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


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WORST_CASE(Omega(n^1), O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 4 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 512 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:

   minus(0, y) -> 0
   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   mod(x, 0) -> 0
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
   if1(true, x, y) -> x
   if1(false, x, y) -> mod(minus(x, y), y)
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   lt(x, 0) -> false
   lt(0, s(x)) -> true
   lt(s(x), s(y)) -> lt(x, 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^3).


The TRS R consists of the following rules:

   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if(gt(s(x), y), x, y) [1]
   if(true, x, y) -> s(minus(x, y)) [1]
   if(false, x, y) -> 0 [1]
   mod(x, 0) -> 0 [1]
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y)) [1]
   if1(true, x, y) -> x [1]
   if1(false, x, y) -> mod(minus(x, y), y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(x)) -> true [1]
   lt(s(x), s(y)) -> lt(x, 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:

   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if(gt(s(x), y), x, y) [1]
   if(true, x, y) -> s(minus(x, y)) [1]
   if(false, x, y) -> 0 [1]
   mod(x, 0) -> 0 [1]
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y)) [1]
   if1(true, x, y) -> x [1]
   if1(false, x, y) -> mod(minus(x, y), y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(x)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s
gt :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
mod :: 0:s -> 0:s -> 0:s
if1 :: true:false -> 0:s -> 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false

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

(5) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

(6)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(0, y) -> 0 [1]
   minus(s(x), y) -> if(gt(s(x), y), x, y) [1]
   if(true, x, y) -> s(minus(x, y)) [1]
   if(false, x, y) -> 0 [1]
   mod(x, 0) -> 0 [1]
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y)) [1]
   if1(true, x, y) -> x [1]
   if1(false, x, y) -> mod(minus(x, y), y) [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(x)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s
gt :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
mod :: 0:s -> 0:s -> 0:s
if1 :: true:false -> 0:s -> 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false

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:

   gt(z, z') -{ 1 }-> gt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gt(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gt(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   if(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> mod(minus(x, y), y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> if(gt(1 + x, y), x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   mod(z, z') -{ 1 }-> if1(lt(x, 1 + y), x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0

Only complete derivations are relevant for the runtime complexity.

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

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

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


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

#### Computed strongly connected components 
0. recursive  : [gt/3]
1. recursive  : [if/4,minus/3]
2. recursive  : [lt/3]
3. recursive  : [if1/4,(mod)/3]
4. non_recursive  : [start/3]

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

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

### Specialization of cost equations gt/3 
* CE 17 is refined into CE [21] 
* CE 16 is refined into CE [22] 
* CE 15 is refined into CE [23] 


### Cost equations --> "Loop" of gt/3 
* CEs [22] --> Loop 15 
* CEs [23] --> Loop 16 
* CEs [21] --> Loop 17 

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

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


### Specialization of cost equations minus/3 
* CE 9 is refined into CE [24] 
* CE 11 is refined into CE [25] 
* CE 10 is refined into CE [26,27] 


### Cost equations --> "Loop" of minus/3 
* CEs [27] --> Loop 18 
* CEs [26] --> Loop 19 
* CEs [24] --> Loop 20 
* CEs [25] --> Loop 21 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [18]: [V1-1,V1-V]
* RF of phase [19]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V1-1
    V1-V
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V1


### Specialization of cost equations lt/3 
* CE 20 is refined into CE [28] 
* CE 18 is refined into CE [29] 
* CE 19 is refined into CE [30] 


### Cost equations --> "Loop" of lt/3 
* CEs [29] --> Loop 22 
* CEs [30] --> Loop 23 
* CEs [28] --> Loop 24 

### Ranking functions of CR lt(V1,V,Out) 
* RF of phase [24]: [V,V1]

#### Partial ranking functions of CR lt(V1,V,Out) 
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V
    V1


### Specialization of cost equations (mod)/3 
* CE 13 is refined into CE [31,32] 
* CE 14 is refined into CE [33] 
* CE 12 is refined into CE [34,35] 


### Cost equations --> "Loop" of (mod)/3 
* CEs [35] --> Loop 25 
* CEs [34] --> Loop 26 
* CEs [32] --> Loop 27 
* CEs [33] --> Loop 28 
* CEs [31] --> Loop 29 

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

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


### Specialization of cost equations start/3 
* CE 2 is refined into CE [36] 
* CE 4 is refined into CE [37,38,39,40] 
* CE 1 is refined into CE [41,42,43,44,45,46,47,48] 
* CE 3 is refined into CE [49] 
* CE 5 is refined into CE [50,51,52,53] 
* CE 6 is refined into CE [54,55,56,57,58,59] 
* CE 7 is refined into CE [60,61,62,63] 
* CE 8 is refined into CE [64,65,66,67] 


### Cost equations --> "Loop" of start/3 
* CEs [57] --> Loop 30 
* CEs [53,56,59,63,66] --> Loop 31 
* CEs [38,52,58,62,67] --> Loop 32 
* CEs [36,37,39,40] --> Loop 33 
* CEs [45] --> Loop 34 
* CEs [43] --> Loop 35 
* CEs [41,51,55,61,65] --> Loop 36 
* CEs [42,44,46,47,48,49,50,54,60,64] --> Loop 37 

### Ranking functions of CR start(V1,V,V7) 

#### Partial ranking functions of CR start(V1,V,V7) 


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

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

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

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

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

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

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[19],21]: 3*it(19)+1
  Such that:it(19) =< Out

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

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

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

* Chain [21]: 1
  with precondition: [V1=0,Out=0,V>=0] 

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

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


#### Cost of chains of lt(V1,V,Out):
* Chain [[24],23]: 1*it(24)+1
  Such that:it(24) =< V1

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

* Chain [[24],22]: 1*it(24)+1
  Such that:it(24) =< V

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

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

* Chain [22]: 1
  with precondition: [V=0,Out=0,V1>=0] 


#### Cost of chains of mod(V1,V,Out):
* Chain [[25],27]: 6*it(25)+1*s(5)+2*s(15)+3*s(16)+1*s(17)+3
  Such that:aux(5) =< V1
aux(6) =< V1-V
s(12) =< V
s(5) =< Out
aux(8) =< V1-Out
it(25) =< aux(8)
it(25) =< aux(5)
s(18) =< aux(5)
it(25) =< aux(6)
s(18) =< aux(8)
s(16) =< it(25)*aux(6)
s(15) =< s(18)
s(17) =< s(16)*s(12)

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

* Chain [[25],26,29]: 6*it(25)+2*s(15)+3*s(16)+1*s(17)+2*s(19)+9
  Such that:aux(5) =< V1
aux(10) =< V1-V
aux(11) =< V
it(25) =< aux(10)
s(19) =< aux(11)
it(25) =< aux(5)
s(18) =< aux(5)
s(18) =< aux(10)
s(16) =< it(25)*aux(10)
s(15) =< s(18)
s(17) =< s(16)*aux(11)

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

* Chain [29]: 3
  with precondition: [V1=0,Out=0,V>=1] 

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

* Chain [27]: 1*s(5)+3
  Such that:s(5) =< V1

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

* Chain [26,29]: 2*s(19)+9
  Such that:aux(9) =< V1
s(19) =< aux(9)

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


#### Cost of chains of start(V1,V,V7):
* Chain [37]: 1*s(21)+13*s(22)+5*s(24)+3*s(25)+12*s(29)+6*s(32)+2*s(33)+2*s(34)+13
  Such that:s(21) =< V
aux(17) =< V-2*V7
aux(18) =< V-V7
aux(19) =< V7
s(22) =< aux(18)
s(29) =< aux(17)
s(24) =< aux(19)
s(29) =< aux(18)
s(31) =< aux(18)
s(31) =< aux(17)
s(32) =< s(29)*aux(17)
s(33) =< s(31)
s(34) =< s(32)*aux(19)
s(25) =< s(22)*aux(19)

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

* Chain [36]: 3*s(54)+5
  Such that:s(54) =< V1

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

* Chain [35]: 3*s(55)+3
  Such that:s(55) =< V

  with precondition: [V1=0,V7=0,V>=1] 

* Chain [34]: 6*s(56)+1*s(59)+13
  Such that:aux(20) =< V7
s(56) =< aux(20)
s(59) =< s(56)*aux(20)

  with precondition: [V1=0,V=2*V7,V>=2] 

* Chain [33]: 1*s(62)+3*s(63)+1*s(65)+1*s(66)+4
  Such that:s(62) =< V
s(63) =< V-V7
s(64) =< V7
s(65) =< s(64)
s(66) =< s(63)*s(64)

  with precondition: [V1=1,V>=0,V7>=0] 

* Chain [32]: 3*s(67)+4*s(68)+3
  Such that:s(67) =< V
aux(21) =< V1
s(68) =< aux(21)

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

* Chain [31]: 3*s(72)+5*s(74)+1*s(75)+3*s(81)+6*s(83)+3*s(85)+1*s(87)+9
  Such that:aux(22) =< V1
aux(23) =< V1-V
aux(24) =< V
s(72) =< aux(23)
s(74) =< aux(24)
s(75) =< s(72)*aux(24)
s(81) =< aux(22)
s(83) =< aux(22)
s(83) =< aux(23)
s(85) =< s(83)*aux(23)
s(87) =< s(85)*aux(24)

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

* Chain [30]: 6*s(93)+2*s(94)+3*s(96)+2*s(97)+1*s(98)+9
  Such that:s(90) =< V1
s(91) =< V1-V
s(92) =< V
s(93) =< s(91)
s(94) =< s(92)
s(93) =< s(90)
s(95) =< s(90)
s(95) =< s(91)
s(96) =< s(93)*s(91)
s(97) =< s(95)
s(98) =< s(96)*s(92)

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


Closed-form bounds of start(V1,V,V7): 
-------------------------------------
* Chain [37] with precondition: [V1=0,V>=0] 
    - Upper bound: V+13+nat(V7)*5+nat(V7)*3*nat(V-V7)+nat(V7)*2*nat(V-2*V7)*nat(V-2*V7)+nat(V-V7)*15+nat(V-2*V7)*12+nat(V-2*V7)*6*nat(V-2*V7) 
    - Complexity: n^3 
* Chain [36] with precondition: [V=0,V1>=0] 
    - Upper bound: 3*V1+5 
    - Complexity: n 
* Chain [35] with precondition: [V1=0,V7=0,V>=1] 
    - Upper bound: 3*V+3 
    - Complexity: n 
* Chain [34] with precondition: [V1=0,V=2*V7,V>=2] 
    - Upper bound: 6*V7+13+V7*V7 
    - Complexity: n^2 
* Chain [33] with precondition: [V1=1,V>=0,V7>=0] 
    - Upper bound: V+V7+4+nat(V-V7)*V7+nat(V-V7)*3 
    - Complexity: n^2 
* Chain [32] with precondition: [V1>=1,V>=V1] 
    - Upper bound: 4*V1+3*V+3 
    - Complexity: n 
* Chain [31] with precondition: [V>=1,V1>=V] 
    - Upper bound: 3*V1-3*V+(9*V1+9+(V1-V)*(V*V1)+(V1-V)*(3*V1)+5*V+(V1-V)*V) 
    - Complexity: n^3 
* Chain [30] with precondition: [V>=1,V1>=2*V] 
    - Upper bound: 6*V1-6*V+(2*V1+2*V+9+(V1-V)*((V1-V)*V))+(3*V1-3*V)*(V1-V) 
    - Complexity: n^3 

### Maximum cost of start(V1,V,V7): max([max([3*V1,nat(V7)*6+8+nat(V7)*nat(V7)])+2,max([max([V,2*V1+max([nat(V1-V)*V*nat(V1-V)+6+nat(V1-V)*6+nat(V1-V)*3*nat(V1-V),5*V1+6+V*V1*nat(V1-V)+3*V1*nat(V1-V)+2*V+nat(V1-V)*V+nat(V1-V)*3+(2*V1+V)])])+V,nat(V7)*4+9+nat(V7)*2*nat(V-V7)+nat(V7)*2*nat(V-2*V7)*nat(V-2*V7)+nat(V-V7)*12+nat(V-2*V7)*12+nat(V-2*V7)*6*nat(V-2*V7)+(nat(V7)+1+nat(V-V7)*nat(V7)+nat(V-V7)*3)])+V])+3 
Asymptotic class: n^3 
* Total analysis performed in 415 ms.


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(10)
BOUNDS(1, n^3)

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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^3).


The TRS R consists of the following rules:

   minus(0, y) -> 0
   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   mod(x, 0) -> 0
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
   if1(true, x, y) -> x
   if1(false, x, y) -> mod(minus(x, y), y)
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   lt(x, 0) -> false
   lt(0, s(x)) -> true
   lt(s(x), s(y)) -> lt(x, y)

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

lt(s(x), s(y)) ->^+ lt(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 [ ].




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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^3).


The TRS R consists of the following rules:

   minus(0, y) -> 0
   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   mod(x, 0) -> 0
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
   if1(true, x, y) -> x
   if1(false, x, y) -> mod(minus(x, y), y)
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   lt(x, 0) -> false
   lt(0, s(x)) -> true
   lt(s(x), s(y)) -> lt(x, y)

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^3).


The TRS R consists of the following rules:

   minus(0, y) -> 0
   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   mod(x, 0) -> 0
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
   if1(true, x, y) -> x
   if1(false, x, y) -> mod(minus(x, y), y)
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   lt(x, 0) -> false
   lt(0, s(x)) -> true
   lt(s(x), s(y)) -> lt(x, y)

S is empty.
Rewrite Strategy: INNERMOST