/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 544 ms]
    (10) BOUNDS(1, n^1)
(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^1).


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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

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:

   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [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]
   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [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
pred :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
mod :: 0:s -> 0:s -> 0:s
if_mod :: true:false -> 0:s -> 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:

   pred(v0) -> null_pred [0]
   if_mod(v0, v1, v2) -> null_if_mod [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   mod(v0, v1) -> null_mod [0]

And the following fresh constants:    null_pred, null_if_mod, null_le, null_minus, null_mod

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

(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]
   pred(s(x)) -> x [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> pred(minus(x, y)) [1]
   mod(0, y) -> 0 [1]
   mod(s(x), 0) -> 0 [1]
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) [1]
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y)) [1]
   if_mod(false, s(x), s(y)) -> s(x) [1]
   pred(v0) -> null_pred [0]
   if_mod(v0, v1, v2) -> null_if_mod [0]
   le(v0, v1) -> null_le [0]
   minus(v0, v1) -> null_minus [0]
   mod(v0, v1) -> null_mod [0]

The TRS has the following type information:
le :: 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod -> true:false:null_le
0 :: 0:s:null_pred:null_if_mod:null_minus:null_mod
true :: true:false:null_le
s :: 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod
false :: true:false:null_le
pred :: 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod
minus :: 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod
mod :: 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod
if_mod :: true:false:null_le -> 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod -> 0:s:null_pred:null_if_mod:null_minus:null_mod
null_pred :: 0:s:null_pred:null_if_mod:null_minus:null_mod
null_if_mod :: 0:s:null_pred:null_if_mod:null_minus:null_mod
null_le :: true:false:null_le
null_minus :: 0:s:null_pred:null_if_mod:null_minus:null_mod
null_mod :: 0:s:null_pred:null_if_mod:null_minus:null_mod

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_pred => 0
   null_if_mod => 0
   null_le => 0
   null_minus => 0
   null_mod => 0

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

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

   if_mod(z, z', z'') -{ 1 }-> mod(minus(x, y), 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   if_mod(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if_mod(z, z', z'') -{ 1 }-> 1 + x :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y
   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
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> pred(minus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   mod(z, z') -{ 1 }-> if_mod(le(y, x), 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   mod(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   mod(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   pred(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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),0,[le(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[pred(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V15),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[mod(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[fun(V1, V, V15, Out)],[V1 >= 0,V >= 0,V15 >= 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(pred(V1, Out),1,[],[Out = V6,V6 >= 0,V1 = 1 + V6]).
eq(minus(V1, V, Out),1,[],[Out = V7,V7 >= 0,V1 = V7,V = 0]).
eq(minus(V1, V, Out),1,[minus(V8, V9, Ret0),pred(Ret0, Ret1)],[Out = Ret1,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = V8]).
eq(mod(V1, V, Out),1,[],[Out = 0,V10 >= 0,V1 = 0,V = V10]).
eq(mod(V1, V, Out),1,[],[Out = 0,V11 >= 0,V1 = 1 + V11,V = 0]).
eq(mod(V1, V, Out),1,[le(V12, V13, Ret01),fun(Ret01, 1 + V13, 1 + V12, Ret2)],[Out = Ret2,V = 1 + V12,V13 >= 0,V12 >= 0,V1 = 1 + V13]).
eq(fun(V1, V, V15, Out),1,[minus(V16, V14, Ret02),mod(Ret02, 1 + V14, Ret3)],[Out = Ret3,V1 = 2,V = 1 + V16,V16 >= 0,V14 >= 0,V15 = 1 + V14]).
eq(fun(V1, V, V15, Out),1,[],[Out = 1 + V17,V = 1 + V17,V1 = 1,V17 >= 0,V18 >= 0,V15 = 1 + V18]).
eq(pred(V1, Out),0,[],[Out = 0,V19 >= 0,V1 = V19]).
eq(fun(V1, V, V15, Out),0,[],[Out = 0,V21 >= 0,V15 = V22,V20 >= 0,V1 = V21,V = V20,V22 >= 0]).
eq(le(V1, V, Out),0,[],[Out = 0,V24 >= 0,V23 >= 0,V1 = V24,V = V23]).
eq(minus(V1, V, Out),0,[],[Out = 0,V25 >= 0,V26 >= 0,V1 = V25,V = V26]).
eq(mod(V1, V, Out),0,[],[Out = 0,V27 >= 0,V28 >= 0,V1 = V27,V = V28]).
input_output_vars(le(V1,V,Out),[V1,V],[Out]).
input_output_vars(pred(V1,Out),[V1],[Out]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(mod(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,V15,Out),[V1,V,V15],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [pred/2]
1. recursive [non_tail] : [minus/3]
2. recursive  : [le/3]
3. recursive  : [fun/4,(mod)/3]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into pred/2
1. SCC is partially evaluated into minus/3
2. SCC is partially evaluated into le/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 pred/2 
* CE 21 is refined into CE [23] 
* CE 22 is refined into CE [24] 


### Cost equations --> "Loop" of pred/2 
* CEs [23] --> Loop 16 
* CEs [24] --> Loop 17 

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

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


### Specialization of cost equations minus/3 
* CE 10 is refined into CE [25] 
* CE 8 is refined into CE [26] 
* CE 9 is refined into CE [27,28] 


### Cost equations --> "Loop" of minus/3 
* CEs [28] --> Loop 18 
* CEs [27] --> Loop 19 
* CEs [25] --> Loop 20 
* CEs [26] --> Loop 21 

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

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


### Specialization of cost equations le/3 
* CE 20 is refined into CE [29] 
* CE 18 is refined into CE [30] 
* CE 17 is refined into CE [31] 
* CE 19 is refined into CE [32] 


### Cost equations --> "Loop" of le/3 
* CEs [32] --> Loop 22 
* CEs [29] --> Loop 23 
* CEs [30] --> Loop 24 
* CEs [31] --> Loop 25 

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

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


### Specialization of cost equations (mod)/3 
* CE 12 is refined into CE [33,34] 
* CE 15 is refined into CE [35] 
* CE 11 is refined into CE [36,37,38,39,40] 
* CE 14 is refined into CE [41] 
* CE 16 is refined into CE [42] 
* CE 13 is refined into CE [43,44,45,46] 


### Cost equations --> "Loop" of (mod)/3 
* CEs [46] --> Loop 26 
* CEs [45] --> Loop 27 
* CEs [43] --> Loop 28 
* CEs [44] --> Loop 29 
* CEs [34] --> Loop 30 
* CEs [36] --> Loop 31 
* CEs [35] --> Loop 32 
* CEs [33] --> Loop 33 
* CEs [37] --> Loop 34 
* CEs [38,39,40,41,42] --> Loop 35 

### Ranking functions of CR mod(V1,V,Out) 
* RF of phase [26]: [V1-1,V1-V+1]
* RF of phase [28]: [V1]

#### Partial ranking functions of CR mod(V1,V,Out) 
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1-1
    V1-V+1
* Partial RF of phase [28]:
  - RF of loop [28:1]:
    V1


### Specialization of cost equations start/3 
* CE 3 is refined into CE [47,48,49,50,51,52,53,54] 
* CE 1 is refined into CE [55] 
* CE 2 is refined into CE [56] 
* CE 4 is refined into CE [57,58,59,60,61] 
* CE 5 is refined into CE [62,63] 
* CE 6 is refined into CE [64,65,66] 
* CE 7 is refined into CE [67,68,69,70,71,72,73] 


### Cost equations --> "Loop" of start/3 
* CEs [70] --> Loop 36 
* CEs [58,64,69] --> Loop 37 
* CEs [51] --> Loop 38 
* CEs [47,48,49,50,52,53,54] --> Loop 39 
* CEs [68] --> Loop 40 
* CEs [56] --> Loop 41 
* CEs [55,57,59,60,61,62,63,65,66,67,71,72,73] --> Loop 42 

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

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


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

#### Cost of chains of pred(V1,Out):
* Chain [17]: 0
  with precondition: [Out=0,V1>=0] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[19],[18],21]: 3*it(18)+1
  Such that:aux(1) =< V
it(18) =< aux(1)

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

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

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

* Chain [[19],20]: 1*it(19)+0
  Such that:it(19) =< V

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

* Chain [[18],21]: 2*it(18)+1
  Such that:it(18) =< V

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

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

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


#### Cost of chains of le(V1,V,Out):
* Chain [[22],25]: 1*it(22)+1
  Such that:it(22) =< V1

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

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

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

* Chain [[22],23]: 1*it(22)+0
  Such that:it(22) =< V

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

* Chain [25]: 1
  with precondition: [V1=0,Out=2,V>=0] 

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

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


#### Cost of chains of mod(V1,V,Out):
* Chain [[28],35]: 6*it(28)+1*s(8)+2
  Such that:s(8) =< 1
aux(4) =< V1
it(28) =< aux(4)

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

* Chain [[28],31]: 4*it(28)+2
  Such that:it(28) =< V1

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

* Chain [[28],29,35]: 4*it(28)+1*s(8)+6
  Such that:s(8) =< 1
it(28) =< V1

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

* Chain [[26],35]: 9*it(26)+1*s(8)+2
  Such that:s(8) =< V
aux(8) =< V1
it(26) =< aux(8)

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

* Chain [[26],34]: 4*it(26)+3*s(15)+2
  Such that:it(26) =< V1-V+1
aux(9) =< V1
it(26) =< aux(9)
s(15) =< aux(9)

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

* Chain [[26],33]: 4*it(26)+3*s(15)+3
  Such that:it(26) =< V1-V+1
aux(10) =< V1
it(26) =< aux(10)
s(15) =< aux(10)

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

* Chain [[26],30]: 4*it(26)+3*s(15)+1*s(17)+3
  Such that:aux(6) =< V1
it(26) =< V1-V+1
aux(7) =< V1-Out
s(17) =< Out
it(26) =< aux(6)
s(16) =< aux(6)
it(26) =< aux(7)
s(16) =< aux(7)
s(15) =< s(16)

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

* Chain [[26],27,35]: 4*it(26)+7*s(8)+3*s(15)+6
  Such that:aux(6) =< V1
aux(12) =< V
aux(13) =< V1-V
it(26) =< aux(13)
s(8) =< aux(12)
it(26) =< aux(6)
s(16) =< aux(6)
s(16) =< aux(13)
s(15) =< s(16)

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

* Chain [35]: 2*s(6)+1*s(8)+2
  Such that:s(8) =< V
aux(3) =< V1
s(6) =< aux(3)

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

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

* Chain [33]: 3
  with precondition: [V1=1,Out=1,V>=2] 

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

* Chain [31]: 2
  with precondition: [V=1,Out=0,V1>=1] 

* Chain [30]: 1*s(17)+3
  Such that:s(17) =< V1

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

* Chain [29,35]: 1*s(8)+6
  Such that:s(8) =< 1

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

* Chain [27,35]: 7*s(8)+6
  Such that:aux(12) =< V
s(8) =< aux(12)

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


#### Cost of chains of start(V1,V,V15):
* Chain [42]: 25*s(46)+23*s(48)+12*s(53)+4*s(57)+3*s(59)+6
  Such that:s(52) =< V1-V
aux(19) =< V1
aux(20) =< V1-V+1
aux(21) =< V
s(48) =< aux(19)
s(53) =< aux(20)
s(46) =< aux(21)
s(57) =< s(52)
s(57) =< aux(19)
s(58) =< aux(19)
s(58) =< s(52)
s(59) =< s(58)
s(53) =< aux(19)

  with precondition: [V1>=0] 

* Chain [41]: 1
  with precondition: [V1=1,V>=1,V15>=1] 

* Chain [40]: 3
  with precondition: [V1=1,V>=2] 

* Chain [39]: 39*s(72)+19*s(75)+45*s(85)+12*s(97)+4*s(101)+3*s(103)+22*s(104)+8
  Such that:s(96) =< V-2*V15
aux(26) =< 1
aux(27) =< V
aux(28) =< V-2*V15+1
aux(29) =< V-V15
aux(30) =< V15
s(97) =< aux(28)
s(104) =< aux(29)
s(85) =< aux(30)
s(72) =< aux(27)
s(75) =< aux(26)
s(101) =< s(96)
s(101) =< aux(29)
s(102) =< aux(29)
s(102) =< s(96)
s(103) =< s(102)
s(97) =< aux(29)

  with precondition: [V1=2,V>=1,V15>=1] 

* Chain [38]: 2*s(118)+5
  Such that:s(118) =< V15

  with precondition: [V1=2,V=V15+1,V>=3] 

* Chain [37]: 1
  with precondition: [V=0,V1>=0] 

* Chain [36]: 3*s(121)+14*s(122)+6
  Such that:s(119) =< 1
s(120) =< V1
s(121) =< s(119)
s(122) =< s(120)

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


Closed-form bounds of start(V1,V,V15): 
-------------------------------------
* Chain [42] with precondition: [V1>=0] 
    - Upper bound: 26*V1+6+nat(V)*25+nat(V1-V+1)*12+nat(V1-V)*4 
    - Complexity: n 
* Chain [41] with precondition: [V1=1,V>=1,V15>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [40] with precondition: [V1=1,V>=2] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [39] with precondition: [V1=2,V>=1,V15>=1] 
    - Upper bound: 39*V+45*V15+27+nat(V-2*V15+1)*12+nat(V-V15)*25+nat(V-2*V15)*4 
    - Complexity: n 
* Chain [38] with precondition: [V1=2,V=V15+1,V>=3] 
    - Upper bound: 2*V15+5 
    - Complexity: n 
* Chain [37] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [36] with precondition: [V=1,V1>=1] 
    - Upper bound: 14*V1+9 
    - Complexity: n 

### Maximum cost of start(V1,V,V15): max([14*V1+3+max([3,nat(V)*25+12*V1+nat(V1-V+1)*12+nat(V1-V)*4]),nat(V)*39+22+nat(V15)*43+nat(V-2*V15+1)*12+nat(V-V15)*25+nat(V-2*V15)*4+(nat(V15)*2+2)])+3 
Asymptotic class: n 
* Total analysis performed in 447 ms.


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(10)
BOUNDS(1, n^1)

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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^1).


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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

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

minus(x, s(y)) ->^+ pred(minus(x, y))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [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^1).


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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

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^1).


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)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(x)

S is empty.
Rewrite Strategy: INNERMOST