/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 1 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 3063 ms]
    (10) BOUNDS(1, n^2)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   division(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(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^2).


The TRS R consists of the following rules:

   division(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(lt(x, y), x, y, inc(z)) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   inc(0) -> s(0) [1]
   inc(s(x)) -> s(inc(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:

   division(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(lt(x, y), x, y, inc(z)) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   inc(0) -> s(0) [1]
   inc(s(x)) -> s(inc(x)) [1]

The TRS has the following type information:
division :: 0:s -> 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s -> 0:s
0 :: 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
inc :: 0:s -> 0:s
true :: true:false
false :: true:false
s :: 0:s -> 0:s
minus :: 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(v0, v1, v2, v3) -> null_if [0]
   minus(v0, v1) -> null_minus [0]
   lt(v0, v1) -> null_lt [0]
   inc(v0) -> null_inc [0]

And the following fresh constants:    null_if, null_minus, null_lt, null_inc

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

(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:

   division(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(lt(x, y), x, y, inc(z)) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   inc(0) -> s(0) [1]
   inc(s(x)) -> s(inc(x)) [1]
   if(v0, v1, v2, v3) -> null_if [0]
   minus(v0, v1) -> null_minus [0]
   lt(v0, v1) -> null_lt [0]
   inc(v0) -> null_inc [0]

The TRS has the following type information:
division :: 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc
div :: 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc
0 :: 0:s:null_if:null_minus:null_inc
if :: true:false:null_lt -> 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc
lt :: 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc -> true:false:null_lt
inc :: 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc
true :: true:false:null_lt
false :: true:false:null_lt
s :: 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc
minus :: 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc -> 0:s:null_if:null_minus:null_inc
null_if :: 0:s:null_if:null_minus:null_inc
null_minus :: 0:s:null_if:null_minus:null_inc
null_lt :: true:false:null_lt
null_inc :: 0:s:null_if:null_minus:null_inc

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 => 0
   null_minus => 0
   null_lt => 0
   null_inc => 0

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

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

   div(z', z'', z1) -{ 1 }-> if(lt(x, y), x, y, inc(z)) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   division(z', z'') -{ 1 }-> div(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   if(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   if(z', z'', z1, z2) -{ 1 }-> div(minus(x, 1 + y), 1 + y, z) :|: z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z1 = 1 + y, z' = 1
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   inc(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   inc(z') -{ 1 }-> 1 + inc(x) :|: z' = 1 + x, x >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 1 }-> lt(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   lt(z', z'') -{ 1 }-> 2 :|: y >= 0, z'' = 1 + y, z' = 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   lt(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   minus(z', z'') -{ 1 }-> minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0

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, V7, V11),0,[division(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V7, V11),0,[div(V, V1, V7, Out)],[V >= 0,V1 >= 0,V7 >= 0]).
eq(start(V, V1, V7, V11),0,[if(V, V1, V7, V11, Out)],[V >= 0,V1 >= 0,V7 >= 0,V11 >= 0]).
eq(start(V, V1, V7, V11),0,[minus(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V7, V11),0,[lt(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V7, V11),0,[inc(V, Out)],[V >= 0]).
eq(division(V, V1, Out),1,[div(V3, V2, 0, Ret)],[Out = Ret,V = V3,V1 = V2,V3 >= 0,V2 >= 0]).
eq(div(V, V1, V7, Out),1,[lt(V4, V5, Ret0),inc(V6, Ret3),if(Ret0, V4, V5, Ret3, Ret1)],[Out = Ret1,V7 = V6,V6 >= 0,V = V4,V1 = V5,V4 >= 0,V5 >= 0]).
eq(if(V, V1, V7, V11, Out),1,[],[Out = V8,V7 = V9,V8 >= 0,V = 2,V11 = V8,V10 >= 0,V9 >= 0,V1 = V10]).
eq(if(V, V1, V7, V11, Out),1,[minus(V12, 1 + V13, Ret01),div(Ret01, 1 + V13, V14, Ret2)],[Out = Ret2,V14 >= 0,V11 = V14,V12 >= 0,V13 >= 0,V1 = V12,V7 = 1 + V13,V = 1]).
eq(minus(V, V1, Out),1,[],[Out = V15,V1 = 0,V = V15,V15 >= 0]).
eq(minus(V, V1, Out),1,[minus(V17, V16, Ret4)],[Out = Ret4,V = 1 + V17,V17 >= 0,V16 >= 0,V1 = 1 + V16]).
eq(lt(V, V1, Out),1,[],[Out = 1,V1 = 0,V = V18,V18 >= 0]).
eq(lt(V, V1, Out),1,[],[Out = 2,V19 >= 0,V1 = 1 + V19,V = 0]).
eq(lt(V, V1, Out),1,[lt(V20, V21, Ret5)],[Out = Ret5,V = 1 + V20,V20 >= 0,V21 >= 0,V1 = 1 + V21]).
eq(inc(V, Out),1,[],[Out = 1,V = 0]).
eq(inc(V, Out),1,[inc(V22, Ret11)],[Out = 1 + Ret11,V = 1 + V22,V22 >= 0]).
eq(if(V, V1, V7, V11, Out),0,[],[Out = 0,V11 = V25,V24 >= 0,V7 = V26,V23 >= 0,V1 = V23,V26 >= 0,V25 >= 0,V = V24]).
eq(minus(V, V1, Out),0,[],[Out = 0,V28 >= 0,V27 >= 0,V1 = V27,V = V28]).
eq(lt(V, V1, Out),0,[],[Out = 0,V30 >= 0,V29 >= 0,V1 = V29,V = V30]).
eq(inc(V, Out),0,[],[Out = 0,V31 >= 0,V = V31]).
input_output_vars(division(V,V1,Out),[V,V1],[Out]).
input_output_vars(div(V,V1,V7,Out),[V,V1,V7],[Out]).
input_output_vars(if(V,V1,V7,V11,Out),[V,V1,V7,V11],[Out]).
input_output_vars(minus(V,V1,Out),[V,V1],[Out]).
input_output_vars(lt(V,V1,Out),[V,V1],[Out]).
input_output_vars(inc(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [minus/3]
1. recursive  : [inc/2]
2. recursive  : [lt/3]
3. recursive  : [(div)/4,if/5]
4. non_recursive  : [division/3]
5. non_recursive  : [start/4]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into minus/3
1. SCC is partially evaluated into inc/2
2. SCC is partially evaluated into lt/3
3. SCC is partially evaluated into (div)/4
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into start/4

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

### Specialization of cost equations minus/3 
* CE 11 is refined into CE [22] 
* CE 9 is refined into CE [23] 
* CE 10 is refined into CE [24] 


### Cost equations --> "Loop" of minus/3 
* CEs [24] --> Loop 15 
* CEs [22] --> Loop 16 
* CEs [23] --> Loop 17 

### Ranking functions of CR minus(V,V1,Out) 
* RF of phase [15]: [V,V1]

#### Partial ranking functions of CR minus(V,V1,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V
    V1


### Specialization of cost equations inc/2 
* CE 21 is refined into CE [25] 
* CE 19 is refined into CE [26] 
* CE 20 is refined into CE [27] 


### Cost equations --> "Loop" of inc/2 
* CEs [27] --> Loop 18 
* CEs [25] --> Loop 19 
* CEs [26] --> Loop 20 

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

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


### Specialization of cost equations lt/3 
* CE 18 is refined into CE [28] 
* CE 15 is refined into CE [29] 
* CE 16 is refined into CE [30] 
* CE 17 is refined into CE [31] 


### Cost equations --> "Loop" of lt/3 
* CEs [31] --> Loop 21 
* CEs [28] --> Loop 22 
* CEs [29] --> Loop 23 
* CEs [30] --> Loop 24 

### Ranking functions of CR lt(V,V1,Out) 
* RF of phase [21]: [V,V1]

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


### Specialization of cost equations (div)/4 
* CE 12 is refined into CE [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51] 
* CE 14 is refined into CE [52,53,54,55,56,57,58,59] 
* CE 13 is refined into CE [60,61,62,63,64,65,66,67] 


### Cost equations --> "Loop" of (div)/4 
* CEs [67] --> Loop 25 
* CEs [65] --> Loop 26 
* CEs [63] --> Loop 27 
* CEs [66] --> Loop 28 
* CEs [64] --> Loop 29 
* CEs [62] --> Loop 30 
* CEs [61] --> Loop 31 
* CEs [60] --> Loop 32 
* CEs [59] --> Loop 33 
* CEs [58] --> Loop 34 
* CEs [56] --> Loop 35 
* CEs [40,44,48] --> Loop 36 
* CEs [36,37,38,39] --> Loop 37 
* CEs [55] --> Loop 38 
* CEs [54] --> Loop 39 
* CEs [52] --> Loop 40 
* CEs [32,33,34,35,41,42,43,45,46,47,49,50,51,53,57] --> Loop 41 

### Ranking functions of CR div(V,V1,V7,Out) 
* RF of phase [25,26,27,31]: [V,V-V1+1]

#### Partial ranking functions of CR div(V,V1,V7,Out) 
* Partial RF of phase [25,26,27,31]:
  - RF of loop [25:1,26:1,27:1,31:1]:
    V
    V-V1+1
  - RF of loop [31:1]:
    -V7+1 depends on loops [25:1,27:1] 


### Specialization of cost equations start/4 
* CE 3 is refined into CE [68] 
* CE 1 is refined into CE [69] 
* CE 2 is refined into CE [70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87] 
* CE 4 is refined into CE [88,89,90,91,92,93,94,95] 
* CE 5 is refined into CE [96,97,98,99,100,101,102,103,104,105,106,107,108] 
* CE 6 is refined into CE [109,110,111] 
* CE 7 is refined into CE [112,113,114,115,116] 
* CE 8 is refined into CE [117,118,119,120] 


### Cost equations --> "Loop" of start/4 
* CEs [100,101] --> Loop 42 
* CEs [109,113] --> Loop 43 
* CEs [68] --> Loop 44 
* CEs [76,77] --> Loop 45 
* CEs [70,71,72,73,74,75,78,79,80,81,82,83,84,85,86,87] --> Loop 46 
* CEs [69,88,89,90,91,92,93,94,95,96,97,98,99,102,103,104,105,106,107,108,110,111,112,114,115,116,117,118,119,120] --> Loop 47 

### Ranking functions of CR start(V,V1,V7,V11) 

#### Partial ranking functions of CR start(V,V1,V7,V11) 


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

#### Cost of chains of minus(V,V1,Out):
* Chain [[15],17]: 1*it(15)+1
  Such that:it(15) =< V1

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

* Chain [[15],16]: 1*it(15)+0
  Such that:it(15) =< V1

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

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

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


#### Cost of chains of inc(V,Out):
* Chain [[18],20]: 1*it(18)+1
  Such that:it(18) =< Out

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

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

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

* Chain [20]: 1
  with precondition: [V=0,Out=1] 

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


#### Cost of chains of lt(V,V1,Out):
* Chain [[21],24]: 1*it(21)+1
  Such that:it(21) =< V

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

* Chain [[21],23]: 1*it(21)+1
  Such that:it(21) =< V1

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

* Chain [[21],22]: 1*it(21)+0
  Such that:it(21) =< V1

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

* Chain [24]: 1
  with precondition: [V=0,Out=2,V1>=1] 

* Chain [23]: 1
  with precondition: [V1=0,Out=1,V>=0] 

* Chain [22]: 0
  with precondition: [Out=0,V>=0,V1>=0] 


#### Cost of chains of div(V,V1,V7,Out):
* Chain [[25,26,27,31],41]: 9*it(25)+4*it(27)+5*it(31)+10*s(3)+4*s(4)+6*s(5)+1*s(42)+6*s(44)+1*s(45)+3
  Such that:aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(3) =< V+V7
aux(2) =< V1
aux(13) =< -V7+1
aux(21) =< V
aux(22) =< V+V7+1
s(3) =< aux(22)
s(5) =< aux(2)
s(4) =< aux(3)
aux(14) =< aux(21)
it(25) =< aux(21)
it(27) =< aux(21)
it(31) =< aux(21)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
it(27) =< aux(22)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< aux(21)

  with precondition: [Out=0,V1>=1,V7>=0,V>=V1] 

* Chain [[25,26,27,31],40]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+4
  Such that:aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(19) =< V+V7
aux(18) =< V+V7+1
aux(13) =< -V7+1
aux(23) =< V
aux(14) =< aux(23)
it(25) =< aux(23)
it(27) =< aux(23)
it(31) =< aux(23)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(19)
s(43) =< aux(19)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< aux(23)
s(41) =< s(43)

  with precondition: [Out=1,V1>=1,V7>=0,V>=V1] 

* Chain [[25,26,27,31],39]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+1*s(51)+4
  Such that:aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(18) =< V+V7+1
aux(19) =< V+V7-Out+1
aux(13) =< -V7+1
s(51) =< Out
aux(24) =< V
aux(14) =< aux(24)
it(25) =< aux(24)
it(27) =< aux(24)
it(31) =< aux(24)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(19)
s(43) =< aux(19)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< aux(24)
s(41) =< s(43)

  with precondition: [V1>=1,V7>=0,Out>=2,V>=V1,V+V7+2>=Out+V1] 

* Chain [[25,26,27,31],38]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+1*s(52)+3
  Such that:aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(18) =< V+V7+1
aux(13) =< -V7+1
aux(25) =< V
aux(26) =< V+V7
s(52) =< aux(26)
aux(14) =< aux(25)
it(25) =< aux(25)
it(27) =< aux(25)
it(31) =< aux(25)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(26)
s(43) =< aux(26)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< aux(25)
s(41) =< s(43)

  with precondition: [V1>=1,V7>=0,Out>=1,V>=V1,V+V7+1>=Out+V1] 

* Chain [[25,26,27,31],36]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+7*s(44)+1*s(45)+2*s(53)+3
  Such that:aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(19) =< V+V7
aux(18) =< V+V7+1
aux(27) =< V1
aux(13) =< -V7+1
aux(28) =< V
s(44) =< aux(28)
s(53) =< aux(27)
aux(14) =< aux(28)
it(25) =< aux(28)
it(27) =< aux(28)
it(31) =< aux(28)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(19)
s(43) =< aux(19)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(41) =< s(43)

  with precondition: [Out=0,V1>=1,V7>=0,V>=V1] 

* Chain [[25,26,27,31],35]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+7*s(44)+1*s(45)+4
  Such that:aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(19) =< V+V7
aux(18) =< V+V7+1
aux(13) =< -V7+1
aux(29) =< V
s(44) =< aux(29)
aux(14) =< aux(29)
it(25) =< aux(29)
it(27) =< aux(29)
it(31) =< aux(29)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(19)
s(43) =< aux(19)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(41) =< s(43)

  with precondition: [Out=1,V1>=2,V7>=0,V>=V1+1] 

* Chain [[25,26,27,31],34]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+7*s(44)+1*s(45)+1*s(58)+4
  Such that:aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(18) =< V+V7+1
aux(19) =< V+V7-Out+1
aux(13) =< -V7+1
s(58) =< Out
aux(30) =< V
s(44) =< aux(30)
aux(14) =< aux(30)
it(25) =< aux(30)
it(27) =< aux(30)
it(31) =< aux(30)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(19)
s(43) =< aux(19)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(41) =< s(43)

  with precondition: [V1>=2,V7>=0,Out>=2,V>=V1+1,V+2*V7+3>=2*Out+V1] 

* Chain [[25,26,27,31],33]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+2*s(59)+3
  Such that:aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(18) =< V+V7+1
aux(13) =< -V7+1
aux(31) =< V
aux(32) =< V+V7
s(59) =< aux(32)
aux(14) =< aux(31)
it(25) =< aux(31)
it(27) =< aux(31)
it(31) =< aux(31)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(32)
s(43) =< aux(32)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< aux(31)
s(41) =< s(43)

  with precondition: [V1>=2,V7>=0,Out>=1,V>=V1+1,V+2*V7+1>=2*Out+V1] 

* Chain [[25,26,27,31],32,41]: 9*it(25)+4*it(27)+5*it(31)+4*s(3)+4*s(4)+8*s(5)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+7
  Such that:aux(3) =< 1
aux(4) =< 2
aux(15) =< V
aux(16) =< V-V1+1
aux(18) =< V+V7+1
aux(34) =< V1
aux(13) =< -V7+1
aux(35) =< V-V1
aux(36) =< V-V1+V7
s(5) =< aux(34)
s(4) =< aux(3)
s(3) =< aux(4)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(35)
it(25) =< aux(35)
it(27) =< aux(35)
it(31) =< aux(35)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(36)
s(43) =< aux(36)
s(46) =< aux(35)
aux(10) =< aux(36)+1
s(42) =< it(25)*aux(36)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [Out=0,V1>=1,V7>=0,V>=2*V1] 

* Chain [[25,26,27,31],32,39]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+1*s(51)+2*s(61)+8
  Such that:s(51) =< 2
aux(15) =< V
aux(16) =< V-V1+1
aux(18) =< V+V7+1
aux(33) =< V1
aux(13) =< -V7+1
aux(37) =< V-V1
aux(38) =< V-V1+V7
s(61) =< aux(33)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(37)
it(25) =< aux(37)
it(27) =< aux(37)
it(31) =< aux(37)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(38)
s(43) =< aux(38)
s(46) =< aux(37)
aux(10) =< aux(38)+1
s(42) =< it(25)*aux(38)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [Out=2,V1>=1,V7>=0,V>=2*V1] 

* Chain [[25,26,27,31],32,38]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+1*s(52)+2*s(61)+7
  Such that:s(52) =< 1
aux(15) =< V
aux(16) =< V-V1+1
aux(18) =< V+V7+1
aux(33) =< V1
aux(13) =< -V7+1
aux(39) =< V-V1
aux(40) =< V-V1+V7
s(61) =< aux(33)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(39)
it(25) =< aux(39)
it(27) =< aux(39)
it(31) =< aux(39)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(40)
s(43) =< aux(40)
s(46) =< aux(39)
aux(10) =< aux(40)+1
s(42) =< it(25)*aux(40)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [Out=1,V1>=1,V7>=0,V>=2*V1] 

* Chain [[25,26,27,31],30,41]: 9*it(25)+4*it(27)+5*it(31)+4*s(3)+8*s(5)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+6
  Such that:aux(4) =< 1
aux(15) =< V
aux(16) =< V-V1+1
aux(18) =< V+V7+1
aux(43) =< V1
aux(13) =< -V7+1
aux(44) =< V-V1
aux(45) =< V-V1+V7
s(5) =< aux(43)
s(3) =< aux(4)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(44)
it(25) =< aux(44)
it(27) =< aux(44)
it(31) =< aux(44)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(45)
s(43) =< aux(45)
s(46) =< aux(44)
aux(10) =< aux(45)+1
s(42) =< it(25)*aux(45)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [Out=0,V1>=1,V7>=0,V>=2*V1] 

* Chain [[25,26,27,31],30,40]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+2*s(63)+7
  Such that:aux(15) =< V
aux(16) =< V-V1+1
aux(18) =< V+V7+1
aux(41) =< V1
aux(13) =< -V7+1
aux(46) =< V-V1
aux(47) =< V-V1+V7
s(63) =< aux(41)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(46)
it(25) =< aux(46)
it(27) =< aux(46)
it(31) =< aux(46)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(47)
s(43) =< aux(47)
s(46) =< aux(46)
aux(10) =< aux(47)+1
s(42) =< it(25)*aux(47)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [Out=1,V1>=1,V7>=0,V>=2*V1] 

* Chain [[25,26,27,31],30,36]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+4*s(53)+6
  Such that:aux(15) =< V
aux(16) =< V-V1+1
aux(18) =< V+V7+1
aux(48) =< V1
aux(13) =< -V7+1
aux(49) =< V-V1
aux(50) =< V-V1+V7
s(53) =< aux(48)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(49)
it(25) =< aux(49)
it(27) =< aux(49)
it(31) =< aux(49)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(50)
s(43) =< aux(50)
s(46) =< aux(49)
aux(10) =< aux(50)+1
s(42) =< it(25)*aux(50)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [Out=0,V1>=1,V7>=0,V>=2*V1] 

* Chain [[25,26,27,31],29,41]: 9*it(25)+4*it(27)+5*it(31)+4*s(3)+5*s(4)+8*s(5)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+7
  Such that:aux(15) =< V
aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(4) =< V-V1+V7+2
aux(18) =< V+V7+1
aux(52) =< V1
aux(13) =< -V7+1
aux(54) =< V-V1
aux(55) =< V-V1+V7+1
s(4) =< aux(55)
s(5) =< aux(52)
s(3) =< aux(4)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(54)
it(25) =< aux(54)
it(27) =< aux(54)
it(31) =< aux(54)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(55)
s(43) =< aux(55)
s(46) =< aux(54)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [Out=0,V1>=1,V7>=0,V>=2*V1] 

* Chain [[25,26,27,31],29,39]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+2*s(51)+2*s(65)+8
  Such that:aux(15) =< V
aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(19) =< V-V1+V7-Out+2
aux(18) =< V+V7+1
aux(51) =< V1
aux(13) =< -V7+1
aux(56) =< Out
aux(57) =< V-V1
s(51) =< aux(56)
s(65) =< aux(51)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(57)
it(25) =< aux(57)
it(27) =< aux(57)
it(31) =< aux(57)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(19)
s(43) =< aux(19)
s(46) =< aux(57)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [V1>=1,V7>=0,Out>=3,V>=2*V1,V+V7+3>=2*V1+Out] 

* Chain [[25,26,27,31],29,38]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+2*s(52)+2*s(65)+7
  Such that:aux(15) =< V
aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(18) =< V+V7+1
aux(51) =< V1
aux(13) =< -V7+1
aux(59) =< V-V1
aux(60) =< V-V1+V7+1
s(52) =< aux(60)
s(65) =< aux(51)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(59)
it(25) =< aux(59)
it(27) =< aux(59)
it(31) =< aux(59)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(60)
s(43) =< aux(60)
s(46) =< aux(59)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [V1>=1,V7>=0,Out>=1,V>=2*V1,V+V7+2>=2*V1+Out] 

* Chain [[25,26,27,31],28,41]: 9*it(25)+4*it(27)+5*it(31)+4*s(3)+5*s(4)+8*s(5)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+6
  Such that:aux(15) =< V
aux(16) =< V-V1+1
aux(4) =< V-V1+V7+1
aux(18) =< V+V7+1
aux(62) =< V1
aux(13) =< -V7+1
aux(64) =< V-V1
aux(65) =< V-V1+V7
s(4) =< aux(65)
s(5) =< aux(62)
s(3) =< aux(4)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(64)
it(25) =< aux(64)
it(27) =< aux(64)
it(31) =< aux(64)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(65)
s(43) =< aux(65)
s(46) =< aux(64)
aux(10) =< aux(65)+1
s(42) =< it(25)*aux(65)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [Out=0,V1>=1,V7>=0,V>=2*V1] 

* Chain [[25,26,27,31],28,39]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+2*s(51)+2*s(68)+7
  Such that:aux(15) =< V
aux(16) =< V-V1+1
aux(9) =< V-V1+V7
aux(19) =< V-V1+V7-Out+1
aux(18) =< V+V7+1
aux(61) =< V1
aux(13) =< -V7+1
aux(66) =< Out
aux(67) =< V-V1
s(51) =< aux(66)
s(68) =< aux(61)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(67)
it(25) =< aux(67)
it(27) =< aux(67)
it(31) =< aux(67)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(19)
s(43) =< aux(19)
s(46) =< aux(67)
aux(10) =< aux(9)+1
s(42) =< it(25)*aux(9)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [V1>=1,V7>=0,Out>=2,V>=2*V1,V+V7+2>=2*V1+Out] 

* Chain [[25,26,27,31],28,38]: 9*it(25)+4*it(27)+5*it(31)+2*s(41)+1*s(42)+6*s(44)+1*s(45)+2*s(52)+2*s(68)+6
  Such that:aux(15) =< V
aux(16) =< V-V1+1
aux(18) =< V+V7+1
aux(61) =< V1
aux(13) =< -V7+1
aux(69) =< V-V1
aux(70) =< V-V1+V7
s(52) =< aux(70)
s(68) =< aux(61)
aux(14) =< aux(15)
it(25) =< aux(15)
it(27) =< aux(15)
it(31) =< aux(15)
s(46) =< aux(15)
aux(14) =< aux(16)
it(25) =< aux(16)
it(27) =< aux(16)
it(31) =< aux(16)
aux(14) =< aux(69)
it(25) =< aux(69)
it(27) =< aux(69)
it(31) =< aux(69)
it(27) =< aux(18)
s(43) =< aux(18)
it(27) =< aux(70)
s(43) =< aux(70)
s(46) =< aux(69)
aux(10) =< aux(70)+1
s(42) =< it(25)*aux(70)
it(31) =< aux(14)+aux(13)
s(45) =< it(25)*aux(10)
s(44) =< s(46)
s(41) =< s(43)

  with precondition: [V1>=1,V7>=0,Out>=1,V>=2*V1,V+V7+1>=2*V1+Out] 

* Chain [41]: 4*s(3)+4*s(4)+6*s(5)+4*s(15)+3
  Such that:aux(1) =< V
aux(2) =< V1
aux(3) =< V7
aux(4) =< V7+1
s(15) =< aux(1)
s(5) =< aux(2)
s(4) =< aux(3)
s(3) =< aux(4)

  with precondition: [Out=0,V>=0,V1>=0,V7>=0] 

* Chain [40]: 4
  with precondition: [V=0,V7=0,Out=1,V1>=1] 

* Chain [39]: 1*s(51)+4
  Such that:s(51) =< V7+1

  with precondition: [V=0,V7+1=Out,V1>=1,V7>=1] 

* Chain [38]: 1*s(52)+3
  Such that:s(52) =< V7

  with precondition: [V=0,V1>=1,Out>=1,V7>=Out] 

* Chain [37]: 1*s(71)+1*s(72)+3
  Such that:s(72) =< V7
s(71) =< V7+1

  with precondition: [V1=0,Out=0,V>=0,V7>=0] 

* Chain [36]: 2*s(53)+1*s(55)+3
  Such that:s(55) =< V
aux(27) =< V1
s(53) =< aux(27)

  with precondition: [V7=0,Out=0,V>=0,V1>=0] 

* Chain [35]: 1*s(56)+4
  Such that:s(56) =< V

  with precondition: [V7=0,Out=1,V>=1,V1>=V+1] 

* Chain [34]: 1*s(57)+1*s(58)+4
  Such that:s(57) =< V
s(58) =< V7+1

  with precondition: [V7+1=Out,V>=1,V7>=1,V1>=V+1] 

* Chain [33]: 1*s(59)+1*s(60)+3
  Such that:s(59) =< V
s(60) =< V7

  with precondition: [V>=1,Out>=1,V1>=V+1,V7>=Out] 

* Chain [32,41]: 4*s(3)+4*s(4)+8*s(5)+7
  Such that:aux(3) =< 1
aux(4) =< 2
aux(34) =< V1
s(5) =< aux(34)
s(4) =< aux(3)
s(3) =< aux(4)

  with precondition: [V7=0,Out=0,V1>=1,V>=V1] 

* Chain [32,39]: 1*s(51)+2*s(61)+8
  Such that:s(51) =< 2
aux(33) =< V1
s(61) =< aux(33)

  with precondition: [V7=0,Out=2,V1>=1,V>=V1] 

* Chain [32,38]: 1*s(52)+2*s(61)+7
  Such that:s(52) =< 1
aux(33) =< V1
s(61) =< aux(33)

  with precondition: [V7=0,Out=1,V1>=1,V>=V1] 

* Chain [30,41]: 4*s(3)+8*s(5)+6
  Such that:aux(4) =< 1
aux(43) =< V1
s(5) =< aux(43)
s(3) =< aux(4)

  with precondition: [Out=0,V1>=1,V7>=0,V>=V1] 

* Chain [30,40]: 2*s(63)+7
  Such that:aux(41) =< V1
s(63) =< aux(41)

  with precondition: [Out=1,V1>=1,V7>=0,V>=V1] 

* Chain [30,36]: 4*s(53)+6
  Such that:aux(48) =< V1
s(53) =< aux(48)

  with precondition: [Out=0,V1>=1,V7>=0,V>=V1] 

* Chain [29,41]: 4*s(3)+5*s(4)+8*s(5)+7
  Such that:aux(4) =< V7+2
aux(52) =< V1
aux(53) =< V7+1
s(4) =< aux(53)
s(5) =< aux(52)
s(3) =< aux(4)

  with precondition: [Out=0,V1>=1,V7>=1,V>=V1] 

* Chain [29,39]: 2*s(51)+2*s(65)+8
  Such that:aux(51) =< V1
aux(56) =< Out
s(51) =< aux(56)
s(65) =< aux(51)

  with precondition: [V7+2=Out,V1>=1,V7>=1,V>=V1] 

* Chain [29,38]: 2*s(52)+2*s(65)+7
  Such that:aux(51) =< V1
aux(58) =< V7+1
s(52) =< aux(58)
s(65) =< aux(51)

  with precondition: [V1>=1,V7>=1,Out>=1,V>=V1,V7+1>=Out] 

* Chain [28,41]: 4*s(3)+5*s(4)+8*s(5)+6
  Such that:aux(4) =< V7+1
aux(62) =< V1
aux(63) =< V7
s(4) =< aux(63)
s(5) =< aux(62)
s(3) =< aux(4)

  with precondition: [Out=0,V1>=1,V7>=1,V>=V1] 

* Chain [28,39]: 2*s(51)+2*s(68)+7
  Such that:aux(61) =< V1
aux(66) =< Out
s(51) =< aux(66)
s(68) =< aux(61)

  with precondition: [V1>=1,Out>=2,V>=V1,V7+1>=Out] 

* Chain [28,38]: 2*s(52)+2*s(68)+6
  Such that:aux(61) =< V1
aux(68) =< V7
s(52) =< aux(68)
s(68) =< aux(61)

  with precondition: [V1>=1,Out>=1,V>=V1,V7>=Out] 


#### Cost of chains of start(V,V1,V7,V11):
* Chain [47]: 48*s(514)+224*s(515)+216*s(518)+44*s(519)+60*s(520)+12*s(524)+12*s(525)+144*s(526)+16*s(527)+20*s(528)+123*s(529)+7*s(530)+11*s(531)+7*s(532)+6*s(535)+144*s(537)+32*s(538)+40*s(539)+8*s(541)+8*s(542)+10*s(543)+20*s(544)+4*s(613)+2*s(621)+18*s(677)+14*s(678)+32*s(700)+60*s(701)+12*s(705)+12*s(706)+16*s(708)+7*s(711)+11*s(712)+4*s(713)+12*s(714)+6*s(716)+20*s(719)+40*s(720)+8*s(722)+8*s(723)+10*s(724)+22*s(725)+7*s(726)+12*s(727)+6*s(728)+4*s(786)+2*s(794)+9
  Such that:aux(125) =< 1
aux(126) =< 2
aux(127) =< V
aux(128) =< V+1
aux(129) =< V-V1
aux(130) =< V-V1+1
aux(131) =< V-V1+2
aux(132) =< V-V1+V7
aux(133) =< V-V1+V7+1
aux(134) =< V-V1+V7+2
aux(135) =< V+V7
aux(136) =< V+V7+1
aux(137) =< V1
aux(138) =< -V7+1
aux(139) =< V7
aux(140) =< V7+1
aux(141) =< V7+2
s(514) =< aux(125)
s(528) =< aux(126)
s(529) =< aux(127)
s(544) =< aux(128)
s(515) =< aux(137)
s(678) =< aux(139)
s(677) =< aux(140)
s(532) =< aux(131)
s(517) =< aux(127)
s(518) =< aux(127)
s(613) =< aux(127)
s(520) =< aux(127)
s(521) =< aux(127)
s(517) =< aux(130)
s(518) =< aux(130)
s(613) =< aux(130)
s(520) =< aux(130)
s(517) =< aux(129)
s(518) =< aux(129)
s(613) =< aux(129)
s(520) =< aux(129)
s(613) =< aux(128)
s(616) =< aux(128)
s(613) =< aux(131)
s(616) =< aux(131)
s(521) =< aux(129)
s(523) =< aux(129)+1
s(524) =< s(518)*aux(129)
s(520) =< s(517)+aux(125)
s(525) =< s(518)*s(523)
s(526) =< s(521)
s(621) =< s(616)
s(536) =< aux(127)
s(537) =< aux(127)
s(538) =< aux(127)
s(539) =< aux(127)
s(536) =< aux(130)
s(537) =< aux(130)
s(538) =< aux(130)
s(539) =< aux(130)
s(538) =< aux(128)
s(541) =< s(537)*aux(129)
s(539) =< s(536)+aux(125)
s(542) =< s(537)*s(523)
s(519) =< aux(127)
s(519) =< aux(130)
s(519) =< aux(129)
s(519) =< aux(128)
s(522) =< aux(128)
s(522) =< aux(129)
s(527) =< s(522)
s(700) =< aux(127)
s(701) =< aux(127)
s(700) =< aux(130)
s(701) =< aux(130)
s(700) =< aux(129)
s(701) =< aux(129)
s(700) =< aux(136)
s(703) =< aux(136)
s(700) =< aux(132)
s(703) =< aux(132)
s(704) =< aux(132)+1
s(705) =< s(518)*aux(132)
s(701) =< s(517)+aux(138)
s(706) =< s(518)*s(704)
s(708) =< s(703)
s(711) =< aux(132)
s(712) =< aux(133)
s(713) =< aux(134)
s(714) =< aux(127)
s(714) =< aux(130)
s(714) =< aux(129)
s(714) =< aux(136)
s(715) =< aux(136)
s(714) =< aux(133)
s(715) =< aux(133)
s(716) =< s(715)
s(719) =< aux(127)
s(720) =< aux(127)
s(719) =< aux(130)
s(720) =< aux(130)
s(719) =< aux(136)
s(721) =< aux(136)
s(719) =< aux(135)
s(721) =< aux(135)
s(722) =< s(537)*aux(132)
s(720) =< s(536)+aux(138)
s(723) =< s(537)*s(704)
s(724) =< s(721)
s(725) =< aux(136)
s(726) =< aux(135)
s(727) =< aux(127)
s(727) =< aux(130)
s(727) =< aux(136)
s(728) =< aux(141)
s(786) =< aux(127)
s(786) =< aux(130)
s(786) =< aux(129)
s(786) =< aux(136)
s(789) =< aux(136)
s(786) =< aux(134)
s(789) =< aux(134)
s(794) =< s(789)
s(530) =< aux(129)
s(531) =< aux(130)
s(534) =< aux(128)
s(534) =< aux(130)
s(535) =< s(534)
s(540) =< aux(128)
s(540) =< aux(127)
s(543) =< s(540)

  with precondition: [V>=0] 

* Chain [46]: 219*s(865)+42*s(867)+28*s(869)+34*s(889)+8*s(900)+18*s(901)+5*s(903)+9*s(904)+4*s(905)+2*s(908)+2*s(916)+10*s(920)+1*s(929)+108*s(951)+32*s(952)+60*s(953)+12*s(957)+12*s(958)+72*s(959)+16*s(960)+59*s(962)+7*s(963)+11*s(964)+4*s(965)+12*s(966)+6*s(968)+72*s(970)+20*s(971)+40*s(972)+8*s(974)+8*s(975)+10*s(976)+22*s(977)+7*s(978)+12*s(979)+4*s(1050)+2*s(1058)+10
  Such that:s(929) =< V1
s(878) =< -V7+V11
s(879) =< -V7+V11+1
s(872) =< -V7+V11+2
aux(155) =< 1
aux(156) =< 2
aux(157) =< V1-2*V7
aux(158) =< V1-2*V7+1
aux(159) =< V1-2*V7+V11
aux(160) =< V1-2*V7+V11+1
aux(161) =< V1-2*V7+V11+2
aux(162) =< V1-V7
aux(163) =< V1-V7+V11
aux(164) =< V1-V7+V11+1
aux(165) =< V7
aux(166) =< -V11+1
aux(167) =< V11
aux(168) =< V11+1
aux(169) =< V11+2
s(889) =< aux(155)
s(901) =< aux(156)
s(962) =< aux(162)
s(865) =< aux(165)
s(869) =< aux(167)
s(867) =< aux(168)
s(895) =< aux(168)
s(895) =< s(878)
s(900) =< s(895)
s(903) =< s(878)
s(904) =< s(879)
s(905) =< s(872)
s(907) =< aux(168)
s(907) =< s(879)
s(908) =< s(907)
s(913) =< aux(168)
s(913) =< aux(167)
s(916) =< s(913)
s(920) =< aux(169)
s(950) =< aux(162)
s(951) =< aux(162)
s(952) =< aux(162)
s(953) =< aux(162)
s(954) =< aux(162)
s(950) =< aux(158)
s(951) =< aux(158)
s(952) =< aux(158)
s(953) =< aux(158)
s(950) =< aux(157)
s(951) =< aux(157)
s(952) =< aux(157)
s(953) =< aux(157)
s(952) =< aux(164)
s(955) =< aux(164)
s(952) =< aux(159)
s(955) =< aux(159)
s(954) =< aux(157)
s(956) =< aux(159)+1
s(957) =< s(951)*aux(159)
s(953) =< s(950)+aux(166)
s(958) =< s(951)*s(956)
s(959) =< s(954)
s(960) =< s(955)
s(963) =< aux(159)
s(964) =< aux(160)
s(965) =< aux(161)
s(966) =< aux(162)
s(966) =< aux(158)
s(966) =< aux(157)
s(966) =< aux(164)
s(967) =< aux(164)
s(966) =< aux(160)
s(967) =< aux(160)
s(968) =< s(967)
s(969) =< aux(162)
s(970) =< aux(162)
s(971) =< aux(162)
s(972) =< aux(162)
s(969) =< aux(158)
s(970) =< aux(158)
s(971) =< aux(158)
s(972) =< aux(158)
s(971) =< aux(164)
s(973) =< aux(164)
s(971) =< aux(163)
s(973) =< aux(163)
s(974) =< s(970)*aux(159)
s(972) =< s(969)+aux(166)
s(975) =< s(970)*s(956)
s(976) =< s(973)
s(977) =< aux(164)
s(978) =< aux(163)
s(979) =< aux(162)
s(979) =< aux(158)
s(979) =< aux(164)
s(1050) =< aux(162)
s(1050) =< aux(158)
s(1050) =< aux(157)
s(1050) =< aux(164)
s(1053) =< aux(164)
s(1050) =< aux(161)
s(1053) =< aux(161)
s(1058) =< s(1053)

  with precondition: [V=1,V1>=0,V7>=1,V11>=0] 

* Chain [45]: 2*s(1127)+1*s(1128)+1*s(1130)+6
  Such that:s(1130) =< V11
s(1128) =< V11+1
aux(170) =< V1
s(1127) =< aux(170)

  with precondition: [V=1,V1=V7,V1>=1,V11>=1] 

* Chain [44]: 1
  with precondition: [V=2,V1>=0,V7>=0,V11>=0] 

* Chain [43]: 1
  with precondition: [V1=0,V>=0] 

* Chain [42]: 2*s(1133)+10*s(1135)+4*s(1136)+4*s(1137)+7
  Such that:s(1131) =< 1
s(1132) =< 2
s(1134) =< V1
aux(171) =< V
s(1133) =< aux(171)
s(1135) =< s(1134)
s(1136) =< s(1131)
s(1137) =< s(1132)

  with precondition: [V7=0,V>=0,V1>=0] 


Closed-form bounds of start(V,V1,V7,V11): 
-------------------------------------
* Chain [47] with precondition: [V>=0] 
    - Upper bound: 1027*V+97+40*V*nat(V-V1+V7)+40*V*nat(V-V1)+nat(V1)*224+nat(V7)*14+nat(V+V7)*7+(54*V+54)+nat(V7+1)*18+nat(V7+2)*6+nat(V+V7+1)*56+nat(V-V1+V7+1)*11+nat(V-V1+V7+2)*4+nat(V-V1+V7)*7+nat(V-V1+1)*11+nat(V-V1+2)*7+nat(V-V1)*7 
    - Complexity: n^2 
* Chain [46] with precondition: [V=1,V1>=0,V7>=1,V11>=0] 
    - Upper bound: V1+219*V7+92*V11+154+nat(-V7+V11)*5+nat(-V7+V11+1)*9+nat(-V7+V11+2)*4+nat(V1-V7+V11+1)*56+nat(V1-2*V7+V11+1)*11+nat(V1-2*V7+V11+2)*4+nat(V1-V7+V11)*7+nat(V1-2*V7+V11)*7+nat(V1-2*V7+V11)*40*nat(V1-V7)+nat(V1-V7)*511 
    - Complexity: n^2 
* Chain [45] with precondition: [V=1,V1=V7,V1>=1,V11>=1] 
    - Upper bound: 2*V1+2*V11+7 
    - Complexity: n 
* Chain [44] with precondition: [V=2,V1>=0,V7>=0,V11>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [43] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [42] with precondition: [V7=0,V>=0,V1>=0] 
    - Upper bound: 2*V+10*V1+19 
    - Complexity: n 

### Maximum cost of start(V,V1,V7,V11): nat(V1)+5+max([nat(V1)+max([nat(V11+1)+nat(V11),1025*V+78+40*V*nat(V-V1+V7)+40*V*nat(V-V1)+nat(V1)*214+nat(V7)*14+nat(V+V7)*7+(54*V+54)+nat(V7+1)*18+nat(V7+2)*6+nat(V+V7+1)*56+nat(V-V1+V7+1)*11+nat(V-V1+V7+2)*4+nat(V-V1+V7)*7+nat(V-V1+1)*11+nat(V-V1+2)*7+nat(V-V1)*7+(2*V+13+nat(V1)*8)]),nat(V7)*219+74+nat(V11)*28+nat(V11+1)*54+nat(V11+2)*10+nat(-V7+V11)*5+nat(-V7+V11+1)*9+nat(-V7+V11+2)*4+nat(V1-V7+V11+1)*56+nat(V1-2*V7+V11+1)*11+nat(V1-2*V7+V11+2)*4+nat(V1-V7+V11)*7+nat(V1-2*V7+V11)*7+nat(V1-2*V7+V11)*40*nat(V1-V7)+nat(V1-V7)*511])+1 
Asymptotic class: n^2 
* Total analysis performed in 2747 ms.


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

(10)
BOUNDS(1, n^2)

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

(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^2).


The TRS R consists of the following rules:

   division(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(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

minus(s(x), s(y)) ->^+ minus(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^2).


The TRS R consists of the following rules:

   division(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(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^2).


The TRS R consists of the following rules:

   division(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))

S is empty.
Rewrite Strategy: INNERMOST