WORST_CASE(?,O(n^3))
proof of input_Kt1b0umqFr.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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, 687 ms]
(10) BOUNDS(1, n^3)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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^3).


The TRS R consists of the following rules:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
mod :: 0:s -> 0:s -> 0:s
if_mod :: true:false -> 0:s -> 0:s -> 0:s

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:

   if_minus(v0, v1, v2) -> null_if_minus [0]
   if_mod(v0, v1, v2) -> null_if_mod [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   mod(v0, v1) -> null_mod [0]

And the following fresh constants:    null_if_minus, null_if_mod, null_le, null_minus, null_mod

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

(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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   if_mod(v0, v1, v2) -> null_if_mod [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   mod(v0, v1) -> null_mod [0]

The TRS has the following type information:
le :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> true:false:null_le
0 :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod
true :: true:false:null_le
s :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod
false :: true:false:null_le
minus :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod
if_minus :: true:false:null_le -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod
mod :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod
if_mod :: true:false:null_le -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod -> 0:s:null_if_minus:null_if_mod:null_minus:null_mod
null_if_minus :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod
null_if_mod :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod
null_le :: true:false:null_le
null_minus :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod
null_mod :: 0:s:null_if_minus:null_if_mod:null_minus:null_mod

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:

   0 => 0
   true => 2
   false => 1
   null_if_minus => 0
   null_if_mod => 0
   null_le => 0
   null_minus => 0
   null_mod => 0

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = 1 + x, z'' = y, x >= 0, y >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0
   if_mod(z, z', z'') -{ 1 }-> mod(minus(x, y), 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   if_mod(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + x :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   minus(z, z') -{ 1 }-> if_minus(le(1 + x, y), 1 + x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   mod(z, z') -{ 1 }-> if_mod(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   mod(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

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(V1, V, V11),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V11),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V11),0,[fun(V1, V, V11, Out)],[V1 >= 0,V >= 0,V11 >= 0]).
eq(start(V1, V, V11),0,[mod(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V11),0,[fun1(V1, V, V11, Out)],[V1 >= 0,V >= 0,V11 >= 0]).
eq(le(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = 0,V = V2]).
eq(le(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = 1 + V3,V = 0]).
eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(minus(V1, V, Out),1,[],[Out = 0,V6 >= 0,V1 = 0,V = V6]).
eq(minus(V1, V, Out),1,[le(1 + V7, V8, Ret0),fun(Ret0, 1 + V7, V8, Ret1)],[Out = Ret1,V7 >= 0,V8 >= 0,V1 = 1 + V7,V = V8]).
eq(fun(V1, V, V11, Out),1,[],[Out = 0,V1 = 2,V = 1 + V9,V11 = V10,V9 >= 0,V10 >= 0]).
eq(fun(V1, V, V11, Out),1,[minus(V12, V13, Ret11)],[Out = 1 + Ret11,V = 1 + V12,V11 = V13,V1 = 1,V12 >= 0,V13 >= 0]).
eq(mod(V1, V, Out),1,[],[Out = 0,V14 >= 0,V1 = 0,V = V14]).
eq(mod(V1, V, Out),1,[],[Out = 0,V15 >= 0,V1 = 1 + V15,V = 0]).
eq(mod(V1, V, Out),1,[le(V16, V17, Ret01),fun1(Ret01, 1 + V17, 1 + V16, Ret2)],[Out = Ret2,V = 1 + V16,V17 >= 0,V16 >= 0,V1 = 1 + V17]).
eq(fun1(V1, V, V11, Out),1,[minus(V18, V19, Ret02),mod(Ret02, 1 + V19, Ret3)],[Out = Ret3,V1 = 2,V = 1 + V18,V18 >= 0,V19 >= 0,V11 = 1 + V19]).
eq(fun1(V1, V, V11, Out),1,[],[Out = 1 + V20,V = 1 + V20,V1 = 1,V20 >= 0,V21 >= 0,V11 = 1 + V21]).
eq(fun(V1, V, V11, Out),0,[],[Out = 0,V23 >= 0,V11 = V24,V22 >= 0,V1 = V23,V = V22,V24 >= 0]).
eq(fun1(V1, V, V11, Out),0,[],[Out = 0,V27 >= 0,V11 = V25,V26 >= 0,V1 = V27,V = V26,V25 >= 0]).
eq(le(V1, V, Out),0,[],[Out = 0,V29 >= 0,V28 >= 0,V1 = V29,V = V28]).
eq(minus(V1, V, Out),0,[],[Out = 0,V30 >= 0,V31 >= 0,V1 = V30,V = V31]).
eq(mod(V1, V, Out),0,[],[Out = 0,V33 >= 0,V32 >= 0,V1 = V33,V = V32]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,V11,Out),[V1,V,V11],[Out]).
input_output_vars(mod(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V11,Out),[V1,V,V11],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [le/3]
1. recursive  : [fun/4,minus/3]
2. recursive  : [fun1/4,(mod)/3]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into le/3
1. SCC is partially evaluated into minus/3
2. SCC is partially evaluated into (mod)/3
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations le/3 
* CE 23 is refined into CE [24] 
* CE 21 is refined into CE [25] 
* CE 20 is refined into CE [26] 
* CE 22 is refined into CE [27] 


### Cost equations --> "Loop" of le/3 
* CEs [27] --> Loop 13 
* CEs [24] --> Loop 14 
* CEs [25] --> Loop 15 
* CEs [26] --> Loop 16 

### Ranking functions of CR le(V1,V,Out) 
* RF of phase [13]: [V,V1]

#### Partial ranking functions of CR le(V1,V,Out) 
* Partial RF of phase [13]:
  - RF of loop [13:1]:
    V
    V1


### Specialization of cost equations minus/3 
* CE 9 is refined into CE [28,29,30,31] 
* CE 11 is refined into CE [32] 
* CE 12 is refined into CE [33] 
* CE 13 is refined into CE [34] 
* CE 10 is refined into CE [35,36] 


### Cost equations --> "Loop" of minus/3 
* CEs [36] --> Loop 17 
* CEs [35] --> Loop 18 
* CEs [28] --> Loop 19 
* CEs [29,30,31,32,33,34] --> Loop 20 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [17]: [V1-1,V1-V]
* RF of phase [18]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V1-1
    V1-V
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V1


### Specialization of cost equations (mod)/3 
* CE 15 is refined into CE [37,38] 
* CE 18 is refined into CE [39] 
* CE 14 is refined into CE [40,41,42,43,44] 
* CE 17 is refined into CE [45] 
* CE 19 is refined into CE [46] 
* CE 16 is refined into CE [47,48,49,50] 


### Cost equations --> "Loop" of (mod)/3 
* CEs [50] --> Loop 21 
* CEs [49] --> Loop 22 
* CEs [48] --> Loop 23 
* CEs [47] --> Loop 24 
* CEs [38] --> Loop 25 
* CEs [40] --> Loop 26 
* CEs [39] --> Loop 27 
* CEs [37] --> Loop 28 
* CEs [41] --> Loop 29 
* CEs [42,43,44,45,46] --> Loop 30 

### Ranking functions of CR mod(V1,V,Out) 
* RF of phase [21]: [V1/2-1,V1/2-V/2]
* RF of phase [23]: [V1-1]

#### Partial ranking functions of CR mod(V1,V,Out) 
* Partial RF of phase [21]:
  - RF of loop [21:1]:
    V1/2-1
    V1/2-V/2
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V1-1


### Specialization of cost equations start/3 
* CE 3 is refined into CE [51,52,53,54,55,56,57,58] 
* CE 5 is refined into CE [59] 
* CE 1 is refined into CE [60] 
* CE 2 is refined into CE [61] 
* CE 4 is refined into CE [62,63,64] 
* CE 6 is refined into CE [65,66,67,68,69] 
* CE 7 is refined into CE [70,71,72] 
* CE 8 is refined into CE [73,74,75,76,77,78,79] 


### Cost equations --> "Loop" of start/3 
* CEs [76] --> Loop 31 
* CEs [66,71,75] --> Loop 32 
* CEs [51,52,53,54,55,56,57,58,59] --> Loop 33 
* CEs [74] --> Loop 34 
* CEs [61,62,63,64] --> Loop 35 
* CEs [60,65,67,68,69,70,72,73,77,78,79] --> Loop 36 

### Ranking functions of CR start(V1,V,V11) 

#### Partial ranking functions of CR start(V1,V,V11) 


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

#### Cost of chains of le(V1,V,Out):
* Chain [[13],16]: 1*it(13)+1
  Such that:it(13) =< V1

  with precondition: [Out=2,V1>=1,V>=V1] 

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

  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [[13],14]: 1*it(13)+0
  Such that:it(13) =< V

  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [16]: 1
  with precondition: [V1=0,Out=2,V>=0] 

* Chain [15]: 1
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [14]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[18],20]: 3*it(18)+2*s(4)+3
  Such that:aux(1) =< V1-Out
it(18) =< Out
s(4) =< aux(1)

  with precondition: [V=0,Out>=1,V1>=Out] 

* Chain [[18],19]: 3*it(18)+2
  Such that:it(18) =< Out

  with precondition: [V=0,Out>=1,V1>=Out+1] 

* Chain [[17],20]: 3*it(17)+2*s(2)+2*s(4)+1*s(8)+3
  Such that:aux(1) =< V1-Out
it(17) =< Out
aux(4) =< V
s(4) =< aux(1)
s(2) =< aux(4)
s(8) =< it(17)*aux(4)

  with precondition: [V>=1,Out>=1,V1>=Out+V] 

* Chain [20]: 2*s(2)+2*s(4)+3
  Such that:aux(1) =< V1
aux(2) =< V
s(4) =< aux(1)
s(2) =< aux(2)

  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [19]: 2
  with precondition: [V=0,Out=0,V1>=1] 


#### Cost of chains of mod(V1,V,Out):
* Chain [[23],30]: 10*it(23)+1*s(19)+6*s(28)+2
  Such that:s(19) =< 1
aux(10) =< V1
it(23) =< aux(10)
s(31) =< it(23)*aux(10)
s(28) =< s(31)

  with precondition: [V=1,Out=0,V1>=2] 

* Chain [[23],26]: 8*it(23)+6*s(28)+2
  Such that:aux(11) =< V1
it(23) =< aux(11)
s(31) =< it(23)*aux(11)
s(28) =< s(31)

  with precondition: [V=1,Out=0,V1>=2] 

* Chain [[23],24,30]: 10*it(23)+1*s(19)+6*s(28)+8
  Such that:s(19) =< 1
aux(12) =< V1
it(23) =< aux(12)
s(31) =< it(23)*aux(12)
s(28) =< s(31)

  with precondition: [V=1,Out=0,V1>=2] 

* Chain [[21],30]: 13*it(21)+1*s(19)+3*s(50)+1*s(52)+2
  Such that:aux(14) =< V1-V
aux(17) =< V1
aux(18) =< V
it(21) =< aux(17)
s(19) =< aux(18)
s(50) =< it(21)*aux(14)
s(52) =< s(50)*aux(18)

  with precondition: [Out=0,V>=2,V1>=V+1] 

* Chain [[21],29]: 6*it(21)+5*s(49)+3*s(50)+1*s(52)+2
  Such that:aux(14) =< V1-V
it(21) =< V1/2-V/2
s(45) =< V
aux(19) =< V1
s(50) =< it(21)*aux(14)
s(49) =< aux(19)
s(52) =< s(50)*s(45)

  with precondition: [Out=0,V>=2,V1>=V+1] 

* Chain [[21],28]: 6*it(21)+5*s(49)+3*s(50)+1*s(52)+3
  Such that:aux(14) =< V1-V
it(21) =< V1/2-V/2
s(45) =< V
aux(20) =< V1
s(50) =< it(21)*aux(14)
s(49) =< aux(20)
s(52) =< s(50)*s(45)

  with precondition: [Out=1,V>=2,V1>=V+1] 

* Chain [[21],25]: 6*it(21)+5*s(49)+3*s(50)+1*s(52)+1*s(55)+3
  Such that:aux(15) =< V1
aux(14) =< V1-V
aux(16) =< V1-Out
it(21) =< V1/2-V/2
s(45) =< V
s(55) =< Out
s(53) =< aux(15)
s(53) =< aux(16)
s(50) =< it(21)*aux(14)
s(49) =< s(53)
s(52) =< s(50)*s(45)

  with precondition: [Out>=2,V>=Out+1,V1>=Out+V] 

* Chain [[21],22,30]: 13*it(21)+4*s(19)+3*s(50)+1*s(52)+8
  Such that:aux(14) =< V1-V
aux(23) =< V1
aux(24) =< V
it(21) =< aux(23)
s(19) =< aux(24)
s(50) =< it(21)*aux(14)
s(52) =< s(50)*aux(24)

  with precondition: [Out=0,V>=2,V1>=2*V] 

* Chain [30]: 2*s(17)+1*s(19)+2
  Such that:s(19) =< V
aux(6) =< V1
s(17) =< aux(6)

  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [29]: 2
  with precondition: [V1=1,Out=0,V>=2] 

* Chain [28]: 3
  with precondition: [V1=1,Out=1,V>=2] 

* Chain [27]: 1
  with precondition: [V=0,Out=0,V1>=1] 

* Chain [26]: 2
  with precondition: [V=1,Out=0,V1>=1] 

* Chain [25]: 1*s(55)+3
  Such that:s(55) =< V1

  with precondition: [V1=Out,V1>=2,V>=V1+1] 

* Chain [24,30]: 1*s(19)+2*s(34)+8
  Such that:s(19) =< 1
s(32) =< V1
s(34) =< s(32)

  with precondition: [V=1,Out=0,V1>=1] 

* Chain [22,30]: 4*s(19)+2*s(59)+8
  Such that:s(57) =< V1
aux(22) =< V
s(19) =< aux(22)
s(59) =< s(57)

  with precondition: [Out=0,V>=2,V1>=V] 


#### Cost of chains of start(V1,V,V11):
* Chain [36]: 16*s(106)+55*s(108)+1*s(118)+18*s(119)+6*s(125)+2*s(126)+9*s(127)+3*s(128)+8
  Such that:aux(32) =< V1
aux(33) =< V1-V
aux(34) =< V1/2-V/2
aux(35) =< V
s(108) =< aux(32)
s(119) =< aux(34)
s(106) =< aux(35)
s(125) =< s(108)*aux(33)
s(126) =< s(125)*aux(35)
s(127) =< s(119)*aux(33)
s(128) =< s(127)*aux(35)
s(118) =< s(108)*aux(35)

  with precondition: [V1>=0,V>=0] 

* Chain [35]: 15*s(149)+4*s(150)+1*s(160)+4
  Such that:aux(38) =< V
aux(39) =< V11
s(149) =< aux(38)
s(150) =< aux(39)
s(160) =< s(149)*aux(39)

  with precondition: [V1=1,V>=1,V11>=0] 

* Chain [34]: 3
  with precondition: [V1=1,V>=2] 

* Chain [33]: 158*s(163)+32*s(164)+6*s(165)+3*s(173)+1*s(174)+16*s(183)+9*s(185)+3*s(186)+18*s(198)+4*s(204)+6*s(205)+6*s(211)+2*s(212)+3*s(213)+1*s(214)+1*s(220)+12*s(228)+6*s(231)+2*s(233)+12
  Such that:s(207) =< V-2*V11
s(205) =< V/2-V11
s(167) =< -V11
s(165) =< -V11/2
aux(52) =< 1
aux(53) =< V
aux(54) =< V-V11
aux(55) =< V11
s(183) =< aux(52)
s(163) =< aux(53)
s(185) =< s(163)*aux(53)
s(186) =< s(185)*aux(52)
s(197) =< s(163)*aux(53)
s(198) =< s(197)
s(164) =< aux(55)
s(220) =< s(183)*aux(55)
s(228) =< aux(54)
s(231) =< s(228)*aux(54)
s(233) =< s(231)*aux(55)
s(204) =< s(163)*aux(55)
s(173) =< s(165)*s(167)
s(174) =< s(173)*aux(55)
s(211) =< s(163)*s(207)
s(212) =< s(211)*aux(55)
s(213) =< s(205)*s(207)
s(214) =< s(213)*aux(55)

  with precondition: [V1=2,V>=1,V11>=0] 

* Chain [32]: 8*s(259)+3
  Such that:aux(56) =< V1
s(259) =< aux(56)

  with precondition: [V=0,V1>=1] 

* Chain [31]: 3*s(263)+30*s(264)+18*s(266)+8
  Such that:s(261) =< 1
s(262) =< V1
s(263) =< s(261)
s(264) =< s(262)
s(265) =< s(264)*s(262)
s(266) =< s(265)

  with precondition: [V=1,V1>=1] 


Closed-form bounds of start(V1,V,V11): 
-------------------------------------
* Chain [36] with precondition: [V1>=0,V>=0] 
    - Upper bound: 55*V1+8+V*V1+2*V1*V*nat(V1-V)+6*V1*nat(V1-V)+16*V+3*V*nat(V1-V)*nat(V1/2-V/2)+nat(V1-V)*9*nat(V1/2-V/2)+nat(V1/2-V/2)*18 
    - Complexity: n^3 
* Chain [35] with precondition: [V1=1,V>=1,V11>=0] 
    - Upper bound: 15*V+4+V11*V+4*V11 
    - Complexity: n^2 
* Chain [34] with precondition: [V1=1,V>=2] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [33] with precondition: [V1=2,V>=1,V11>=0] 
    - Upper bound: 158*V+28+30*V*V+4*V*V11+2*V*V11*nat(V-2*V11)+6*V*nat(V-2*V11)+33*V11+2*V11*nat(V-V11)*nat(V-V11)+nat(V-2*V11)*V11*nat(V/2-V11)+nat(V-V11)*12+nat(V-V11)*6*nat(V-V11)+nat(V-2*V11)*3*nat(V/2-V11)+nat(V/2-V11)*6 
    - Complexity: n^3 
* Chain [32] with precondition: [V=0,V1>=1] 
    - Upper bound: 8*V1+3 
    - Complexity: n 
* Chain [31] with precondition: [V=1,V1>=1] 
    - Upper bound: 30*V1+11+18*V1*V1 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V11): max([22*V1+5+max([18*V1*V1+3,25*V1+V*V1+2*V1*V*nat(V1-V)+6*V1*nat(V1-V)+16*V+3*V*nat(V1-V)*nat(V1/2-V/2)+nat(V1-V)*9*nat(V1/2-V/2)+nat(V1/2-V/2)*18])+8*V1,143*V+24+30*V*V+3*V*nat(V11)+2*V*nat(V11)*nat(V-2*V11)+6*V*nat(V-2*V11)+nat(V11)*29+nat(V11)*2*nat(V-V11)*nat(V-V11)+nat(V-2*V11)*nat(V11)*nat(V/2-V11)+nat(V-V11)*12+nat(V-V11)*6*nat(V-V11)+nat(V-2*V11)*3*nat(V/2-V11)+nat(V/2-V11)*6+(15*V+1+nat(V11)*V+nat(V11)*4)])+3 
Asymptotic class: n^3 
* Total analysis performed in 587 ms.


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

(10)
BOUNDS(1, n^3)