/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^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 616 ms]
    (10) BOUNDS(1, n^3)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 292 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 42 ms]
            (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   minus(x, y) -> if(gt(x, y), x, y)
   if(true, x, y) -> s(minus(p(x), y))
   if(false, x, y) -> 0
   p(0) -> 0
   p(s(x)) -> x
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   div(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0), x, y)
   if1(false, x, y) -> 0
   if2(true, x, y) -> s(div(minus(x, y), y))
   if2(false, x, y) -> 0

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

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

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


The TRS R consists of the following rules:

   minus(x, y) -> if(gt(x, y), x, y) [1]
   if(true, x, y) -> s(minus(p(x), y)) [1]
   if(false, x, y) -> 0 [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   div(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> 0 [1]
   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
   if2(false, x, y) -> 0 [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:

   minus(x, y) -> if(gt(x, y), x, y) [1]
   if(true, x, y) -> s(minus(p(x), y)) [1]
   if(false, x, y) -> 0 [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   div(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> 0 [1]
   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
   if2(false, x, y) -> 0 [1]

The TRS has the following type information:
minus :: s:0 -> s:0 -> s:0
if :: true:false -> s:0 -> s:0 -> s:0
gt :: s:0 -> s:0 -> true:false
true :: true:false
s :: s:0 -> s:0
p :: s:0 -> s:0
false :: true:false
0 :: s:0
ge :: s:0 -> s:0 -> true:false
div :: s:0 -> s:0 -> s:0
if1 :: true:false -> s:0 -> s:0 -> s:0
if2 :: true:false -> s:0 -> s:0 -> s:0

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:
none

And the following fresh constants: none

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

(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:

   minus(x, y) -> if(gt(x, y), x, y) [1]
   if(true, x, y) -> s(minus(p(x), y)) [1]
   if(false, x, y) -> 0 [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   div(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> 0 [1]
   if2(true, x, y) -> s(div(minus(x, y), y)) [1]
   if2(false, x, y) -> 0 [1]

The TRS has the following type information:
minus :: s:0 -> s:0 -> s:0
if :: true:false -> s:0 -> s:0 -> s:0
gt :: s:0 -> s:0 -> true:false
true :: true:false
s :: s:0 -> s:0
p :: s:0 -> s:0
false :: true:false
0 :: s:0
ge :: s:0 -> s:0 -> true:false
div :: s:0 -> s:0 -> s:0
if1 :: true:false -> s:0 -> s:0 -> s:0
if2 :: true:false -> s:0 -> s:0 -> s:0

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

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

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

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

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

   div(z, z') -{ 1 }-> if1(ge(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0
   gt(z, z') -{ 1 }-> gt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gt(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gt(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   if(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + minus(p(x), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> if2(gt(y, 0), x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if2(z, z', z'') -{ 1 }-> 1 + div(minus(x, y), y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   minus(z, z') -{ 1 }-> if(gt(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V5),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[if(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[p(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V5),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[if1(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[if2(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(minus(V1, V, Out),1,[gt(V3, V2, Ret0),if(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
eq(if(V1, V, V5, Out),1,[p(V4, Ret10),minus(Ret10, V6, Ret1)],[Out = 1 + Ret1,V = V4,V5 = V6,V1 = 1,V4 >= 0,V6 >= 0]).
eq(if(V1, V, V5, Out),1,[],[Out = 0,V = V8,V5 = V7,V8 >= 0,V7 >= 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(p(V1, Out),1,[],[Out = V9,V9 >= 0,V1 = 1 + V9]).
eq(ge(V1, V, Out),1,[],[Out = 1,V10 >= 0,V1 = V10,V = 0]).
eq(ge(V1, V, Out),1,[],[Out = 0,V = 1 + V11,V11 >= 0,V1 = 0]).
eq(ge(V1, V, Out),1,[ge(V13, V12, Ret2)],[Out = Ret2,V = 1 + V12,V13 >= 0,V12 >= 0,V1 = 1 + V13]).
eq(gt(V1, V, Out),1,[],[Out = 0,V14 >= 0,V1 = 0,V = V14]).
eq(gt(V1, V, Out),1,[],[Out = 1,V15 >= 0,V1 = 1 + V15,V = 0]).
eq(gt(V1, V, Out),1,[gt(V17, V16, Ret3)],[Out = Ret3,V = 1 + V16,V17 >= 0,V16 >= 0,V1 = 1 + V17]).
eq(div(V1, V, Out),1,[ge(V19, V18, Ret01),if1(Ret01, V19, V18, Ret4)],[Out = Ret4,V19 >= 0,V18 >= 0,V1 = V19,V = V18]).
eq(if1(V1, V, V5, Out),1,[gt(V20, 0, Ret02),if2(Ret02, V21, V20, Ret5)],[Out = Ret5,V = V21,V5 = V20,V1 = 1,V21 >= 0,V20 >= 0]).
eq(if1(V1, V, V5, Out),1,[],[Out = 0,V = V23,V5 = V22,V23 >= 0,V22 >= 0,V1 = 0]).
eq(if2(V1, V, V5, Out),1,[minus(V24, V25, Ret101),div(Ret101, V25, Ret11)],[Out = 1 + Ret11,V = V24,V5 = V25,V1 = 1,V24 >= 0,V25 >= 0]).
eq(if2(V1, V, V5, Out),1,[],[Out = 0,V = V26,V5 = V27,V26 >= 0,V27 >= 0,V1 = 0]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).
input_output_vars(div(V1,V,Out),[V1,V],[Out]).
input_output_vars(if1(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(if2(V1,V,V5,Out),[V1,V,V5],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [ge/3]
1. recursive  : [gt/3]
2. non_recursive  : [p/2]
3. recursive  : [if/4,minus/3]
4. recursive  : [(div)/3,if1/4,if2/4]
5. non_recursive  : [start/3]

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

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

### Specialization of cost equations ge/3 
* CE 23 is refined into CE [24] 
* CE 21 is refined into CE [25] 
* CE 22 is refined into CE [26] 


### Cost equations --> "Loop" of ge/3 
* CEs [25] --> Loop 15 
* CEs [26] --> Loop 16 
* CEs [24] --> Loop 17 

### Ranking functions of CR ge(V1,V,Out) 
* RF of phase [17]: [V,V1]

#### Partial ranking functions of CR ge(V1,V,Out) 
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V
    V1


### Specialization of cost equations gt/3 
* CE 13 is refined into CE [27] 
* CE 12 is refined into CE [28] 
* CE 11 is refined into CE [29] 


### Cost equations --> "Loop" of gt/3 
* CEs [28] --> Loop 18 
* CEs [29] --> Loop 19 
* CEs [27] --> Loop 20 

### Ranking functions of CR gt(V1,V,Out) 
* RF of phase [20]: [V,V1]

#### Partial ranking functions of CR gt(V1,V,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V
    V1


### Specialization of cost equations p/2 
* CE 20 is refined into CE [30] 
* CE 19 is refined into CE [31] 


### Cost equations --> "Loop" of p/2 
* CEs [30] --> Loop 21 
* CEs [31] --> Loop 22 

### Ranking functions of CR p(V1,Out) 

#### Partial ranking functions of CR p(V1,Out) 


### Specialization of cost equations minus/3 
* CE 15 is refined into CE [32,33] 
* CE 14 is refined into CE [34,35] 


### Cost equations --> "Loop" of minus/3 
* CEs [35] --> Loop 23 
* CEs [34] --> Loop 24 
* CEs [33] --> Loop 25 
* CEs [32] --> Loop 26 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [25]: [V1-1,V1-V]
* RF of phase [26]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V1-1
    V1-V
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1


### Specialization of cost equations (div)/3 
* CE 16 is refined into CE [36,37] 
* CE 18 is refined into CE [38] 
* CE 17 is refined into CE [39,40] 


### Cost equations --> "Loop" of (div)/3 
* CEs [40] --> Loop 27 
* CEs [39] --> Loop 28 
* CEs [37] --> Loop 29 
* CEs [38] --> Loop 30 
* CEs [36] --> Loop 31 

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

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


### Specialization of cost equations start/3 
* CE 2 is refined into CE [41,42,43,44,45,46] 
* CE 3 is refined into CE [47] 
* CE 4 is refined into CE [48,49,50,51,52,53,54,55] 
* CE 5 is refined into CE [56,57,58,59,60] 
* CE 1 is refined into CE [61] 
* CE 6 is refined into CE [62,63,64,65] 
* CE 7 is refined into CE [66,67] 
* CE 8 is refined into CE [68,69,70,71] 
* CE 9 is refined into CE [72,73,74,75] 
* CE 10 is refined into CE [76,77,78,79,80,81] 


### Cost equations --> "Loop" of start/3 
* CEs [43,52] --> Loop 32 
* CEs [45,46,54,55,60] --> Loop 33 
* CEs [59] --> Loop 34 
* CEs [42,51] --> Loop 35 
* CEs [44,53] --> Loop 36 
* CEs [47,50,58] --> Loop 37 
* CEs [57,64,65,67,70,71,74,75,78,79,80,81] --> Loop 38 
* CEs [41,48,49,56,63,69,73,77] --> Loop 39 
* CEs [61,62,66,68,72,76] --> Loop 40 

### Ranking functions of CR start(V1,V,V5) 

#### Partial ranking functions of CR start(V1,V,V5) 


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

#### Cost of chains of ge(V1,V,Out):
* Chain [[17],16]: 1*it(17)+1
  Such that:it(17) =< V1

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

* Chain [[17],15]: 1*it(17)+1
  Such that:it(17) =< V

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

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

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


#### Cost of chains of gt(V1,V,Out):
* Chain [[20],19]: 1*it(20)+1
  Such that:it(20) =< V1

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

* Chain [[20],18]: 1*it(20)+1
  Such that:it(20) =< V

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

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

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


#### Cost of chains of p(V1,Out):
* Chain [22]: 1
  with precondition: [V1=0,Out=0] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[26],24]: 4*it(26)+3
  Such that:it(26) =< Out

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

* Chain [[25],23]: 4*it(25)+1*s(1)+1*s(4)+3
  Such that:it(25) =< Out
aux(2) =< V1-Out
s(1) =< aux(2)
s(4) =< it(25)*aux(2)

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

* Chain [24]: 3
  with precondition: [V1=0,Out=0,V>=0] 

* Chain [23]: 1*s(1)+3
  Such that:s(1) =< V1

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


#### Cost of chains of div(V1,V,Out):
* Chain [[27],29]: 8*it(27)+3*s(5)+4*s(16)+1*s(17)+3
  Such that:aux(7) =< V1-V
s(12) =< V
aux(9) =< V1
it(27) =< aux(9)
s(5) =< aux(9)
it(27) =< aux(7)
s(16) =< it(27)*aux(7)
s(17) =< s(16)*s(12)

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

* Chain [[27],28,31]: 8*it(27)+2*s(15)+4*s(16)+1*s(17)+2*s(19)+11
  Such that:aux(6) =< V1
aux(11) =< V1-V
aux(12) =< V
it(27) =< aux(11)
s(19) =< aux(12)
it(27) =< aux(6)
s(18) =< aux(6)
s(18) =< aux(11)
s(16) =< it(27)*aux(11)
s(15) =< s(18)
s(17) =< s(16)*aux(12)

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

* Chain [31]: 3
  with precondition: [V1=0,Out=0,V>=1] 

* Chain [30]: 5
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [29]: 1*s(5)+3
  Such that:s(5) =< V1

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

* Chain [28,31]: 2*s(19)+11
  Such that:aux(10) =< V1
s(19) =< aux(10)

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


#### Cost of chains of start(V1,V,V5):
* Chain [40]: 3
  with precondition: [V1=0] 

* Chain [39]: 4*s(21)+9
  Such that:s(21) =< V1

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

* Chain [38]: 7*s(22)+4*s(23)+7*s(25)+1*s(26)+16*s(37)+8*s(39)+2*s(40)+2*s(48)+11
  Such that:aux(13) =< V1
aux(14) =< V1-V
aux(15) =< V
s(22) =< aux(13)
s(23) =< aux(14)
s(25) =< aux(15)
s(37) =< aux(13)
s(37) =< aux(14)
s(39) =< s(37)*aux(14)
s(40) =< s(39)*aux(15)
s(46) =< aux(13)
s(46) =< aux(14)
s(48) =< s(46)
s(26) =< s(23)*aux(15)

  with precondition: [V1>=1] 

* Chain [37]: 8*s(50)+9
  Such that:aux(16) =< V
s(50) =< aux(16)

  with precondition: [V1=1,V5=0,V>=0] 

* Chain [36]: 14*s(52)+2*s(55)+17
  Such that:aux(19) =< V5
s(52) =< aux(19)
s(55) =< s(52)*aux(19)

  with precondition: [V1=1,V=2*V5,V>=2] 

* Chain [35]: 2*s(64)+9
  Such that:aux(20) =< V
s(64) =< aux(20)

  with precondition: [V1=1,V>=1,V5>=V] 

* Chain [34]: 1*s(66)+5
  Such that:s(66) =< V

  with precondition: [V1=1,V>=2,V5+1>=V] 

* Chain [33]: 26*s(67)+9*s(69)+5*s(70)+32*s(74)+16*s(76)+4*s(77)+4*s(89)+17
  Such that:aux(29) =< V-2*V5
aux(30) =< V-V5
aux(31) =< V5
s(67) =< aux(30)
s(74) =< aux(30)
s(74) =< aux(29)
s(76) =< s(74)*aux(29)
s(77) =< s(76)*aux(31)
s(69) =< aux(31)
s(70) =< s(67)*aux(31)
s(87) =< aux(30)
s(87) =< aux(29)
s(89) =< s(87)

  with precondition: [V1=1,V5>=1,V>=V5+2] 

* Chain [32]: 10*s(119)+2*s(121)+2*s(122)+9
  Such that:aux(34) =< V-V5
aux(35) =< V5
s(119) =< aux(34)
s(121) =< aux(35)
s(122) =< s(119)*aux(35)

  with precondition: [V1=1,2*V5>=V+1,V>=V5+1] 


Closed-form bounds of start(V1,V,V5): 
-------------------------------------
* Chain [40] with precondition: [V1=0] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [39] with precondition: [V=0,V1>=0] 
    - Upper bound: 4*V1+9 
    - Complexity: n 
* Chain [38] with precondition: [V1>=1] 
    - Upper bound: 25*V1+11+2*V1*nat(V)*nat(V1-V)+8*V1*nat(V1-V)+nat(V)*7+nat(V1-V)*nat(V)+nat(V1-V)*4 
    - Complexity: n^3 
* Chain [37] with precondition: [V1=1,V5=0,V>=0] 
    - Upper bound: 8*V+9 
    - Complexity: n 
* Chain [36] with precondition: [V1=1,V=2*V5,V>=2] 
    - Upper bound: 14*V5+17+2*V5*V5 
    - Complexity: n^2 
* Chain [35] with precondition: [V1=1,V>=1,V5>=V] 
    - Upper bound: 2*V+9 
    - Complexity: n 
* Chain [34] with precondition: [V1=1,V>=2,V5+1>=V] 
    - Upper bound: V+5 
    - Complexity: n 
* Chain [33] with precondition: [V1=1,V5>=1,V>=V5+2] 
    - Upper bound: 62*V-62*V5+(9*V5+17+(V-V5)*(5*V5)+(V-V5)*(4*V5)*nat(V-2*V5))+(16*V-16*V5)*nat(V-2*V5) 
    - Complexity: n^3 
* Chain [32] with precondition: [V1=1,2*V5>=V+1,V>=V5+1] 
    - Upper bound: 10*V-10*V5+(2*V5+9+(V-V5)*(2*V5)) 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V5): max([max([4*V1+6,nat(V5)*2+6+max([nat(V5)*2*nat(V-V5)+nat(V-V5)*10,nat(V5)*7+8+max([nat(V5)*2*nat(V5)+nat(V5)*5,nat(V5)*4*nat(V-V5)*nat(V-2*V5)+nat(V5)*5*nat(V-V5)+nat(V-V5)*62+nat(V-V5)*16*nat(V-2*V5)])])]),nat(V)*5+max([nat(V),25*V1+2+2*V1*nat(V)*nat(V1-V)+8*V1*nat(V1-V)+nat(V1-V)*nat(V)+nat(V1-V)*4])+(nat(V)+4)+(nat(V)+2)])+3 
Asymptotic class: n^3 
* Total analysis performed in 509 ms.


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

(10)
BOUNDS(1, n^3)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   minus(x, y) -> if(gt(x, y), x, y)
   if(true, x, y) -> s(minus(p(x), y))
   if(false, x, y) -> 0'
   p(0') -> 0'
   p(s(x)) -> x
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   gt(0', y) -> false
   gt(s(x), 0') -> true
   gt(s(x), s(y)) -> gt(x, y)
   div(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0'), x, y)
   if1(false, x, y) -> 0'
   if2(true, x, y) -> s(div(minus(x, y), y))
   if2(false, x, y) -> 0'

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

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

(14)
Obligation:
Innermost TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, gt, ge, div

They will be analysed ascendingly in the following order:
gt < minus
minus < div
gt < div
ge < div

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

(16)
Obligation:
Innermost TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
gt, minus, ge, div

They will be analysed ascendingly in the following order:
gt < minus
minus < div
gt < div
ge < div

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> false, rt in Omega(1 + n5_0)

Induction Base:
gt(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1)
false

Induction Step:
gt(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
gt, minus, ge, div

They will be analysed ascendingly in the following order:
gt < minus
minus < div
gt < div
ge < div

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> false, rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
minus, ge, div

They will be analysed ascendingly in the following order:
minus < div
ge < div

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_s:0'3_0(n395_0), gen_s:0'3_0(n395_0)) -> true, rt in Omega(1 + n395_0)

Induction Base:
ge(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_s:0'3_0(+(n395_0, 1)), gen_s:0'3_0(+(n395_0, 1))) ->_R^Omega(1)
ge(gen_s:0'3_0(n395_0), gen_s:0'3_0(n395_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
ge(gen_s:0'3_0(n395_0), gen_s:0'3_0(n395_0)) -> true, rt in Omega(1 + n395_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
div