WORST_CASE(?,O(n^2))
proof of input_mR0HPQBlrn.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(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, 1182 ms]
(10) BOUNDS(1, n^2)


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

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


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   p(0) -> 0
   p(s(x)) -> 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^2).


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z) [1]
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z) [1]
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> 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:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z) [1]
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z) [1]
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2
gr :: 0:s -> 0:s -> true:false
p :: 0:s -> 0:s
false :: true:false
0 :: 0:s
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:

   cond1(v0, v1, v2, v3) -> null_cond1 [0]

And the following fresh constants:    null_cond1

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

(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:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z) [1]
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z) [1]
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   cond1(v0, v1, v2, v3) -> null_cond1 [0]

The TRS has the following type information:
cond1 :: true:false -> 0:s -> 0:s -> 0:s -> null_cond1
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s -> null_cond1
gr :: 0:s -> 0:s -> true:false
p :: 0:s -> 0:s
false :: true:false
0 :: 0:s
s :: 0:s -> 0:s
null_cond1 :: null_cond1

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:

   true => 1
   false => 0
   0 => 0
   null_cond1 => 0

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

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

   cond1(z', z'', z1, z2) -{ 1 }-> cond2(gr(y, z), x, y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   cond1(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   cond2(z', z'', z1, z2) -{ 1 }-> cond2(gr(y, z), x, p(y), z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   cond2(z', z'', z1, z2) -{ 1 }-> cond1(gr(x, z), p(x), y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0
   gr(z', z'') -{ 1 }-> gr(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   gr(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   p(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, V1, V6, V2),0,[cond1(V, V1, V6, V2, Out)],[V >= 0,V1 >= 0,V6 >= 0,V2 >= 0]).
eq(start(V, V1, V6, V2),0,[cond2(V, V1, V6, V2, Out)],[V >= 0,V1 >= 0,V6 >= 0,V2 >= 0]).
eq(start(V, V1, V6, V2),0,[gr(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V6, V2),0,[p(V, Out)],[V >= 0]).
eq(cond1(V, V1, V6, V2, Out),1,[gr(V3, V4, Ret0),cond2(Ret0, V5, V3, V4, Ret)],[Out = Ret,V6 = V3,V4 >= 0,V2 = V4,V5 >= 0,V3 >= 0,V1 = V5,V = 1]).
eq(cond2(V, V1, V6, V2, Out),1,[gr(V9, V7, Ret01),p(V9, Ret2),cond2(Ret01, V8, Ret2, V7, Ret1)],[Out = Ret1,V6 = V9,V7 >= 0,V2 = V7,V8 >= 0,V9 >= 0,V1 = V8,V = 1]).
eq(cond2(V, V1, V6, V2, Out),1,[gr(V11, V12, Ret02),p(V11, Ret11),cond1(Ret02, Ret11, V10, V12, Ret3)],[Out = Ret3,V6 = V10,V12 >= 0,V2 = V12,V11 >= 0,V10 >= 0,V1 = V11,V = 0]).
eq(gr(V, V1, Out),1,[],[Out = 0,V13 >= 0,V1 = V13,V = 0]).
eq(gr(V, V1, Out),1,[],[Out = 1,V1 = 0,V = 1 + V14,V14 >= 0]).
eq(gr(V, V1, Out),1,[gr(V16, V15, Ret4)],[Out = Ret4,V = 1 + V16,V16 >= 0,V15 >= 0,V1 = 1 + V15]).
eq(p(V, Out),1,[],[Out = 0,V = 0]).
eq(p(V, Out),1,[],[Out = V17,V = 1 + V17,V17 >= 0]).
eq(cond1(V, V1, V6, V2, Out),0,[],[Out = 0,V2 = V20,V19 >= 0,V6 = V21,V18 >= 0,V1 = V18,V21 >= 0,V20 >= 0,V = V19]).
input_output_vars(cond1(V,V1,V6,V2,Out),[V,V1,V6,V2],[Out]).
input_output_vars(cond2(V,V1,V6,V2,Out),[V,V1,V6,V2],[Out]).
input_output_vars(gr(V,V1,Out),[V,V1],[Out]).
input_output_vars(p(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [gr/3]
1. non_recursive  : [p/2]
2. recursive  : [cond1/5,cond2/5]
3. non_recursive  : [start/4]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into gr/3
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into cond2/5
3. SCC is partially evaluated into start/4

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

### Specialization of cost equations gr/3 
* CE 8 is refined into CE [14] 
* CE 7 is refined into CE [15] 
* CE 6 is refined into CE [16] 


### Cost equations --> "Loop" of gr/3 
* CEs [15] --> Loop 10 
* CEs [16] --> Loop 11 
* CEs [14] --> Loop 12 

### Ranking functions of CR gr(V,V1,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR gr(V,V1,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations p/2 
* CE 13 is refined into CE [17] 
* CE 12 is refined into CE [18] 


### Cost equations --> "Loop" of p/2 
* CEs [17] --> Loop 13 
* CEs [18] --> Loop 14 

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

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


### Specialization of cost equations cond2/5 
* CE 11 is refined into CE [19,20,21,22] 
* CE 10 is refined into CE [23,24,25,26,27] 
* CE 9 is refined into CE [28,29,30,31] 


### Cost equations --> "Loop" of cond2/5 
* CEs [31] --> Loop 15 
* CEs [30] --> Loop 16 
* CEs [29] --> Loop 17 
* CEs [28] --> Loop 18 
* CEs [22] --> Loop 19 
* CEs [21] --> Loop 20 
* CEs [20] --> Loop 21 
* CEs [19] --> Loop 22 
* CEs [27] --> Loop 23 
* CEs [26] --> Loop 24 
* CEs [24] --> Loop 25 
* CEs [25] --> Loop 26 
* CEs [23] --> Loop 27 

### Ranking functions of CR cond2(V,V1,V6,V2,Out) 
* RF of phase [19]: [V6-1,V6-V2]
* RF of phase [21]: [V6]
* RF of phase [24]: [V1-1,V1-V2,V1-V6]
* RF of phase [26]: [V1-1,V1-V2]
* RF of phase [27]: [V1]

#### Partial ranking functions of CR cond2(V,V1,V6,V2,Out) 
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V6-1
    V6-V2
* Partial RF of phase [21]:
  - RF of loop [21:1]:
    V6
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V1-1
    V1-V2
    V1-V6
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1-1
    V1-V2
* Partial RF of phase [27]:
  - RF of loop [27:1]:
    V1


### Specialization of cost equations start/4 
* CE 1 is refined into CE [32] 
* CE 2 is refined into CE [33,34,35,36,37,38,39,40,41,42,43,44,45] 
* CE 3 is refined into CE [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64] 
* CE 4 is refined into CE [65,66,67,68] 
* CE 5 is refined into CE [69,70] 


### Cost equations --> "Loop" of start/4 
* CEs [44,62] --> Loop 28 
* CEs [41,60] --> Loop 29 
* CEs [45,64] --> Loop 30 
* CEs [42,63] --> Loop 31 
* CEs [39,61] --> Loop 32 
* CEs [36,59] --> Loop 33 
* CEs [37,58] --> Loop 34 
* CEs [34,35,57] --> Loop 35 
* CEs [43,56] --> Loop 36 
* CEs [40,55] --> Loop 37 
* CEs [32,38,54] --> Loop 38 
* CEs [33,53,66,67,68,70] --> Loop 39 
* CEs [46,47,48,49,50,51,52,65,69] --> Loop 40 

### Ranking functions of CR start(V,V1,V6,V2) 

#### Partial ranking functions of CR start(V,V1,V6,V2) 


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

#### Cost of chains of gr(V,V1,Out):
* Chain [[12],11]: 1*it(12)+1
  Such that:it(12) =< V

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

* Chain [[12],10]: 1*it(12)+1
  Such that:it(12) =< V1

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

* Chain [11]: 1
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [10]: 1
  with precondition: [V1=0,Out=1,V>=1] 


#### Cost of chains of p(V,Out):
* Chain [14]: 1
  with precondition: [V=0,Out=0] 

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


#### Cost of chains of cond2(V,V1,V6,V2,Out):
* Chain [[27],18]: 5*it(27)+3
  Such that:it(27) =< V1

  with precondition: [V=0,V6=0,V2=0,Out=0,V1>=1] 

* Chain [[27],17]: 5*it(27)+3
  Such that:it(27) =< V1

  with precondition: [V=0,V6=0,V2=0,Out=0,V1>=2] 

* Chain [[26],16]: 5*it(26)+1*s(1)+1*s(4)+3
  Such that:it(26) =< V1-V2
aux(2) =< V2
s(1) =< aux(2)
s(4) =< it(26)*aux(2)

  with precondition: [V=0,V6=0,Out=0,V2>=1,V1>=V2+1] 

* Chain [[26],15]: 5*it(26)+1*s(4)+1*s(5)+3
  Such that:it(26) =< V1-V2
aux(3) =< V2
s(5) =< aux(3)
s(4) =< it(26)*aux(3)

  with precondition: [V=0,V6=0,Out=0,V2>=1,V1>=V2+2] 

* Chain [[24],16]: 5*it(24)+1*s(1)+1*s(10)+1*s(11)+3
  Such that:it(24) =< V1-V2
aux(4) =< V6
aux(6) =< V2
s(1) =< aux(6)
s(10) =< it(24)*aux(6)
s(11) =< it(24)*aux(4)

  with precondition: [V=0,Out=0,V6>=1,V2>=V6,V1>=V2+1] 

* Chain [[24],15]: 5*it(24)+1*s(5)+1*s(10)+1*s(11)+3
  Such that:it(24) =< V1-V2
aux(4) =< V6
aux(7) =< V2
s(5) =< aux(7)
s(10) =< it(24)*aux(7)
s(11) =< it(24)*aux(4)

  with precondition: [V=0,Out=0,V6>=1,V2>=V6,V1>=V2+2] 

* Chain [[21],22,[27],18]: 3*it(21)+5*it(27)+6
  Such that:it(27) =< V1
it(21) =< V6

  with precondition: [V=1,V2=0,Out=0,V1>=1,V6>=1] 

* Chain [[21],22,[27],17]: 3*it(21)+5*it(27)+6
  Such that:it(27) =< V1
it(21) =< V6

  with precondition: [V=1,V2=0,Out=0,V1>=2,V6>=1] 

* Chain [[21],22,18]: 3*it(21)+6
  Such that:it(21) =< V6

  with precondition: [V=1,V1=0,V2=0,Out=0,V6>=1] 

* Chain [[21],22,17]: 3*it(21)+6
  Such that:it(21) =< V6

  with precondition: [V=1,V2=0,Out=0,V1>=1,V6>=1] 

* Chain [[19],20,[26],16]: 3*it(19)+5*it(26)+2*s(1)+1*s(4)+1*s(15)+6
  Such that:it(26) =< V1
it(19) =< V6
aux(9) =< 1
s(1) =< aux(9)
s(4) =< it(26)*aux(9)
s(15) =< it(19)*aux(9)

  with precondition: [V=1,V2=1,Out=0,V1>=2,V6>=2] 

* Chain [[19],20,[26],15]: 3*it(19)+5*it(26)+1*s(4)+2*s(5)+1*s(15)+6
  Such that:it(26) =< V1
it(19) =< V6
aux(10) =< 1
s(5) =< aux(10)
s(4) =< it(26)*aux(10)
s(15) =< it(19)*aux(10)

  with precondition: [V=1,V2=1,Out=0,V1>=3,V6>=2] 

* Chain [[19],20,[24],16]: 3*it(19)+5*it(24)+2*s(1)+2*s(10)+1*s(15)+6
  Such that:it(24) =< V1-V2
it(19) =< V6-V2
aux(12) =< V2
s(1) =< aux(12)
s(10) =< it(24)*aux(12)
s(15) =< it(19)*aux(12)

  with precondition: [V=1,Out=0,V2>=2,V1>=V2+1,V6>=V2+1] 

* Chain [[19],20,[24],15]: 3*it(19)+5*it(24)+2*s(5)+2*s(10)+1*s(15)+6
  Such that:it(24) =< V1-V2
it(19) =< V6-V2
aux(14) =< V2
s(5) =< aux(14)
s(10) =< it(24)*aux(14)
s(15) =< it(19)*aux(14)

  with precondition: [V=1,Out=0,V2>=2,V1>=V2+2,V6>=V2+1] 

* Chain [[19],20,18]: 3*it(19)+1*s(12)+1*s(15)+6
  Such that:it(19) =< V6-V2
aux(15) =< V2
s(12) =< aux(15)
s(15) =< it(19)*aux(15)

  with precondition: [V=1,V1=0,Out=0,V2>=1,V6>=V2+1] 

* Chain [[19],20,16]: 3*it(19)+1*s(1)+1*s(12)+1*s(15)+6
  Such that:s(1) =< V1
it(19) =< V6-V2
aux(16) =< V2
s(12) =< aux(16)
s(15) =< it(19)*aux(16)

  with precondition: [V=1,Out=0,V1>=1,V2>=V1,V6>=V2+1] 

* Chain [[19],20,15]: 3*it(19)+2*s(5)+1*s(15)+6
  Such that:it(19) =< V6-V2
aux(17) =< V2
s(5) =< aux(17)
s(15) =< it(19)*aux(17)

  with precondition: [V=1,Out=0,V2>=1,V1>=V2+1,V6>=V2+1] 

* Chain [25,[21],22,[27],18]: 3*it(21)+5*it(27)+11
  Such that:it(27) =< V1
it(21) =< V6

  with precondition: [V=0,V2=0,Out=0,V1>=2,V6>=1] 

* Chain [25,[21],22,[27],17]: 3*it(21)+5*it(27)+11
  Such that:it(27) =< V1
it(21) =< V6

  with precondition: [V=0,V2=0,Out=0,V1>=3,V6>=1] 

* Chain [25,[21],22,18]: 3*it(21)+11
  Such that:it(21) =< V6

  with precondition: [V=0,V1=1,V2=0,Out=0,V6>=1] 

* Chain [25,[21],22,17]: 3*it(21)+11
  Such that:it(21) =< V6

  with precondition: [V=0,V2=0,Out=0,V1>=2,V6>=1] 

* Chain [23,[19],20,[26],16]: 3*it(19)+5*it(26)+4*s(1)+1*s(4)+1*s(15)+11
  Such that:it(26) =< V1
it(19) =< V6
aux(19) =< 1
s(1) =< aux(19)
s(4) =< it(26)*aux(19)
s(15) =< it(19)*aux(19)

  with precondition: [V=0,V2=1,Out=0,V1>=3,V6>=2] 

* Chain [23,[19],20,[26],15]: 3*it(19)+5*it(26)+1*s(4)+4*s(5)+1*s(15)+11
  Such that:it(26) =< V1
it(19) =< V6
aux(20) =< 1
s(5) =< aux(20)
s(4) =< it(26)*aux(20)
s(15) =< it(19)*aux(20)

  with precondition: [V=0,V2=1,Out=0,V1>=4,V6>=2] 

* Chain [23,[19],20,[24],16]: 3*it(19)+5*it(24)+4*s(1)+2*s(10)+1*s(15)+11
  Such that:it(24) =< V1-V2
it(19) =< V6-V2
aux(21) =< V2
s(1) =< aux(21)
s(10) =< it(24)*aux(21)
s(15) =< it(19)*aux(21)

  with precondition: [V=0,Out=0,V2>=2,V1>=V2+2,V6>=V2+1] 

* Chain [23,[19],20,[24],15]: 3*it(19)+5*it(24)+4*s(5)+2*s(10)+1*s(15)+11
  Such that:it(24) =< V1-V2
it(19) =< V6-V2
aux(22) =< V2
s(5) =< aux(22)
s(10) =< it(24)*aux(22)
s(15) =< it(19)*aux(22)

  with precondition: [V=0,Out=0,V2>=2,V1>=V2+3,V6>=V2+1] 

* Chain [23,[19],20,16]: 3*it(19)+4*s(1)+1*s(15)+11
  Such that:it(19) =< V6-V2
aux(23) =< V2
s(1) =< aux(23)
s(15) =< it(19)*aux(23)

  with precondition: [V=0,Out=0,V1=V2+1,V1>=2,V6>=V1] 

* Chain [23,[19],20,15]: 3*it(19)+4*s(5)+1*s(15)+11
  Such that:it(19) =< V6-V2
aux(24) =< V2
s(5) =< aux(24)
s(15) =< it(19)*aux(24)

  with precondition: [V=0,Out=0,V2>=1,V1>=V2+2,V6>=V2+1] 

* Chain [22,[27],18]: 5*it(27)+6
  Such that:it(27) =< V1

  with precondition: [V=1,V6=0,V2=0,Out=0,V1>=1] 

* Chain [22,[27],17]: 5*it(27)+6
  Such that:it(27) =< V1

  with precondition: [V=1,V6=0,V2=0,Out=0,V1>=2] 

* Chain [22,[26],16]: 5*it(26)+1*s(1)+1*s(4)+6
  Such that:it(26) =< V1-V2
aux(2) =< V2
s(1) =< aux(2)
s(4) =< it(26)*aux(2)

  with precondition: [V=1,V6=0,Out=0,V2>=1,V1>=V2+1] 

* Chain [22,[26],15]: 5*it(26)+1*s(4)+1*s(5)+6
  Such that:it(26) =< V1-V2
aux(3) =< V2
s(5) =< aux(3)
s(4) =< it(26)*aux(3)

  with precondition: [V=1,V6=0,Out=0,V2>=1,V1>=V2+2] 

* Chain [22,18]: 6
  with precondition: [V=1,V1=0,V6=0,Out=0,V2>=0] 

* Chain [22,17]: 6
  with precondition: [V=1,V6=0,V2=0,Out=0,V1>=1] 

* Chain [22,16]: 1*s(1)+6
  Such that:s(1) =< V1

  with precondition: [V=1,V6=0,Out=0,V1>=1,V2>=V1] 

* Chain [22,15]: 1*s(5)+6
  Such that:s(5) =< V2

  with precondition: [V=1,V6=0,Out=0,V2>=1,V1>=V2+1] 

* Chain [20,[26],16]: 5*it(26)+1*s(1)+1*s(4)+1*s(12)+6
  Such that:s(12) =< 1
it(26) =< V1-V2
aux(2) =< V2
s(1) =< aux(2)
s(4) =< it(26)*aux(2)

  with precondition: [V=1,V6=1,Out=0,V2>=1,V1>=V2+1] 

* Chain [20,[26],15]: 5*it(26)+1*s(4)+1*s(5)+1*s(12)+6
  Such that:s(12) =< 1
it(26) =< V1-V2
aux(3) =< V2
s(5) =< aux(3)
s(4) =< it(26)*aux(3)

  with precondition: [V=1,V6=1,Out=0,V2>=1,V1>=V2+2] 

* Chain [20,[24],16]: 5*it(24)+1*s(1)+1*s(10)+1*s(11)+1*s(12)+6
  Such that:it(24) =< V1-V2
aux(6) =< V2
aux(11) =< V6
s(12) =< aux(11)
s(1) =< aux(6)
s(10) =< it(24)*aux(6)
s(11) =< it(24)*aux(11)

  with precondition: [V=1,Out=0,V6>=2,V2>=V6,V1>=V2+1] 

* Chain [20,[24],15]: 5*it(24)+1*s(5)+1*s(10)+1*s(11)+1*s(12)+6
  Such that:it(24) =< V1-V2
aux(7) =< V2
aux(13) =< V6
s(12) =< aux(13)
s(5) =< aux(7)
s(10) =< it(24)*aux(7)
s(11) =< it(24)*aux(13)

  with precondition: [V=1,Out=0,V6>=2,V2>=V6,V1>=V2+2] 

* Chain [20,18]: 1*s(12)+6
  Such that:s(12) =< V6

  with precondition: [V=1,V1=0,Out=0,V6>=1,V2>=V6] 

* Chain [20,16]: 1*s(1)+1*s(12)+6
  Such that:s(1) =< V1
s(12) =< V6

  with precondition: [V=1,Out=0,V1>=1,V6>=1,V2>=V1,V2>=V6] 

* Chain [20,15]: 1*s(5)+1*s(12)+6
  Such that:s(12) =< V6
s(5) =< V2

  with precondition: [V=1,Out=0,V6>=1,V2>=V6,V1>=V2+1] 

* Chain [18]: 3
  with precondition: [V=0,V1=0,Out=0,V6>=0,V2>=0] 

* Chain [17]: 3
  with precondition: [V=0,V2=0,Out=0,V1>=1,V6>=0] 

* Chain [16]: 1*s(1)+3
  Such that:s(1) =< V1

  with precondition: [V=0,Out=0,V1>=1,V6>=0,V2>=V1] 

* Chain [15]: 1*s(5)+3
  Such that:s(5) =< V2

  with precondition: [V=0,Out=0,V6>=0,V2>=1,V1>=V2+1] 


#### Cost of chains of start(V,V1,V6,V2):
* Chain [40]: 31*s(147)+18*s(148)+30*s(155)+12*s(156)+21*s(157)+8*s(158)+4*s(159)+2*s(160)+8*s(166)+2*s(167)+2*s(168)+11
  Such that:s(161) =< 1
s(151) =< V1-V2
aux(50) =< V1
aux(51) =< V6
aux(52) =< V6-V2
aux(53) =< V2
s(147) =< aux(50)
s(156) =< aux(52)
s(148) =< aux(51)
s(166) =< s(161)
s(167) =< s(147)*s(161)
s(168) =< s(148)*s(161)
s(155) =< s(151)
s(157) =< aux(53)
s(158) =< s(155)*aux(53)
s(159) =< s(156)*aux(53)
s(160) =< s(155)*aux(51)

  with precondition: [V=0] 

* Chain [39]: 1*s(174)+1*s(175)+6
  Such that:s(174) =< V
s(175) =< V1

  with precondition: [V>=1] 

* Chain [38]: 6*s(176)+8
  Such that:aux(54) =< V6
s(176) =< aux(54)

  with precondition: [V>=0,V1>=0,V6>=0,V2>=0] 

* Chain [37]: 2*s(178)+6
  Such that:aux(55) =< V6
s(178) =< aux(55)

  with precondition: [V=1,V1=0,V6>=1,V2>=V6] 

* Chain [36]: 3*s(180)+6*s(181)+2*s(184)+8
  Such that:aux(57) =< V6-V2
aux(58) =< V2
s(181) =< aux(57)
s(180) =< aux(58)
s(184) =< s(181)*aux(58)

  with precondition: [V=1,V1=0,V2>=1,V6>=V2+1] 

* Chain [35]: 30*s(191)+13
  Such that:aux(59) =< V1
s(191) =< aux(59)

  with precondition: [V=1,V6=0,V2=0,V1>=1] 

* Chain [34]: 2*s(197)+6
  Such that:aux(60) =< V1
s(197) =< aux(60)

  with precondition: [V=1,V6=0,V1>=1,V2>=V1] 

* Chain [33]: 40*s(203)+9*s(204)+20*s(205)+10*s(206)+3*s(207)+13
  Such that:s(201) =< -V2
aux(61) =< V1-V2
aux(62) =< V2
s(203) =< aux(61)
s(204) =< s(201)
s(205) =< aux(62)
s(206) =< s(203)*aux(62)
s(207) =< s(204)*aux(62)

  with precondition: [V=1,V6=0,V2>=1,V1>=V2+1] 

* Chain [32]: 20*s(216)+18*s(217)+8
  Such that:aux(63) =< V1
aux(64) =< V6
s(216) =< aux(63)
s(217) =< aux(64)

  with precondition: [V=1,V2=0,V1>=1,V6>=1] 

* Chain [31]: 2*s(222)+2*s(223)+6
  Such that:aux(65) =< V1
aux(66) =< V6
s(223) =< aux(65)
s(222) =< aux(66)

  with precondition: [V=1,V1>=1,V6>=1,V2>=V1,V2>=V6] 

* Chain [30]: 3*s(226)+2*s(227)+6*s(228)+2*s(231)+8
  Such that:aux(68) =< V1
aux(69) =< V6-V2
aux(70) =< V2
s(227) =< aux(68)
s(228) =< aux(69)
s(226) =< aux(70)
s(231) =< s(228)*aux(70)

  with precondition: [V=1,V1>=1,V2>=V1,V6>=V2+1] 

* Chain [29]: 4*s(237)+50*s(242)+9*s(243)+22*s(244)+12*s(245)+3*s(246)+4*s(247)+2*s(252)+13
  Such that:s(248) =< 1
s(240) =< V6-V2
aux(72) =< V1-V2
aux(73) =< V6
aux(74) =< V2
s(252) =< s(248)
s(242) =< aux(72)
s(237) =< aux(73)
s(244) =< aux(74)
s(245) =< s(242)*aux(74)
s(247) =< s(242)*aux(73)
s(243) =< s(240)
s(246) =< s(243)*aux(74)

  with precondition: [V=1,V6>=1,V2>=V6,V1>=V2+1] 

* Chain [28]: 13*s(258)+20*s(265)+20*s(266)+12*s(267)+18*s(268)+8*s(269)+4*s(270)+4*s(271)+8*s(273)+6*s(274)+8
  Such that:aux(76) =< 1
aux(77) =< V1
aux(78) =< V1-V2
aux(79) =< V6
aux(80) =< V6-V2
aux(81) =< V2
s(258) =< aux(81)
s(265) =< aux(77)
s(266) =< aux(78)
s(267) =< aux(79)
s(268) =< aux(80)
s(269) =< aux(76)
s(270) =< s(265)*aux(76)
s(271) =< s(267)*aux(76)
s(273) =< s(266)*aux(81)
s(274) =< s(268)*aux(81)

  with precondition: [V=1,V2>=1,V1>=V2+1,V6>=V2+1] 


Closed-form bounds of start(V,V1,V6,V2): 
-------------------------------------
* Chain [40] with precondition: [V=0] 
    - Upper bound: nat(V1)*33+19+nat(V6)*20+nat(V6)*2*nat(V1-V2)+nat(V2)*21+nat(V2)*8*nat(V1-V2)+nat(V2)*4*nat(V6-V2)+nat(V1-V2)*30+nat(V6-V2)*12 
    - Complexity: n^2 
* Chain [39] with precondition: [V>=1] 
    - Upper bound: V+6+nat(V1) 
    - Complexity: n 
* Chain [38] with precondition: [V>=0,V1>=0,V6>=0,V2>=0] 
    - Upper bound: 6*V6+8 
    - Complexity: n 
* Chain [37] with precondition: [V=1,V1=0,V6>=1,V2>=V6] 
    - Upper bound: 2*V6+6 
    - Complexity: n 
* Chain [36] with precondition: [V=1,V1=0,V2>=1,V6>=V2+1] 
    - Upper bound: 6*V6-6*V2+(3*V2+8+(V6-V2)*(2*V2)) 
    - Complexity: n^2 
* Chain [35] with precondition: [V=1,V6=0,V2=0,V1>=1] 
    - Upper bound: 30*V1+13 
    - Complexity: n 
* Chain [34] with precondition: [V=1,V6=0,V1>=1,V2>=V1] 
    - Upper bound: 2*V1+6 
    - Complexity: n 
* Chain [33] with precondition: [V=1,V6=0,V2>=1,V1>=V2+1] 
    - Upper bound: 40*V1-40*V2+(20*V2+13+(V1-V2)*(10*V2)) 
    - Complexity: n^2 
* Chain [32] with precondition: [V=1,V2=0,V1>=1,V6>=1] 
    - Upper bound: 20*V1+18*V6+8 
    - Complexity: n 
* Chain [31] with precondition: [V=1,V1>=1,V6>=1,V2>=V1,V2>=V6] 
    - Upper bound: 2*V1+2*V6+6 
    - Complexity: n 
* Chain [30] with precondition: [V=1,V1>=1,V2>=V1,V6>=V2+1] 
    - Upper bound: 6*V6-6*V2+(2*V1+3*V2+8+(V6-V2)*(2*V2)) 
    - Complexity: n^2 
* Chain [29] with precondition: [V=1,V6>=1,V2>=V6,V1>=V2+1] 
    - Upper bound: 50*V1-50*V2+(4*V6+15+(V1-V2)*(4*V6)+22*V2+(V1-V2)*(12*V2)) 
    - Complexity: n^2 
* Chain [28] with precondition: [V=1,V2>=1,V1>=V2+1,V6>=V2+1] 
    - Upper bound: 18*V6-18*V2+(20*V1-20*V2+(24*V1+16*V6+13*V2+16+(V1-V2)*(8*V2)+(V6-V2)*(6*V2))) 
    - Complexity: n^2 

### Maximum cost of start(V,V1,V6,V2): max([max([nat(V2)*3+2+max([nat(V2)*2*nat(V6-V2)+nat(V6-V2)*6,nat(V2)*17+5+nat(V2)*10*nat(V1-V2)+nat(V1-V2)*40]),nat(V6)*2+2+max([nat(V6)*2,nat(V6)*4*nat(V1-V2)+7+nat(V2)*22+nat(V2)*12*nat(V1-V2)+nat(V1-V2)*50])+nat(V6)*2]),nat(V1)+max([V,nat(V1)+max([max([nat(V6)*2,nat(V2)*3+2+nat(V2)*2*nat(V6-V2)+nat(V6-V2)*6]),nat(V1)*18+2+max([nat(V6)*18,nat(V1)*4+5+max([nat(V6)*16+3+nat(V2)*13+nat(V2)*8*nat(V1-V2)+nat(V2)*6*nat(V6-V2)+nat(V1-V2)*20+nat(V6-V2)*18,nat(V1)*3+6+nat(V6)*20+nat(V6)*2*nat(V1-V2)+nat(V2)*21+nat(V2)*8*nat(V1-V2)+nat(V2)*4*nat(V6-V2)+nat(V1-V2)*30+nat(V6-V2)*12+nat(V1)*6])])])])])+6 
Asymptotic class: n^2 
* Total analysis performed in 1080 ms.


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

(10)
BOUNDS(1, n^2)