out)],[V >= 0]).
eq(start(V, V2, V6),0,[and(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(a(V, V2, V6, Out),1,[a(V4, V1, V5, Ret01),a(V3, V1, V1, Ret1)],[Out = 1 + Ret01 + Ret1,V = 1 + V3 + V4,V6 = V5,V4 >= 0,V5 >= 0,V2 = V1,V1 >= 0,V3 >= 0]).
eq(a(V, V2, V6, Out),1,[],[Out = 0,V6 = V8,V8 >= 0,V2 = V7,V7 >= 0,V = 0]).
eq(eqZList(V, V2, Out),1,[eqZList(V11, V12, Ret0),eqZList(V9, V10, Ret11),and(Ret0, Ret11, Ret)],[Out = Ret,V12 >= 0,V = 1 + V11 + V9,V11 >= 0,V2 = 1 + V10 + V12,V10 >= 0,V9 >= 0]).
eq(eqZList(V, V2, Out),1,[],[Out = 1,V2 = 0,V = 1 + V13 + V14,V13 >= 0,V14 >= 0]).
eq(eqZList(V, V2, Out),1,[],[Out = 1,V15 >= 0,V2 = 1 + V15 + V16,V16 >= 0,V = 0]).
eq(eqZList(V, V2, Out),1,[],[Out = 2,V2 = 0,V = 0]).
eq(second(V, Out),1,[],[Out = V17,V = 1 + V17 + V18,V18 >= 0,V17 >= 0]).
eq(first(V, Out),1,[],[Out = V20,V = 1 + V19 + V20,V20 >= 0,V19 >= 0]).
eq(and(V, V2, Out),0,[],[Out = 1,V = 1,V2 = 1]).
eq(and(V, V2, Out),0,[],[Out = 1,V = 2,V2 = 1]).
eq(and(V, V2, Out),0,[],[Out = 1,V = 1,V2 = 2]).
eq(and(V, V2, Out),0,[],[Out = 2,V = 2,V2 = 2]).
eq(and(V, V2, Out),0,[],[Out = 0,V22 >= 0,V21 >= 0,V2 = V21,V = V22]).
eq(second(V, Out),0,[],[Out = 0,V23 >= 0,V = V23]).
eq(first(V, Out),0,[],[Out = 0,V24 >= 0,V = V24]).
eq(a(V, V2, V6, Out),0,[],[Out = 0,V25 >= 0,V6 = V27,V26 >= 0,V2 = V26,V27 >= 0,V = V25]).
eq(eqZList(V, V2, Out),0,[],[Out = 0,V28 >= 0,V29 >= 0,V2 = V29,V = V28]).
input_output_vars(a(V,V2,V6,Out),[V,V2,V6],[Out]).
input_output_vars(eqZList(V,V2,Out),[V,V2],[Out]).
input_output_vars(second(V,Out),[V],[Out]).
input_output_vars(first(V,Out),[V],[Out]).
input_output_vars(and(V,V2,Out),[V,V2],[Out]).


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

#### Computed strongly connected components 
0. recursive [multiple] : [a/4]
1. non_recursive  : [and/3]
2. recursive [non_tail,multiple] : [eqZList/3]
3. non_recursive  : [first/2]
4. non_recursive  : [second/2]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into a/4
1. SCC is partially evaluated into and/3
2. SCC is partially evaluated into eqZList/3
3. SCC is partially evaluated into first/2
4. SCC is partially evaluated into second/2
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations a/4 
* CE 7 is refined into CE [23] 
* CE 8 is refined into CE [24] 
* CE 6 is refined into CE [25] 


### Cost equations --> "Loop" of a/4 
* CEs [25] --> Loop 18 
* CEs [23,24] --> Loop 19 

### Ranking functions of CR a(V,V2,V6,Out) 
* RF of phase [18]: [V]

#### Partial ranking functions of CR a(V,V2,V6,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1,18:2]:
    V


### Specialization of cost equations and/3 
* CE 22 is refined into CE [26] 
* CE 21 is refined into CE [27] 
* CE 19 is refined into CE [28] 
* CE 20 is refined into CE [29] 
* CE 18 is refined into CE [30] 


### Cost equations --> "Loop" of and/3 
* CEs [26] --> Loop 20 
* CEs [27] --> Loop 21 
* CEs [28] --> Loop 22 
* CEs [29] --> Loop 23 
* CEs [30] --> Loop 24 

### Ranking functions of CR and(V,V2,Out) 

#### Partial ranking functions of CR and(V,V2,Out) 


### Specialization of cost equations eqZList/3 
* CE 13 is refined into CE [31] 
* CE 10 is refined into CE [32] 
* CE 11 is refined into CE [33] 
* CE 12 is refined into CE [34] 
* CE 9 is refined into CE [35,36,37,38,39] 


### Cost equations --> "Loop" of eqZList/3 
* CEs [38] --> Loop 25 
* CEs [37] --> Loop 26 
* CEs [36] --> Loop 27 
* CEs [35] --> Loop 28 
* CEs [39] --> Loop 29 
* CEs [31] --> Loop 30 
* CEs [32] --> Loop 31 
* CEs [33] --> Loop 32 
* CEs [34] --> Loop 33 

### Ranking functions of CR eqZList(V,V2,Out) 
* RF of phase [25]: [V,V2]
* RF of phase [26,27,28]: [V,V2]
* RF of phase [29]: [V,V2]

#### Partial ranking functions of CR eqZList(V,V2,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1,25:2]:
    V
    V2
* Partial RF of phase [26,27,28]:
  - RF of loop [26:1,26:2,27:1,27:2,28:1,28:2]:
    V
    V2
* Partial RF of phase [29]:
  - RF of loop [29:1,29:2]:
    V
    V2


### Specialization of cost equations first/2 
* CE 16 is refined into CE [40] 
* CE 17 is refined into CE [41] 


### Cost equations --> "Loop" of first/2 
* CEs [40] --> Loop 34 
* CEs [41] --> Loop 35 

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

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


### Specialization of cost equations second/2 
* CE 14 is refined into CE [42] 
* CE 15 is refined into CE [43] 


### Cost equations --> "Loop" of second/2 
* CEs [42] --> Loop 36 
* CEs [43] --> Loop 37 

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

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [44,45] 
* CE 2 is refined into CE [46,47,48,49,50,51] 
* CE 3 is refined into CE [52,53] 
* CE 4 is refined into CE [54,55] 
* CE 5 is refined into CE [56,57,58,59,60] 


### Cost equations --> "Loop" of start/3 
* CEs [48] --> Loop 38 
* CEs [59] --> Loop 39 
* CEs [58] --> Loop 40 
* CEs [57] --> Loop 41 
* CEs [51,56] --> Loop 42 
* CEs [47] --> Loop 43 
* CEs [44,45,46,49,50,52,53,54,55,60] --> Loop 44 

### Ranking functions of CR start(V,V2,V6) 

#### Partial ranking functions of CR start(V,V2,V6) 


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

#### Cost of chains of a(V,V2,V6,Out):
* Chain [19]: 1
  with precondition: [Out=0,V>=0,V2>=0,V6>=0] 

* Chain [multiple([18],[[19]])]: 1*it(18)+1*it([19])+0
  Such that:it(18) =< V
it([19]) =< V+1

  with precondition: [V2>=0,V6>=0,Out>=1,V>=Out] 


#### Cost of chains of and(V,V2,Out):
* Chain [24]: 0
  with precondition: [V=1,V2=1,Out=1] 

* Chain [23]: 0
  with precondition: [V=1,V2=2,Out=1] 

* Chain [22]: 0
  with precondition: [V=2,V2=1,Out=1] 

* Chain [21]: 0
  with precondition: [V=2,V2=2,Out=2] 

* Chain [20]: 0
  with precondition: [Out=0,V>=0,V2>=0] 


#### Cost of chains of eqZList(V,V2,Out):
* Chain [33]: 1
  with precondition: [V=0,V2=0,Out=2] 

* Chain [32]: 1
  with precondition: [V=0,Out=1,V2>=1] 

* Chain [31]: 1
  with precondition: [V2=0,Out=1,V>=1] 

* Chain [30]: 0
  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [multiple([29],[[multiple([26,27,28],[[multiple([25],[[33]])],[33],[32],[31]])],[multiple([25],[[33]])],[33],[32],[31],[30]])]: 1*it(29)+1*it([31])+1*it([32])+1*it([33])+5*s(1)+1*s(3)+1*s(4)+1*s(5)+1*s(6)+1*s(7)+0
  Such that:aux(11) =< V
aux(12) =< V+1
aux(13) =< V/2+1/2
aux(14) =< V/3+V2/3+2/3
aux(15) =< V2+1
aux(16) =< V2/2+1/2
it(29) =< aux(11)
it([31]) =< aux(11)
it([31]) =< aux(12)
it([32]) =< aux(12)
it([33]) =< aux(12)
it([31]) =< aux(13)
it([33]) =< aux(15)
s(1) =< aux(15)
it([32]) =< aux(16)
s(11) =< aux(15)*(1/2)
s(9) =< aux(14)*(6/5)
s(6) =< aux(13)
s(5) =< aux(14)
s(3) =< s(9)
s(4) =< s(9)
s(5) =< s(9)
s(4) =< aux(15)
s(5) =< aux(15)
s(7) =< aux(15)
s(7) =< s(11)
s(5) =< s(1)*(1/3)+aux(14)
s(3) =< s(1)*(1/5)+s(9)
s(4) =< s(1)*(1/5)+s(9)
s(5) =< s(1)*(1/5)+s(9)

  with precondition: [Out=0,V>=1,V2>=1] 

* Chain [multiple([26,27,28],[[multiple([25],[[33]])],[33],[32],[31]])]: 1*it(26)+1*it(27)+1*it(28)+1*it([31])+1*it([32])+3*it([33])+0
  Such that:aux(3) =< V/2+1/2
aux(4) =< V/3+V2/3
aux(5) =< 2/5*V+2/5*V2
aux(6) =< V2
aux(7) =< V2+1
aux(8) =< V2/2+1/2
it([31]) =< aux(3)
it(28) =< aux(4)
it(26) =< aux(5)
it(27) =< aux(5)
it(28) =< aux(5)
it(27) =< aux(6)
it(28) =< aux(6)
it([32]) =< aux(6)
it([33]) =< aux(7)
it([32]) =< aux(8)
it(28) =< it([33])*(1/3)+aux(4)
it(26) =< it([33])*(1/5)+aux(5)
it(27) =< it([33])*(1/5)+aux(5)
it(28) =< it([33])*(1/5)+aux(5)

  with precondition: [Out=1,V>=1,V2>=1,V+V2>=3] 

* Chain [multiple([25],[[33]])]: 1*it(25)+1*it([33])+0
  Such that:it(25) =< V2
it([33]) =< V2+1

  with precondition: [Out=2,V=V2,V>=1] 


#### Cost of chains of first(V,Out):
* Chain [35]: 0
  with precondition: [Out=0,V>=0] 

* Chain [34]: 1
  with precondition: [Out>=0,V>=Out+1] 


#### Cost of chains of second(V,Out):
* Chain [37]: 0
  with precondition: [Out=0,V>=0] 

* Chain [36]: 1
  with precondition: [Out>=0,V>=Out+1] 


#### Cost of chains of start(V,V2,V6):
* Chain [44]: 2*s(33)+1*s(34)+1*s(42)+1*s(43)+1*s(44)+8*s(45)+2*s(48)+1*s(49)+1*s(50)+1*s(51)+1*s(52)+1*s(60)+1*s(61)+1*s(62)+1*s(63)+1
  Such that:s(54) =< V/3+V2/3
s(38) =< V/3+V2/3+2/3
s(55) =< 2/5*V+2/5*V2
s(56) =< V2
aux(17) =< V
aux(18) =< V+1
aux(19) =< V/2+1/2
aux(20) =< V2+1
aux(21) =< V2/2+1/2
s(33) =< aux(17)
s(34) =< aux(18)
s(42) =< aux(17)
s(42) =< aux(18)
s(43) =< aux(18)
s(44) =< aux(18)
s(42) =< aux(19)
s(44) =< aux(20)
s(45) =< aux(20)
s(43) =< aux(21)
s(46) =< aux(20)*(1/2)
s(47) =< s(38)*(6/5)
s(48) =< aux(19)
s(49) =< s(38)
s(50) =< s(47)
s(51) =< s(47)
s(49) =< s(47)
s(51) =< aux(20)
s(49) =< aux(20)
s(52) =< aux(20)
s(52) =< s(46)
s(49) =< s(45)*(1/3)+s(38)
s(50) =< s(45)*(1/5)+s(47)
s(51) =< s(45)*(1/5)+s(47)
s(49) =< s(45)*(1/5)+s(47)
s(60) =< s(54)
s(61) =< s(55)
s(62) =< s(55)
s(60) =< s(55)
s(62) =< s(56)
s(60) =< s(56)
s(63) =< s(56)
s(63) =< aux(21)
s(60) =< s(45)*(1/3)+s(54)
s(61) =< s(45)*(1/5)+s(55)
s(62) =< s(45)*(1/5)+s(55)
s(60) =< s(45)*(1/5)+s(55)

  with precondition: [V>=0] 

* Chain [43]: 1
  with precondition: [V=0,V2>=1] 

* Chain [42]: 1*s(65)+1*s(66)+0
  Such that:s(65) =< V2
s(66) =< V2+1

  with precondition: [V=V2,V>=1] 

* Chain [41]: 0
  with precondition: [V=1,V2=2] 

* Chain [40]: 0
  with precondition: [V=2,V2=1] 

* Chain [39]: 0
  with precondition: [V=2,V2=2] 

* Chain [38]: 1
  with precondition: [V2=0,V>=1] 


Closed-form bounds of start(V,V2,V6): 
-------------------------------------
* Chain [44] with precondition: [V>=0] 
    - Upper bound: 3*V+1+nat(V2)+(3*V+3)+nat(V2+1)*9+nat(2/5*V+2/5*V2)*2+17/5*nat(V/3+V2/3+2/3)+(V+1)+nat(V/3+V2/3) 
    - Complexity: n 
* Chain [43] with precondition: [V=0,V2>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [42] with precondition: [V=V2,V>=1] 
    - Upper bound: 2*V2+1 
    - Complexity: n 
* Chain [41] with precondition: [V=1,V2=2] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [40] with precondition: [V=2,V2=1] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [39] with precondition: [V=2,V2=2] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [38] with precondition: [V2=0,V>=1] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V,V2,V6): max([1,6*V+4+nat(V2+1)*8+nat(2/5*V+2/5*V2)*2+17/5*nat(V/3+V2/3+2/3)+(V+1)+nat(V/3+V2/3)+(nat(V2+1)+nat(V2))]) 
Asymptotic class: n 
* Total analysis performed in 560 ms.


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(12)
BOUNDS(1, n^1)

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

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y))
   a(Z, y, z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

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

The rewrite sequence

a(C(x1, x2), y, z) ->^+ C(a(x1, y, z), a(x2, y, y))

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

The pumping substitution is [x1 / C(x1, x2)].

The result substitution is [ ].




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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y))
   a(Z, y, z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

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

(19)
BOUNDS(n^1, INF)

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

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

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


The TRS R consists of the following rules:

   a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y))
   a(Z, y, z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: INNERMOST