/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^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 502 ms]
    (12) BOUNDS(1, n^1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 276 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 29 ms]
            (26) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) 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)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> 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: FULL
----------------------------------------

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
mod(s(x), s([]))
mod(s([]), s(y))
if_mod(true, s(x), s([]))


The defined contexts are:
if_mod([], s(x1), s(x2))
mod([], s(x1))
le(x0, [])
if_mod(x0, s([]), s(x2))
minus([], x1)


[] just represents basic- or constructor-terms in the following defined contexts:
if_mod([], s(x1), s(x2))
mod([], s(x1))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(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)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> 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
----------------------------------------

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(4)
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]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> 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
----------------------------------------

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

(6)
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(x, 0) -> x [1]
   minus(s(x), s(y)) -> 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
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
----------------------------------------

(7) 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:

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

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

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

(8)
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(x, 0) -> x [1]
   minus(s(x), s(y)) -> 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]
   minus(v0, v1) -> null_minus [0]
   if_mod(v0, v1, v2) -> null_if_mod [0]
   le(v0, v1) -> null_le [0]
   mod(v0, v1) -> null_mod [0]

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

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

(9) 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_minus => 0
   null_if_mod => 0
   null_le => 0
   null_mod => 0

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

(10)
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 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + 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

Only complete derivations are relevant for the runtime complexity.

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

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

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


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

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

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

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

### Specialization of cost equations minus/3 
* CE 9 is refined into CE [20] 
* CE 7 is refined into CE [21] 
* CE 8 is refined into CE [22] 


### Cost equations --> "Loop" of minus/3 
* CEs [22] --> Loop 14 
* CEs [20] --> Loop 15 
* CEs [21] --> Loop 16 

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

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


### Specialization of cost equations le/3 
* CE 19 is refined into CE [23] 
* CE 17 is refined into CE [24] 
* CE 16 is refined into CE [25] 
* CE 18 is refined into CE [26] 


### Cost equations --> "Loop" of le/3 
* CEs [26] --> Loop 17 
* CEs [23] --> Loop 18 
* CEs [24] --> Loop 19 
* CEs [25] --> Loop 20 

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

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


### Specialization of cost equations (mod)/3 
* CE 11 is refined into CE [27,28] 
* CE 14 is refined into CE [29] 
* CE 10 is refined into CE [30,31,32,33,34] 
* CE 13 is refined into CE [35] 
* CE 15 is refined into CE [36] 
* CE 12 is refined into CE [37,38,39,40] 


### Cost equations --> "Loop" of (mod)/3 
* CEs [40] --> Loop 21 
* CEs [39] --> Loop 22 
* CEs [37] --> Loop 23 
* CEs [38] --> Loop 24 
* CEs [28] --> Loop 25 
* CEs [30] --> Loop 26 
* CEs [29] --> Loop 27 
* CEs [27] --> Loop 28 
* CEs [31] --> Loop 29 
* CEs [32,33,34,35,36] --> Loop 30 

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

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


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


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

### Ranking functions of CR start(V1,V,V14) 

#### Partial ranking functions of CR start(V1,V,V14) 


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

#### Cost of chains of minus(V1,V,Out):
* Chain [[14],16]: 1*it(14)+1
  Such that:it(14) =< V

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

* Chain [[14],15]: 1*it(14)+0
  Such that:it(14) =< V

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

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

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


#### Cost of chains of le(V1,V,Out):
* Chain [[17],20]: 1*it(17)+1
  Such that:it(17) =< V1

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

* Chain [[17],19]: 1*it(17)+1
  Such that:it(17) =< V

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

* Chain [[17],18]: 1*it(17)+0
  Such that:it(17) =< V

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

* Chain [20]: 1
  with precondition: [V1=0,Out=2,V>=0] 

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

* Chain [18]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of mod(V1,V,Out):
* Chain [[23],30]: 6*it(23)+1*s(5)+2
  Such that:s(5) =< 1
aux(2) =< V1
it(23) =< aux(2)

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

* Chain [[23],26]: 4*it(23)+2
  Such that:it(23) =< V1

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

* Chain [[23],24,30]: 4*it(23)+1*s(5)+5
  Such that:s(5) =< 1
it(23) =< V1

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

* Chain [[21],30]: 8*it(21)+1*s(5)+2
  Such that:s(5) =< V
aux(6) =< V1
it(21) =< aux(6)

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

* Chain [[21],29]: 4*it(21)+2*s(11)+2
  Such that:it(21) =< V1-V+1
aux(7) =< V1
it(21) =< aux(7)
s(11) =< aux(7)

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

* Chain [[21],28]: 4*it(21)+2*s(11)+3
  Such that:it(21) =< V1-V+1
aux(8) =< V1
it(21) =< aux(8)
s(11) =< aux(8)

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

* Chain [[21],25]: 4*it(21)+2*s(11)+1*s(13)+3
  Such that:aux(4) =< V1
it(21) =< V1-V+1
aux(5) =< V1-Out
s(13) =< Out
it(21) =< aux(4)
s(12) =< aux(4)
it(21) =< aux(5)
s(12) =< aux(5)
s(11) =< s(12)

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

* Chain [[21],22,30]: 4*it(21)+3*s(5)+2*s(11)+5
  Such that:aux(4) =< V1
aux(10) =< V
aux(11) =< V1-V
it(21) =< aux(11)
s(5) =< aux(10)
it(21) =< aux(4)
s(12) =< aux(4)
s(12) =< aux(11)
s(11) =< s(12)

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

* Chain [30]: 2*s(3)+1*s(5)+2
  Such that:s(5) =< V
aux(1) =< V1
s(3) =< aux(1)

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

* Chain [29]: 2
  with precondition: [V1=1,Out=0,V>=2] 

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

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

* Chain [26]: 2
  with precondition: [V=1,Out=0,V1>=1] 

* Chain [25]: 1*s(13)+3
  Such that:s(13) =< V1

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

* Chain [24,30]: 1*s(5)+5
  Such that:s(5) =< 1

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

* Chain [22,30]: 3*s(5)+5
  Such that:aux(10) =< V
s(5) =< aux(10)

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


#### Cost of chains of start(V1,V,V14):
* Chain [37]: 12*s(41)+19*s(43)+12*s(47)+4*s(51)+2*s(53)+5
  Such that:s(46) =< V1-V
aux(17) =< V1
aux(18) =< V1-V+1
aux(19) =< V
s(43) =< aux(17)
s(47) =< aux(18)
s(41) =< aux(19)
s(51) =< s(46)
s(51) =< aux(17)
s(52) =< aux(17)
s(52) =< s(46)
s(53) =< s(52)
s(47) =< aux(17)

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

* Chain [36]: 1
  with precondition: [V1=1,V>=1,V14>=1] 

* Chain [35]: 3
  with precondition: [V1=1,V>=2] 

* Chain [34]: 36*s(66)+11*s(69)+21*s(78)+12*s(90)+4*s(94)+2*s(96)+18*s(97)+7
  Such that:s(89) =< V-2*V14
aux(24) =< 1
aux(25) =< V
aux(26) =< V-2*V14+1
aux(27) =< V-V14
aux(28) =< V14
s(90) =< aux(26)
s(97) =< aux(27)
s(78) =< aux(28)
s(66) =< aux(25)
s(69) =< aux(24)
s(94) =< s(89)
s(94) =< aux(27)
s(95) =< aux(27)
s(95) =< s(89)
s(96) =< s(95)
s(90) =< aux(27)

  with precondition: [V1=2,V>=1,V14>=1] 

* Chain [33]: 1*s(111)+5
  Such that:s(111) =< V14

  with precondition: [V1=2,V=V14+1,V>=3] 

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

* Chain [31]: 3*s(114)+14*s(115)+5
  Such that:s(112) =< 1
s(113) =< V1
s(114) =< s(112)
s(115) =< s(113)

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


Closed-form bounds of start(V1,V,V14): 
-------------------------------------
* Chain [37] with precondition: [V1>=0,V>=0] 
    - Upper bound: 21*V1+12*V+5+nat(V1-V+1)*12+nat(V1-V)*4 
    - Complexity: n 
* Chain [36] with precondition: [V1=1,V>=1,V14>=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [35] with precondition: [V1=1,V>=2] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [34] with precondition: [V1=2,V>=1,V14>=1] 
    - Upper bound: 36*V+21*V14+18+nat(V-2*V14+1)*12+nat(V-V14)*20+nat(V-2*V14)*4 
    - Complexity: n 
* Chain [33] with precondition: [V1=2,V=V14+1,V>=3] 
    - Upper bound: V14+5 
    - Complexity: n 
* Chain [32] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [31] with precondition: [V=1,V1>=1] 
    - Upper bound: 14*V1+8 
    - Complexity: n 

### Maximum cost of start(V1,V,V14): max([14*V1+2+max([3,7*V1+12*V+nat(V1-V+1)*12+nat(V1-V)*4]),36*V+13+nat(V14)*20+nat(V-2*V14+1)*12+nat(V-V14)*20+nat(V-2*V14)*4+(nat(V14)+2)])+3 
Asymptotic class: n 
* Total analysis performed in 408 ms.


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

(12)
BOUNDS(1, n^1)

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

(13) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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(x, 0') -> x
   minus(s(x), s(y)) -> 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: FULL
----------------------------------------

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

(16)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> 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)

Types:
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
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, minus, mod

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

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

(18)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> 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)

Types:
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
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, minus, mod

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_0':s3_0(n5_0), gen_0':s3_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).
----------------------------------------

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> 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)

Types:
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
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, minus, mod

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

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> 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)

Types:
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
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, mod

They will be analysed ascendingly in the following order:
minus < mod

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(n252_0), gen_0':s3_0(n252_0)) -> gen_0':s3_0(0), rt in Omega(1 + n252_0)

Induction Base:
minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
minus(gen_0':s3_0(+(n252_0, 1)), gen_0':s3_0(+(n252_0, 1))) ->_R^Omega(1)
minus(gen_0':s3_0(n252_0), gen_0':s3_0(n252_0)) ->_IH
gen_0':s3_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(26)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(s(x), s(y)) -> 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)

Types:
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
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
minus(gen_0':s3_0(n252_0), gen_0':s3_0(n252_0)) -> gen_0':s3_0(0), rt in Omega(1 + n252_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
mod