/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^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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 1 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 1404 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)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), 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]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), 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]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), 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
gcd :: 0:s -> 0:s -> 0:s
if_gcd :: 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:

   minus(v0, v1) -> null_minus [0]
   if_gcd(v0, v1, v2) -> null_if_gcd [0]
   le(v0, v1) -> null_le [0]
   gcd(v0, v1) -> null_gcd [0]

And the following fresh constants:    null_minus, null_if_gcd, null_le, null_gcd

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

(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(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   gcd(0, y) -> y [1]
   gcd(s(x), 0) -> s(x) [1]
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1]
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1]
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1]
   minus(v0, v1) -> null_minus [0]
   if_gcd(v0, v1, v2) -> null_if_gcd [0]
   le(v0, v1) -> null_le [0]
   gcd(v0, v1) -> null_gcd [0]

The TRS has the following type information:
le :: 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd -> true:false:null_le
0 :: 0:s:null_minus:null_if_gcd:null_gcd
true :: true:false:null_le
s :: 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd
false :: true:false:null_le
minus :: 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd
gcd :: 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd
if_gcd :: true:false:null_le -> 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd -> 0:s:null_minus:null_if_gcd:null_gcd
null_minus :: 0:s:null_minus:null_if_gcd:null_gcd
null_if_gcd :: 0:s:null_minus:null_if_gcd:null_gcd
null_le :: true:false:null_le
null_gcd :: 0:s:null_minus:null_if_gcd:null_gcd

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_minus => 0
   null_if_gcd => 0
   null_le => 0
   null_gcd => 0

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

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

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

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, 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,[gcd(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(gcd(V1, V, Out),1,[],[Out = V9,V9 >= 0,V1 = 0,V = V9]).
eq(gcd(V1, V, Out),1,[],[Out = 1 + V10,V10 >= 0,V1 = 1 + V10,V = 0]).
eq(gcd(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),gcd(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,[minus(V17, V16, Ret02),gcd(Ret02, 1 + V16, Ret4)],[Out = Ret4,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(gcd(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(gcd(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  : [le/3]
1. recursive  : [minus/3]
2. recursive  : [fun/4,gcd/3]
3. 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 gcd/3
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations le/3 
* CE 19 is refined into CE [20] 
* CE 17 is refined into CE [21] 
* CE 16 is refined into CE [22] 
* CE 18 is refined into CE [23] 


### Cost equations --> "Loop" of le/3 
* CEs [23] --> Loop 15 
* CEs [20] --> Loop 16 
* CEs [21] --> Loop 17 
* CEs [22] --> Loop 18 

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

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


### Specialization of cost equations minus/3 
* CE 9 is refined into CE [24] 
* CE 7 is refined into CE [25] 
* CE 8 is refined into CE [26] 


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

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

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


### Specialization of cost equations gcd/3 
* CE 10 is refined into CE [27,28,29,30,31] 
* CE 15 is refined into CE [32] 
* CE 14 is refined into CE [33] 
* CE 13 is refined into CE [34] 
* CE 12 is refined into CE [35,36,37,38] 
* CE 11 is refined into CE [39,40,41,42] 


### Cost equations --> "Loop" of gcd/3 
* CEs [42] --> Loop 22 
* CEs [38] --> Loop 23 
* CEs [41] --> Loop 24 
* CEs [37] --> Loop 25 
* CEs [35] --> Loop 26 
* CEs [36] --> Loop 27 
* CEs [39] --> Loop 28 
* CEs [40] --> Loop 29 
* CEs [27] --> Loop 30 
* CEs [33] --> Loop 31 
* CEs [28,29,30,31,32] --> Loop 32 
* CEs [34] --> Loop 33 

### Ranking functions of CR gcd(V1,V,Out) 
* RF of phase [22,23]: [V1+V-3]
* RF of phase [26]: [V1]

#### Partial ranking functions of CR gcd(V1,V,Out) 
* Partial RF of phase [22,23]:
  - RF of loop [22:1]:
    V-2
    V1/2+V/2-2
  - RF of loop [23:1]:
    V1-1 depends on loops [22:1] 
    V1-V+1 depends on loops [22:1] 
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1


### Specialization of cost equations start/3 
* CE 3 is refined into CE [43,44,45,46,47,48,49,50,51,52,53,54] 
* CE 1 is refined into CE [55] 
* CE 2 is refined into CE [56,57,58,59,60,61,62,63,64,65,66,67] 
* CE 4 is refined into CE [68,69,70,71,72] 
* CE 5 is refined into CE [73,74,75] 
* CE 6 is refined into CE [76,77,78,79,80,81,82,83,84] 


### Cost equations --> "Loop" of start/3 
* CEs [80,81] --> Loop 34 
* CEs [69,73,79] --> Loop 35 
* CEs [51] --> Loop 36 
* CEs [49] --> Loop 37 
* CEs [43,44,45,46,47,48,50,52,53,54] --> Loop 38 
* CEs [62] --> Loop 39 
* CEs [64,78] --> Loop 40 
* CEs [56,57,58,59,60,61,63,65,66,67] --> Loop 41 
* CEs [55,68,70,71,72,74,75,76,77,82,83,84] --> Loop 42 

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

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


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

#### Cost of chains of le(V1,V,Out):
* Chain [[15],18]: 1*it(15)+1
  Such that:it(15) =< V1

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

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

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

* Chain [[15],16]: 1*it(15)+0
  Such that:it(15) =< V

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

* Chain [18]: 1
  with precondition: [V1=0,Out=2,V>=0] 

* Chain [17]: 1
  with precondition: [V=0,Out=1,V1>=1] 

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


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

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

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

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

* 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 gcd(V1,V,Out):
* Chain [[26],33]: 4*it(26)+1
  Such that:it(26) =< V1

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

* Chain [[26],32]: 6*it(26)+1*s(5)+2
  Such that:s(5) =< 1
aux(2) =< V1
it(26) =< aux(2)

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

* Chain [[26],30]: 4*it(26)+2
  Such that:it(26) =< V1

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

* Chain [[26],27,33]: 4*it(26)+4
  Such that:it(26) =< V1

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

* Chain [[26],27,32]: 4*it(26)+1*s(5)+5
  Such that:s(5) =< 1
it(26) =< V1

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

* Chain [[22,23],33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+1
  Such that:aux(8) =< V1-V+1
aux(20) =< V1+V
aux(21) =< V1+V-Out
it(22) =< V1/2+V/2
aux(23) =< V1/2+V/2-Out/2
aux(24) =< V
aux(25) =< V-Out
aux(7) =< 2*V-2*Out
aux(26) =< V1
it(22) =< aux(20)
it(23) =< aux(20)
s(16) =< aux(20)
it(22) =< aux(21)
it(23) =< aux(21)
s(16) =< aux(21)
it(22) =< aux(23)
it(23) =< aux(23)
aux(5) =< aux(24)
it(22) =< aux(24)
aux(5) =< aux(25)
it(22) =< aux(25)
it(23) =< aux(7)+aux(8)
it(23) =< aux(5)+aux(26)
s(18) =< aux(5)+aux(26)
s(18) =< it(23)*aux(24)
s(17) =< s(18)
s(15) =< s(16)

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

* Chain [[22,23],32]: 4*it(22)+4*it(23)+5*s(3)+2*s(17)+2
  Such that:aux(8) =< V1-V+1
it(22) =< V1/2+V/2
aux(7) =< 2*V
aux(27) =< V1
aux(28) =< V1+V
aux(29) =< V
s(3) =< aux(28)
it(22) =< aux(28)
it(23) =< aux(28)
it(22) =< aux(29)
it(23) =< aux(7)+aux(8)
it(23) =< aux(29)+aux(27)
s(18) =< aux(29)+aux(27)
s(18) =< it(23)*aux(29)
s(17) =< s(18)

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

* Chain [[22,23],29,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+4
  Such that:aux(8) =< V1-V+1
aux(7) =< 2*V
aux(30) =< V1
aux(31) =< V1+V
aux(32) =< V1/2+V/2
aux(33) =< V
it(22) =< aux(32)
it(22) =< aux(31)
it(23) =< aux(31)
it(23) =< aux(32)
it(22) =< aux(33)
it(23) =< aux(7)+aux(8)
it(23) =< aux(33)+aux(30)
s(18) =< aux(33)+aux(30)
s(18) =< it(23)*aux(33)
s(17) =< s(18)
s(15) =< aux(31)

  with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],29,32]: 4*it(22)+4*it(23)+1*s(5)+2*s(15)+2*s(17)+5
  Such that:s(5) =< 1
aux(8) =< V1-V+1
aux(7) =< 2*V
aux(34) =< V1
aux(35) =< V1+V
aux(36) =< V1/2+V/2
aux(37) =< V
it(22) =< aux(36)
it(22) =< aux(35)
it(23) =< aux(35)
it(23) =< aux(36)
it(22) =< aux(37)
it(23) =< aux(7)+aux(8)
it(23) =< aux(37)+aux(34)
s(18) =< aux(37)+aux(34)
s(18) =< it(23)*aux(37)
s(17) =< s(18)
s(15) =< aux(35)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,[26],33]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+5
  Such that:aux(8) =< V1-V+1
it(22) =< V1/2+V/2
aux(7) =< 2*V
aux(38) =< V1
aux(39) =< V1+V
aux(40) =< V
it(26) =< aux(40)
it(22) =< aux(39)
it(23) =< aux(39)
it(22) =< aux(40)
it(23) =< aux(7)+aux(8)
it(23) =< aux(40)+aux(38)
s(18) =< aux(40)+aux(38)
s(18) =< it(23)*aux(40)
s(17) =< s(18)
s(15) =< aux(39)

  with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,[26],32]: 4*it(22)+4*it(23)+6*it(26)+1*s(5)+2*s(15)+2*s(17)+6
  Such that:s(5) =< 1
aux(8) =< V1-V+1
it(22) =< V1/2+V/2
aux(41) =< V1
aux(42) =< V1+V
aux(43) =< V
aux(44) =< 2*V
it(26) =< aux(44)
it(22) =< aux(42)
it(23) =< aux(42)
it(22) =< aux(43)
it(23) =< aux(44)+aux(8)
it(23) =< aux(43)+aux(41)
s(18) =< aux(43)+aux(41)
s(18) =< it(23)*aux(43)
s(17) =< s(18)
s(15) =< aux(42)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,[26],30]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+6
  Such that:aux(8) =< V1-V+1
it(22) =< V1/2+V/2
aux(7) =< 2*V
aux(45) =< V1
aux(46) =< V1+V
aux(47) =< V
it(26) =< aux(47)
it(22) =< aux(46)
it(23) =< aux(46)
it(22) =< aux(47)
it(23) =< aux(7)+aux(8)
it(23) =< aux(47)+aux(45)
s(18) =< aux(47)+aux(45)
s(18) =< it(23)*aux(47)
s(17) =< s(18)
s(15) =< aux(46)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 

* Chain [[22,23],28,[26],27,33]: 4*it(22)+4*it(23)+4*it(26)+2*s(15)+2*s(17)+8
  Such that:aux(8) =< V1-V+1
it(22) =< V1/2+V/2
aux(7) =< 2*V
aux(48) =< V1
aux(49) =< V1+V
aux(50) =< V
it(26) =< aux(50)
it(22) =< aux(49)
it(23) =< aux(49)
it(22) =< aux(50)
it(23) =< aux(7)+aux(8)
it(23) =< aux(50)+aux(48)
s(18) =< aux(50)+aux(48)
s(18) =< it(23)*aux(50)
s(17) =< s(18)
s(15) =< aux(49)

  with precondition: [Out=1,V1>=3,V>=3,V+V1>=7] 

* Chain [[22,23],28,[26],27,32]: 4*it(22)+4*it(23)+4*it(26)+1*s(5)+2*s(15)+2*s(17)+9
  Such that:s(5) =< 1
aux(8) =< V1-V+1
it(22) =< V1/2+V/2
aux(7) =< 2*V
aux(51) =< V1
aux(52) =< V1+V
aux(53) =< V
it(26) =< aux(53)
it(22) =< aux(52)
it(23) =< aux(52)
it(22) =< aux(53)
it(23) =< aux(7)+aux(8)
it(23) =< aux(53)+aux(51)
s(18) =< aux(53)+aux(51)
s(18) =< it(23)*aux(53)
s(17) =< s(18)
s(15) =< aux(52)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=7] 

* Chain [[22,23],28,32]: 4*it(22)+4*it(23)+2*s(3)+1*s(5)+2*s(15)+2*s(17)+6
  Such that:s(5) =< 1
aux(8) =< V1-V+1
it(22) =< V1/2+V/2
aux(54) =< V1
aux(55) =< V1+V
aux(56) =< V
aux(57) =< 2*V
s(3) =< aux(57)
it(22) =< aux(55)
it(23) =< aux(55)
it(22) =< aux(56)
it(23) =< aux(57)+aux(8)
it(23) =< aux(56)+aux(54)
s(18) =< aux(56)+aux(54)
s(18) =< it(23)*aux(56)
s(17) =< s(18)
s(15) =< aux(55)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,30]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+6
  Such that:aux(8) =< V1-V+1
aux(7) =< 2*V
aux(58) =< V1
aux(59) =< V1+V
aux(60) =< V1/2+V/2
aux(61) =< V
it(22) =< aux(60)
it(22) =< aux(59)
it(23) =< aux(59)
it(23) =< aux(60)
it(22) =< aux(61)
it(23) =< aux(7)+aux(8)
it(23) =< aux(61)+aux(58)
s(18) =< aux(61)+aux(58)
s(18) =< it(23)*aux(61)
s(17) =< s(18)
s(15) =< aux(59)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,27,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+8
  Such that:aux(8) =< V1-V+1
aux(7) =< 2*V
aux(62) =< V1
aux(63) =< V1+V
aux(64) =< V1/2+V/2
aux(65) =< V
it(22) =< aux(64)
it(22) =< aux(63)
it(23) =< aux(63)
it(23) =< aux(64)
it(22) =< aux(65)
it(23) =< aux(7)+aux(8)
it(23) =< aux(65)+aux(62)
s(18) =< aux(65)+aux(62)
s(18) =< it(23)*aux(65)
s(17) =< s(18)
s(15) =< aux(63)

  with precondition: [Out=1,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],28,27,32]: 4*it(22)+4*it(23)+1*s(5)+2*s(15)+2*s(17)+9
  Such that:s(5) =< 1
aux(8) =< V1-V+1
aux(7) =< 2*V
aux(66) =< V1
aux(67) =< V1+V
aux(68) =< V1/2+V/2
aux(69) =< V
it(22) =< aux(68)
it(22) =< aux(67)
it(23) =< aux(67)
it(23) =< aux(68)
it(22) =< aux(69)
it(23) =< aux(7)+aux(8)
it(23) =< aux(69)+aux(66)
s(18) =< aux(69)+aux(66)
s(18) =< it(23)*aux(69)
s(17) =< s(18)
s(15) =< aux(67)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=5] 

* Chain [[22,23],25,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+2*s(20)+4
  Such that:aux(19) =< V1
aux(8) =< V1-V+1
aux(20) =< V1+V
aux(21) =< V1+V-2*Out
aux(22) =< V1-Out
it(22) =< V1/2+V/2
aux(23) =< V1/2+V/2-Out
aux(24) =< V
aux(25) =< V-Out
aux(7) =< 2*V-2*Out
aux(70) =< Out
s(20) =< aux(70)
it(22) =< aux(20)
it(23) =< aux(20)
s(16) =< aux(20)
it(22) =< aux(21)
it(23) =< aux(21)
s(16) =< aux(21)
it(22) =< aux(23)
it(23) =< aux(23)
aux(5) =< aux(24)
it(22) =< aux(24)
aux(5) =< aux(25)
it(22) =< aux(25)
it(23) =< aux(7)+aux(8)
it(23) =< aux(5)+aux(19)
s(18) =< aux(5)+aux(22)
s(18) =< aux(5)+aux(19)
it(23) =< aux(5)+aux(22)
s(18) =< it(23)*aux(24)
s(17) =< s(18)
s(15) =< s(16)

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

* Chain [[22,23],25,32]: 4*it(22)+4*it(23)+3*s(5)+2*s(15)+2*s(17)+5
  Such that:aux(8) =< V1-V+1
aux(7) =< 2*V
aux(72) =< V1
aux(73) =< V1+V
aux(74) =< V1/2+V/2
aux(75) =< V
it(22) =< aux(74)
s(5) =< aux(75)
it(22) =< aux(73)
it(23) =< aux(73)
it(23) =< aux(74)
it(22) =< aux(75)
it(23) =< aux(7)+aux(8)
it(23) =< aux(75)+aux(72)
s(18) =< aux(75)+aux(72)
s(18) =< it(23)*aux(75)
s(17) =< s(18)
s(15) =< aux(73)

  with precondition: [Out=0,V1>=2,V>=2,V+V1>=6] 

* Chain [[22,23],24,33]: 4*it(22)+4*it(23)+2*s(15)+2*s(17)+2*s(22)+4
  Such that:aux(19) =< V1
aux(8) =< V1-V+1
aux(20) =< V1+V
aux(21) =< V1+V-2*Out
aux(22) =< V1-Out
it(22) =< V1/2+V/2
aux(23) =< V1/2+V/2-Out
aux(24) =< V
aux(25) =< V-Out
aux(7) =< 2*V-2*Out
aux(76) =< Out
s(22) =< aux(76)
it(22) =< aux(20)
it(23) =< aux(20)
s(16) =< aux(20)
it(22) =< aux(21)
it(23) =< aux(21)
s(16) =< aux(21)
it(22) =< aux(23)
it(23) =< aux(23)
aux(5) =< aux(24)
it(22) =< aux(24)
aux(5) =< aux(25)
it(22) =< aux(25)
it(23) =< aux(7)+aux(8)
it(23) =< aux(5)+aux(19)
s(18) =< aux(5)+aux(22)
s(18) =< aux(5)+aux(19)
it(23) =< aux(5)+aux(22)
s(18) =< it(23)*aux(24)
s(17) =< s(18)
s(15) =< s(16)

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

* Chain [[22,23],24,32]: 4*it(22)+4*it(23)+3*s(5)+2*s(15)+2*s(17)+5
  Such that:aux(8) =< V1-V+1
aux(7) =< 2*V
aux(78) =< V1
aux(79) =< V1+V
aux(80) =< V1/2+V/2
aux(81) =< V
it(22) =< aux(80)
s(5) =< aux(78)
it(22) =< aux(79)
it(23) =< aux(79)
it(23) =< aux(80)
it(22) =< aux(81)
it(23) =< aux(7)+aux(8)
it(23) =< aux(81)+aux(78)
s(18) =< aux(81)+aux(78)
s(18) =< it(23)*aux(81)
s(17) =< s(18)
s(15) =< aux(79)

  with precondition: [Out=0,V1>=3,V>=3,V+V1>=8] 

* Chain [33]: 1
  with precondition: [V1=0,V=Out,V>=0] 

* Chain [32]: 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 [31]: 1
  with precondition: [V=0,V1=Out,V1>=1] 

* Chain [30]: 2
  with precondition: [V=1,Out=0,V1>=1] 

* Chain [29,33]: 4
  with precondition: [V1=1,Out=1,V>=2] 

* Chain [29,32]: 1*s(5)+5
  Such that:s(5) =< 1

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

* Chain [28,[26],33]: 4*it(26)+5
  Such that:it(26) =< V

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

* Chain [28,[26],32]: 6*it(26)+1*s(5)+6
  Such that:s(5) =< 1
aux(2) =< V
it(26) =< aux(2)

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

* Chain [28,[26],30]: 4*it(26)+6
  Such that:it(26) =< V

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

* Chain [28,[26],27,33]: 4*it(26)+8
  Such that:it(26) =< V

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

* Chain [28,[26],27,32]: 4*it(26)+1*s(5)+9
  Such that:s(5) =< 1
it(26) =< V

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

* Chain [28,32]: 2*s(3)+1*s(5)+6
  Such that:s(5) =< 1
aux(1) =< V
s(3) =< aux(1)

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

* Chain [28,30]: 6
  with precondition: [V1=1,Out=0,V>=2] 

* Chain [28,27,33]: 8
  with precondition: [V1=1,Out=1,V>=2] 

* Chain [28,27,32]: 1*s(5)+9
  Such that:s(5) =< 1

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

* Chain [27,33]: 4
  with precondition: [V=1,Out=1,V1>=1] 

* Chain [27,32]: 1*s(5)+5
  Such that:s(5) =< 1

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

* Chain [25,33]: 2*s(20)+4
  Such that:aux(70) =< Out
s(20) =< aux(70)

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

* Chain [25,32]: 3*s(5)+5
  Such that:aux(71) =< V
s(5) =< aux(71)

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

* Chain [24,33]: 2*s(22)+4
  Such that:aux(76) =< Out
s(22) =< aux(76)

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

* Chain [24,32]: 3*s(5)+5
  Such that:aux(77) =< V1
s(5) =< aux(77)

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


#### Cost of chains of start(V1,V,V14):
* Chain [42]: 45*s(266)+9*s(268)+10*s(278)+68*s(279)+36*s(281)+18*s(283)+43*s(284)+8*s(285)+32*s(286)+16*s(288)+9
  Such that:s(271) =< 1
aux(114) =< V1
aux(115) =< V1-V+1
aux(116) =< V1+V
aux(117) =< V1/2+V/2
aux(118) =< V
aux(119) =< 2*V
s(268) =< aux(114)
s(266) =< aux(118)
s(278) =< s(271)
s(279) =< aux(117)
s(279) =< aux(116)
s(281) =< aux(116)
s(281) =< aux(117)
s(279) =< aux(118)
s(281) =< aux(119)+aux(115)
s(281) =< aux(118)+aux(114)
s(282) =< aux(118)+aux(114)
s(282) =< s(281)*aux(118)
s(283) =< s(282)
s(284) =< aux(116)
s(285) =< aux(119)
s(286) =< aux(116)
s(286) =< aux(119)+aux(115)
s(286) =< aux(118)+aux(114)
s(287) =< aux(118)+aux(114)
s(287) =< s(286)*aux(118)
s(288) =< s(287)

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

* Chain [41]: 64*s(341)+40*s(342)+20*s(344)+10*s(346)+96*s(347)+8*s(348)+20*s(349)+10*s(351)+101*s(359)+40*s(369)+20*s(371)+10*s(373)+16*s(375)+20*s(376)+10*s(378)+68*s(389)+36*s(391)+18*s(393)+32*s(396)+16*s(398)+8*s(399)+11
  Such that:s(340) =< 2
s(363) =< -V+1
s(365) =< V/2
aux(130) =< 1
aux(131) =< -2*V+V14+1
aux(132) =< -V+V14
aux(133) =< V
aux(134) =< 2*V
aux(135) =< V14
aux(136) =< V14/2
s(359) =< aux(133)
s(341) =< aux(130)
s(347) =< aux(135)
s(369) =< s(365)
s(369) =< aux(133)
s(371) =< aux(133)
s(371) =< s(365)
s(371) =< aux(134)+s(363)
s(372) =< aux(133)
s(372) =< s(371)*aux(133)
s(373) =< s(372)
s(375) =< aux(134)
s(376) =< aux(133)
s(376) =< aux(134)+s(363)
s(377) =< aux(133)
s(377) =< s(376)*aux(133)
s(378) =< s(377)
s(389) =< aux(136)
s(389) =< aux(135)
s(391) =< aux(135)
s(391) =< aux(136)
s(389) =< aux(133)
s(391) =< aux(134)+aux(131)
s(391) =< aux(133)+aux(132)
s(392) =< aux(133)+aux(132)
s(392) =< s(391)*aux(133)
s(393) =< s(392)
s(396) =< aux(135)
s(396) =< aux(134)+aux(131)
s(396) =< aux(133)+aux(132)
s(397) =< aux(133)+aux(132)
s(397) =< s(396)*aux(133)
s(398) =< s(397)
s(399) =< aux(132)
s(342) =< aux(136)
s(342) =< aux(135)
s(344) =< aux(135)
s(344) =< aux(136)
s(342) =< aux(130)
s(344) =< s(340)+aux(135)
s(344) =< aux(130)+aux(135)
s(345) =< aux(130)+aux(135)
s(345) =< s(344)*aux(130)
s(346) =< s(345)
s(348) =< s(340)
s(349) =< aux(135)
s(349) =< s(340)+aux(135)
s(349) =< aux(130)+aux(135)
s(350) =< aux(130)+aux(135)
s(350) =< s(349)*aux(130)
s(351) =< s(350)

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

* Chain [40]: 9*s(447)+8*s(451)+10
  Such that:s(450) =< V
aux(137) =< V14
s(451) =< s(450)
s(447) =< aux(137)

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

* Chain [39]: 1*s(452)+3
  Such that:s(452) =< V14

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

* Chain [38]: 64*s(460)+40*s(461)+20*s(463)+10*s(465)+96*s(466)+8*s(467)+20*s(468)+10*s(470)+101*s(478)+40*s(488)+20*s(490)+10*s(492)+16*s(494)+20*s(495)+10*s(497)+68*s(508)+36*s(510)+18*s(512)+32*s(515)+16*s(517)+8*s(518)+11
  Such that:s(459) =< 2
s(482) =< -V14+1
s(484) =< V14/2
aux(148) =< 1
aux(149) =< V
aux(150) =< V-2*V14+1
aux(151) =< V-V14
aux(152) =< V/2
aux(153) =< V14
aux(154) =< 2*V14
s(478) =< aux(153)
s(460) =< aux(148)
s(466) =< aux(149)
s(488) =< s(484)
s(488) =< aux(153)
s(490) =< aux(153)
s(490) =< s(484)
s(490) =< aux(154)+s(482)
s(491) =< aux(153)
s(491) =< s(490)*aux(153)
s(492) =< s(491)
s(494) =< aux(154)
s(495) =< aux(153)
s(495) =< aux(154)+s(482)
s(496) =< aux(153)
s(496) =< s(495)*aux(153)
s(497) =< s(496)
s(508) =< aux(152)
s(508) =< aux(149)
s(510) =< aux(149)
s(510) =< aux(152)
s(508) =< aux(153)
s(510) =< aux(154)+aux(150)
s(510) =< aux(153)+aux(151)
s(511) =< aux(153)+aux(151)
s(511) =< s(510)*aux(153)
s(512) =< s(511)
s(515) =< aux(149)
s(515) =< aux(154)+aux(150)
s(515) =< aux(153)+aux(151)
s(516) =< aux(153)+aux(151)
s(516) =< s(515)*aux(153)
s(517) =< s(516)
s(518) =< aux(151)
s(461) =< aux(152)
s(461) =< aux(149)
s(463) =< aux(149)
s(463) =< aux(152)
s(461) =< aux(148)
s(463) =< s(459)+aux(149)
s(463) =< aux(148)+aux(149)
s(464) =< aux(148)+aux(149)
s(464) =< s(463)*aux(148)
s(465) =< s(464)
s(467) =< s(459)
s(468) =< aux(149)
s(468) =< s(459)+aux(149)
s(468) =< aux(148)+aux(149)
s(469) =< aux(148)+aux(149)
s(469) =< s(468)*aux(148)
s(470) =< s(469)

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

* Chain [37]: 1*s(566)+3
  Such that:s(566) =< V14

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

* Chain [36]: 9*s(567)+10
  Such that:aux(155) =< V14
s(567) =< aux(155)

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

* Chain [35]: 1
  with precondition: [V=0,V1>=0] 

* Chain [34]: 3*s(572)+22*s(573)+5
  Such that:s(570) =< 1
aux(156) =< V1
s(572) =< s(570)
s(573) =< aux(156)

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


Closed-form bounds of start(V1,V,V14): 
-------------------------------------
* Chain [42] with precondition: [V1>=0,V>=0] 
    - Upper bound: 188*V1+240*V+19 
    - Complexity: n 
* Chain [41] with precondition: [V1=1,V>=1,V14>=1] 
    - Upper bound: 227*V+224*V14+151+nat(-V+V14)*42+20*V+34*V14 
    - Complexity: n 
* Chain [40] with precondition: [V1=1,V>=2] 
    - Upper bound: 8*V+10+nat(V14)*9 
    - Complexity: n 
* Chain [39] with precondition: [V1=1,V=V14,V>=2] 
    - Upper bound: V14+3 
    - Complexity: n 
* Chain [38] with precondition: [V1=2,V>=1,V14>=1] 
    - Upper bound: 224*V+227*V14+151+nat(V-V14)*42+34*V+20*V14 
    - Complexity: n 
* Chain [37] with precondition: [V1=2,V=V14,V>=2] 
    - Upper bound: V14+3 
    - Complexity: n 
* Chain [36] with precondition: [V1=2,V=V14+1,V>=3] 
    - Upper bound: 9*V14+10 
    - Complexity: n 
* Chain [35] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [34] with precondition: [V=1,V1>=1] 
    - Upper bound: 22*V1+8 
    - Complexity: n 

### Maximum cost of start(V1,V,V14): max([188*V1+240*V+18,187*V+141+nat(V14)*186+20*V+nat(V14/2)*40+max([nat(V14)*29+32*V+nat(-V+V14)*42+nat(V14/2)*28,nat(2*V14)*16+29*V+nat(V-V14)*42+14*V])+8*V+(nat(V14)*8+7)+(nat(V14)+2)])+1 
Asymptotic class: n 
* Total analysis performed in 1253 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.
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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^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)
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), 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(s(x), s(y)) ->^+ minus(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^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)
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

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^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)
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

S is empty.
Rewrite Strategy: INNERMOST