/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^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(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), 5 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 632 ms]
    (10) BOUNDS(1, n^1)
(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), 279 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), 32 ms]
            (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y)
   if(false, b, x, y) -> div_by_zero
   if(true, false, x, y) -> 0
   if(true, true, x, y) -> id_inc(div(minus(x, y), y))

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^1).


The TRS R consists of the following rules:

   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y) [1]
   if(false, b, x, y) -> div_by_zero [1]
   if(true, false, x, y) -> 0 [1]
   if(true, true, x, y) -> id_inc(div(minus(x, y), y)) [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:

   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y) [1]
   if(false, b, x, y) -> div_by_zero [1]
   if(true, false, x, y) -> 0 [1]
   if(true, true, x, y) -> id_inc(div(minus(x, y), y)) [1]

The TRS has the following type information:
ge :: 0:s:div_by_zero -> 0:s:div_by_zero -> true:false
0 :: 0:s:div_by_zero
true :: true:false
s :: 0:s:div_by_zero -> 0:s:div_by_zero
false :: true:false
minus :: 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
id_inc :: 0:s:div_by_zero -> 0:s:div_by_zero
div :: 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
if :: true:false -> true:false -> 0:s:div_by_zero -> 0:s:div_by_zero -> 0:s:div_by_zero
div_by_zero :: 0:s:div_by_zero

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:

   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   if(v0, v1, v2, v3) -> null_if [0]

And the following fresh constants:    null_ge, null_minus, null_if

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

(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:

   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, y) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y) [1]
   if(false, b, x, y) -> div_by_zero [1]
   if(true, false, x, y) -> 0 [1]
   if(true, true, x, y) -> id_inc(div(minus(x, y), y)) [1]
   ge(v0, v1) -> null_ge [0]
   minus(v0, v1) -> null_minus [0]
   if(v0, v1, v2, v3) -> null_if [0]

The TRS has the following type information:
ge :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> true:false:null_ge
0 :: 0:s:div_by_zero:null_minus:null_if
true :: true:false:null_ge
s :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
false :: true:false:null_ge
minus :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
id_inc :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
div :: 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
if :: true:false:null_ge -> true:false:null_ge -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if -> 0:s:div_by_zero:null_minus:null_if
div_by_zero :: 0:s:div_by_zero:null_minus:null_if
null_ge :: true:false:null_ge
null_minus :: 0:s:div_by_zero:null_minus:null_if
null_if :: 0:s:div_by_zero:null_minus:null_if

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

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

   0 => 0
   true => 2
   false => 1
   div_by_zero => 1
   null_ge => 0
   null_minus => 0
   null_if => 0

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

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

   div(z, z') -{ 1 }-> if(ge(y, 1 + 0), 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 }-> 2 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   id_inc(z) -{ 1 }-> x :|: x >= 0, z = x
   id_inc(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   if(z, z', z'', z1) -{ 1 }-> id_inc(div(minus(x, y), y)) :|: z = 2, z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   if(z, z', z'', z1) -{ 1 }-> 1 :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z = 2, z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, 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

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, V15, V18),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15, V18),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15, V18),0,[fun(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V15, V18),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15, V18),0,[if(V1, V, V15, V18, Out)],[V1 >= 0,V >= 0,V15 >= 0,V18 >= 0]).
eq(ge(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = V2,V = 0]).
eq(ge(V1, V, Out),1,[],[Out = 1,V = 1 + V3,V3 >= 0,V1 = 0]).
eq(ge(V1, V, Out),1,[ge(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(minus(V1, V, Out),1,[],[Out = V6,V6 >= 0,V1 = V6,V = 0]).
eq(minus(V1, V, Out),1,[],[Out = 0,V7 >= 0,V1 = 0,V = V7]).
eq(minus(V1, V, Out),1,[minus(V8, V9, Ret1)],[Out = Ret1,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = 1 + V8]).
eq(fun(V1, Out),1,[],[Out = V10,V10 >= 0,V1 = V10]).
eq(fun(V1, Out),1,[],[Out = 1 + V11,V11 >= 0,V1 = V11]).
eq(div(V1, V, Out),1,[ge(V12, 1 + 0, Ret0),ge(V13, V12, Ret11),if(Ret0, Ret11, V13, V12, Ret2)],[Out = Ret2,V13 >= 0,V12 >= 0,V1 = V13,V = V12]).
eq(if(V1, V, V15, V18, Out),1,[],[Out = 1,V17 >= 0,V18 = V14,V1 = 1,V16 >= 0,V14 >= 0,V = V17,V15 = V16]).
eq(if(V1, V, V15, V18, Out),1,[],[Out = 0,V1 = 2,V18 = V20,V19 >= 0,V20 >= 0,V15 = V19,V = 1]).
eq(if(V1, V, V15, V18, Out),1,[minus(V22, V21, Ret00),div(Ret00, V21, Ret01),fun(Ret01, Ret3)],[Out = Ret3,V1 = 2,V18 = V21,V = 2,V22 >= 0,V21 >= 0,V15 = V22]).
eq(ge(V1, V, Out),0,[],[Out = 0,V24 >= 0,V23 >= 0,V1 = V24,V = V23]).
eq(minus(V1, V, Out),0,[],[Out = 0,V26 >= 0,V25 >= 0,V1 = V26,V = V25]).
eq(if(V1, V, V15, V18, Out),0,[],[Out = 0,V18 = V29,V28 >= 0,V15 = V30,V27 >= 0,V1 = V28,V = V27,V30 >= 0,V29 >= 0]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,Out),[V1],[Out]).
input_output_vars(div(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V15,V18,Out),[V1,V,V15,V18],[Out]).


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

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

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

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

### Specialization of cost equations ge/3 
* CE 22 is refined into CE [23] 
* CE 19 is refined into CE [24] 
* CE 20 is refined into CE [25] 
* CE 21 is refined into CE [26] 


### Cost equations --> "Loop" of ge/3 
* CEs [26] --> Loop 16 
* CEs [23] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [25] --> Loop 19 

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

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


### Specialization of cost equations fun/2 
* CE 17 is refined into CE [27] 
* CE 18 is refined into CE [28] 


### Cost equations --> "Loop" of fun/2 
* CEs [27] --> Loop 20 
* CEs [28] --> Loop 21 

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

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


### Specialization of cost equations minus/3 
* CE 9 is refined into CE [29] 
* CE 10 is refined into CE [30] 
* CE 12 is refined into CE [31] 
* CE 11 is refined into CE [32] 


### Cost equations --> "Loop" of minus/3 
* CEs [32] --> Loop 22 
* CEs [29] --> Loop 23 
* CEs [30,31] --> Loop 24 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [22]: [V,V1]

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


### Specialization of cost equations (div)/3 
* CE 16 is refined into CE [33,34] 
* CE 13 is refined into CE [35,36,37,38,39,40,41,42,43,44,45] 
* CE 15 is refined into CE [46,47] 
* CE 14 is refined into CE [48,49,50,51] 


### Cost equations --> "Loop" of (div)/3 
* CEs [51] --> Loop 25 
* CEs [50] --> Loop 26 
* CEs [49] --> Loop 27 
* CEs [48] --> Loop 28 
* CEs [33,34] --> Loop 29 
* CEs [35,36,38] --> Loop 30 
* CEs [37,39,40,41,42,43,44,45,46,47] --> Loop 31 

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

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


### Specialization of cost equations start/4 
* CE 2 is refined into CE [52,53,54,55,56,57,58,59,60,61,62,63,64,65] 
* CE 3 is refined into CE [66] 
* CE 1 is refined into CE [67] 
* CE 4 is refined into CE [68] 
* CE 5 is refined into CE [69,70,71,72,73] 
* CE 6 is refined into CE [74,75,76] 
* CE 7 is refined into CE [77,78] 
* CE 8 is refined into CE [79,80,81,82] 


### Cost equations --> "Loop" of start/4 
* CEs [70,74,80] --> Loop 32 
* CEs [52,53,54,55,56,57,58,59,60,61,62,63,64,65] --> Loop 33 
* CEs [66] --> Loop 34 
* CEs [68] --> Loop 35 
* CEs [67,69,71,72,73,75,76,77,78,79,81,82] --> Loop 36 

### Ranking functions of CR start(V1,V,V15,V18) 

#### Partial ranking functions of CR start(V1,V,V15,V18) 


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

#### Cost of chains of ge(V1,V,Out):
* Chain [[16],19]: 1*it(16)+1
  Such that:it(16) =< V1

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

* Chain [[16],18]: 1*it(16)+1
  Such that:it(16) =< V

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

* Chain [[16],17]: 1*it(16)+0
  Such that:it(16) =< V

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

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

* Chain [18]: 1
  with precondition: [V=0,Out=2,V1>=0] 

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


#### Cost of chains of fun(V1,Out):
* Chain [21]: 1
  with precondition: [V1+1=Out,V1>=0] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[22],24]: 1*it(22)+1
  Such that:it(22) =< V

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

* Chain [[22],23]: 1*it(22)+1
  Such that:it(22) =< V

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

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

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


#### Cost of chains of div(V1,V,Out):
* Chain [[25,26],31]: 12*it(25)+10*s(3)+4*s(5)+7*s(7)+2*s(32)+4
  Such that:aux(1) =< 1
aux(9) =< V1-V+1
aux(3) =< V
aux(12) =< V1
s(3) =< aux(1)
s(7) =< aux(12)
s(5) =< aux(3)
aux(6) =< aux(12)
it(25) =< aux(12)
aux(6) =< aux(9)
it(25) =< aux(9)
s(32) =< aux(6)

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

* Chain [[25,26],28,31]: 12*it(25)+11*s(3)+6*s(5)+2*s(32)+4*s(33)+10
  Such that:aux(14) =< 1
aux(8) =< V1
aux(9) =< V1-V+1
aux(15) =< V
aux(16) =< V1-V
s(3) =< aux(14)
s(5) =< aux(15)
aux(6) =< aux(8)
it(25) =< aux(8)
s(34) =< aux(8)
aux(6) =< aux(9)
it(25) =< aux(9)
aux(6) =< aux(16)
it(25) =< aux(16)
s(34) =< aux(16)
s(32) =< aux(6)
s(33) =< s(34)

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

* Chain [[25,26],27,31]: 12*it(25)+11*s(3)+6*s(5)+2*s(32)+4*s(33)+10
  Such that:aux(18) =< 1
aux(8) =< V1
aux(9) =< V1-V+1
aux(19) =< V
aux(20) =< V1-V
s(3) =< aux(18)
s(5) =< aux(19)
aux(6) =< aux(8)
it(25) =< aux(8)
s(34) =< aux(8)
aux(6) =< aux(9)
it(25) =< aux(9)
aux(6) =< aux(20)
it(25) =< aux(20)
s(34) =< aux(20)
s(32) =< aux(6)
s(33) =< s(34)

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

* Chain [31]: 10*s(3)+4*s(5)+3*s(7)+4
  Such that:aux(1) =< 1
aux(2) =< V1
aux(3) =< V
s(3) =< aux(1)
s(7) =< aux(2)
s(5) =< aux(3)

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

* Chain [30]: 4
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [29]: 4
  with precondition: [V=0,Out=1,V1>=0] 

* Chain [28,31]: 11*s(3)+6*s(5)+10
  Such that:aux(14) =< 1
aux(15) =< V
s(3) =< aux(14)
s(5) =< aux(15)

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

* Chain [27,31]: 11*s(3)+6*s(5)+10
  Such that:aux(18) =< 1
aux(19) =< V
s(3) =< aux(18)
s(5) =< aux(19)

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


#### Cost of chains of start(V1,V,V15,V18):
* Chain [36]: 36*s(83)+11*s(84)+64*s(91)+24*s(102)+4*s(104)+8*s(105)+12*s(108)+2*s(109)+10
  Such that:aux(27) =< 1
aux(28) =< V1
aux(29) =< V1-V
aux(30) =< V1-V+1
aux(31) =< V
s(84) =< aux(28)
s(83) =< aux(31)
s(91) =< aux(27)
s(101) =< aux(28)
s(102) =< aux(28)
s(103) =< aux(28)
s(101) =< aux(30)
s(102) =< aux(30)
s(101) =< aux(29)
s(102) =< aux(29)
s(103) =< aux(29)
s(104) =< s(101)
s(105) =< s(103)
s(107) =< aux(28)
s(108) =< aux(28)
s(107) =< aux(30)
s(108) =< aux(30)
s(109) =< s(107)

  with precondition: [V1>=0] 

* Chain [35]: 1
  with precondition: [V1=1,V>=0,V15>=0,V18>=0] 

* Chain [34]: 1
  with precondition: [V1=2,V=1,V15>=0,V18>=0] 

* Chain [33]: 212*s(125)+6*s(127)+92*s(134)+20*s(156)+48*s(173)+8*s(175)+16*s(176)+24*s(179)+4*s(180)+13
  Such that:aux(40) =< 1
aux(41) =< V15
aux(42) =< V15-2*V18
aux(43) =< V15-2*V18+1
aux(44) =< V15-V18
aux(45) =< V18
s(125) =< aux(40)
s(127) =< aux(41)
s(134) =< aux(45)
s(156) =< aux(44)
s(172) =< aux(44)
s(173) =< aux(44)
s(174) =< aux(44)
s(172) =< aux(43)
s(173) =< aux(43)
s(172) =< aux(42)
s(173) =< aux(42)
s(174) =< aux(42)
s(175) =< s(172)
s(176) =< s(174)
s(178) =< aux(44)
s(179) =< aux(44)
s(178) =< aux(43)
s(179) =< aux(43)
s(180) =< s(178)

  with precondition: [V1=2,V=2,V15>=0,V18>=0] 

* Chain [32]: 4
  with precondition: [V=0,V1>=0] 


Closed-form bounds of start(V1,V,V15,V18): 
-------------------------------------
* Chain [36] with precondition: [V1>=0] 
    - Upper bound: 61*V1+74+nat(V)*36 
    - Complexity: n 
* Chain [35] with precondition: [V1=1,V>=0,V15>=0,V18>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [34] with precondition: [V1=2,V=1,V15>=0,V18>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [33] with precondition: [V1=2,V=2,V15>=0,V18>=0] 
    - Upper bound: 6*V15+92*V18+225+nat(V15-V18)*120 
    - Complexity: n 
* Chain [32] with precondition: [V=0,V1>=0] 
    - Upper bound: 4 
    - Complexity: constant 

### Maximum cost of start(V1,V,V15,V18): max([3,61*V1+73+nat(V)*36,nat(V15)*6+224+nat(V18)*92+nat(V15-V18)*120])+1 
Asymptotic class: n 
* Total analysis performed in 536 ms.


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

(10)
BOUNDS(1, n^1)

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

(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:

   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0') -> x
   minus(0', y) -> 0'
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
   if(false, b, x, y) -> div_by_zero
   if(true, false, x, y) -> 0'
   if(true, true, x, y) -> id_inc(div(minus(x, y), y))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
if(false, b, x, y) -> div_by_zero
if(true, false, x, y) -> 0'
if(true, true, x, y) -> id_inc(div(minus(x, y), y))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero

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

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

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

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

(16)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
if(false, b, x, y) -> div_by_zero
if(true, false, x, y) -> 0'
if(true, true, x, y) -> id_inc(div(minus(x, y), y))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero


Generator Equations:
gen_0':s:div_by_zero3_0(0) <=> 0'
gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
ge(gen_0':s:div_by_zero3_0(0), gen_0':s:div_by_zero3_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s:div_by_zero3_0(+(n5_0, 1)), gen_0':s:div_by_zero3_0(+(n5_0, 1))) ->_R^Omega(1)
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) ->_IH
true

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:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
if(false, b, x, y) -> div_by_zero
if(true, false, x, y) -> 0'
if(true, true, x, y) -> id_inc(div(minus(x, y), y))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero


Generator Equations:
gen_0':s:div_by_zero3_0(0) <=> 0'
gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))


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

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

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
if(false, b, x, y) -> div_by_zero
if(true, false, x, y) -> 0'
if(true, true, x, y) -> id_inc(div(minus(x, y), y))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero


Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s:div_by_zero3_0(0) <=> 0'
gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))


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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s:div_by_zero3_0(n317_0), gen_0':s:div_by_zero3_0(n317_0)) -> gen_0':s:div_by_zero3_0(0), rt in Omega(1 + n317_0)

Induction Base:
minus(gen_0':s:div_by_zero3_0(0), gen_0':s:div_by_zero3_0(0)) ->_R^Omega(1)
gen_0':s:div_by_zero3_0(0)

Induction Step:
minus(gen_0':s:div_by_zero3_0(+(n317_0, 1)), gen_0':s:div_by_zero3_0(+(n317_0, 1))) ->_R^Omega(1)
minus(gen_0':s:div_by_zero3_0(n317_0), gen_0':s:div_by_zero3_0(n317_0)) ->_IH
gen_0':s:div_by_zero3_0(0)

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:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
id_inc(x) -> x
id_inc(x) -> s(x)
div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
if(false, b, x, y) -> div_by_zero
if(true, false, x, y) -> 0'
if(true, true, x, y) -> id_inc(div(minus(x, y), y))

Types:
ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
0' :: 0':s:div_by_zero
true :: true:false
s :: 0':s:div_by_zero -> 0':s:div_by_zero
false :: true:false
minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
div_by_zero :: 0':s:div_by_zero
hole_true:false1_0 :: true:false
hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero


Lemmas:
ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
minus(gen_0':s:div_by_zero3_0(n317_0), gen_0':s:div_by_zero3_0(n317_0)) -> gen_0':s:div_by_zero3_0(0), rt in Omega(1 + n317_0)


Generator Equations:
gen_0':s:div_by_zero3_0(0) <=> 0'
gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))


The following defined symbols remain to be analysed:
div