WORST_CASE(Omega(n^1),O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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), 12 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 824 ms]
    (10) BOUNDS(1, n^3)
(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^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if_minus :: true:false -> 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
log :: 0:s -> 0:s

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:

   if_minus(v0, v1, v2) -> null_if_minus [0]
   quot(v0, v1) -> null_quot [0]
   log(v0) -> null_log [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]

And the following fresh constants:    null_if_minus, null_quot, null_log, null_le, null_minus

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

(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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1]
   if_minus(true, s(x), y) -> 0 [1]
   if_minus(false, s(x), y) -> s(minus(x, y)) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(quot(x, s(s(0)))))) [1]
   if_minus(v0, v1, v2) -> null_if_minus [0]
   quot(v0, v1) -> null_quot [0]
   log(v0) -> null_log [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]

The TRS has the following type information:
le :: 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus -> true:false:null_le
0 :: 0:s:null_if_minus:null_quot:null_log:null_minus
true :: true:false:null_le
s :: 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus
false :: true:false:null_le
minus :: 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus
if_minus :: true:false:null_le -> 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus
quot :: 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus
log :: 0:s:null_if_minus:null_quot:null_log:null_minus -> 0:s:null_if_minus:null_quot:null_log:null_minus
null_if_minus :: 0:s:null_if_minus:null_quot:null_log:null_minus
null_quot :: 0:s:null_if_minus:null_quot:null_log:null_minus
null_log :: 0:s:null_if_minus:null_quot:null_log:null_minus
null_le :: true:false:null_le
null_minus :: 0:s:null_if_minus:null_quot:null_log:null_minus

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
   null_if_minus => 0
   null_quot => 0
   null_log => 0
   null_le => 0
   null_minus => 0

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

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

   if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = 1 + x, z'' = y, x >= 0, y >= 0
   if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + 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 }-> 2 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   log(z) -{ 1 }-> 1 + log(1 + quot(x, 1 + (1 + 0))) :|: x >= 0, z = 1 + (1 + x)
   minus(z, z') -{ 1 }-> if_minus(le(1 + x, y), 1 + x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x

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, V11),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V11),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V11),0,[fun(V1, V, V11, Out)],[V1 >= 0,V >= 0,V11 >= 0]).
eq(start(V1, V, V11),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V11),0,[log(V1, Out)],[V1 >= 0]).
eq(le(V1, V, Out),1,[],[Out = 2,V2 >= 0,V1 = 0,V = V2]).
eq(le(V1, V, Out),1,[],[Out = 1,V3 >= 0,V1 = 1 + V3,V = 0]).
eq(le(V1, V, Out),1,[le(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(minus(V1, V, Out),1,[],[Out = 0,V6 >= 0,V1 = 0,V = V6]).
eq(minus(V1, V, Out),1,[le(1 + V7, V8, Ret0),fun(Ret0, 1 + V7, V8, Ret1)],[Out = Ret1,V7 >= 0,V8 >= 0,V1 = 1 + V7,V = V8]).
eq(fun(V1, V, V11, Out),1,[],[Out = 0,V1 = 2,V = 1 + V9,V11 = V10,V9 >= 0,V10 >= 0]).
eq(fun(V1, V, V11, Out),1,[minus(V12, V13, Ret11)],[Out = 1 + Ret11,V = 1 + V12,V11 = V13,V1 = 1,V12 >= 0,V13 >= 0]).
eq(quot(V1, V, Out),1,[],[Out = 0,V = 1 + V14,V14 >= 0,V1 = 0]).
eq(quot(V1, V, Out),1,[minus(V15, V16, Ret10),quot(Ret10, 1 + V16, Ret12)],[Out = 1 + Ret12,V = 1 + V16,V15 >= 0,V16 >= 0,V1 = 1 + V15]).
eq(log(V1, Out),1,[],[Out = 0,V1 = 1]).
eq(log(V1, Out),1,[quot(V17, 1 + (1 + 0), Ret101),log(1 + Ret101, Ret13)],[Out = 1 + Ret13,V17 >= 0,V1 = 2 + V17]).
eq(fun(V1, V, V11, Out),0,[],[Out = 0,V19 >= 0,V11 = V20,V18 >= 0,V1 = V19,V = V18,V20 >= 0]).
eq(quot(V1, V, Out),0,[],[Out = 0,V22 >= 0,V21 >= 0,V1 = V22,V = V21]).
eq(log(V1, Out),0,[],[Out = 0,V23 >= 0,V1 = V23]).
eq(le(V1, V, Out),0,[],[Out = 0,V24 >= 0,V25 >= 0,V1 = V24,V = V25]).
eq(minus(V1, V, Out),0,[],[Out = 0,V26 >= 0,V27 >= 0,V1 = V26,V = V27]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,V11,Out),[V1,V,V11],[Out]).
input_output_vars(quot(V1,V,Out),[V1,V],[Out]).
input_output_vars(log(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [le/3]
1. recursive  : [fun/4,minus/3]
2. recursive  : [quot/3]
3. recursive  : [log/2]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into le/3
1. SCC is partially evaluated into minus/3
2. SCC is partially evaluated into quot/3
3. SCC is partially evaluated into log/2
4. SCC is partially evaluated into start/3

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

### Specialization of cost equations le/3 
* CE 16 is refined into CE [23] 
* CE 14 is refined into CE [24] 
* CE 13 is refined into CE [25] 
* CE 15 is refined into CE [26] 


### Cost equations --> "Loop" of le/3 
* CEs [26] --> Loop 13 
* CEs [23] --> Loop 14 
* CEs [24] --> Loop 15 
* CEs [25] --> Loop 16 

### Ranking functions of CR le(V1,V,Out) 
* RF of phase [13]: [V,V1]

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


### Specialization of cost equations minus/3 
* CE 8 is refined into CE [27,28,29,30] 
* CE 10 is refined into CE [31] 
* CE 11 is refined into CE [32] 
* CE 12 is refined into CE [33] 
* CE 9 is refined into CE [34,35] 


### Cost equations --> "Loop" of minus/3 
* CEs [35] --> Loop 17 
* CEs [34] --> Loop 18 
* CEs [27] --> Loop 19 
* CEs [28,29,30,31,32,33] --> Loop 20 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [17]: [V1-1,V1-V]
* RF of phase [18]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V1-1
    V1-V
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V1


### Specialization of cost equations quot/3 
* CE 17 is refined into CE [36] 
* CE 19 is refined into CE [37] 
* CE 18 is refined into CE [38,39,40] 


### Cost equations --> "Loop" of quot/3 
* CEs [40] --> Loop 21 
* CEs [38] --> Loop 22 
* CEs [39] --> Loop 23 
* CEs [36,37] --> Loop 24 

### Ranking functions of CR quot(V1,V,Out) 
* RF of phase [21]: [V1/2-1,V1/2-V/2]
* RF of phase [23]: [V1-1]

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


### Specialization of cost equations log/2 
* CE 20 is refined into CE [41] 
* CE 22 is refined into CE [42] 
* CE 21 is refined into CE [43,44,45,46] 


### Cost equations --> "Loop" of log/2 
* CEs [46] --> Loop 25 
* CEs [45] --> Loop 26 
* CEs [44] --> Loop 27 
* CEs [43] --> Loop 28 
* CEs [41,42] --> Loop 29 

### Ranking functions of CR log(V1,Out) 
* RF of phase [25,26]: [V1-4,V1/2-2]

#### Partial ranking functions of CR log(V1,Out) 
* Partial RF of phase [25,26]:
  - RF of loop [25:1]:
    V1/2-2
  - RF of loop [26:1]:
    V1/3-4/3


### Specialization of cost equations start/3 
* CE 3 is refined into CE [47] 
* CE 1 is refined into CE [48] 
* CE 2 is refined into CE [49,50,51] 
* CE 4 is refined into CE [52,53,54,55,56] 
* CE 5 is refined into CE [57,58,59] 
* CE 6 is refined into CE [60,61,62,63,64,65] 
* CE 7 is refined into CE [66,67,68,69,70,71] 


### Cost equations --> "Loop" of start/3 
* CEs [60,61] --> Loop 30 
* CEs [53,58] --> Loop 31 
* CEs [47] --> Loop 32 
* CEs [49,50,51] --> Loop 33 
* CEs [48,52,54,55,56,57,59,62,63,64,65,66,67,68,69,70,71] --> Loop 34 

### Ranking functions of CR start(V1,V,V11) 

#### Partial ranking functions of CR start(V1,V,V11) 


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

#### Cost of chains of le(V1,V,Out):
* Chain [[13],16]: 1*it(13)+1
  Such that:it(13) =< V1

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

* Chain [[13],15]: 1*it(13)+1
  Such that:it(13) =< V

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

* Chain [[13],14]: 1*it(13)+0
  Such that:it(13) =< V

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

* Chain [16]: 1
  with precondition: [V1=0,Out=2,V>=0] 

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

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[18],20]: 3*it(18)+2*s(4)+3
  Such that:aux(1) =< V1-Out
it(18) =< Out
s(4) =< aux(1)

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

* Chain [[18],19]: 3*it(18)+2
  Such that:it(18) =< Out

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

* Chain [[17],20]: 3*it(17)+2*s(2)+2*s(4)+1*s(8)+3
  Such that:aux(1) =< V1-Out
it(17) =< Out
aux(4) =< V
s(4) =< aux(1)
s(2) =< aux(4)
s(8) =< it(17)*aux(4)

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

* Chain [20]: 2*s(2)+2*s(4)+3
  Such that:aux(1) =< V1
aux(2) =< V
s(4) =< aux(1)
s(2) =< aux(2)

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

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


#### Cost of chains of quot(V1,V,Out):
* Chain [[23],24]: 6*it(23)+6*s(25)+1
  Such that:aux(9) =< V1
it(23) =< aux(9)
s(28) =< it(23)*aux(9)
s(25) =< s(28)

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

* Chain [[23],22,24]: 8*it(23)+6*s(25)+2*s(32)+5
  Such that:s(30) =< 1
aux(10) =< V1
it(23) =< aux(10)
s(32) =< s(30)
s(28) =< it(23)*aux(10)
s(25) =< s(28)

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

* Chain [[21],24]: 4*it(21)+3*s(45)+4*s(46)+1*s(48)+1
  Such that:aux(13) =< V1
aux(12) =< V1+1
aux(11) =< V1-V
it(21) =< V1/2-V/2
s(41) =< V
s(49) =< aux(12)
s(49) =< aux(13)
s(45) =< it(21)*aux(11)
s(46) =< s(49)
s(48) =< s(45)*s(41)

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

* Chain [[21],22,24]: 6*it(21)+2*s(32)+3*s(45)+4*s(46)+1*s(48)+5
  Such that:aux(12) =< V1+1
aux(11) =< V1-V
aux(14) =< V1
aux(15) =< V
it(21) =< aux(14)
s(32) =< aux(15)
s(49) =< aux(12)
s(49) =< aux(14)
s(45) =< it(21)*aux(11)
s(46) =< s(49)
s(48) =< s(45)*aux(15)

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

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

* Chain [22,24]: 2*s(31)+2*s(32)+5
  Such that:s(29) =< V1
s(30) =< V
s(31) =< s(29)
s(32) =< s(30)

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


#### Cost of chains of log(V1,Out):
* Chain [[25,26],29]: 6*it(25)+2*it(26)+10*s(82)+2*s(83)+3*s(84)+1*s(85)+4*s(88)+3*s(89)+4*s(90)+1*s(91)+1
  Such that:aux(20) =< 2
it(26) =< V1/3
aux(26) =< 5/2*V1
aux(25) =< 5/2*V1+45/2
aux(28) =< 13/6*V1
aux(27) =< 13/6*V1+91/6
aux(29) =< V1
aux(30) =< V1/2
aux(18) =< aux(29)
it(25) =< aux(29)
it(26) =< aux(29)
aux(18) =< aux(30)
it(25) =< aux(30)
it(26) =< aux(30)
it(26) =< aux(25)
s(87) =< aux(25)
it(26) =< aux(26)
s(87) =< aux(26)
aux(19) =< aux(27)
aux(19) =< aux(28)
s(78) =< aux(29)
s(88) =< aux(19)*(1/2)
s(86) =< aux(18)*2
s(89) =< s(88)*s(78)
s(90) =< aux(19)
s(91) =< s(89)*aux(20)
s(82) =< s(87)
s(83) =< s(86)
s(84) =< s(82)*aux(29)
s(85) =< s(84)*aux(20)

  with precondition: [Out>=1,V1>=4*Out+1] 

* Chain [[25,26],28,29]: 6*it(25)+2*it(26)+10*s(82)+2*s(83)+3*s(84)+1*s(85)+4*s(88)+3*s(89)+4*s(90)+1*s(91)+3
  Such that:aux(20) =< 2
it(26) =< V1/3
aux(26) =< 5/2*V1
aux(25) =< 5/2*V1+45/2
aux(28) =< 13/6*V1
aux(27) =< 13/6*V1+91/6
aux(31) =< V1
aux(32) =< V1/2
aux(18) =< aux(31)
it(25) =< aux(31)
it(26) =< aux(31)
aux(18) =< aux(32)
it(25) =< aux(32)
it(26) =< aux(32)
it(26) =< aux(25)
s(87) =< aux(25)
it(26) =< aux(26)
s(87) =< aux(26)
aux(19) =< aux(27)
aux(19) =< aux(28)
s(78) =< aux(31)
s(88) =< aux(19)*(1/2)
s(86) =< aux(18)*2
s(89) =< s(88)*s(78)
s(90) =< aux(19)
s(91) =< s(89)*aux(20)
s(82) =< s(87)
s(83) =< s(86)
s(84) =< s(82)*aux(31)
s(85) =< s(84)*aux(20)

  with precondition: [Out>=2,V1+3>=4*Out] 

* Chain [[25,26],27,29]: 6*it(25)+2*it(26)+10*s(82)+2*s(83)+3*s(84)+1*s(85)+4*s(88)+3*s(89)+4*s(90)+1*s(91)+2*s(95)+2*s(96)+7
  Such that:aux(26) =< 5*V1
aux(28) =< 13*V1
aux(23) =< V1/2
aux(25) =< 5/2*V1+45/2
aux(27) =< 13/6*V1+91/6
aux(33) =< 2
aux(34) =< V1
it(26) =< aux(34)
s(95) =< aux(34)
s(96) =< aux(33)
aux(18) =< aux(34)
it(25) =< aux(34)
aux(18) =< aux(23)
it(25) =< aux(23)
it(26) =< aux(23)
it(26) =< aux(25)
s(87) =< aux(25)
it(26) =< aux(26)
s(87) =< aux(26)
aux(19) =< aux(27)
aux(19) =< aux(28)
s(78) =< aux(34)
s(88) =< aux(19)*(1/2)
s(86) =< aux(18)*2
s(89) =< s(88)*s(78)
s(90) =< aux(19)
s(91) =< s(89)*aux(33)
s(82) =< s(87)
s(83) =< s(86)
s(84) =< s(82)*aux(34)
s(85) =< s(84)*aux(33)

  with precondition: [Out>=2,V1+3>=4*Out] 

* Chain [[25,26],27,28,29]: 6*it(25)+2*it(26)+10*s(82)+2*s(83)+3*s(84)+1*s(85)+4*s(88)+3*s(89)+4*s(90)+1*s(91)+2*s(95)+2*s(96)+9
  Such that:aux(26) =< 5*V1
aux(28) =< 13*V1
aux(23) =< V1/2
aux(25) =< 5/2*V1+45/2
aux(27) =< 13/6*V1+91/6
aux(35) =< 2
aux(36) =< V1
it(26) =< aux(36)
s(95) =< aux(36)
s(96) =< aux(35)
aux(18) =< aux(36)
it(25) =< aux(36)
aux(18) =< aux(23)
it(25) =< aux(23)
it(26) =< aux(23)
it(26) =< aux(25)
s(87) =< aux(25)
it(26) =< aux(26)
s(87) =< aux(26)
aux(19) =< aux(27)
aux(19) =< aux(28)
s(78) =< aux(36)
s(88) =< aux(19)*(1/2)
s(86) =< aux(18)*2
s(89) =< s(88)*s(78)
s(90) =< aux(19)
s(91) =< s(89)*aux(35)
s(82) =< s(87)
s(83) =< s(86)
s(84) =< s(82)*aux(36)
s(85) =< s(84)*aux(35)

  with precondition: [Out>=3,V1+7>=4*Out] 

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

* Chain [28,29]: 3
  with precondition: [Out=1,V1>=2] 

* Chain [27,29]: 2*s(95)+2*s(96)+7
  Such that:s(94) =< 2
s(93) =< V1
s(95) =< s(93)
s(96) =< s(94)

  with precondition: [Out=1,V1>=3] 

* Chain [27,28,29]: 2*s(95)+2*s(96)+9
  Such that:s(94) =< 2
s(93) =< V1
s(95) =< s(93)
s(96) =< s(94)

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


#### Cost of chains of start(V1,V,V11):
* Chain [34]: 10*s(147)+24*s(149)+1*s(159)+4*s(167)+3*s(170)+8*s(171)+1*s(172)+3*s(180)+1*s(182)+8*s(186)+4*s(192)+24*s(200)+8*s(204)+6*s(206)+8*s(207)+2*s(208)+20*s(209)+8*s(210)+6*s(211)+2*s(212)+4*s(237)+8*s(242)+6*s(243)+8*s(244)+2*s(245)+20*s(246)+6*s(247)+2*s(248)+9
  Such that:s(167) =< V1/2-V/2
aux(43) =< 2
aux(44) =< V1
aux(45) =< V1+1
aux(46) =< V1-V
aux(47) =< 5*V1
aux(48) =< 13*V1
aux(49) =< V1/2
aux(50) =< V1/3
aux(51) =< 5/2*V1
aux(52) =< 5/2*V1+45/2
aux(53) =< 13/6*V1
aux(54) =< 13/6*V1+91/6
aux(55) =< V
s(149) =< aux(44)
s(192) =< aux(50)
s(147) =< aux(55)
s(186) =< aux(43)
s(199) =< aux(44)
s(200) =< aux(44)
s(192) =< aux(44)
s(199) =< aux(49)
s(200) =< aux(49)
s(192) =< aux(49)
s(192) =< aux(52)
s(201) =< aux(52)
s(192) =< aux(51)
s(201) =< aux(51)
s(202) =< aux(54)
s(202) =< aux(53)
s(203) =< aux(44)
s(204) =< s(202)*(1/2)
s(205) =< s(199)*2
s(206) =< s(204)*s(203)
s(207) =< s(202)
s(208) =< s(206)*aux(43)
s(209) =< s(201)
s(210) =< s(205)
s(211) =< s(209)*aux(44)
s(212) =< s(211)*aux(43)
s(237) =< aux(44)
s(237) =< aux(49)
s(237) =< aux(52)
s(240) =< aux(52)
s(237) =< aux(47)
s(240) =< aux(47)
s(241) =< aux(54)
s(241) =< aux(48)
s(242) =< s(241)*(1/2)
s(243) =< s(242)*s(203)
s(244) =< s(241)
s(245) =< s(243)*aux(43)
s(246) =< s(240)
s(247) =< s(246)*aux(44)
s(248) =< s(247)*aux(43)
s(169) =< aux(45)
s(169) =< aux(44)
s(170) =< s(167)*aux(46)
s(171) =< s(169)
s(172) =< s(170)*aux(55)
s(180) =< s(149)*aux(46)
s(182) =< s(180)*aux(55)
s(159) =< s(149)*aux(55)

  with precondition: [V1>=0] 

* Chain [33]: 15*s(275)+4*s(276)+1*s(286)+4
  Such that:aux(58) =< V
aux(59) =< V11
s(275) =< aux(58)
s(276) =< aux(59)
s(286) =< s(275)*aux(59)

  with precondition: [V1=1,V>=1,V11>=0] 

* Chain [32]: 1
  with precondition: [V1=2,V>=1,V11>=0] 

* Chain [31]: 8*s(289)+3
  Such that:aux(60) =< V1
s(289) =< aux(60)

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

* Chain [30]: 14*s(292)+12*s(294)+2*s(298)+5
  Such that:s(295) =< 1
aux(61) =< V1
s(292) =< aux(61)
s(298) =< s(295)
s(293) =< s(292)*aux(61)
s(294) =< s(293)

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


Closed-form bounds of start(V1,V,V11): 
-------------------------------------
* Chain [34] with precondition: [V1>=0] 
    - Upper bound: 68*V1+25+nat(V)*V1+nat(V)*V1*nat(V1-V)+(5/2*V1+45/2)*(20*V1)+(13/6*V1+91/6)*(10*V1)+3*V1*nat(V1-V)+nat(V)*10+nat(V1-V)*nat(V)*nat(V1/2-V/2)+(8*V1+8)+(100*V1+900)+(52*V1+364)+nat(V1-V)*3*nat(V1/2-V/2)+nat(V1/2-V/2)*4+4/3*V1 
    - Complexity: n^3 
* Chain [33] with precondition: [V1=1,V>=1,V11>=0] 
    - Upper bound: 15*V+4+V11*V+4*V11 
    - Complexity: n^2 
* Chain [32] with precondition: [V1=2,V>=1,V11>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [31] with precondition: [V=0,V1>=1] 
    - Upper bound: 8*V1+3 
    - Complexity: n 
* Chain [30] with precondition: [V=1,V1>=2] 
    - Upper bound: 14*V1+7+12*V1*V1 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V11): max([6*V1+4+max([12*V1*V1,54*V1+18+nat(V)*V1+nat(V)*V1*nat(V1-V)+(5/2*V1+45/2)*(20*V1)+(13/6*V1+91/6)*(10*V1)+3*V1*nat(V1-V)+nat(V)*10+nat(V1-V)*nat(V)*nat(V1/2-V/2)+(8*V1+8)+(100*V1+900)+(52*V1+364)+nat(V1-V)*3*nat(V1/2-V/2)+nat(V1/2-V/2)*4+4/3*V1])+(8*V1+2),nat(V)*15+3+nat(V11)*nat(V)+nat(V11)*4])+1 
Asymptotic class: n^3 
* Total analysis performed in 668 ms.


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

(10)
BOUNDS(1, n^3)

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

(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^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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

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

The rewrite sequence

le(s(x), s(y)) ->^+ le(x, y)

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

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

The result substitution is [ ].




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

(14)
Complex Obligation (BEST)

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

(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^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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

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

(17)
BOUNDS(n^1, INF)

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

(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^3).


The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(0, y) -> 0
   minus(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

S is empty.
Rewrite Strategy: INNERMOST