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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(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, 501 ms]
    (10) BOUNDS(1, n^2)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2))
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil)
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
   leq#2(0, x8) -> True
   leq#2(S(x12), 0) -> False
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
   main(x3) -> fold#3(insert_ord(leq), x3)

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^2).


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil [1]
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2)) [1]
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2)) [1]
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil) [1]
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
   leq#2(0, x8) -> True [1]
   leq#2(S(x12), 0) -> False [1]
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
   main(x3) -> fold#3(insert_ord(leq), x3) [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:

   fold#3(insert_ord(x2), Nil) -> Nil [1]
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2)) [1]
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2)) [1]
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil) [1]
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
   leq#2(0, x8) -> True [1]
   leq#2(S(x12), 0) -> False [1]
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
   main(x3) -> fold#3(insert_ord(leq), x3) [1]

The TRS has the following type information:
fold#3 :: insert_ord -> Nil:Cons -> Nil:Cons
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons
Cons :: 0:S -> Nil:Cons -> Nil:Cons
insert_ord#2 :: leq -> 0:S -> Nil:Cons -> Nil:Cons
cond_insert_ord_x_ys_1 :: True:False -> 0:S -> 0:S -> Nil:Cons -> Nil:Cons
True :: True:False
False :: True:False
leq :: leq
leq#2 :: 0:S -> 0:S -> True:False
0 :: 0:S
S :: 0:S -> 0:S
main :: Nil:Cons -> Nil:Cons

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:

   fold#3(v0, v1) -> null_fold#3 [0]
   insert_ord#2(v0, v1, v2) -> null_insert_ord#2 [0]

And the following fresh constants:    null_fold#3, null_insert_ord#2, const

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

(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:

   fold#3(insert_ord(x2), Nil) -> Nil [1]
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2)) [1]
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1)) [1]
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2)) [1]
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil) [1]
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) [1]
   leq#2(0, x8) -> True [1]
   leq#2(S(x12), 0) -> False [1]
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2) [1]
   main(x3) -> fold#3(insert_ord(leq), x3) [1]
   fold#3(v0, v1) -> null_fold#3 [0]
   insert_ord#2(v0, v1, v2) -> null_insert_ord#2 [0]

The TRS has the following type information:
fold#3 :: insert_ord -> Nil:Cons:null_fold#3:null_insert_ord#2 -> Nil:Cons:null_fold#3:null_insert_ord#2
insert_ord :: leq -> insert_ord
Nil :: Nil:Cons:null_fold#3:null_insert_ord#2
Cons :: 0:S -> Nil:Cons:null_fold#3:null_insert_ord#2 -> Nil:Cons:null_fold#3:null_insert_ord#2
insert_ord#2 :: leq -> 0:S -> Nil:Cons:null_fold#3:null_insert_ord#2 -> Nil:Cons:null_fold#3:null_insert_ord#2
cond_insert_ord_x_ys_1 :: True:False -> 0:S -> 0:S -> Nil:Cons:null_fold#3:null_insert_ord#2 -> Nil:Cons:null_fold#3:null_insert_ord#2
True :: True:False
False :: True:False
leq :: leq
leq#2 :: 0:S -> 0:S -> True:False
0 :: 0:S
S :: 0:S -> 0:S
main :: Nil:Cons:null_fold#3:null_insert_ord#2 -> Nil:Cons:null_fold#3:null_insert_ord#2
null_fold#3 :: Nil:Cons:null_fold#3:null_insert_ord#2
null_insert_ord#2 :: Nil:Cons:null_fold#3:null_insert_ord#2
const :: insert_ord

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:

   Nil => 0
   True => 1
   False => 0
   leq => 0
   0 => 0
   null_fold#3 => 0
   null_insert_ord#2 => 0
   const => 0

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

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

   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + x3 + (1 + x2 + x1) :|: x1 >= 0, z = 1, z' = x3, z1 = x1, z'' = x2, x3 >= 0, x2 >= 0
   cond_insert_ord_x_ys_1(z, z', z'', z1) -{ 1 }-> 1 + x5 + insert_ord#2(0, x0, x2) :|: x0 >= 0, x5 >= 0, z1 = x2, z'' = x5, z = 0, x2 >= 0, z' = x0
   fold#3(z, z') -{ 1 }-> insert_ord#2(x6, x4, fold#3(1 + x6, x2)) :|: x4 >= 0, z = 1 + x6, z' = 1 + x4 + x2, x6 >= 0, x2 >= 0
   fold#3(z, z') -{ 1 }-> 0 :|: z = 1 + x2, x2 >= 0, z' = 0
   fold#3(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   insert_ord#2(z, z', z'') -{ 1 }-> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2) :|: z' = x6, x4 >= 0, z'' = 1 + x4 + x2, x6 >= 0, z = 0, x2 >= 0
   insert_ord#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   insert_ord#2(z, z', z'') -{ 1 }-> 1 + x2 + 0 :|: z'' = 0, z' = x2, z = 0, x2 >= 0
   leq#2(z, z') -{ 1 }-> leq#2(x4, x2) :|: x4 >= 0, z' = 1 + x2, z = 1 + x4, x2 >= 0
   leq#2(z, z') -{ 1 }-> 1 :|: x8 >= 0, z = 0, z' = x8
   leq#2(z, z') -{ 1 }-> 0 :|: z = 1 + x12, x12 >= 0, z' = 0
   main(z) -{ 1 }-> fold#3(1 + 0, x3) :|: z = x3, x3 >= 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, V6, V10),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6, V10),0,[fun2(V1, V, V6, V10, Out)],[V1 >= 0,V >= 0,V6 >= 0,V10 >= 0]).
eq(start(V1, V, V6, V10),0,[fun1(V1, V, V6, Out)],[V1 >= 0,V >= 0,V6 >= 0]).
eq(start(V1, V, V6, V10),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6, V10),0,[main(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, Out),1,[],[Out = 0,V1 = 1 + V2,V2 >= 0,V = 0]).
eq(fun(V1, V, Out),1,[fun(1 + V4, V3, Ret2),fun1(V4, V5, Ret2, Ret)],[Out = Ret,V5 >= 0,V1 = 1 + V4,V = 1 + V3 + V5,V4 >= 0,V3 >= 0]).
eq(fun2(V1, V, V6, V10, Out),1,[],[Out = 2 + V7 + V8 + V9,V8 >= 0,V1 = 1,V = V7,V10 = V8,V6 = V9,V7 >= 0,V9 >= 0]).
eq(fun2(V1, V, V6, V10, Out),1,[fun1(0, V12, V11, Ret1)],[Out = 1 + Ret1 + V13,V12 >= 0,V13 >= 0,V10 = V11,V6 = V13,V1 = 0,V11 >= 0,V = V12]).
eq(fun1(V1, V, V6, Out),1,[],[Out = 1 + V14,V6 = 0,V = V14,V1 = 0,V14 >= 0]).
eq(fun1(V1, V, V6, Out),1,[fun3(V17, V16, Ret0),fun2(Ret0, V17, V16, V15, Ret3)],[Out = Ret3,V = V17,V16 >= 0,V6 = 1 + V15 + V16,V17 >= 0,V1 = 0,V15 >= 0]).
eq(fun3(V1, V, Out),1,[],[Out = 1,V18 >= 0,V1 = 0,V = V18]).
eq(fun3(V1, V, Out),1,[],[Out = 0,V1 = 1 + V19,V19 >= 0,V = 0]).
eq(fun3(V1, V, Out),1,[fun3(V21, V20, Ret4)],[Out = Ret4,V21 >= 0,V = 1 + V20,V1 = 1 + V21,V20 >= 0]).
eq(main(V1, Out),1,[fun(1 + 0, V22, Ret5)],[Out = Ret5,V1 = V22,V22 >= 0]).
eq(fun(V1, V, Out),0,[],[Out = 0,V24 >= 0,V23 >= 0,V1 = V24,V = V23]).
eq(fun1(V1, V, V6, Out),0,[],[Out = 0,V26 >= 0,V6 = V27,V25 >= 0,V1 = V26,V = V25,V27 >= 0]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,V6,V10,Out),[V1,V,V6,V10],[Out]).
input_output_vars(fun1(V1,V,V6,Out),[V1,V,V6],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).
input_output_vars(main(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [fun3/3]
1. recursive  : [fun1/4,fun2/5]
2. recursive [non_tail] : [fun/3]
3. non_recursive  : [main/2]
4. non_recursive  : [start/4]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun3/3
1. SCC is partially evaluated into fun1/4
2. SCC is partially evaluated into fun/3
3. SCC is completely evaluated into other SCCs
4. SCC is partially evaluated into start/4

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

### Specialization of cost equations fun3/3 
* CE 16 is refined into CE [17] 
* CE 15 is refined into CE [18] 
* CE 14 is refined into CE [19] 


### Cost equations --> "Loop" of fun3/3 
* CEs [18] --> Loop 12 
* CEs [19] --> Loop 13 
* CEs [17] --> Loop 14 

### Ranking functions of CR fun3(V1,V,Out) 
* RF of phase [14]: [V,V1]

#### Partial ranking functions of CR fun3(V1,V,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V
    V1


### Specialization of cost equations fun1/4 
* CE 10 is refined into CE [20] 
* CE 8 is refined into CE [21,22] 
* CE 9 is refined into CE [23] 
* CE 7 is refined into CE [24,25] 


### Cost equations --> "Loop" of fun1/4 
* CEs [25] --> Loop 15 
* CEs [24] --> Loop 16 
* CEs [20] --> Loop 17 
* CEs [22] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [21] --> Loop 20 

### Ranking functions of CR fun1(V1,V,V6,Out) 
* RF of phase [15,16]: [V6]

#### Partial ranking functions of CR fun1(V1,V,V6,Out) 
* Partial RF of phase [15,16]:
  - RF of loop [15:1]:
    V6-1
  - RF of loop [16:1]:
    V6


### Specialization of cost equations fun/3 
* CE 11 is refined into CE [26] 
* CE 13 is refined into CE [27] 
* CE 12 is refined into CE [28,29,30,31,32] 


### Cost equations --> "Loop" of fun/3 
* CEs [32] --> Loop 21 
* CEs [31] --> Loop 22 
* CEs [30] --> Loop 23 
* CEs [28] --> Loop 24 
* CEs [29] --> Loop 25 
* CEs [26,27] --> Loop 26 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [21,22,23,24,25]: [V]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [21,22,23,24,25]:
  - RF of loop [21:1,24:1,25:1]:
    V
  - RF of loop [22:1,23:1]:
    V/2-1/2


### Specialization of cost equations start/4 
* CE 2 is refined into CE [33] 
* CE 1 is refined into CE [34,35,36,37,38] 
* CE 3 is refined into CE [39,40] 
* CE 4 is refined into CE [41,42,43,44,45] 
* CE 5 is refined into CE [46,47,48,49] 
* CE 6 is refined into CE [50,51] 


### Cost equations --> "Loop" of start/4 
* CEs [47] --> Loop 27 
* CEs [33] --> Loop 28 
* CEs [35] --> Loop 29 
* CEs [42] --> Loop 30 
* CEs [41] --> Loop 31 
* CEs [34,36,37,38,39,40,43,44,45,46,48,49,50,51] --> Loop 32 

### Ranking functions of CR start(V1,V,V6,V10) 

#### Partial ranking functions of CR start(V1,V,V6,V10) 


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

#### Cost of chains of fun3(V1,V,Out):
* Chain [[14],13]: 1*it(14)+1
  Such that:it(14) =< V1

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

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

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

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

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


#### Cost of chains of fun1(V1,V,V6,Out):
* Chain [[15,16],19]: 7*it(15)+1
  Such that:aux(3) =< -V+Out
it(15) =< aux(3)

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

* Chain [[15,16],18]: 7*it(15)+1*s(4)+3
  Such that:s(4) =< -V6+Out
aux(1) =< V6
aux(2) =< 2*V6-Out
it(15) =< aux(1)
it(15) =< aux(2)

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

* Chain [[15,16],17]: 7*it(15)+0
  Such that:aux(1) =< V6
aux(2) =< Out
it(15) =< aux(1)
it(15) =< aux(2)

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

* Chain [20]: 3
  with precondition: [V1=0,V=0,V6+1=Out,V6>=1] 

* Chain [19]: 1
  with precondition: [V1=0,V6=0,V+1=Out,V>=0] 

* Chain [18]: 1*s(4)+3
  Such that:s(4) =< V

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

* Chain [17]: 0
  with precondition: [Out=0,V1>=0,V>=0,V6>=0] 


#### Cost of chains of fun(V1,V,Out):
* Chain [[21,22,23,24,25],26]: 9*it(21)+5*it(22)+7*s(31)+7*s(34)+7*s(35)+1
  Such that:aux(14) =< V
aux(15) =< V/2
it(21) =< aux(14)
it(22) =< aux(14)
it(22) =< aux(15)
aux(9) =< aux(14)+1
aux(8) =< aux(14)
aux(7) =< aux(14)-2
s(32) =< it(22)*aux(14)
s(38) =< it(22)*aux(9)
s(37) =< it(22)*aux(8)
s(36) =< it(22)*aux(7)
s(34) =< s(38)
s(35) =< s(37)
s(35) =< s(36)
s(31) =< s(32)

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

* Chain [26]: 1
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of start(V1,V,V6,V10):
* Chain [32]: 12*s(41)+2*s(42)+7*s(45)+7*s(46)+7*s(49)+5*s(53)+7*s(61)+7*s(62)+7*s(63)+7*s(69)+7*s(70)+7*s(73)+10*s(75)+5*s(79)+7*s(87)+7*s(88)+7*s(89)+4
  Such that:s(77) =< V1/2
s(68) =< -V+V6
s(44) =< -V+V10
s(51) =< V/2
s(64) =< V6+1
s(40) =< V10+1
aux(18) =< V1
aux(19) =< V
aux(20) =< V+1
aux(21) =< V6
aux(22) =< V10
s(75) =< aux(18)
s(41) =< aux(19)
s(42) =< aux(20)
s(79) =< aux(18)
s(79) =< s(77)
s(80) =< aux(18)+1
s(81) =< aux(18)
s(82) =< aux(18)-2
s(83) =< s(79)*aux(18)
s(84) =< s(79)*s(80)
s(85) =< s(79)*s(81)
s(86) =< s(79)*s(82)
s(87) =< s(84)
s(88) =< s(85)
s(88) =< s(86)
s(89) =< s(83)
s(69) =< s(64)
s(70) =< aux(21)
s(70) =< s(68)
s(45) =< s(40)
s(46) =< aux(22)
s(46) =< s(44)
s(53) =< aux(19)
s(53) =< s(51)
s(54) =< aux(19)+1
s(55) =< aux(19)
s(56) =< aux(19)-2
s(57) =< s(53)*aux(19)
s(58) =< s(53)*s(54)
s(59) =< s(53)*s(55)
s(60) =< s(53)*s(56)
s(61) =< s(58)
s(62) =< s(59)
s(62) =< s(60)
s(63) =< s(57)
s(73) =< aux(21)
s(49) =< aux(22)

  with precondition: [V1>=0] 

* Chain [31]: 3
  with precondition: [V1=0,V=0,V6>=1] 

* Chain [30]: 1
  with precondition: [V1=0,V6=0,V>=0] 

* Chain [29]: 2
  with precondition: [V1=0,V10=0,V>=0,V6>=0] 

* Chain [28]: 1
  with precondition: [V1=1,V>=0,V6>=0,V10>=0] 

* Chain [27]: 1
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V,V6,V10): 
-------------------------------------
* Chain [32] with precondition: [V1>=0] 
    - Upper bound: 22*V1+4+21*V1*V1+nat(V)*24+nat(V)*21*nat(V)+nat(V6)*14+nat(V10)*14+nat(V+1)*2+nat(V6+1)*7+nat(V10+1)*7 
    - Complexity: n^2 
* Chain [31] with precondition: [V1=0,V=0,V6>=1] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [30] with precondition: [V1=0,V6=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [29] with precondition: [V1=0,V10=0,V>=0,V6>=0] 
    - Upper bound: 2 
    - Complexity: constant 
* Chain [28] with precondition: [V1=1,V>=0,V6>=0,V10>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [27] with precondition: [V=0,V1>=1] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V1,V,V6,V10): max([2,22*V1+3+21*V1*V1+nat(V)*24+nat(V)*21*nat(V)+nat(V6)*14+nat(V10)*14+nat(V+1)*2+nat(V6+1)*7+nat(V10+1)*7])+1 
Asymptotic class: n^2 
* Total analysis performed in 401 ms.


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(10)
BOUNDS(1, n^2)

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(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2))
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil)
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
   leq#2(0, x8) -> True
   leq#2(S(x12), 0) -> False
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
   main(x3) -> fold#3(insert_ord(leq), x3)

S is empty.
Rewrite Strategy: INNERMOST
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(13) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

leq#2(S(x4), S(x2)) ->^+ leq#2(x4, x2)

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

The pumping substitution is [x4 / S(x4), x2 / S(x2)].

The result substitution is [ ].




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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2))
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil)
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
   leq#2(0, x8) -> True
   leq#2(S(x12), 0) -> False
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
   main(x3) -> fold#3(insert_ord(leq), x3)

S is empty.
Rewrite Strategy: INNERMOST
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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   fold#3(insert_ord(x2), Nil) -> Nil
   fold#3(insert_ord(x6), Cons(x4, x2)) -> insert_ord#2(x6, x4, fold#3(insert_ord(x6), x2))
   cond_insert_ord_x_ys_1(True, x3, x2, x1) -> Cons(x3, Cons(x2, x1))
   cond_insert_ord_x_ys_1(False, x0, x5, x2) -> Cons(x5, insert_ord#2(leq, x0, x2))
   insert_ord#2(leq, x2, Nil) -> Cons(x2, Nil)
   insert_ord#2(leq, x6, Cons(x4, x2)) -> cond_insert_ord_x_ys_1(leq#2(x6, x4), x6, x4, x2)
   leq#2(0, x8) -> True
   leq#2(S(x12), 0) -> False
   leq#2(S(x4), S(x2)) -> leq#2(x4, x2)
   main(x3) -> fold#3(insert_ord(leq), x3)

S is empty.
Rewrite Strategy: INNERMOST