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


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 1991 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 283 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
f(true, [], y, z)
f(true, x, [], z)
f(true, x, y, [])


The defined contexts are:
f([], x1, s(x2), x3)
and([], x1)
and(x0, [])
f([], x1, x2, s(x3))


[] just represents basic- or constructor-terms in the following defined contexts:
and([], x1)
and(x0, [])


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) [1]
   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]
   and(x, true) -> x [1]
   and(x, false) -> false [1]

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

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

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

   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) [1]
   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]
   and(x, true) -> x [1]
   and(x, false) -> false [1]

The TRS has the following type information:
f :: true:false -> s:0 -> s:0 -> s:0 -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: s:0 -> s:0 -> true:false
s :: s:0 -> s:0
0 :: s:0
false :: true:false

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

(7) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   f(v0, v1, v2, v3) -> null_f [0]

And the following fresh constants:    null_f

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

(8)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) [1]
   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) [1]
   gt(0, v) -> false [1]
   gt(s(u), 0) -> true [1]
   gt(s(u), s(v)) -> gt(u, v) [1]
   and(x, true) -> x [1]
   and(x, false) -> false [1]
   f(v0, v1, v2, v3) -> null_f [0]

The TRS has the following type information:
f :: true:false -> s:0 -> s:0 -> s:0 -> null_f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: s:0 -> s:0 -> true:false
s :: s:0 -> s:0
0 :: s:0
false :: true:false
null_f :: null_f

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

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

   true => 1
   0 => 0
   false => 0
   null_f => 0

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

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

   and(z', z'') -{ 1 }-> x :|: z' = x, x >= 0, z'' = 1
   and(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
   f(z', z'', z1, z2) -{ 1 }-> f(and(gt(x, y), gt(x, z)), x, y, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   f(z', z'', z1, z2) -{ 1 }-> f(and(gt(x, y), gt(x, z)), x, 1 + y, z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   f(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   gt(z', z'') -{ 1 }-> gt(u, v) :|: v >= 0, z' = 1 + u, z'' = 1 + v, u >= 0
   gt(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + u, u >= 0
   gt(z', z'') -{ 1 }-> 0 :|: z'' = v, v >= 0, z' = 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V1, V6, V2),0,[f(V, V1, V6, V2, Out)],[V >= 0,V1 >= 0,V6 >= 0,V2 >= 0]).
eq(start(V, V1, V6, V2),0,[gt(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V6, V2),0,[and(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(f(V, V1, V6, V2, Out),1,[gt(V4, V3, Ret00),gt(V4, V5, Ret01),and(Ret00, Ret01, Ret0),f(Ret0, V4, 1 + V3, V5, Ret)],[Out = Ret,V6 = V3,V5 >= 0,V2 = V5,V4 >= 0,V3 >= 0,V1 = V4,V = 1]).
eq(f(V, V1, V6, V2, Out),1,[gt(V7, V9, Ret001),gt(V7, V8, Ret011),and(Ret001, Ret011, Ret02),f(Ret02, V7, V9, 1 + V8, Ret1)],[Out = Ret1,V6 = V9,V8 >= 0,V2 = V8,V7 >= 0,V9 >= 0,V1 = V7,V = 1]).
eq(gt(V, V1, Out),1,[],[Out = 0,V1 = V10,V10 >= 0,V = 0]).
eq(gt(V, V1, Out),1,[],[Out = 1,V1 = 0,V = 1 + V11,V11 >= 0]).
eq(gt(V, V1, Out),1,[gt(V12, V13, Ret2)],[Out = Ret2,V13 >= 0,V = 1 + V12,V1 = 1 + V13,V12 >= 0]).
eq(and(V, V1, Out),1,[],[Out = V14,V = V14,V14 >= 0,V1 = 1]).
eq(and(V, V1, Out),1,[],[Out = 0,V1 = 0,V = V15,V15 >= 0]).
eq(f(V, V1, V6, V2, Out),0,[],[Out = 0,V2 = V18,V17 >= 0,V6 = V19,V16 >= 0,V1 = V16,V19 >= 0,V18 >= 0,V = V17]).
input_output_vars(f(V,V1,V6,V2,Out),[V,V1,V6,V2],[Out]).
input_output_vars(gt(V,V1,Out),[V,V1],[Out]).
input_output_vars(and(V,V1,Out),[V,V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [and/3]
1. recursive  : [gt/3]
2. recursive  : [f/5]
3. non_recursive  : [start/4]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into and/3
1. SCC is partially evaluated into gt/3
2. SCC is partially evaluated into f/5
3. SCC is partially evaluated into start/4

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

### Specialization of cost equations and/3 
* CE 10 is refined into CE [12] 
* CE 11 is refined into CE [13] 


### Cost equations --> "Loop" of and/3 
* CEs [12] --> Loop 10 
* CEs [13] --> Loop 11 

### Ranking functions of CR and(V,V1,Out) 

#### Partial ranking functions of CR and(V,V1,Out) 


### Specialization of cost equations gt/3 
* CE 9 is refined into CE [14] 
* CE 8 is refined into CE [15] 
* CE 7 is refined into CE [16] 


### Cost equations --> "Loop" of gt/3 
* CEs [15] --> Loop 12 
* CEs [16] --> Loop 13 
* CEs [14] --> Loop 14 

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

#### Partial ranking functions of CR gt(V,V1,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V
    V1


### Specialization of cost equations f/5 
* CE 6 is refined into CE [17] 
* CE 5 is refined into CE [18,19,20,21,22,23,24,25,26,27] 
* CE 4 is refined into CE [28,29,30,31,32,33,34,35,36,37] 


### Cost equations --> "Loop" of f/5 
* CEs [27] --> Loop 15 
* CEs [37] --> Loop 16 
* CEs [24] --> Loop 17 
* CEs [26] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [34] --> Loop 20 
* CEs [36] --> Loop 21 
* CEs [33] --> Loop 22 
* CEs [25] --> Loop 23 
* CEs [35] --> Loop 24 
* CEs [22] --> Loop 25 
* CEs [32] --> Loop 26 
* CEs [31] --> Loop 27 
* CEs [21] --> Loop 28 
* CEs [30] --> Loop 29 
* CEs [20] --> Loop 30 
* CEs [29] --> Loop 31 
* CEs [19] --> Loop 32 
* CEs [18] --> Loop 33 
* CEs [28] --> Loop 34 
* CEs [17] --> Loop 35 

### Ranking functions of CR f(V,V1,V6,V2,Out) 
* RF of phase [15,16]: [2*V1-V6-V2-1]
* RF of phase [24]: [V1-V6]
* RF of phase [28]: [V1-V2]

#### Partial ranking functions of CR f(V,V1,V6,V2,Out) 
* Partial RF of phase [15,16]:
  - RF of loop [15:1]:
    V1-V2
  - RF of loop [16:1]:
    V1-V6
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V1-V6
* Partial RF of phase [28]:
  - RF of loop [28:1]:
    V1-V2


### Specialization of cost equations start/4 
* CE 1 is refined into CE [38,39,40,41,42,43,44,45,46,47] 
* CE 2 is refined into CE [48,49,50,51] 
* CE 3 is refined into CE [52,53] 


### Cost equations --> "Loop" of start/4 
* CEs [51] --> Loop 36 
* CEs [53] --> Loop 37 
* CEs [49,52] --> Loop 38 
* CEs [47] --> Loop 39 
* CEs [46] --> Loop 40 
* CEs [45] --> Loop 41 
* CEs [44] --> Loop 42 
* CEs [43] --> Loop 43 
* CEs [42] --> Loop 44 
* CEs [41] --> Loop 45 
* CEs [40,50] --> Loop 46 
* CEs [38,39] --> Loop 47 
* CEs [48] --> Loop 48 

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

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


#### Cost of chains of gt(V,V1,Out):
* Chain [[14],13]: 1*it(14)+1
  Such that:it(14) =< V

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

* Chain [[14],12]: 1*it(14)+1
  Such that:it(14) =< V1

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

* Chain [13]: 1
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [12]: 1
  with precondition: [V1=0,Out=1,V>=1] 


#### Cost of chains of f(V,V1,V6,V2,Out):
* Chain [[28],35]: 4*it(28)+1*s(3)+0
  Such that:aux(1) =< V1
it(28) =< V1-V2
s(3) =< it(28)*aux(1)

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

* Chain [[28],30,35]: 4*it(28)+1*s(3)+1*s(4)+4
  Such that:it(28) =< V1-V2
aux(2) =< V1
s(4) =< aux(2)
s(3) =< it(28)*aux(2)

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

* Chain [[28],29,35]: 4*it(28)+1*s(3)+1*s(5)+4
  Such that:it(28) =< V1-V2
aux(3) =< V1
s(5) =< aux(3)
s(3) =< it(28)*aux(3)

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

* Chain [[28],27,[15,16],35]: 9*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+4
  Such that:it(28) =< 2*V1-V2
aux(12) =< V1
aux(13) =< 2*V1
it(15) =< aux(13)
aux(7) =< aux(12)-1
aux(6) =< aux(12)+1
s(14) =< it(15)*aux(12)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(12)

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

* Chain [[28],27,[15,16],21,35]: 8*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3*s(18)+8
  Such that:aux(15) =< 2*V1
it(28) =< 2*V1-V2
aux(17) =< V1
it(15) =< aux(17)
s(18) =< aux(17)
it(15) =< aux(15)
aux(7) =< aux(17)-1
aux(6) =< aux(17)+1
s(14) =< it(15)*aux(17)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(17)

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

* Chain [[28],27,[15,16],20,35]: 8*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3*s(18)+8
  Such that:aux(19) =< 2*V1
it(28) =< 2*V1-V2
aux(21) =< V1
it(15) =< aux(21)
s(18) =< aux(21)
it(15) =< aux(19)
aux(7) =< aux(21)-1
aux(6) =< aux(21)+1
s(14) =< it(15)*aux(21)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(21)

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

* Chain [[28],27,[15,16],18,35]: 8*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3*s(18)+8
  Such that:aux(23) =< 2*V1
it(28) =< 2*V1-V2
aux(25) =< V1
it(15) =< aux(25)
s(18) =< aux(25)
it(15) =< aux(23)
aux(7) =< aux(25)-1
aux(6) =< aux(25)+1
s(14) =< it(15)*aux(25)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(25)

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

* Chain [[28],27,[15,16],17,35]: 8*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3*s(18)+8
  Such that:aux(27) =< 2*V1
it(28) =< 2*V1-V2
aux(29) =< V1
it(15) =< aux(29)
s(18) =< aux(29)
it(15) =< aux(27)
aux(7) =< aux(29)-1
aux(6) =< aux(29)+1
s(14) =< it(15)*aux(29)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(29)

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

* Chain [[28],27,35]: 4*it(28)+1*s(3)+1*s(18)+4
  Such that:it(28) =< V1-V2
aux(30) =< V1
s(18) =< aux(30)
s(3) =< it(28)*aux(30)

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

* Chain [[24],35]: 4*it(24)+1*s(29)+0
  Such that:aux(31) =< V1
it(24) =< V1-V6
s(29) =< it(24)*aux(31)

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

* Chain [[24],26,35]: 4*it(24)+1*s(29)+1*s(30)+4
  Such that:it(24) =< V1-V6
aux(32) =< V1
s(30) =< aux(32)
s(29) =< it(24)*aux(32)

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

* Chain [[24],25,35]: 4*it(24)+1*s(29)+1*s(31)+4
  Such that:it(24) =< V1-V6
aux(33) =< V1
s(31) =< aux(33)
s(29) =< it(24)*aux(33)

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

* Chain [[24],23,[15,16],35]: 9*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+1*s(29)+4
  Such that:it(24) =< 2*V1-V6
aux(34) =< V1
aux(35) =< 2*V1
it(15) =< aux(35)
aux(7) =< aux(34)-1
aux(6) =< aux(34)+1
s(14) =< it(15)*aux(34)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(34)

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

* Chain [[24],23,[15,16],21,35]: 8*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+3*s(19)+1*s(29)+8
  Such that:aux(15) =< 2*V1
it(24) =< 2*V1-V6
aux(37) =< V1
it(15) =< aux(37)
s(19) =< aux(37)
it(15) =< aux(15)
aux(7) =< aux(37)-1
aux(6) =< aux(37)+1
s(14) =< it(15)*aux(37)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(37)

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

* Chain [[24],23,[15,16],20,35]: 8*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+3*s(21)+1*s(29)+8
  Such that:aux(19) =< 2*V1
it(24) =< 2*V1-V6
aux(39) =< V1
it(15) =< aux(39)
s(21) =< aux(39)
it(15) =< aux(19)
aux(7) =< aux(39)-1
aux(6) =< aux(39)+1
s(14) =< it(15)*aux(39)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(39)

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

* Chain [[24],23,[15,16],18,35]: 8*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+3*s(23)+1*s(29)+8
  Such that:aux(23) =< 2*V1
it(24) =< 2*V1-V6
aux(41) =< V1
it(15) =< aux(41)
s(23) =< aux(41)
it(15) =< aux(23)
aux(7) =< aux(41)-1
aux(6) =< aux(41)+1
s(14) =< it(15)*aux(41)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(41)

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

* Chain [[24],23,[15,16],17,35]: 8*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+3*s(25)+1*s(29)+8
  Such that:aux(27) =< 2*V1
it(24) =< 2*V1-V6
aux(43) =< V1
it(15) =< aux(43)
s(25) =< aux(43)
it(15) =< aux(27)
aux(7) =< aux(43)-1
aux(6) =< aux(43)+1
s(14) =< it(15)*aux(43)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(43)

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

* Chain [[24],23,35]: 4*it(24)+1*s(29)+1*s(32)+4
  Such that:it(24) =< V1-V6
aux(44) =< V1
s(32) =< aux(44)
s(29) =< it(24)*aux(44)

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

* Chain [[15,16],35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+0
  Such that:aux(8) =< V1
aux(11) =< 2*V1-V6-V2
it(15) =< aux(11)
aux(7) =< aux(8)-1
aux(6) =< aux(8)+1
s(14) =< it(15)*aux(8)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [[15,16],21,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+2*s(19)+4
  Such that:it(16) =< V1-V6
it(15) =< V1-V2
aux(14) =< V1
aux(15) =< 2*V1-V6-V2
s(19) =< aux(14)
it(15) =< aux(15)
it(16) =< aux(15)
aux(7) =< aux(14)-1
aux(6) =< aux(14)+1
s(14) =< it(15)*aux(14)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [[15,16],20,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+2*s(21)+4
  Such that:it(16) =< V1-V6
it(15) =< V1-V2
aux(18) =< V1
aux(19) =< 2*V1-V6-V2
s(21) =< aux(18)
it(15) =< aux(19)
it(16) =< aux(19)
aux(7) =< aux(18)-1
aux(6) =< aux(18)+1
s(14) =< it(15)*aux(18)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [[15,16],18,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+2*s(23)+4
  Such that:it(16) =< V1-V6
it(15) =< V1-V2
aux(22) =< V1
aux(23) =< 2*V1-V6-V2
s(23) =< aux(22)
it(15) =< aux(23)
it(16) =< aux(23)
aux(7) =< aux(22)-1
aux(6) =< aux(22)+1
s(14) =< it(15)*aux(22)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [[15,16],17,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+2*s(25)+4
  Such that:it(16) =< V1-V6
it(15) =< V1-V2
aux(26) =< V1
aux(27) =< 2*V1-V6-V2
s(25) =< aux(26)
it(15) =< aux(27)
it(16) =< aux(27)
aux(7) =< aux(26)-1
aux(6) =< aux(26)+1
s(14) =< it(15)*aux(26)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [35]: 0
  with precondition: [Out=0,V>=0,V1>=0,V6>=0,V2>=0] 

* Chain [34,35]: 4
  with precondition: [V=1,V1=0,Out=0,V6>=0,V2>=0] 

* Chain [33,35]: 4
  with precondition: [V=1,V1=0,Out=0,V6>=0,V2>=0] 

* Chain [32,[28],35]: 4*it(28)+1*s(3)+4
  Such that:aux(45) =< V1
it(28) =< aux(45)
s(3) =< it(28)*aux(45)

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

* Chain [32,[28],30,35]: 5*it(28)+1*s(3)+8
  Such that:aux(46) =< V1
it(28) =< aux(46)
s(3) =< it(28)*aux(46)

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

* Chain [32,[28],29,35]: 5*it(28)+1*s(3)+8
  Such that:aux(47) =< V1
it(28) =< aux(47)
s(3) =< it(28)*aux(47)

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

* Chain [32,[28],27,[15,16],35]: 13*it(15)+3*s(3)+1*s(16)+1*s(17)+8
  Such that:aux(12) =< V1
aux(48) =< 2*V1
it(15) =< aux(48)
aux(7) =< aux(12)-1
aux(6) =< aux(12)+1
s(3) =< it(15)*aux(12)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [32,[28],27,[15,16],21,35]: 8*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3*s(18)+12
  Such that:aux(17) =< V1
aux(49) =< 2*V1
it(28) =< aux(49)
it(15) =< aux(17)
s(18) =< aux(17)
it(15) =< aux(49)
aux(7) =< aux(17)-1
aux(6) =< aux(17)+1
s(14) =< it(15)*aux(17)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(17)

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

* Chain [32,[28],27,[15,16],20,35]: 8*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3*s(18)+12
  Such that:aux(21) =< V1
aux(50) =< 2*V1
it(28) =< aux(50)
it(15) =< aux(21)
s(18) =< aux(21)
it(15) =< aux(50)
aux(7) =< aux(21)-1
aux(6) =< aux(21)+1
s(14) =< it(15)*aux(21)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(21)

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

* Chain [32,[28],27,[15,16],18,35]: 8*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3*s(18)+12
  Such that:aux(25) =< V1
aux(51) =< 2*V1
it(28) =< aux(51)
it(15) =< aux(25)
s(18) =< aux(25)
it(15) =< aux(51)
aux(7) =< aux(25)-1
aux(6) =< aux(25)+1
s(14) =< it(15)*aux(25)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(25)

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

* Chain [32,[28],27,[15,16],17,35]: 8*it(15)+4*it(28)+1*s(3)+2*s(14)+1*s(16)+1*s(17)+3*s(18)+12
  Such that:aux(29) =< V1
aux(52) =< 2*V1
it(28) =< aux(52)
it(15) =< aux(29)
s(18) =< aux(29)
it(15) =< aux(52)
aux(7) =< aux(29)-1
aux(6) =< aux(29)+1
s(14) =< it(15)*aux(29)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(3) =< it(28)*aux(29)

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

* Chain [32,[28],27,35]: 5*it(28)+1*s(3)+8
  Such that:aux(53) =< V1
it(28) =< aux(53)
s(3) =< it(28)*aux(53)

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

* Chain [32,35]: 4
  with precondition: [V=1,V6=0,V2=0,Out=0,V1>=1] 

* Chain [32,30,35]: 1*s(4)+8
  Such that:s(4) =< 1

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

* Chain [32,29,35]: 1*s(5)+8
  Such that:s(5) =< 1

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

* Chain [32,27,[15,16],35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+8
  Such that:s(18) =< 1
aux(8) =< V1
aux(11) =< 2*V1
it(15) =< aux(11)
aux(7) =< aux(8)-1
aux(6) =< aux(8)+1
s(14) =< it(15)*aux(8)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [32,27,[15,16],21,35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+2*s(19)+12
  Such that:s(18) =< 1
aux(15) =< 2*V1
aux(54) =< V1
it(15) =< aux(54)
s(19) =< aux(54)
it(15) =< aux(15)
aux(7) =< aux(54)-1
aux(6) =< aux(54)+1
s(14) =< it(15)*aux(54)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [32,27,[15,16],20,35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+2*s(21)+12
  Such that:s(18) =< 1
aux(19) =< 2*V1
aux(55) =< V1
it(15) =< aux(55)
s(21) =< aux(55)
it(15) =< aux(19)
aux(7) =< aux(55)-1
aux(6) =< aux(55)+1
s(14) =< it(15)*aux(55)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [32,27,[15,16],18,35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+2*s(23)+12
  Such that:s(18) =< 1
aux(23) =< 2*V1
aux(56) =< V1
it(15) =< aux(56)
s(23) =< aux(56)
it(15) =< aux(23)
aux(7) =< aux(56)-1
aux(6) =< aux(56)+1
s(14) =< it(15)*aux(56)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [32,27,[15,16],17,35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+2*s(25)+12
  Such that:s(18) =< 1
aux(27) =< 2*V1
aux(57) =< V1
it(15) =< aux(57)
s(25) =< aux(57)
it(15) =< aux(27)
aux(7) =< aux(57)-1
aux(6) =< aux(57)+1
s(14) =< it(15)*aux(57)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [32,27,35]: 1*s(18)+8
  Such that:s(18) =< 1

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

* Chain [31,[24],35]: 4*it(24)+1*s(29)+4
  Such that:aux(58) =< V1
it(24) =< aux(58)
s(29) =< it(24)*aux(58)

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

* Chain [31,[24],26,35]: 5*it(24)+1*s(29)+8
  Such that:aux(59) =< V1
it(24) =< aux(59)
s(29) =< it(24)*aux(59)

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

* Chain [31,[24],25,35]: 5*it(24)+1*s(29)+8
  Such that:aux(60) =< V1
it(24) =< aux(60)
s(29) =< it(24)*aux(60)

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

* Chain [31,[24],23,[15,16],35]: 13*it(15)+3*s(14)+1*s(16)+1*s(17)+8
  Such that:aux(34) =< V1
aux(61) =< 2*V1
it(15) =< aux(61)
aux(7) =< aux(34)-1
aux(6) =< aux(34)+1
s(14) =< it(15)*aux(34)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [31,[24],23,[15,16],21,35]: 8*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+3*s(19)+1*s(29)+12
  Such that:aux(37) =< V1
aux(62) =< 2*V1
it(24) =< aux(62)
it(15) =< aux(37)
s(19) =< aux(37)
it(15) =< aux(62)
aux(7) =< aux(37)-1
aux(6) =< aux(37)+1
s(14) =< it(15)*aux(37)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(37)

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

* Chain [31,[24],23,[15,16],20,35]: 8*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+3*s(21)+1*s(29)+12
  Such that:aux(39) =< V1
aux(63) =< 2*V1
it(24) =< aux(63)
it(15) =< aux(39)
s(21) =< aux(39)
it(15) =< aux(63)
aux(7) =< aux(39)-1
aux(6) =< aux(39)+1
s(14) =< it(15)*aux(39)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(39)

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

* Chain [31,[24],23,[15,16],18,35]: 8*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+3*s(23)+1*s(29)+12
  Such that:aux(41) =< V1
aux(64) =< 2*V1
it(24) =< aux(64)
it(15) =< aux(41)
s(23) =< aux(41)
it(15) =< aux(64)
aux(7) =< aux(41)-1
aux(6) =< aux(41)+1
s(14) =< it(15)*aux(41)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(41)

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

* Chain [31,[24],23,[15,16],17,35]: 8*it(15)+4*it(24)+2*s(14)+1*s(16)+1*s(17)+3*s(25)+1*s(29)+12
  Such that:aux(43) =< V1
aux(65) =< 2*V1
it(24) =< aux(65)
it(15) =< aux(43)
s(25) =< aux(43)
it(15) =< aux(65)
aux(7) =< aux(43)-1
aux(6) =< aux(43)+1
s(14) =< it(15)*aux(43)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)
s(29) =< it(24)*aux(43)

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

* Chain [31,[24],23,35]: 5*it(24)+1*s(29)+8
  Such that:aux(66) =< V1
it(24) =< aux(66)
s(29) =< it(24)*aux(66)

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

* Chain [31,35]: 4
  with precondition: [V=1,V6=0,V2=0,Out=0,V1>=1] 

* Chain [31,26,35]: 1*s(30)+8
  Such that:s(30) =< 1

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

* Chain [31,25,35]: 1*s(31)+8
  Such that:s(31) =< 1

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

* Chain [31,23,[15,16],35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(32)+8
  Such that:s(32) =< 1
aux(8) =< V1
aux(11) =< 2*V1
it(15) =< aux(11)
aux(7) =< aux(8)-1
aux(6) =< aux(8)+1
s(14) =< it(15)*aux(8)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [31,23,[15,16],21,35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+2*s(19)+1*s(32)+12
  Such that:s(32) =< 1
aux(15) =< 2*V1
aux(67) =< V1
it(15) =< aux(67)
s(19) =< aux(67)
it(15) =< aux(15)
aux(7) =< aux(67)-1
aux(6) =< aux(67)+1
s(14) =< it(15)*aux(67)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [31,23,[15,16],20,35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+2*s(21)+1*s(32)+12
  Such that:s(32) =< 1
aux(19) =< 2*V1
aux(68) =< V1
it(15) =< aux(68)
s(21) =< aux(68)
it(15) =< aux(19)
aux(7) =< aux(68)-1
aux(6) =< aux(68)+1
s(14) =< it(15)*aux(68)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [31,23,[15,16],18,35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+2*s(23)+1*s(32)+12
  Such that:s(32) =< 1
aux(23) =< 2*V1
aux(69) =< V1
it(15) =< aux(69)
s(23) =< aux(69)
it(15) =< aux(23)
aux(7) =< aux(69)-1
aux(6) =< aux(69)+1
s(14) =< it(15)*aux(69)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [31,23,[15,16],17,35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+2*s(25)+1*s(32)+12
  Such that:s(32) =< 1
aux(27) =< 2*V1
aux(70) =< V1
it(15) =< aux(70)
s(25) =< aux(70)
it(15) =< aux(27)
aux(7) =< aux(70)-1
aux(6) =< aux(70)+1
s(14) =< it(15)*aux(70)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [31,23,35]: 1*s(32)+8
  Such that:s(32) =< 1

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

* Chain [30,35]: 1*s(4)+4
  Such that:s(4) =< V1

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

* Chain [29,35]: 1*s(5)+4
  Such that:s(5) =< V1

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

* Chain [27,[15,16],35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+4
  Such that:aux(8) =< V1
aux(11) =< 2*V1-V2
s(18) =< V2
it(15) =< aux(11)
aux(7) =< aux(8)-1
aux(6) =< aux(8)+1
s(14) =< it(15)*aux(8)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [27,[15,16],21,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+2*s(19)+8
  Such that:it(15) =< V1-V2
aux(15) =< 2*V1-V2
s(18) =< V2
aux(16) =< V1
it(16) =< aux(16)
s(19) =< aux(16)
it(15) =< aux(15)
it(16) =< aux(15)
aux(7) =< aux(16)-1
aux(6) =< aux(16)+1
s(14) =< it(15)*aux(16)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [27,[15,16],20,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+2*s(21)+8
  Such that:it(15) =< V1-V2
aux(19) =< 2*V1-V2
s(18) =< V2
aux(20) =< V1
it(16) =< aux(20)
s(21) =< aux(20)
it(15) =< aux(19)
it(16) =< aux(19)
aux(7) =< aux(20)-1
aux(6) =< aux(20)+1
s(14) =< it(15)*aux(20)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [27,[15,16],18,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+2*s(23)+8
  Such that:it(15) =< V1-V2
aux(23) =< 2*V1-V2
s(18) =< V2
aux(24) =< V1
it(16) =< aux(24)
s(23) =< aux(24)
it(15) =< aux(23)
it(16) =< aux(23)
aux(7) =< aux(24)-1
aux(6) =< aux(24)+1
s(14) =< it(15)*aux(24)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [27,[15,16],17,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+1*s(18)+2*s(25)+8
  Such that:it(15) =< V1-V2
aux(27) =< 2*V1-V2
s(18) =< V2
aux(28) =< V1
it(16) =< aux(28)
s(25) =< aux(28)
it(15) =< aux(27)
it(16) =< aux(27)
aux(7) =< aux(28)-1
aux(6) =< aux(28)+1
s(14) =< it(15)*aux(28)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [27,35]: 1*s(18)+4
  Such that:s(18) =< V2

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

* Chain [26,35]: 1*s(30)+4
  Such that:s(30) =< V1

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

* Chain [25,35]: 1*s(31)+4
  Such that:s(31) =< V1

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

* Chain [23,[15,16],35]: 8*it(15)+2*s(14)+1*s(16)+1*s(17)+1*s(32)+4
  Such that:aux(8) =< V1
aux(11) =< 2*V1-V6
s(32) =< V6
it(15) =< aux(11)
aux(7) =< aux(8)-1
aux(6) =< aux(8)+1
s(14) =< it(15)*aux(8)
s(17) =< it(15)*aux(7)
s(16) =< it(15)*aux(6)

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

* Chain [23,[15,16],21,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+2*s(19)+1*s(32)+8
  Such that:it(16) =< V1-V6
aux(15) =< 2*V1-V6
s(32) =< V6
aux(36) =< V1
it(15) =< aux(36)
s(19) =< aux(36)
it(15) =< aux(15)
it(16) =< aux(15)
aux(7) =< aux(36)-1
aux(6) =< aux(36)+1
s(14) =< it(15)*aux(36)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [23,[15,16],20,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+2*s(21)+1*s(32)+8
  Such that:it(16) =< V1-V6
aux(19) =< 2*V1-V6
s(32) =< V6
aux(38) =< V1
it(15) =< aux(38)
s(21) =< aux(38)
it(15) =< aux(19)
it(16) =< aux(19)
aux(7) =< aux(38)-1
aux(6) =< aux(38)+1
s(14) =< it(15)*aux(38)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [23,[15,16],18,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+2*s(23)+1*s(32)+8
  Such that:it(16) =< V1-V6
aux(23) =< 2*V1-V6
s(32) =< V6
aux(40) =< V1
it(15) =< aux(40)
s(23) =< aux(40)
it(15) =< aux(23)
it(16) =< aux(23)
aux(7) =< aux(40)-1
aux(6) =< aux(40)+1
s(14) =< it(15)*aux(40)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [23,[15,16],17,35]: 4*it(15)+4*it(16)+2*s(14)+1*s(16)+1*s(17)+2*s(25)+1*s(32)+8
  Such that:it(16) =< V1-V6
aux(27) =< 2*V1-V6
s(32) =< V6
aux(42) =< V1
it(15) =< aux(42)
s(25) =< aux(42)
it(15) =< aux(27)
it(16) =< aux(27)
aux(7) =< aux(42)-1
aux(6) =< aux(42)+1
s(14) =< it(15)*aux(42)
s(17) =< it(16)*aux(7)
s(16) =< it(16)*aux(6)

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

* Chain [23,35]: 1*s(32)+4
  Such that:s(32) =< V6

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

* Chain [22,35]: 2*s(33)+4
  Such that:aux(71) =< V1
s(33) =< aux(71)

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

* Chain [21,35]: 1*s(19)+1*s(20)+4
  Such that:s(20) =< V1
s(19) =< V6+1

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

* Chain [20,35]: 1*s(21)+1*s(22)+4
  Such that:s(21) =< V1
s(22) =< V2

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

* Chain [19,35]: 2*s(35)+4
  Such that:aux(72) =< V1
s(35) =< aux(72)

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

* Chain [18,35]: 1*s(23)+1*s(24)+4
  Such that:s(24) =< V1
s(23) =< V6

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

* Chain [17,35]: 1*s(25)+1*s(26)+4
  Such that:s(25) =< V1
s(26) =< V2+1

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


#### Cost of chains of start(V,V1,V6,V2):
* Chain [48]: 1
  with precondition: [V=0,V1>=0] 

* Chain [47]: 16*s(582)+128*s(583)+78*s(584)+32*s(587)+16*s(588)+16*s(589)+74*s(590)+18*s(591)+4*s(592)+4*s(593)+8*s(594)+12
  Such that:s(579) =< 1
s(580) =< V1
s(581) =< 2*V1
s(582) =< s(579)
s(583) =< s(580)
s(584) =< s(580)
s(583) =< s(581)
s(585) =< s(580)-1
s(586) =< s(580)+1
s(587) =< s(583)*s(580)
s(588) =< s(583)*s(585)
s(589) =< s(583)*s(586)
s(590) =< s(581)
s(591) =< s(590)*s(580)
s(592) =< s(590)*s(585)
s(593) =< s(590)*s(586)
s(594) =< s(584)*s(580)

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

* Chain [46]: 2*s(596)+1*s(597)+4
  Such that:s(597) =< V
s(595) =< V1
s(596) =< s(595)

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

* Chain [45]: 16*s(603)+16*s(604)+28*s(605)+6*s(606)+7*s(609)+1*s(610)+1*s(611)+4*s(612)+16*s(613)+23*s(614)+8*s(615)+4*s(616)+4*s(617)+32*s(618)+8*s(619)+4*s(620)+4*s(621)+9*s(622)+2*s(623)+1*s(624)+1*s(625)+8
  Such that:s(598) =< V1
s(599) =< V1-V2
s(600) =< 2*V1
s(601) =< 2*V1-V2
s(602) =< V2
s(603) =< s(599)
s(604) =< s(599)
s(605) =< s(601)
s(606) =< s(602)
s(607) =< s(598)-1
s(608) =< s(598)+1
s(609) =< s(605)*s(598)
s(610) =< s(605)*s(607)
s(611) =< s(605)*s(608)
s(612) =< s(604)*s(598)
s(613) =< s(598)
s(614) =< s(598)
s(603) =< s(601)
s(613) =< s(601)
s(615) =< s(603)*s(598)
s(616) =< s(613)*s(607)
s(617) =< s(613)*s(608)
s(618) =< s(598)
s(618) =< s(600)
s(619) =< s(618)*s(598)
s(620) =< s(618)*s(607)
s(621) =< s(618)*s(608)
s(622) =< s(600)
s(623) =< s(622)*s(598)
s(624) =< s(622)*s(607)
s(625) =< s(622)*s(608)

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

* Chain [44]: 2*s(627)+4
  Such that:s(626) =< V1
s(627) =< s(626)

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

* Chain [43]: 16*s(633)+16*s(634)+28*s(635)+6*s(636)+7*s(639)+1*s(640)+1*s(641)+4*s(642)+16*s(643)+23*s(644)+8*s(645)+4*s(646)+4*s(647)+32*s(648)+8*s(649)+4*s(650)+4*s(651)+9*s(652)+2*s(653)+1*s(654)+1*s(655)+8
  Such that:s(628) =< V1
s(629) =< V1-V6
s(630) =< 2*V1
s(631) =< 2*V1-V6
s(632) =< V6
s(633) =< s(629)
s(634) =< s(629)
s(635) =< s(631)
s(636) =< s(632)
s(637) =< s(628)-1
s(638) =< s(628)+1
s(639) =< s(635)*s(628)
s(640) =< s(635)*s(637)
s(641) =< s(635)*s(638)
s(642) =< s(634)*s(628)
s(643) =< s(628)
s(644) =< s(628)
s(643) =< s(631)
s(633) =< s(631)
s(645) =< s(643)*s(628)
s(646) =< s(633)*s(637)
s(647) =< s(633)*s(638)
s(648) =< s(628)
s(648) =< s(630)
s(649) =< s(648)*s(628)
s(650) =< s(648)*s(637)
s(651) =< s(648)*s(638)
s(652) =< s(630)
s(653) =< s(652)*s(628)
s(654) =< s(652)*s(637)
s(655) =< s(652)*s(638)

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

* Chain [42]: 4*s(657)+4
  Such that:s(656) =< V1
s(657) =< s(656)

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

* Chain [41]: 16*s(662)+16*s(663)+8*s(664)+2*s(667)+1*s(668)+1*s(669)+8*s(670)+8*s(671)+4*s(672)+4*s(673)+4
  Such that:s(658) =< V1
s(659) =< V1-V6
s(660) =< V1-V2
s(661) =< 2*V1-V6-V2
s(662) =< s(659)
s(663) =< s(660)
s(664) =< s(661)
s(665) =< s(658)-1
s(666) =< s(658)+1
s(667) =< s(664)*s(658)
s(668) =< s(664)*s(665)
s(669) =< s(664)*s(666)
s(670) =< s(658)
s(663) =< s(661)
s(662) =< s(661)
s(671) =< s(663)*s(658)
s(672) =< s(662)*s(665)
s(673) =< s(662)*s(666)

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

* Chain [40]: 1*s(674)+1*s(675)+2*s(677)+4
  Such that:s(676) =< V1
s(674) =< V6
s(675) =< V6+1
s(677) =< s(676)

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

* Chain [39]: 1*s(678)+1*s(679)+2*s(681)+4
  Such that:s(680) =< V1
s(678) =< V2
s(679) =< V2+1
s(681) =< s(680)

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

* Chain [38]: 1
  with precondition: [V1=0,V>=0] 

* Chain [37]: 1
  with precondition: [V1=1,V>=0] 

* Chain [36]: 1*s(682)+1
  Such that:s(682) =< V1

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


Closed-form bounds of start(V,V1,V6,V2): 
-------------------------------------
* Chain [48] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [47] with precondition: [V>=0,V1>=0,V6>=0,V2>=0] 
    - Upper bound: 222*V1+28+56*V1*V1+16*V1*nat(V1-1)+22*V1*(2*V1)+nat(V1-1)*4*(2*V1)+156*V1 
    - Complexity: n^2 
* Chain [46] with precondition: [V>=1,V1>=V] 
    - Upper bound: V+2*V1+4 
    - Complexity: n 
* Chain [45] with precondition: [V=1,V6=0,V2>=1,V1>=V2+1] 
    - Upper bound: 58*V1-29*V2+(32*V1-32*V2+(79*V1+8+16*V1*V1+(V1-1)*(8*V1)+3*V1*(2*V1)+(V1-V2)*(12*V1)+(2*V1-V2)*(8*V1)+6*V2+(V1-1)*(2*V1)+(2*V1-V2)*(V1-1)+20*V1)) 
    - Complexity: n^2 
* Chain [44] with precondition: [V=1,V2=0,V1>=1,V6>=V1] 
    - Upper bound: 2*V1+4 
    - Complexity: n 
* Chain [43] with precondition: [V=1,V2=0,V6>=1,V1>=V6+1] 
    - Upper bound: 58*V1-29*V6+(36*V1-36*V6+(75*V1+8+20*V1*V1+(V1-1)*(4*V1)+3*V1*(2*V1)+(V1-V6)*(8*V1)+(2*V1-V6)*(8*V1)+6*V6+(V1-1)*(2*V1)+(4*V1-4)*(V1-V6)+(2*V1-V6)*(V1-1)+20*V1)) 
    - Complexity: n^2 
* Chain [42] with precondition: [V=1,V1>=1,V6>=V1,V2>=V1] 
    - Upper bound: 4*V1+4 
    - Complexity: n 
* Chain [41] with precondition: [V=1,V6>=1,V2>=1,V1>=V6+1,V1>=V2+1] 
    - Upper bound: 18*V1-9*V6-9*V2+(16*V1-16*V2+(20*V1-20*V6+(8*V1+4+(V1-V6)*(4*V1)+(V1-V2)*(8*V1)+(2*V1-V6-V2)*(3*V1)+(4*V1-4)*(V1-V6)+(2*V1-V6-V2)*(V1-1)))) 
    - Complexity: n^2 
* Chain [40] with precondition: [V=1,V6>=1,V2>=V1,V1>=V6+1] 
    - Upper bound: 2*V1+2*V6+5 
    - Complexity: n 
* Chain [39] with precondition: [V=1,V2>=1,V6>=V1,V1>=V2+1] 
    - Upper bound: 2*V1+2*V2+5 
    - Complexity: n 
* Chain [38] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [37] with precondition: [V1=1,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [36] with precondition: [V1>=1,V>=V1+1] 
    - Upper bound: V1+1 
    - Complexity: n 

### Maximum cost of start(V,V1,V6,V2): V1+3+max([max([V,nat(V6+1)+nat(V6),nat(V2+1)+nat(V2)]),4*V1+max([67*V1+4+16*V1*V1+4*V1*nat(V1-1)+3*V1*(2*V1)+2*V1*nat(V1-1)+20*V1+max([4*V1*nat(V1-1)+4*V1+max([12*V1*nat(V1-V2)+8*V1*nat(2*V1-V2)+nat(V2)*6+nat(2*V1-V2)*nat(V1-1)+nat(V1-V2)*32+nat(2*V1-V2)*29,143*V1+20+40*V1*V1+8*V1*nat(V1-1)+19*V1*(2*V1)+nat(V1-1)*3*(2*V1)+136*V1]),8*V1*nat(V1-V6)+4*V1*V1+8*V1*nat(2*V1-V6)+nat(V6)*6+nat(V1-1)*4*nat(V1-V6)+nat(2*V1-V6)*nat(V1-1)+nat(V1-V6)*36+nat(2*V1-V6)*29]),8*V1*nat(V1-V2)+4*V1*nat(V1-V6)+3*V1*nat(2*V1-V6-V2)+nat(V1-1)*4*nat(V1-V6)+nat(2*V1-V6-V2)*nat(V1-1)+nat(V1-V6)*20+nat(V1-V2)*16+nat(2*V1-V6-V2)*9])+2*V1])+V1+1 
Asymptotic class: n^2 
* Total analysis performed in 1833 ms.


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

(12)
BOUNDS(1, n^2)

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

(13) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
   f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
   gt(0', v) -> false
   gt(s(u), 0') -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false

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

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

(16)
Obligation:
TRS:
Rules:
f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false

Types:
f :: true:false -> s:0' -> s:0' -> s:0' -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, gt

They will be analysed ascendingly in the following order:
gt < f

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

(18)
Obligation:
TRS:
Rules:
f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false

Types:
f :: true:false -> s:0' -> s:0' -> s:0' -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
gt, f

They will be analysed ascendingly in the following order:
gt < f

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gt(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
false

Induction Step:
gt(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(20)
Complex Obligation (BEST)

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(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false

Types:
f :: true:false -> s:0' -> s:0' -> s:0' -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
gt, f

They will be analysed ascendingly in the following order:
gt < f

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(22) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(23)
BOUNDS(n^1, INF)

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(24)
Obligation:
TRS:
Rules:
f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z)
f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false

Types:
f :: true:false -> s:0' -> s:0' -> s:0' -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
f