/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^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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 7 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 719 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), 309 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) 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:

   p(s(x)) -> x
   p(0) -> 0
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   average(x, y) -> if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y)
   if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y)
   if(false, b1, b2, b3, x, y) -> average(p(x), s(y))
   if2(true, b2, b3, x, y) -> 0
   if2(false, b2, b3, x, y) -> if3(b2, b3, x, y)
   if3(true, b3, x, y) -> 0
   if3(false, b3, x, y) -> if4(b3, x, y)
   if4(true, x, y) -> s(0)
   if4(false, x, y) -> average(s(x), p(p(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:

   p(s(x)) -> x [1]
   p(0) -> 0 [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   average(x, y) -> if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) [1]
   if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) [1]
   if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) [1]
   if2(true, b2, b3, x, y) -> 0 [1]
   if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) [1]
   if3(true, b3, x, y) -> 0 [1]
   if3(false, b3, x, y) -> if4(b3, x, y) [1]
   if4(true, x, y) -> s(0) [1]
   if4(false, x, y) -> average(s(x), p(p(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:

   p(s(x)) -> x [1]
   p(0) -> 0 [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   average(x, y) -> if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) [1]
   if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) [1]
   if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) [1]
   if2(true, b2, b3, x, y) -> 0 [1]
   if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) [1]
   if3(true, b3, x, y) -> 0 [1]
   if3(false, b3, x, y) -> if4(b3, x, y) [1]
   if4(true, x, y) -> s(0) [1]
   if4(false, x, y) -> average(s(x), p(p(y))) [1]

The TRS has the following type information:
p :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
le :: s:0 -> s:0 -> true:false
true :: true:false
false :: true:false
average :: s:0 -> s:0 -> s:0
if :: true:false -> true:false -> true:false -> true:false -> s:0 -> s:0 -> s:0
if2 :: true:false -> true:false -> true:false -> s:0 -> s:0 -> s:0
if3 :: true:false -> true:false -> s:0 -> s:0 -> s:0
if4 :: 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:

   p(s(x)) -> x [1]
   p(0) -> 0 [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   average(x, y) -> if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) [1]
   if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) [1]
   if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) [1]
   if2(true, b2, b3, x, y) -> 0 [1]
   if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) [1]
   if3(true, b3, x, y) -> 0 [1]
   if3(false, b3, x, y) -> if4(b3, x, y) [1]
   if4(true, x, y) -> s(0) [1]
   if4(false, x, y) -> average(s(x), p(p(y))) [1]

The TRS has the following type information:
p :: s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0
le :: s:0 -> s:0 -> true:false
true :: true:false
false :: true:false
average :: s:0 -> s:0 -> s:0
if :: true:false -> true:false -> true:false -> true:false -> s:0 -> s:0 -> s:0
if2 :: true:false -> true:false -> true:false -> s:0 -> s:0 -> s:0
if3 :: true:false -> true:false -> s:0 -> s:0 -> s:0
if4 :: 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:

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

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

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

   average(z, z') -{ 1 }-> if(le(x, 0), le(y, 0), le(y, 1 + 0), le(y, 1 + (1 + 0)), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   if(z, z', z'', z1, z2, z3) -{ 1 }-> if2(b1, b2, b3, x, y) :|: b2 >= 0, z2 = x, b1 >= 0, z = 1, z1 = b3, z3 = y, x >= 0, y >= 0, z' = b1, z'' = b2, b3 >= 0
   if(z, z', z'', z1, z2, z3) -{ 1 }-> average(p(x), 1 + y) :|: b2 >= 0, z2 = x, b1 >= 0, z1 = b3, z3 = y, x >= 0, y >= 0, z' = b1, z = 0, z'' = b2, b3 >= 0
   if2(z, z', z'', z1, z2) -{ 1 }-> if3(b2, b3, x, y) :|: b2 >= 0, z2 = y, x >= 0, y >= 0, z' = b2, z = 0, z'' = b3, z1 = x, b3 >= 0
   if2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: b2 >= 0, z2 = y, z = 1, x >= 0, y >= 0, z' = b2, z'' = b3, z1 = x, b3 >= 0
   if3(z, z', z'', z1) -{ 1 }-> if4(b3, x, y) :|: z1 = y, z' = b3, x >= 0, y >= 0, z'' = x, z = 0, b3 >= 0
   if3(z, z', z'', z1) -{ 1 }-> 0 :|: z1 = y, z' = b3, z = 1, x >= 0, y >= 0, z'' = x, b3 >= 0
   if4(z, z', z'') -{ 1 }-> average(1 + x, p(p(y))) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if4(z, z', z'') -{ 1 }-> 1 + 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   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(V, V2, V11, V17, V15, V16),0,[p(V, Out)],[V >= 0]).
eq(start(V, V2, V11, V17, V15, V16),0,[le(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2, V11, V17, V15, V16),0,[average(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2, V11, V17, V15, V16),0,[if(V, V2, V11, V17, V15, V16, Out)],[V >= 0,V2 >= 0,V11 >= 0,V17 >= 0,V15 >= 0,V16 >= 0]).
eq(start(V, V2, V11, V17, V15, V16),0,[if2(V, V2, V11, V17, V15, Out)],[V >= 0,V2 >= 0,V11 >= 0,V17 >= 0,V15 >= 0]).
eq(start(V, V2, V11, V17, V15, V16),0,[if3(V, V2, V11, V17, Out)],[V >= 0,V2 >= 0,V11 >= 0,V17 >= 0]).
eq(start(V, V2, V11, V17, V15, V16),0,[if4(V, V2, V11, Out)],[V >= 0,V2 >= 0,V11 >= 0]).
eq(p(V, Out),1,[],[Out = V1,V1 >= 0,V = 1 + V1]).
eq(p(V, Out),1,[],[Out = 0,V = 0]).
eq(le(V, V2, Out),1,[],[Out = 1,V3 >= 0,V = 0,V2 = V3]).
eq(le(V, V2, Out),1,[],[Out = 0,V4 >= 0,V = 1 + V4,V2 = 0]).
eq(le(V, V2, Out),1,[le(V5, V6, Ret)],[Out = Ret,V2 = 1 + V6,V5 >= 0,V6 >= 0,V = 1 + V5]).
eq(average(V, V2, Out),1,[le(V7, 0, Ret0),le(V8, 0, Ret1),le(V8, 1 + 0, Ret2),le(V8, 1 + (1 + 0), Ret3),if(Ret0, Ret1, Ret2, Ret3, V7, V8, Ret4)],[Out = Ret4,V7 >= 0,V8 >= 0,V = V7,V2 = V8]).
eq(if(V, V2, V11, V17, V15, V16, Out),1,[if2(V14, V12, V13, V9, V10, Ret5)],[Out = Ret5,V12 >= 0,V15 = V9,V14 >= 0,V = 1,V17 = V13,V16 = V10,V9 >= 0,V10 >= 0,V2 = V14,V11 = V12,V13 >= 0]).
eq(if(V, V2, V11, V17, V15, V16, Out),1,[p(V20, Ret01),average(Ret01, 1 + V18, Ret6)],[Out = Ret6,V22 >= 0,V15 = V20,V21 >= 0,V17 = V19,V16 = V18,V20 >= 0,V18 >= 0,V2 = V21,V = 0,V11 = V22,V19 >= 0]).
eq(if2(V, V2, V11, V17, V15, Out),1,[],[Out = 0,V25 >= 0,V15 = V23,V = 1,V24 >= 0,V23 >= 0,V2 = V25,V11 = V26,V17 = V24,V26 >= 0]).
eq(if2(V, V2, V11, V17, V15, Out),1,[if3(V29, V30, V27, V28, Ret7)],[Out = Ret7,V29 >= 0,V15 = V28,V27 >= 0,V28 >= 0,V2 = V29,V = 0,V11 = V30,V17 = V27,V30 >= 0]).
eq(if3(V, V2, V11, V17, Out),1,[],[Out = 0,V17 = V32,V2 = V33,V = 1,V31 >= 0,V32 >= 0,V11 = V31,V33 >= 0]).
eq(if3(V, V2, V11, V17, Out),1,[if4(V35, V36, V34, Ret8)],[Out = Ret8,V17 = V34,V2 = V35,V36 >= 0,V34 >= 0,V11 = V36,V = 0,V35 >= 0]).
eq(if4(V, V2, V11, Out),1,[],[Out = 1,V2 = V37,V11 = V38,V = 1,V37 >= 0,V38 >= 0]).
eq(if4(V, V2, V11, Out),1,[p(V40, Ret10),p(Ret10, Ret11),average(1 + V39, Ret11, Ret9)],[Out = Ret9,V2 = V39,V11 = V40,V39 >= 0,V40 >= 0,V = 0]).
input_output_vars(p(V,Out),[V],[Out]).
input_output_vars(le(V,V2,Out),[V,V2],[Out]).
input_output_vars(average(V,V2,Out),[V,V2],[Out]).
input_output_vars(if(V,V2,V11,V17,V15,V16,Out),[V,V2,V11,V17,V15,V16],[Out]).
input_output_vars(if2(V,V2,V11,V17,V15,Out),[V,V2,V11,V17,V15],[Out]).
input_output_vars(if3(V,V2,V11,V17,Out),[V,V2,V11,V17],[Out]).
input_output_vars(if4(V,V2,V11,Out),[V,V2,V11],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [p/2]
1. recursive  : [le/3]
2. recursive  : [average/3,if/7,if2/6,if3/5,if4/4]
3. non_recursive  : [start/6]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into p/2
1. SCC is partially evaluated into le/3
2. SCC is partially evaluated into average/3
3. SCC is partially evaluated into start/6

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

### Specialization of cost equations p/2 
* CE 15 is refined into CE [25] 
* CE 16 is refined into CE [26] 


### Cost equations --> "Loop" of p/2 
* CEs [25] --> Loop 17 
* CEs [26] --> Loop 18 

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

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


### Specialization of cost equations le/3 
* CE 24 is refined into CE [27] 
* CE 23 is refined into CE [28] 
* CE 22 is refined into CE [29] 


### Cost equations --> "Loop" of le/3 
* CEs [28] --> Loop 19 
* CEs [29] --> Loop 20 
* CEs [27] --> Loop 21 

### Ranking functions of CR le(V,V2,Out) 
* RF of phase [21]: [V,V2]

#### Partial ranking functions of CR le(V,V2,Out) 
* Partial RF of phase [21]:
  - RF of loop [21:1]:
    V
    V2


### Specialization of cost equations average/3 
* CE 18 is refined into CE [30] 
* CE 17 is refined into CE [31] 
* CE 21 is refined into CE [32] 
* CE 20 is refined into CE [33,34,35,36] 
* CE 19 is refined into CE [37] 


### Cost equations --> "Loop" of average/3 
* CEs [34] --> Loop 22 
* CEs [35] --> Loop 23 
* CEs [36] --> Loop 24 
* CEs [33] --> Loop 25 
* CEs [37] --> Loop 26 
* CEs [30] --> Loop 27 
* CEs [31] --> Loop 28 
* CEs [32] --> Loop 29 

### Ranking functions of CR average(V,V2,Out) 
* RF of phase [22,23,24,26]: [3*V+2*V2-4]

#### Partial ranking functions of CR average(V,V2,Out) 
* Partial RF of phase [22,23,24,26]:
  - RF of loop [22:1,23:1,24:1]:
    V depends on loops [26:1] 
  - RF of loop [23:1]:
    -V2+3 depends on loops [26:1] 
  - RF of loop [24:1]:
    -V2+2 depends on loops [26:1] 
  - RF of loop [26:1]:
    -V+1 depends on loops [22:1,23:1,24:1] 
    V2/2-1 depends on loops [22:1,23:1,24:1] 


### Specialization of cost equations start/6 
* CE 10 is refined into CE [38] 
* CE 1 is refined into CE [39] 
* CE 4 is refined into CE [40] 
* CE 3 is refined into CE [41] 
* CE 6 is refined into CE [42,43,44,45,46,47,48] 
* CE 2 is refined into CE [49] 
* CE 5 is refined into CE [50] 
* CE 7 is refined into CE [51,52,53,54,55,56,57] 
* CE 8 is refined into CE [58,59,60,61,62,63,64] 
* CE 9 is refined into CE [65,66,67,68,69,70] 
* CE 11 is refined into CE [71,72,73,74,75,76,77] 
* CE 12 is refined into CE [78,79] 
* CE 13 is refined into CE [80,81,82,83] 
* CE 14 is refined into CE [84,85,86,87,88,89] 


### Cost equations --> "Loop" of start/6 
* CEs [89] --> Loop 30 
* CEs [38,41] --> Loop 31 
* CEs [39,40,42,43,44,45,46,47,48,79,81,82,83,87,88] --> Loop 32 
* CEs [49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,80,84,85,86] --> Loop 33 

### Ranking functions of CR start(V,V2,V11,V17,V15,V16) 

#### Partial ranking functions of CR start(V,V2,V11,V17,V15,V16) 


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

#### Cost of chains of p(V,Out):
* Chain [18]: 1
  with precondition: [V=0,Out=0] 

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


#### Cost of chains of le(V,V2,Out):
* Chain [[21],20]: 1*it(21)+1
  Such that:it(21) =< V

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

* Chain [[21],19]: 1*it(21)+1
  Such that:it(21) =< V2

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

* Chain [20]: 1
  with precondition: [V=0,Out=1,V2>=0] 

* Chain [19]: 1
  with precondition: [V2=0,Out=0,V>=1] 


#### Cost of chains of average(V,V2,Out):
* Chain [[22,23,24,26],27]: 37*it(22)+1*s(1)+1*s(2)+3*s(20)+9
  Such that:s(1) =< 1
s(2) =< 2
aux(28) =< 3*V+2*V2
it(22) =< aux(28)
s(20) =< aux(28)*2

  with precondition: [Out=1,V>=0,V2>=1,V2+2*V>=3] 

* Chain [29]: 7
  with precondition: [V=0,V2=0,Out=0] 

* Chain [28]: 2*s(27)+8
  Such that:aux(29) =< 1
s(27) =< aux(29)

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

* Chain [27]: 1*s(1)+1*s(2)+9
  Such that:s(1) =< 1
s(2) =< 2

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

* Chain [25,[22,23,24,26],27]: 37*it(22)+1*s(1)+1*s(2)+3*s(20)+16
  Such that:s(1) =< 1
s(2) =< 2
aux(28) =< 3*V
it(22) =< aux(28)
s(20) =< aux(28)*2

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

* Chain [25,28]: 2*s(27)+15
  Such that:aux(29) =< 1
s(27) =< aux(29)

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


#### Cost of chains of start(V,V2,V11,V17,V15,V16):
* Chain [33]: 41*s(30)+17*s(32)+111*s(34)+9*s(35)+37*s(53)+3*s(54)+111*s(60)+9*s(61)+37*s(79)+3*s(80)+37*s(88)+3*s(89)+37*s(97)+3*s(98)+111*s(104)+9*s(105)+37*s(123)+3*s(124)+21
  Such that:s(122) =< 3*V2+2*V11
s(78) =< 3*V11+2*V17
s(52) =< 3*V17+2*V15
s(96) =< 3*V15+2*V16
s(87) =< 2*V16+2
aux(30) =< 1
aux(31) =< 2
aux(32) =< 3*V2+3
aux(33) =< 3*V11+3
aux(34) =< 3*V17+3
s(30) =< aux(30)
s(32) =< aux(31)
s(34) =< aux(34)
s(35) =< aux(34)*2
s(53) =< s(52)
s(54) =< s(52)*2
s(60) =< aux(33)
s(61) =< aux(33)*2
s(79) =< s(78)
s(80) =< s(78)*2
s(88) =< s(87)
s(89) =< s(87)*2
s(97) =< s(96)
s(98) =< s(96)*2
s(104) =< aux(32)
s(105) =< aux(32)*2
s(123) =< s(122)
s(124) =< s(122)*2

  with precondition: [V=0] 

* Chain [32]: 13*s(130)+5*s(132)+111*s(134)+9*s(135)+37*s(153)+3*s(154)+1*s(155)+1*s(156)+37*s(162)+3*s(163)+22
  Such that:s(156) =< V
s(161) =< 3*V
s(155) =< V2
s(152) =< 3*V15+2*V16
aux(35) =< 1
aux(36) =< 2
aux(37) =< 3*V15+3
s(130) =< aux(35)
s(132) =< aux(36)
s(134) =< aux(37)
s(135) =< aux(37)*2
s(153) =< s(152)
s(154) =< s(152)*2
s(162) =< s(161)
s(163) =< s(161)*2

  with precondition: [V>=1] 

* Chain [31]: 2
  with precondition: [V=1,V2>=0,V11>=0] 

* Chain [30]: 1*s(164)+1*s(165)+37*s(167)+3*s(168)+9
  Such that:s(164) =< 1
s(165) =< 2
s(166) =< 3*V+2*V2
s(167) =< s(166)
s(168) =< s(166)*2

  with precondition: [V>=0,V2>=1,V2+2*V>=3] 


Closed-form bounds of start(V,V2,V11,V17,V15,V16): 
-------------------------------------
* Chain [33] with precondition: [V=0] 
    - Upper bound: nat(2*V16+2)*43+96+nat(3*V2+3)*129+nat(3*V2+2*V11)*43+nat(3*V11+3)*129+nat(3*V11+2*V17)*43+nat(3*V17+3)*129+nat(3*V17+2*V15)*43+nat(3*V15+2*V16)*43 
    - Complexity: n 
* Chain [32] with precondition: [V>=1] 
    - Upper bound: V+45+nat(V2)+129*V+nat(3*V15+3)*129+nat(3*V15+2*V16)*43 
    - Complexity: n 
* Chain [31] with precondition: [V=1,V2>=0,V11>=0] 
    - Upper bound: 2 
    - Complexity: constant 
* Chain [30] with precondition: [V>=0,V2>=1,V2+2*V>=3] 
    - Upper bound: 129*V+86*V2+12 
    - Complexity: n 

### Maximum cost of start(V,V2,V11,V17,V15,V16): max([nat(3*V+2*V2)*43+10,nat(3*V15+2*V16)*43+43+max([nat(V2)+V+129*V+nat(3*V15+3)*129,nat(2*V16+2)*43+51+nat(3*V2+3)*129+nat(3*V2+2*V11)*43+nat(3*V11+3)*129+nat(3*V11+2*V17)*43+nat(3*V17+3)*129+nat(3*V17+2*V15)*43])])+2 
Asymptotic class: n 
* Total analysis performed in 597 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:

   p(s(x)) -> x
   p(0') -> 0'
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
   if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y)
   if(false, b1, b2, b3, x, y) -> average(p(x), s(y))
   if2(true, b2, b3, x, y) -> 0'
   if2(false, b2, b3, x, y) -> if3(b2, b3, x, y)
   if3(true, b3, x, y) -> 0'
   if3(false, b3, x, y) -> if4(b3, x, y)
   if4(true, x, y) -> s(0')
   if4(false, x, y) -> average(s(x), p(p(y)))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
p(0') -> 0'
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) -> average(p(x), s(y))
if2(true, b2, b3, x, y) -> 0'
if2(false, b2, b3, x, y) -> if3(b2, b3, x, y)
if3(true, b3, x, y) -> 0'
if3(false, b3, x, y) -> if4(b3, x, y)
if4(true, x, y) -> s(0')
if4(false, x, y) -> average(s(x), p(p(y)))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
average :: s:0' -> s:0' -> s:0'
if :: true:false -> true:false -> true:false -> true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> true:false -> true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> true:false -> s:0' -> s:0' -> s:0'
if4 :: 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:
le, average

They will be analysed ascendingly in the following order:
le < average

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

(16)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
p(0') -> 0'
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) -> average(p(x), s(y))
if2(true, b2, b3, x, y) -> 0'
if2(false, b2, b3, x, y) -> if3(b2, b3, x, y)
if3(true, b3, x, y) -> 0'
if3(false, b3, x, y) -> if4(b3, x, y)
if4(true, x, y) -> s(0')
if4(false, x, y) -> average(s(x), p(p(y)))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
average :: s:0' -> s:0' -> s:0'
if :: true:false -> true:false -> true:false -> true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> true:false -> true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> true:false -> s:0' -> s:0' -> s:0'
if4 :: 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:
le, average

They will be analysed ascendingly in the following order:
le < average

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

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

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

Induction Step:
le(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_s:0'3_0(n5_0), gen_s:0'3_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:
p(s(x)) -> x
p(0') -> 0'
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) -> average(p(x), s(y))
if2(true, b2, b3, x, y) -> 0'
if2(false, b2, b3, x, y) -> if3(b2, b3, x, y)
if3(true, b3, x, y) -> 0'
if3(false, b3, x, y) -> if4(b3, x, y)
if4(true, x, y) -> s(0')
if4(false, x, y) -> average(s(x), p(p(y)))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
average :: s:0' -> s:0' -> s:0'
if :: true:false -> true:false -> true:false -> true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> true:false -> true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> true:false -> s:0' -> s:0' -> s:0'
if4 :: 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:
le, average

They will be analysed ascendingly in the following order:
le < average

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
p(s(x)) -> x
p(0') -> 0'
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) -> average(p(x), s(y))
if2(true, b2, b3, x, y) -> 0'
if2(false, b2, b3, x, y) -> if3(b2, b3, x, y)
if3(true, b3, x, y) -> 0'
if3(false, b3, x, y) -> if4(b3, x, y)
if4(true, x, y) -> s(0')
if4(false, x, y) -> average(s(x), p(p(y)))

Types:
p :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> true:false
true :: true:false
false :: true:false
average :: s:0' -> s:0' -> s:0'
if :: true:false -> true:false -> true:false -> true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> true:false -> true:false -> s:0' -> s:0' -> s:0'
if3 :: true:false -> true:false -> s:0' -> s:0' -> s:0'
if4 :: 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:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, 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:
average