WORST_CASE(Omega(n^1), O(n^3))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3).

(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), 6 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 2212 ms]
    (10) BOUNDS(1, n^3)
(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^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))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

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^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]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), 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]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), 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
gcd :: 0:s -> 0:s -> 0:s
if_gcd :: 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_gcd(v0, v1, v2) -> null_if_gcd [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   gcd(v0, v1) -> null_gcd [0]

And the following fresh constants:    null_if_minus, null_if_gcd, null_le, null_minus, null_gcd

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

(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]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   if_gcd(v0, v1, v2) -> null_if_gcd [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   gcd(v0, v1) -> null_gcd [0]

The TRS has the following type information:
le :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> true:false:null_le
0 :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
true :: true:false:null_le
s :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
false :: true:false:null_le
minus :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
if_minus :: true:false:null_le -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
gcd :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
if_gcd :: true:false:null_le -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd -> 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
null_if_minus :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
null_if_gcd :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
null_le :: true:false:null_le
null_minus :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd
null_gcd :: 0:s:null_if_minus:null_if_gcd:null_minus:null_gcd

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_gcd => 0
   null_le => 0
   null_minus => 0
   null_gcd => 0

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

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

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

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,[gcd(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(gcd(V1, V, Out),1,[],[Out = V14,V14 >= 0,V1 = 0,V = V14]).
eq(gcd(V1, V, Out),1,[],[Out = 1 + V15,V15 >= 0,V1 = 1 + V15,V = 0]).
eq(gcd(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),gcd(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,[minus(V21, V20, Ret03),gcd(Ret03, 1 + V20, Ret4)],[Out = Ret4,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(gcd(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(gcd(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,gcd/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 gcd/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 14 
* CEs [24] --> Loop 15 
* CEs [25] --> Loop 16 
* CEs [26] --> Loop 17 

### Ranking functions of CR le(V1,V,Out) 
* RF of phase [14]: [V,V1]

#### Partial ranking functions of CR le(V1,V,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    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 18 
* CEs [35] --> Loop 19 
* CEs [28] --> Loop 20 
* CEs [29,30,31,32,33,34] --> Loop 21 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [18]: [V1-1,V1-V]
* RF of phase [19]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V1-1
    V1-V
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V1


### Specialization of cost equations gcd/3 
* CE 14 is refined into CE [37,38,39,40,41] 
* CE 19 is refined into CE [42] 
* CE 18 is refined into CE [43] 
* CE 17 is refined into CE [44] 
* CE 16 is refined into CE [45,46,47,48] 
* CE 15 is refined into CE [49,50,51,52] 


### Cost equations --> "Loop" of gcd/3 
* CEs [52] --> Loop 22 
* CEs [48] --> Loop 23 
* CEs [51] --> Loop 24 
* CEs [47] --> Loop 25 
* CEs [46] --> Loop 26 
* CEs [45] --> Loop 27 
* CEs [50] --> Loop 28 
* CEs [49] --> Loop 29 
* CEs [37] --> Loop 30 
* CEs [43] --> Loop 31 
* CEs [38,39,40,41,42] --> Loop 32 
* CEs [44] --> Loop 33 

### Ranking functions of CR gcd(V1,V,Out) 
* RF of phase [22,23]: [V1/2+V/2-2]
* RF of phase [26]: [V1-1]

#### Partial ranking functions of CR gcd(V1,V,Out) 
* Partial RF of phase [22,23]:
  - RF of loop [22:1]:
    V-2
    V1/2+V/2-2
  - RF of loop [23:1]:
    V1/2-1 depends on loops [22:1] 
    V1/2-V/2 depends on loops [22:1] 
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1-1


### Specialization of cost equations start/3 
* CE 3 is refined into CE [53,54,55,56,57,58,59,60,61,62,63] 
* CE 5 is refined into CE [64] 
* CE 1 is refined into CE [65] 
* CE 2 is refined into CE [66,67,68,69,70,71,72,73,74,75,76] 
* CE 4 is refined into CE [77,78,79] 
* CE 6 is refined into CE [80,81,82,83,84] 
* CE 7 is refined into CE [85,86,87] 
* CE 8 is refined into CE [88,89,90,91,92,93,94,95,96,97] 


### Cost equations --> "Loop" of start/3 
* CEs [92,93] --> Loop 34 
* CEs [81,86,91] --> Loop 35 
* CEs [53,54,55,56,57,58,59,60,61,62,63,64] --> Loop 36 
* CEs [78,90] --> Loop 37 
* CEs [66,67,68,69,70,71,72,73,74,75,76,77,79] --> Loop 38 
* CEs [65,80,82,83,84,85,87,88,89,94,95,96,97] --> Loop 39 

### 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 [[14],17]: 1*it(14)+1
  Such that:it(14) =< V1

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

* Chain [[14],16]: 1*it(14)+1
  Such that:it(14) =< V

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

* Chain [[14],15]: 1*it(14)+0
  Such that:it(14) =< V

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

* Chain [17]: 1
  with precondition: [V1=0,Out=2,V>=0] 

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

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[19],21]: 3*it(19)+2*s(4)+3
  Such that:aux(1) =< V1-Out
it(19) =< Out
s(4) =< aux(1)

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

* Chain [[19],20]: 3*it(19)+2
  Such that:it(19) =< Out

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

* Chain [[18],21]: 3*it(18)+2*s(2)+2*s(4)+1*s(8)+3
  Such that:aux(1) =< V1-Out
it(18) =< Out
aux(4) =< V
s(4) =< aux(1)
s(2) =< aux(4)
s(8) =< it(18)*aux(4)

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

* Chain [21]: 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 [20]: 2
  with precondition: [V=0,Out=0,V1>=1] 


#### Cost of chains of gcd(V1,V,Out):
* Chain [[26],32]: 10*it(26)+1*s(19)+6*s(28)+2
  Such that:s(19) =< 1
aux(10) =< V1
it(26) =< aux(10)
s(31) =< it(26)*aux(10)
s(28) =< s(31)

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

* Chain [[26],30]: 8*it(26)+6*s(28)+2
  Such that:aux(11) =< V1
it(26) =< aux(11)
s(31) =< it(26)*aux(11)
s(28) =< s(31)

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

* Chain [[26],27,33]: 10*it(26)+6*s(28)+7
  Such that:aux(12) =< V1
it(26) =< aux(12)
s(31) =< it(26)*aux(12)
s(28) =< s(31)

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

* Chain [[26],27,32]: 10*it(26)+1*s(19)+6*s(28)+8
  Such that:s(19) =< 1
aux(13) =< V1
it(26) =< aux(13)
s(31) =< it(26)*aux(13)
s(28) =< s(31)

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

* Chain [[22,23],32]: 6*it(22)+6*it(23)+8*s(17)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+2
  Such that:aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(48) =< V1
aux(49) =< V1+V
aux(50) =< V1/2
aux(51) =< V
s(17) =< aux(49)
it(23) =< aux(49)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(49)
it(22) =< aux(51)
s(63) =< aux(51)
aux(16) =< aux(46)
aux(16) =< aux(51)
aux(35) =< aux(51)
aux(27) =< aux(49)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(50)
s(73) =< aux(23)+aux(48)
s(72) =< aux(23)+aux(48)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(49)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(51)
s(65) =< s(63)*aux(49)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],29,33]: 6*it(22)+6*it(23)+7*s(62)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+7
  Such that:aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(52) =< V1
aux(53) =< V1+V
aux(54) =< V1/2
aux(55) =< V
s(62) =< aux(53)
it(23) =< aux(53)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(53)
it(22) =< aux(55)
s(63) =< aux(55)
aux(16) =< aux(46)
aux(16) =< aux(55)
aux(35) =< aux(55)
aux(27) =< aux(53)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(54)
s(73) =< aux(23)+aux(52)
s(72) =< aux(23)+aux(52)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(53)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(55)
s(65) =< s(63)*aux(53)

  with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],29,32]: 6*it(22)+6*it(23)+1*s(19)+7*s(62)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+8
  Such that:s(19) =< 1
aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(56) =< V1
aux(57) =< V1+V
aux(58) =< V1/2
aux(59) =< V
s(62) =< aux(57)
it(23) =< aux(57)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(57)
it(22) =< aux(59)
s(63) =< aux(59)
aux(16) =< aux(46)
aux(16) =< aux(59)
aux(35) =< aux(59)
aux(27) =< aux(57)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(58)
s(73) =< aux(23)+aux(56)
s(72) =< aux(23)+aux(56)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(57)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(59)
s(65) =< s(63)*aux(57)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,[26],32]: 6*it(22)+6*it(23)+23*it(26)+1*s(19)+6*s(28)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+8
  Such that:s(19) =< 1
aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(61) =< V1
aux(62) =< V1+V
aux(63) =< V1/2
aux(64) =< V
it(26) =< aux(62)
s(31) =< it(26)*aux(62)
s(28) =< s(31)
it(23) =< aux(62)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(62)
it(22) =< aux(64)
s(63) =< aux(64)
aux(16) =< aux(46)
aux(16) =< aux(64)
aux(35) =< aux(64)
aux(27) =< aux(62)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(63)
s(73) =< aux(23)+aux(61)
s(72) =< aux(23)+aux(61)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(62)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(64)
s(65) =< s(63)*aux(62)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 

* Chain [[22,23],28,[26],30]: 6*it(22)+6*it(23)+21*it(26)+6*s(28)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+8
  Such that:aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(66) =< V1
aux(67) =< V1+V
aux(68) =< V1/2
aux(69) =< V
it(26) =< aux(67)
s(31) =< it(26)*aux(67)
s(28) =< s(31)
it(23) =< aux(67)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(67)
it(22) =< aux(69)
s(63) =< aux(69)
aux(16) =< aux(46)
aux(16) =< aux(69)
aux(35) =< aux(69)
aux(27) =< aux(67)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(68)
s(73) =< aux(23)+aux(66)
s(72) =< aux(23)+aux(66)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(67)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(69)
s(65) =< s(63)*aux(67)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 

* Chain [[22,23],28,[26],27,33]: 6*it(22)+6*it(23)+23*it(26)+6*s(28)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+13
  Such that:aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(71) =< V1
aux(72) =< V1+V
aux(73) =< V1/2
aux(74) =< V
it(26) =< aux(72)
s(31) =< it(26)*aux(72)
s(28) =< s(31)
it(23) =< aux(72)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(72)
it(22) =< aux(74)
s(63) =< aux(74)
aux(16) =< aux(46)
aux(16) =< aux(74)
aux(35) =< aux(74)
aux(27) =< aux(72)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(73)
s(73) =< aux(23)+aux(71)
s(72) =< aux(23)+aux(71)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(72)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(74)
s(65) =< s(63)*aux(72)

  with precondition: [Out=1,V1>=3,V>=3,V+V1>=7] 

* Chain [[22,23],28,[26],27,32]: 6*it(22)+6*it(23)+23*it(26)+1*s(19)+6*s(28)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+14
  Such that:s(19) =< 1
aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(76) =< V1
aux(77) =< V1+V
aux(78) =< V1/2
aux(79) =< V
it(26) =< aux(77)
s(31) =< it(26)*aux(77)
s(28) =< s(31)
it(23) =< aux(77)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(77)
it(22) =< aux(79)
s(63) =< aux(79)
aux(16) =< aux(46)
aux(16) =< aux(79)
aux(35) =< aux(79)
aux(27) =< aux(77)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(78)
s(73) =< aux(23)+aux(76)
s(72) =< aux(23)+aux(76)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(77)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(79)
s(65) =< s(63)*aux(77)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 

* Chain [[22,23],28,32]: 6*it(22)+6*it(23)+15*s(17)+1*s(19)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+8
  Such that:s(19) =< 1
aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(81) =< V1
aux(82) =< V1+V
aux(83) =< V1/2
aux(84) =< V
s(17) =< aux(82)
it(23) =< aux(82)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(82)
it(22) =< aux(84)
s(63) =< aux(84)
aux(16) =< aux(46)
aux(16) =< aux(84)
aux(35) =< aux(84)
aux(27) =< aux(82)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(83)
s(73) =< aux(23)+aux(81)
s(72) =< aux(23)+aux(81)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(82)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(84)
s(65) =< s(63)*aux(82)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,30]: 6*it(22)+6*it(23)+13*s(62)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+8
  Such that:aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(86) =< V1
aux(87) =< V1+V
aux(88) =< V1/2
aux(89) =< V
s(62) =< aux(87)
it(23) =< aux(87)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(87)
it(22) =< aux(89)
s(63) =< aux(89)
aux(16) =< aux(46)
aux(16) =< aux(89)
aux(35) =< aux(89)
aux(27) =< aux(87)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(88)
s(73) =< aux(23)+aux(86)
s(72) =< aux(23)+aux(86)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(87)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(89)
s(65) =< s(63)*aux(87)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,27,33]: 6*it(22)+6*it(23)+15*s(34)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+13
  Such that:aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(91) =< V1
aux(92) =< V1+V
aux(93) =< V1/2
aux(94) =< V
s(34) =< aux(92)
it(23) =< aux(92)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(92)
it(22) =< aux(94)
s(63) =< aux(94)
aux(16) =< aux(46)
aux(16) =< aux(94)
aux(35) =< aux(94)
aux(27) =< aux(92)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(93)
s(73) =< aux(23)+aux(91)
s(72) =< aux(23)+aux(91)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(92)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(94)
s(65) =< s(63)*aux(92)

  with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,27,32]: 6*it(22)+6*it(23)+1*s(19)+15*s(34)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+14
  Such that:s(19) =< 1
aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(96) =< V1
aux(97) =< V1+V
aux(98) =< V1/2
aux(99) =< V
s(34) =< aux(97)
it(23) =< aux(97)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(97)
it(22) =< aux(99)
s(63) =< aux(99)
aux(16) =< aux(46)
aux(16) =< aux(99)
aux(35) =< aux(99)
aux(27) =< aux(97)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(98)
s(73) =< aux(23)+aux(96)
s(72) =< aux(23)+aux(96)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(97)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(99)
s(65) =< s(63)*aux(97)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],25,33]: 6*it(22)+6*it(23)+5*s(62)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+3*s(82)+2*s(85)+7
  Such that:aux(38) =< V1
aux(39) =< V1+V
aux(41) =< V1-Out
aux(17) =< V1/2
aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(21) =< V1/2-Out/2
aux(44) =< V
aux(45) =< V-Out
aux(46) =< V/2
aux(47) =< V/2-Out/2
aux(100) =< Out
aux(101) =< V1+V-Out
s(82) =< aux(100)
s(85) =< aux(101)
it(23) =< aux(39)
s(66) =< aux(39)
it(23) =< aux(101)
s(66) =< aux(101)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(101)
it(22) =< aux(44)
s(63) =< aux(44)
it(22) =< aux(45)
s(63) =< aux(45)
aux(16) =< aux(46)
aux(16) =< aux(47)
aux(35) =< aux(44)
aux(27) =< aux(39)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(17)
s(73) =< aux(23)+aux(41)
s(73) =< aux(23)+aux(38)
s(72) =< aux(23)+aux(41)
s(72) =< aux(23)+aux(38)
it(23) =< aux(16)+aux(21)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(39)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(44)
s(62) =< s(66)
s(65) =< s(63)*aux(39)

  with precondition: [Out>=2,V1>=Out,V>=Out,V+V1>=3*Out] 

* Chain [[22,23],25,32]: 6*it(22)+6*it(23)+11*s(19)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+8
  Such that:aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(103) =< V1
aux(104) =< V1+V
aux(105) =< V1/2
aux(106) =< V
s(19) =< aux(104)
it(23) =< aux(104)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(104)
it(22) =< aux(106)
s(63) =< aux(106)
aux(16) =< aux(46)
aux(16) =< aux(106)
aux(35) =< aux(106)
aux(27) =< aux(104)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(105)
s(73) =< aux(23)+aux(103)
s(72) =< aux(23)+aux(103)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(104)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(106)
s(65) =< s(63)*aux(104)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=6] 

* Chain [[22,23],24,33]: 6*it(22)+6*it(23)+5*s(62)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+3*s(87)+2*s(90)+7
  Such that:aux(38) =< V1
aux(39) =< V1+V
aux(41) =< V1-Out
aux(17) =< V1/2
aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(21) =< V1/2-Out/2
aux(46) =< V/2
aux(107) =< Out
aux(108) =< V1+V-Out
aux(109) =< V
s(87) =< aux(107)
s(90) =< aux(108)
it(23) =< aux(39)
s(66) =< aux(39)
it(23) =< aux(108)
s(66) =< aux(108)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(108)
it(22) =< aux(109)
s(63) =< aux(109)
aux(16) =< aux(46)
aux(16) =< aux(109)
aux(35) =< aux(109)
aux(27) =< aux(39)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(17)
s(73) =< aux(23)+aux(41)
s(73) =< aux(23)+aux(38)
s(72) =< aux(23)+aux(41)
s(72) =< aux(23)+aux(38)
it(23) =< aux(16)+aux(21)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(39)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(109)
s(62) =< s(66)
s(65) =< s(63)*aux(39)

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

* Chain [[22,23],24,32]: 6*it(22)+6*it(23)+11*s(19)+3*s(63)+1*s(65)+3*s(68)+3*s(69)+2*s(70)+1*s(71)+8
  Such that:aux(17) =< V1/2
aux(19) =< V1/2-V/2
aux(42) =< V1/2+V/2
aux(46) =< V/2
aux(111) =< V1
aux(112) =< V1+V
aux(113) =< V
s(19) =< aux(112)
it(23) =< aux(112)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(112)
it(22) =< aux(113)
s(63) =< aux(113)
aux(16) =< aux(46)
aux(16) =< aux(113)
aux(35) =< aux(113)
aux(27) =< aux(112)-1
aux(23) =< aux(16)*2
it(23) =< aux(16)+aux(19)
it(23) =< aux(16)+aux(17)
s(73) =< aux(23)+aux(111)
s(72) =< aux(23)+aux(111)
it(23) =< aux(16)+aux(111)
s(73) =< it(23)*aux(35)
s(69) =< it(23)*aux(112)
s(72) =< it(23)*aux(27)
s(68) =< s(73)
s(70) =< s(72)
s(71) =< s(69)*aux(113)
s(65) =< s(63)*aux(112)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=8] 

* Chain [33]: 1
  with precondition: [V1=0,V=Out,V>=0] 

* Chain [32]: 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 [31]: 1
  with precondition: [V=0,V1=Out,V1>=1] 

* Chain [30]: 2
  with precondition: [V=1,Out=0,V1>=1] 

* Chain [29,33]: 2*s(76)+7
  Such that:s(74) =< V
s(76) =< s(74)

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

* Chain [29,32]: 1*s(19)+2*s(76)+8
  Such that:s(19) =< 1
s(74) =< V
s(76) =< s(74)

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

* Chain [28,[26],32]: 18*it(26)+1*s(19)+6*s(28)+8
  Such that:s(19) =< 1
aux(60) =< V
it(26) =< aux(60)
s(31) =< it(26)*aux(60)
s(28) =< s(31)

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

* Chain [28,[26],30]: 16*it(26)+6*s(28)+8
  Such that:aux(65) =< V
it(26) =< aux(65)
s(31) =< it(26)*aux(65)
s(28) =< s(31)

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

* Chain [28,[26],27,33]: 18*it(26)+6*s(28)+13
  Such that:aux(70) =< V
it(26) =< aux(70)
s(31) =< it(26)*aux(70)
s(28) =< s(31)

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

* Chain [28,[26],27,32]: 18*it(26)+1*s(19)+6*s(28)+14
  Such that:s(19) =< 1
aux(75) =< V
it(26) =< aux(75)
s(31) =< it(26)*aux(75)
s(28) =< s(31)

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

* Chain [28,32]: 10*s(17)+1*s(19)+8
  Such that:s(19) =< 1
aux(80) =< V
s(17) =< aux(80)

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

* Chain [28,30]: 8*s(80)+8
  Such that:aux(85) =< V
s(80) =< aux(85)

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

* Chain [28,27,33]: 10*s(34)+13
  Such that:aux(90) =< V
s(34) =< aux(90)

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

* Chain [28,27,32]: 1*s(19)+10*s(34)+14
  Such that:s(19) =< 1
aux(95) =< V
s(34) =< aux(95)

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

* Chain [27,33]: 2*s(34)+7
  Such that:s(32) =< V1
s(34) =< s(32)

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

* Chain [27,32]: 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 [25,33]: 3*s(82)+2*s(85)+7
  Such that:s(83) =< V1
aux(100) =< Out
s(82) =< aux(100)
s(85) =< s(83)

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

* Chain [25,32]: 4*s(19)+2*s(85)+8
  Such that:s(83) =< V1
aux(102) =< V
s(19) =< aux(102)
s(85) =< s(83)

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

* Chain [24,33]: 3*s(87)+2*s(90)+7
  Such that:s(88) =< V
aux(107) =< Out
s(87) =< aux(107)
s(90) =< s(88)

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

* Chain [24,32]: 4*s(19)+2*s(90)+8
  Such that:s(88) =< V
aux(110) =< V1
s(19) =< aux(110)
s(90) =< s(88)

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


#### Cost of chains of start(V1,V,V11):
* Chain [39]: 145*s(518)+21*s(520)+1*s(530)+10*s(539)+18*s(542)+212*s(543)+72*s(544)+90*s(545)+36*s(552)+36*s(553)+24*s(554)+12*s(555)+15*s(556)+24*s(558)+18*s(560)+9*s(563)+9*s(564)+6*s(565)+3*s(566)+14
  Such that:s(531) =< 1
aux(148) =< V1
aux(149) =< V1+V
aux(150) =< V1/2
aux(151) =< V1/2-V/2
aux(152) =< V1/2+V/2
aux(153) =< V
aux(154) =< V/2
s(520) =< aux(148)
s(518) =< aux(153)
s(539) =< s(531)
s(541) =< s(518)*aux(153)
s(542) =< s(541)
s(543) =< aux(149)
s(544) =< aux(149)
s(545) =< aux(152)
s(544) =< aux(152)
s(545) =< aux(149)
s(545) =< aux(153)
s(546) =< aux(154)
s(546) =< aux(153)
s(547) =< aux(153)
s(548) =< aux(149)-1
s(549) =< s(546)*2
s(544) =< s(546)+aux(151)
s(544) =< s(546)+aux(150)
s(550) =< s(549)+aux(148)
s(551) =< s(549)+aux(148)
s(550) =< s(544)*s(547)
s(552) =< s(544)*aux(149)
s(551) =< s(544)*s(548)
s(553) =< s(550)
s(554) =< s(551)
s(555) =< s(552)*aux(153)
s(556) =< s(518)*aux(149)
s(557) =< s(543)*aux(149)
s(558) =< s(557)
s(560) =< aux(149)
s(560) =< aux(152)
s(560) =< s(546)+aux(151)
s(560) =< s(546)+aux(150)
s(561) =< s(549)+aux(148)
s(562) =< s(549)+aux(148)
s(560) =< s(546)+aux(148)
s(561) =< s(560)*s(547)
s(563) =< s(560)*aux(149)
s(562) =< s(560)*s(548)
s(564) =< s(561)
s(565) =< s(562)
s(566) =< s(563)*aux(153)
s(530) =< s(520)*aux(153)

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

* Chain [38]: 267*s(644)+458*s(645)+155*s(658)+60*s(661)+60*s(663)+60*s(664)+30*s(671)+30*s(672)+20*s(673)+10*s(674)+10*s(675)+18*s(701)+60*s(703)+60*s(704)+30*s(711)+30*s(712)+20*s(713)+10*s(714)+10*s(715)+42*s(717)+5*s(749)+214*s(762)+90*s(763)+90*s(764)+45*s(771)+45*s(772)+30*s(773)+15*s(774)+15*s(775)+24*s(777)+1*s(791)+1*s(904)+18
  Such that:s(697) =< 1/2
s(654) =< -V/2
aux(177) =< 1
aux(178) =< -V+V11/2
aux(179) =< V
aux(180) =< V+V11
aux(181) =< V/2
aux(182) =< V11
s(658) =< aux(177)
s(645) =< aux(179)
s(660) =< s(645)*aux(179)
s(661) =< s(660)
s(663) =< aux(179)
s(664) =< aux(181)
s(663) =< aux(181)
s(664) =< aux(179)
s(665) =< aux(181)
s(665) =< aux(179)
s(666) =< aux(179)
s(667) =< aux(179)-1
s(668) =< s(665)*2
s(663) =< s(665)+s(654)
s(663) =< s(665)
s(669) =< s(668)
s(670) =< s(668)
s(669) =< s(663)*s(666)
s(671) =< s(663)*aux(179)
s(670) =< s(663)*s(667)
s(672) =< s(669)
s(673) =< s(670)
s(674) =< s(671)*aux(179)
s(675) =< s(645)*aux(179)
s(644) =< aux(182)
s(716) =< s(644)*aux(182)
s(717) =< s(716)
s(762) =< aux(180)
s(763) =< aux(180)
s(764) =< aux(180)
s(764) =< aux(179)
s(767) =< aux(180)-1
s(763) =< s(665)+aux(178)
s(763) =< s(665)+aux(182)
s(769) =< s(668)+aux(182)
s(770) =< s(668)+aux(182)
s(769) =< s(763)*s(666)
s(771) =< s(763)*aux(180)
s(770) =< s(763)*s(767)
s(772) =< s(769)
s(773) =< s(770)
s(774) =< s(771)*aux(179)
s(775) =< s(645)*aux(180)
s(776) =< s(762)*aux(180)
s(777) =< s(776)
s(749) =< s(644)*aux(179)
s(791) =< s(658)*aux(179)
s(700) =< s(658)*aux(177)
s(701) =< s(700)
s(703) =< aux(182)
s(704) =< aux(182)
s(704) =< aux(177)
s(705) =< s(697)
s(705) =< aux(177)
s(706) =< aux(177)
s(707) =< aux(182)-1
s(708) =< s(705)*2
s(703) =< s(705)+aux(182)
s(709) =< s(708)+aux(182)
s(710) =< s(708)+aux(182)
s(709) =< s(703)*s(706)
s(711) =< s(703)*aux(182)
s(710) =< s(703)*s(707)
s(712) =< s(709)
s(713) =< s(710)
s(714) =< s(711)*aux(177)
s(715) =< s(658)*aux(182)
s(904) =< s(645)*aux(182)

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

* Chain [37]: 38*s(907)+6*s(912)+13
  Such that:aux(184) =< V
s(907) =< aux(184)
s(911) =< s(907)*aux(184)
s(912) =< s(911)

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

* Chain [36]: 263*s(915)+451*s(916)+155*s(929)+60*s(932)+60*s(934)+60*s(935)+30*s(942)+30*s(943)+20*s(944)+10*s(945)+10*s(946)+18*s(972)+60*s(974)+60*s(975)+30*s(982)+30*s(983)+20*s(984)+10*s(985)+10*s(986)+42*s(988)+5*s(1020)+214*s(1033)+90*s(1034)+90*s(1035)+45*s(1042)+45*s(1043)+30*s(1044)+15*s(1045)+15*s(1046)+24*s(1048)+1*s(1062)+18
  Such that:s(968) =< 1/2
s(925) =< -V11/2
aux(206) =< 1
aux(207) =< V
aux(208) =< V+V11
aux(209) =< V/2-V11
aux(210) =< V11
aux(211) =< V11/2
s(929) =< aux(206)
s(916) =< aux(210)
s(931) =< s(916)*aux(210)
s(932) =< s(931)
s(934) =< aux(210)
s(935) =< aux(211)
s(934) =< aux(211)
s(935) =< aux(210)
s(936) =< aux(211)
s(936) =< aux(210)
s(937) =< aux(210)
s(938) =< aux(210)-1
s(939) =< s(936)*2
s(934) =< s(936)+s(925)
s(934) =< s(936)
s(940) =< s(939)
s(941) =< s(939)
s(940) =< s(934)*s(937)
s(942) =< s(934)*aux(210)
s(941) =< s(934)*s(938)
s(943) =< s(940)
s(944) =< s(941)
s(945) =< s(942)*aux(210)
s(946) =< s(916)*aux(210)
s(915) =< aux(207)
s(987) =< s(915)*aux(207)
s(988) =< s(987)
s(1033) =< aux(208)
s(1034) =< aux(208)
s(1035) =< aux(208)
s(1035) =< aux(210)
s(1038) =< aux(208)-1
s(1034) =< s(936)+aux(209)
s(1034) =< s(936)+aux(207)
s(1040) =< s(939)+aux(207)
s(1041) =< s(939)+aux(207)
s(1040) =< s(1034)*s(937)
s(1042) =< s(1034)*aux(208)
s(1041) =< s(1034)*s(1038)
s(1043) =< s(1040)
s(1044) =< s(1041)
s(1045) =< s(1042)*aux(210)
s(1046) =< s(916)*aux(208)
s(1047) =< s(1033)*aux(208)
s(1048) =< s(1047)
s(1020) =< s(915)*aux(210)
s(1062) =< s(929)*aux(210)
s(971) =< s(929)*aux(206)
s(972) =< s(971)
s(974) =< aux(207)
s(975) =< aux(207)
s(975) =< aux(206)
s(976) =< s(968)
s(976) =< aux(206)
s(977) =< aux(206)
s(978) =< aux(207)-1
s(979) =< s(976)*2
s(974) =< s(976)+aux(207)
s(980) =< s(979)+aux(207)
s(981) =< s(979)+aux(207)
s(980) =< s(974)*s(977)
s(982) =< s(974)*aux(207)
s(981) =< s(974)*s(978)
s(983) =< s(980)
s(984) =< s(981)
s(985) =< s(982)*aux(206)
s(986) =< s(929)*aux(207)

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

* Chain [35]: 8*s(1168)+3
  Such that:aux(212) =< V1
s(1168) =< aux(212)

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

* Chain [34]: 3*s(1172)+42*s(1173)+24*s(1175)+8
  Such that:s(1170) =< 1
aux(213) =< V1
s(1172) =< s(1170)
s(1173) =< aux(213)
s(1174) =< s(1173)*aux(213)
s(1175) =< s(1174)

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


Closed-form bounds of start(V1,V,V11): 
-------------------------------------
* Chain [39] with precondition: [V1>=0,V>=0] 
    - Upper bound: 96*V1+24+V*V1+145*V+18*V*V+(V1+V)*(15*V)+(V1+V)*((V1+V)*(15*V))+(302*V1+302*V)+(69*V1+69*V)*(V1+V)+(45*V1+45*V)+75*V 
    - Complexity: n^3 
* Chain [38] with precondition: [V1=1,V>=1,V11>=0] 
    - Upper bound: 519*V+301+100*V*V+10*V*V*V+6*V*V11+(V+V11)*(15*V)+(V+V11)*((V+V11)*(15*V))+462*V11+82*V11*V11+(394*V+394*V11)+(69*V+69*V11)*(V+V11)+155*V 
    - Complexity: n^3 
* Chain [37] with precondition: [V1=1,V>=2] 
    - Upper bound: 38*V+13+6*V*V 
    - Complexity: n^2 
* Chain [36] with precondition: [V1=2,V>=1,V11>=0] 
    - Upper bound: 458*V+301+82*V*V+5*V*V11+512*V11+100*V11*V11+10*V11*V11*V11+(V+V11)*(15*V11)+(V+V11)*((V+V11)*(15*V11))+(394*V+394*V11)+(69*V+69*V11)*(V+V11)+155*V11 
    - Complexity: n^3 
* Chain [35] with precondition: [V=0,V1>=1] 
    - Upper bound: 8*V1+3 
    - Complexity: n 
* Chain [34] with precondition: [V=1,V1>=1] 
    - Upper bound: 42*V1+11+24*V1*V1 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V11): max([34*V1+8+24*V1*V1+8*V1,107*V+11+12*V*V+max([96*V1+V*V1+(V1+V)*(15*V)+(V1+V)*((V1+V)*(15*V))+(302*V1+302*V)+(69*V1+69*V)*(V1+V)+(45*V1+45*V)+75*V,313*V+277+64*V*V+5*V*nat(V11)+nat(V11)*462+nat(V11)*82*nat(V11)+nat(V+V11)*394+nat(V+V11)*69*nat(V+V11)+max([nat(V11)*18*nat(V11)+nat(V11)*50+nat(V11)*10*nat(V11)*nat(V11)+nat(V11)*15*nat(V+V11)+nat(V11)*15*nat(V+V11)*nat(V+V11)+nat(V11/2)*310,18*V*V+61*V+10*V*V*V+nat(V11)*V+15*V*nat(V+V11)+15*V*nat(V+V11)*nat(V+V11)+155*V])])+(38*V+10+6*V*V)])+3 
Asymptotic class: n^3 
* Total analysis performed in 2031 ms.


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

(10)
BOUNDS(1, n^3)

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

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

(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^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))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

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

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

The rewrite sequence

le(s(x), s(y)) ->^+ le(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 [ ].




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

(14)
Complex Obligation (BEST)

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

(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^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))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

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

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

(17)
BOUNDS(n^1, INF)

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

(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^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))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

S is empty.
Rewrite Strategy: INNERMOST