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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 559 ms]
    (14) BOUNDS(1, n^3)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
   minus(x, plus(y, z)) -> minus(minus(x, y), z)
   p(s(s(x))) -> s(p(s(x)))
   p(0) -> s(s(0))
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of minus: p, minus
The following defined symbols can occur below the 1th argument of minus: p
The following defined symbols can occur below the 0th argument of p: p
The following defined symbols can occur below the 0th argument of plus: p, minus
The following defined symbols can occur below the 1th argument of plus: p, minus
The following defined symbols can occur below the 0th argument of div: p, minus

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
minus(x, plus(y, z)) -> minus(minus(x, y), z)
div(plus(x, y), z) -> plus(div(x, z), div(y, z))

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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
   p(s(s(x))) -> s(p(s(x)))
   p(0) -> s(s(0))
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))

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

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

The duplicating contexts are:
div(s(x), s([]))


The defined contexts are:
minus([], x1)
minus(x0, [])
plus(x0, [])
div([], s(x1))
p(s([]))
plus([], x1)
minus(s([]), s(0))


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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
   p(s(s(x))) -> s(p(s(x)))
   p(0) -> s(s(0))
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))

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

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

(6)
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:

   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y))) [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   p(0) -> s(s(0)) [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0)))) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y))) [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   p(0) -> s(s(0)) [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0)))) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

(9) 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:

   p(v0) -> null_p [0]
   div(v0, v1) -> null_div [0]
   minus(v0, v1) -> null_minus [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_p, null_div, null_minus, null_plus

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

(10)
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:

   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y))) [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   p(0) -> s(s(0)) [1]
   div(s(x), s(y)) -> s(div(minus(x, y), s(y))) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0)))) [1]
   p(v0) -> null_p [0]
   div(v0, v1) -> null_div [0]
   minus(v0, v1) -> null_minus [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
minus :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
0 :: 0:s:null_p:null_div:null_minus:null_plus
s :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
p :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
div :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
plus :: 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus -> 0:s:null_p:null_div:null_minus:null_plus
null_p :: 0:s:null_p:null_div:null_minus:null_plus
null_div :: 0:s:null_p:null_div:null_minus:null_plus
null_minus :: 0:s:null_p:null_div:null_minus:null_plus
null_plus :: 0:s:null_p:null_div:null_minus:null_plus

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

(11) 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
   null_p => 0
   null_div => 0
   null_minus => 0
   null_plus => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   div(z, z') -{ 1 }-> 1 + div(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(p(1 + x), p(1 + y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x)
   p(z) -{ 1 }-> 1 + (1 + 0) :|: z = 0
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(y, minus(1 + x, 1 + 0)) :|: x >= 0, y >= 0, z = 1 + x, z' = y

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
eq(minus(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = 0,V = V3]).
eq(minus(V1, V, Out),1,[p(1 + V4, Ret0),p(1 + V5, Ret1),minus(Ret0, Ret1, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(p(V1, Out),1,[p(1 + V6, Ret11)],[Out = 1 + Ret11,V6 >= 0,V1 = 2 + V6]).
eq(p(V1, Out),1,[],[Out = 2,V1 = 0]).
eq(div(V1, V, Out),1,[minus(V7, V8, Ret10),div(Ret10, 1 + V8, Ret12)],[Out = 1 + Ret12,V = 1 + V8,V7 >= 0,V8 >= 0,V1 = 1 + V7]).
eq(plus(V1, V, Out),1,[],[Out = V9,V9 >= 0,V1 = 0,V = V9]).
eq(plus(V1, V, Out),1,[minus(1 + V11, 1 + 0, Ret111),plus(V10, Ret111, Ret13)],[Out = 1 + Ret13,V11 >= 0,V10 >= 0,V1 = 1 + V11,V = V10]).
eq(p(V1, Out),0,[],[Out = 0,V12 >= 0,V1 = V12]).
eq(div(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
eq(minus(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(plus(V1, V, Out),0,[],[Out = 0,V17 >= 0,V18 >= 0,V1 = V17,V = V18]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).
input_output_vars(div(V1,V,Out),[V1,V],[Out]).
input_output_vars(plus(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [p/2]
1. recursive  : [minus/3]
2. recursive  : [(div)/3]
3. recursive  : [plus/3]
4. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into p/2
1. SCC is partially evaluated into minus/3
2. SCC is partially evaluated into (div)/3
3. SCC is partially evaluated into plus/3
4. SCC is partially evaluated into start/2

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

### Specialization of cost equations p/2 
* CE 11 is refined into CE [17] 
* CE 10 is refined into CE [18] 
* CE 9 is refined into CE [19] 


### Cost equations --> "Loop" of p/2 
* CEs [19] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [18] --> Loop 15 

### Ranking functions of CR p(V1,Out) 
* RF of phase [13]: [V1-1]

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


### Specialization of cost equations minus/3 
* CE 5 is refined into CE [20] 
* CE 6 is refined into CE [21] 
* CE 8 is refined into CE [22] 
* CE 7 is refined into CE [23,24,25,26] 


### Cost equations --> "Loop" of minus/3 
* CEs [26] --> Loop 16 
* CEs [25] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [20] --> Loop 20 
* CEs [21,22] --> Loop 21 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [16]: [V-1,V1-1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [16]:
  - RF of loop [16:1]:
    V-1
    V1-1


### Specialization of cost equations (div)/3 
* CE 13 is refined into CE [27] 
* CE 12 is refined into CE [28,29,30] 


### Cost equations --> "Loop" of (div)/3 
* CEs [30] --> Loop 22 
* CEs [29] --> Loop 23 
* CEs [28] --> Loop 24 
* CEs [27] --> Loop 25 

### Ranking functions of CR div(V1,V,Out) 
* RF of phase [22]: [V1/2-1]
* RF of phase [24]: [V1]

#### Partial ranking functions of CR div(V1,V,Out) 
* Partial RF of phase [22]:
  - RF of loop [22:1]:
    V1/2-1
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V1


### Specialization of cost equations plus/3 
* CE 16 is refined into CE [31] 
* CE 14 is refined into CE [32] 
* CE 15 is refined into CE [33,34] 


### Cost equations --> "Loop" of plus/3 
* CEs [34] --> Loop 26 
* CEs [33] --> Loop 27 
* CEs [31] --> Loop 28 
* CEs [32] --> Loop 29 

### Ranking functions of CR plus(V1,V,Out) 
* RF of phase [26,27]: [V1+V]

#### Partial ranking functions of CR plus(V1,V,Out) 
* Partial RF of phase [26,27]:
  - RF of loop [26:1]:
    V1+V-1
  - RF of loop [27:1]:
    V1+V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [35,36,37] 
* CE 2 is refined into CE [38,39,40] 
* CE 3 is refined into CE [41,42,43,44,45] 
* CE 4 is refined into CE [46,47,48] 


### Cost equations --> "Loop" of start/2 
* CEs [41] --> Loop 30 
* CEs [35] --> Loop 31 
* CEs [36,37,38,39,40,42,43,44,45,46,47,48] --> Loop 32 

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

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


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

#### Cost of chains of p(V1,Out):
* Chain [[13],14]: 1*it(13)+0
  Such that:it(13) =< Out

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

* Chain [15]: 1
  with precondition: [V1=0,Out=2] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[16],21]: 1*it(16)+1*s(5)+1*s(6)+1
  Such that:aux(3) =< V1
aux(5) =< V
it(16) =< aux(5)
it(16) =< aux(3)
s(6) =< it(16)*aux(5)
s(5) =< it(16)*aux(3)

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

* Chain [[16],19,21]: 1*it(16)+1*s(5)+1*s(6)+2
  Such that:aux(3) =< V1
aux(6) =< V
it(16) =< aux(6)
it(16) =< aux(3)
s(6) =< it(16)*aux(6)
s(5) =< it(16)*aux(3)

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

* Chain [[16],19,20]: 1*it(16)+1*s(5)+1*s(6)+2
  Such that:aux(3) =< V1
aux(7) =< V
it(16) =< aux(7)
it(16) =< aux(3)
s(6) =< it(16)*aux(7)
s(5) =< it(16)*aux(3)

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

* Chain [[16],18,21]: 1*it(16)+1*s(5)+1*s(6)+1*s(7)+2
  Such that:aux(3) =< V1
aux(8) =< V
it(16) =< aux(8)
s(7) =< aux(8)
it(16) =< aux(3)
s(6) =< it(16)*aux(8)
s(5) =< it(16)*aux(3)

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

* Chain [[16],17,21]: 1*it(16)+1*s(5)+1*s(6)+1*s(8)+2
  Such that:aux(4) =< V
aux(9) =< V1
it(16) =< aux(9)
s(8) =< aux(9)
it(16) =< aux(4)
s(6) =< it(16)*aux(4)
s(5) =< it(16)*aux(9)

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

* Chain [[16],17,20]: 1*it(16)+1*s(5)+1*s(6)+1*s(8)+2
  Such that:aux(3) =< V1
s(8) =< Out
aux(10) =< V
it(16) =< aux(10)
it(16) =< aux(3)
s(6) =< it(16)*aux(10)
s(5) =< it(16)*aux(3)

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

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

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

* Chain [19,21]: 2
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [19,20]: 2
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [18,21]: 1*s(7)+2
  Such that:s(7) =< V

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

* Chain [17,21]: 1*s(8)+2
  Such that:s(8) =< V1

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

* Chain [17,20]: 1*s(8)+2
  Such that:s(8) =< Out

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


#### Cost of chains of div(V1,V,Out):
* Chain [[24],25]: 2*it(24)+0
  Such that:it(24) =< Out

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

* Chain [[24],23,25]: 2*it(24)+2*s(47)+2*s(48)+5*s(49)+5*s(50)+5*s(51)+3
  Such that:s(46) =< 1
s(45) =< V1-Out+1
it(24) =< Out
s(47) =< s(45)
s(48) =< s(46)
s(49) =< s(46)
s(49) =< s(45)
s(50) =< s(49)*s(46)
s(51) =< s(49)*s(45)

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

* Chain [[22],25]: 3*it(22)+2*s(66)+1*s(67)+1*s(68)+1*s(69)+0
  Such that:s(59) =< V1
it(22) =< V1/2
s(61) =< V
aux(16) =< s(61)
aux(15) =< s(59)-2
aux(14) =< s(59)
s(71) =< it(22)*aux(16)
s(72) =< it(22)*aux(15)
s(70) =< it(22)*aux(14)
s(66) =< s(72)
s(67) =< s(71)
s(67) =< s(70)
s(68) =< s(67)*s(61)
s(69) =< s(67)*s(59)

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

* Chain [[22],23,25]: 5*it(22)+2*s(48)+5*s(49)+5*s(50)+5*s(51)+2*s(66)+1*s(67)+1*s(68)+1*s(69)+3
  Such that:aux(17) =< V1
aux(18) =< V
it(22) =< aux(17)
s(48) =< aux(18)
s(49) =< aux(18)
s(49) =< aux(17)
s(50) =< s(49)*aux(18)
s(51) =< s(49)*aux(17)
aux(16) =< aux(18)
aux(15) =< aux(17)-2
aux(14) =< aux(17)
s(71) =< it(22)*aux(16)
s(72) =< it(22)*aux(15)
s(70) =< it(22)*aux(14)
s(66) =< s(72)
s(67) =< s(71)
s(67) =< s(70)
s(68) =< s(67)*aux(18)
s(69) =< s(67)*aux(17)

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

* Chain [25]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [23,25]: 2*s(47)+2*s(48)+5*s(49)+5*s(50)+5*s(51)+3
  Such that:s(45) =< V1
s(46) =< V
s(47) =< s(45)
s(48) =< s(46)
s(49) =< s(46)
s(49) =< s(45)
s(50) =< s(49)*s(46)
s(51) =< s(49)*s(45)

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


#### Cost of chains of plus(V1,V,Out):
* Chain [[26,27],29]: 15*it(26)+2*s(110)+1*s(111)+1*s(112)+1*s(113)+5*s(120)+5*s(121)+1
  Such that:aux(24) =< 1
aux(27) =< V1+V
it(26) =< aux(27)
aux(22) =< aux(27)-1
aux(21) =< aux(27)
s(116) =< it(26)*aux(22)
s(114) =< it(26)*aux(21)
s(120) =< it(26)*aux(24)
s(121) =< it(26)*aux(21)
s(110) =< s(116)
s(111) =< aux(27)
s(111) =< s(114)
s(112) =< s(111)*aux(24)
s(113) =< s(111)*aux(27)

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

* Chain [[26,27],28]: 15*it(26)+2*s(110)+1*s(111)+1*s(112)+1*s(113)+5*s(120)+5*s(121)+0
  Such that:aux(24) =< 1
aux(28) =< V1+V
it(26) =< aux(28)
aux(22) =< aux(28)-1
aux(21) =< aux(28)
s(116) =< it(26)*aux(22)
s(114) =< it(26)*aux(21)
s(120) =< it(26)*aux(24)
s(121) =< it(26)*aux(21)
s(110) =< s(116)
s(111) =< aux(28)
s(111) =< s(114)
s(112) =< s(111)*aux(24)
s(113) =< s(111)*aux(28)

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

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

* Chain [28]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of start(V1,V):
* Chain [32]: 12*s(152)+6*s(153)+16*s(154)+16*s(155)+16*s(156)+3*s(173)+2*s(181)+1*s(182)+1*s(183)+1*s(184)+2*s(198)+1*s(199)+1*s(200)+1*s(201)+30*s(204)+10*s(209)+10*s(210)+4*s(211)+2*s(212)+2*s(213)+2*s(214)+3
  Such that:s(202) =< 1
s(203) =< V1+V
s(173) =< V1/2
aux(32) =< V1
aux(33) =< V
s(152) =< aux(32)
s(204) =< s(203)
s(205) =< s(203)-1
s(206) =< s(203)
s(207) =< s(204)*s(205)
s(208) =< s(204)*s(206)
s(209) =< s(204)*s(202)
s(210) =< s(204)*s(206)
s(211) =< s(207)
s(212) =< s(203)
s(212) =< s(208)
s(213) =< s(212)*s(202)
s(214) =< s(212)*s(203)
s(153) =< aux(33)
s(154) =< aux(33)
s(154) =< aux(32)
s(155) =< s(154)*aux(33)
s(156) =< s(154)*aux(32)
s(175) =< aux(33)
s(176) =< aux(32)-2
s(177) =< aux(32)
s(178) =< s(173)*s(175)
s(179) =< s(173)*s(176)
s(180) =< s(173)*s(177)
s(181) =< s(179)
s(182) =< s(178)
s(182) =< s(180)
s(183) =< s(182)*aux(33)
s(184) =< s(182)*aux(32)
s(195) =< s(152)*s(175)
s(196) =< s(152)*s(176)
s(197) =< s(152)*s(177)
s(198) =< s(196)
s(199) =< s(195)
s(199) =< s(197)
s(200) =< s(199)*aux(33)
s(201) =< s(199)*aux(32)

  with precondition: [V1>=0] 

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

* Chain [30]: 6*s(218)+2*s(220)+5*s(221)+5*s(222)+5*s(223)+3
  Such that:s(215) =< 1
aux(34) =< V1+1
s(218) =< aux(34)
s(220) =< s(215)
s(221) =< s(215)
s(221) =< aux(34)
s(222) =< s(221)*s(215)
s(223) =< s(221)*aux(34)

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [32] with precondition: [V1>=0] 
    - Upper bound: 12*V1+3+V1*V1*nat(V)+17*V1*nat(V)+nat(V)*V1*nat(V)+V1/2*(nat(V)*V1)+2*V1*nat(V1-2)+nat(V)*22+nat(V)*16*nat(V)+V1/2*(nat(V)*nat(V))+V1/2*nat(V)+V1/2*(nat(V1-2)*2)+nat(nat(V1+V)+ -1)*4*nat(V1+V)+nat(V1+V)*44+nat(V1+V)*12*nat(V1+V)+3/2*V1 
    - Complexity: n^3 
* Chain [31] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [30] with precondition: [V=1,V1>=1] 
    - Upper bound: 11*V1+26 
    - Complexity: n 

### Maximum cost of start(V1,V): max([11*V1+25,12*V1+2+V1*V1*nat(V)+17*V1*nat(V)+nat(V)*V1*nat(V)+V1/2*(nat(V)*V1)+2*V1*nat(V1-2)+nat(V)*22+nat(V)*16*nat(V)+V1/2*(nat(V)*nat(V))+V1/2*nat(V)+V1/2*(nat(V1-2)*2)+nat(nat(V1+V)+ -1)*4*nat(V1+V)+nat(V1+V)*44+nat(V1+V)*12*nat(V1+V)+3/2*V1])+1 
Asymptotic class: n^3 
* Total analysis performed in 448 ms.


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(14)
BOUNDS(1, n^3)

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

(16)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
   minus(x, plus(y, z)) -> minus(minus(x, y), z)
   p(s(s(x))) -> s(p(s(x)))
   p(0) -> s(s(0))
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))

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

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

The rewrite sequence

p(s(s(x))) ->^+ s(p(s(x)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x / s(x)].

The result substitution is [ ].




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(18)
Complex Obligation (BEST)

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(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
   minus(x, plus(y, z)) -> minus(minus(x, y), z)
   p(s(s(x))) -> s(p(s(x)))
   p(0) -> s(s(0))
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(p(s(x)), p(s(y)))
   minus(x, plus(y, z)) -> minus(minus(x, y), z)
   p(s(s(x))) -> s(p(s(x)))
   p(0) -> s(s(0))
   div(s(x), s(y)) -> s(div(minus(x, y), s(y)))
   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(y, minus(s(x), s(0))))

S is empty.
Rewrite Strategy: FULL