/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 1616 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(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   gcd(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0), x, y)
   if1(false, x, y) -> gcd(y, x)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(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(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]
   gcd(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> gcd(y, x) [1]
   if2(true, x, y) -> gcd(minus(x, y), y) [1]
   if2(false, x, y) -> x [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(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(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]
   gcd(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> gcd(y, x) [1]
   if2(true, x, y) -> gcd(minus(x, y), y) [1]
   if2(false, x, y) -> x [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]

The TRS has the following type information:
minus :: s:0 -> s:0 -> s:0
s :: s:0 -> s:0
if :: true:false -> s:0 -> s:0 -> s:0
gt :: s:0 -> s:0 -> true:false
true :: true:false
false :: true:false
0 :: s:0
gcd :: s:0 -> s:0 -> s:0
if1 :: true:false -> s:0 -> s:0 -> s:0
ge :: s:0 -> s:0 -> true:false
if2 :: true:false -> s:0 -> s:0 -> s:0

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]
   gt(v0, v1) -> null_gt [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2) -> null_if [0]
   if1(v0, v1, v2) -> null_if1 [0]
   if2(v0, v1, v2) -> null_if2 [0]

And the following fresh constants:    null_minus, null_gt, null_ge, null_if, null_if1, null_if2

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

(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(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]
   gcd(x, y) -> if1(ge(x, y), x, y) [1]
   if1(true, x, y) -> if2(gt(y, 0), x, y) [1]
   if1(false, x, y) -> gcd(y, x) [1]
   if2(true, x, y) -> gcd(minus(x, y), y) [1]
   if2(false, x, y) -> x [1]
   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(v0, v1) -> null_minus [0]
   gt(v0, v1) -> null_gt [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2) -> null_if [0]
   if1(v0, v1, v2) -> null_if1 [0]
   if2(v0, v1, v2) -> null_if2 [0]

The TRS has the following type information:
minus :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
s :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
if :: true:false:null_gt:null_ge -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
gt :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> true:false:null_gt:null_ge
true :: true:false:null_gt:null_ge
false :: true:false:null_gt:null_ge
0 :: s:0:null_minus:null_if:null_if1:null_if2
gcd :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
if1 :: true:false:null_gt:null_ge -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
ge :: s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> true:false:null_gt:null_ge
if2 :: true:false:null_gt:null_ge -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2 -> s:0:null_minus:null_if:null_if1:null_if2
null_minus :: s:0:null_minus:null_if:null_if1:null_if2
null_gt :: true:false:null_gt:null_ge
null_ge :: true:false:null_gt:null_ge
null_if :: s:0:null_minus:null_if:null_if1:null_if2
null_if1 :: s:0:null_minus:null_if:null_if1:null_if2
null_if2 :: s:0:null_minus:null_if:null_if1:null_if2

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:

   true => 2
   false => 1
   0 => 0
   null_minus => 0
   null_gt => 0
   null_ge => 0
   null_if => 0
   null_if1 => 0
   null_if2 => 0

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

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

   gcd(z, z') -{ 1 }-> if1(ge(x, y), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 2 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   gt(z, z') -{ 1 }-> gt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gt(z, z') -{ 1 }-> 2 :|: x >= 0, z = 1 + x, z' = 0
   gt(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   gt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> if2(gt(y, 0), x, y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if1(z, z', z'') -{ 1 }-> gcd(y, x) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if2(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if2(z, z', z'') -{ 1 }-> gcd(minus(x, y), y) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   minus(z, z') -{ 1 }-> if(gt(1 + x, y), x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   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, V5),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[if(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[gcd(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[if1(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[if2(V1, V, V5, Out)],[V1 >= 0,V >= 0,V5 >= 0]).
eq(start(V1, V, V5),0,[gt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V5),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(minus(V1, V, Out),1,[gt(1 + V3, V2, Ret0),if(Ret0, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = 1 + V3,V = V2]).
eq(if(V1, V, V5, Out),1,[minus(V4, V6, Ret1)],[Out = 1 + Ret1,V1 = 2,V = V4,V5 = V6,V4 >= 0,V6 >= 0]).
eq(if(V1, V, V5, Out),1,[],[Out = 0,V = V8,V5 = V7,V1 = 1,V8 >= 0,V7 >= 0]).
eq(gcd(V1, V, Out),1,[ge(V9, V10, Ret01),if1(Ret01, V9, V10, Ret2)],[Out = Ret2,V9 >= 0,V10 >= 0,V1 = V9,V = V10]).
eq(if1(V1, V, V5, Out),1,[gt(V11, 0, Ret02),if2(Ret02, V12, V11, Ret3)],[Out = Ret3,V1 = 2,V = V12,V5 = V11,V12 >= 0,V11 >= 0]).
eq(if1(V1, V, V5, Out),1,[gcd(V13, V14, Ret4)],[Out = Ret4,V = V14,V5 = V13,V1 = 1,V14 >= 0,V13 >= 0]).
eq(if2(V1, V, V5, Out),1,[minus(V16, V15, Ret03),gcd(Ret03, V15, Ret5)],[Out = Ret5,V1 = 2,V = V16,V5 = V15,V16 >= 0,V15 >= 0]).
eq(if2(V1, V, V5, Out),1,[],[Out = V17,V = V17,V5 = V18,V1 = 1,V17 >= 0,V18 >= 0]).
eq(gt(V1, V, Out),1,[],[Out = 1,V19 >= 0,V1 = 0,V = V19]).
eq(gt(V1, V, Out),1,[],[Out = 2,V20 >= 0,V1 = 1 + V20,V = 0]).
eq(gt(V1, V, Out),1,[gt(V21, V22, Ret6)],[Out = Ret6,V = 1 + V22,V21 >= 0,V22 >= 0,V1 = 1 + V21]).
eq(ge(V1, V, Out),1,[],[Out = 2,V23 >= 0,V1 = V23,V = 0]).
eq(ge(V1, V, Out),1,[],[Out = 1,V = 1 + V24,V24 >= 0,V1 = 0]).
eq(ge(V1, V, Out),1,[ge(V25, V26, Ret7)],[Out = Ret7,V = 1 + V26,V25 >= 0,V26 >= 0,V1 = 1 + V25]).
eq(minus(V1, V, Out),0,[],[Out = 0,V28 >= 0,V27 >= 0,V1 = V28,V = V27]).
eq(gt(V1, V, Out),0,[],[Out = 0,V30 >= 0,V29 >= 0,V1 = V30,V = V29]).
eq(ge(V1, V, Out),0,[],[Out = 0,V32 >= 0,V31 >= 0,V1 = V32,V = V31]).
eq(if(V1, V, V5, Out),0,[],[Out = 0,V33 >= 0,V5 = V35,V34 >= 0,V1 = V33,V = V34,V35 >= 0]).
eq(if1(V1, V, V5, Out),0,[],[Out = 0,V37 >= 0,V5 = V38,V36 >= 0,V1 = V37,V = V36,V38 >= 0]).
eq(if2(V1, V, V5, Out),0,[],[Out = 0,V40 >= 0,V5 = V41,V39 >= 0,V1 = V40,V = V39,V41 >= 0]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(gcd(V1,V,Out),[V1,V],[Out]).
input_output_vars(if1(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(if2(V1,V,V5,Out),[V1,V,V5],[Out]).
input_output_vars(gt(V1,V,Out),[V1,V],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [ge/3]
1. recursive  : [gt/3]
2. recursive  : [if/4,minus/3]
3. recursive  : [gcd/3,if1/4,if2/4]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into ge/3
1. SCC is partially evaluated into gt/3
2. SCC is partially evaluated into minus/3
3. SCC is partially evaluated into gcd/3
4. SCC is partially evaluated into start/3

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

### Specialization of cost equations ge/3 
* CE 29 is refined into CE [30] 
* CE 26 is refined into CE [31] 
* CE 27 is refined into CE [32] 
* CE 28 is refined into CE [33] 


### Cost equations --> "Loop" of ge/3 
* CEs [33] --> Loop 17 
* CEs [30] --> Loop 18 
* CEs [31] --> Loop 19 
* CEs [32] --> Loop 20 

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

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


### Specialization of cost equations gt/3 
* CE 16 is refined into CE [34] 
* CE 14 is refined into CE [35] 
* CE 13 is refined into CE [36] 
* CE 15 is refined into CE [37] 


### Cost equations --> "Loop" of gt/3 
* CEs [37] --> Loop 21 
* CEs [34] --> Loop 22 
* CEs [35] --> Loop 23 
* CEs [36] --> Loop 24 

### Ranking functions of CR gt(V1,V,Out) 
* RF of phase [21]: [V,V1]

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


### Specialization of cost equations minus/3 
* CE 17 is refined into CE [38,39,40,41] 
* CE 18 is refined into CE [42] 
* CE 20 is refined into CE [43] 
* CE 19 is refined into CE [44,45] 


### Cost equations --> "Loop" of minus/3 
* CEs [45] --> Loop 25 
* CEs [44] --> Loop 26 
* CEs [38,39,40,41,42,43] --> Loop 27 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [25]: [V1-1,V1-V]
* RF of phase [26]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V1-1
    V1-V
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1


### Specialization of cost equations gcd/3 
* CE 23 is refined into CE [46,47] 
* CE 25 is refined into CE [48,49] 
* CE 22 is refined into CE [50] 
* CE 21 is refined into CE [51,52,53,54] 
* CE 24 is refined into CE [55,56,57,58,59] 


### Cost equations --> "Loop" of gcd/3 
* CEs [50] --> Loop 28 
* CEs [51,52,56] --> Loop 29 
* CEs [53,54,55,57,58,59] --> Loop 30 
* CEs [47] --> Loop 31 
* CEs [49] --> Loop 32 
* CEs [46] --> Loop 33 
* CEs [48] --> Loop 34 

### Ranking functions of CR gcd(V1,V,Out) 
* RF of phase [31,32]: [V1+2*V-3]

#### Partial ranking functions of CR gcd(V1,V,Out) 
* Partial RF of phase [31,32]:
  - RF of loop [31:1]:
    V1-1 depends on loops [32:1] 
    V1-V depends on loops [32:1] 
  - RF of loop [32:1]:
    V-1
    -V1/2+V/2 depends on loops [31:1] 


### Specialization of cost equations start/3 
* CE 1 is refined into CE [60,61,62] 
* CE 3 is refined into CE [63] 
* CE 5 is refined into CE [64,65,66,67,68] 
* CE 6 is refined into CE [69,70,71,72,73,74,75,76,77,78] 
* CE 8 is refined into CE [79,80,81] 
* CE 2 is refined into CE [82] 
* CE 4 is refined into CE [83] 
* CE 7 is refined into CE [84,85,86,87,88,89] 
* CE 9 is refined into CE [90,91,92] 
* CE 10 is refined into CE [93,94,95,96,97,98] 
* CE 11 is refined into CE [99,100,101,102,103] 
* CE 12 is refined into CE [104,105,106,107,108] 


### Cost equations --> "Loop" of start/3 
* CEs [90,95,96,100,105] --> Loop 35 
* CEs [60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81] --> Loop 36 
* CEs [85] --> Loop 37 
* CEs [83,84,86,87,88,89] --> Loop 38 
* CEs [82,91,92,93,94,97,98,99,101,102,103,104,106,107,108] --> Loop 39 

### Ranking functions of CR start(V1,V,V5) 

#### Partial ranking functions of CR start(V1,V,V5) 


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

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

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

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

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

* 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=1,V>=1] 

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

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


#### Cost of chains of gt(V1,V,Out):
* Chain [[21],24]: 1*it(21)+1
  Such that:it(21) =< V1

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

* Chain [[21],23]: 1*it(21)+1
  Such that:it(21) =< V

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

* Chain [[21],22]: 1*it(21)+0
  Such that:it(21) =< V

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

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

* Chain [23]: 1
  with precondition: [V=0,Out=2,V1>=1] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[26],27]: 3*it(26)+2*s(4)+3
  Such that:aux(1) =< V1-Out
it(26) =< Out
s(4) =< aux(1)

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

* Chain [[25],27]: 3*it(25)+2*s(3)+2*s(4)+1*s(9)+3
  Such that:aux(1) =< V1-Out
it(25) =< Out
aux(4) =< V
s(4) =< aux(1)
s(3) =< aux(4)
s(9) =< it(25)*aux(4)

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

* Chain [27]: 2*s(3)+2*s(4)+3
  Such that:aux(1) =< V1
aux(2) =< V
s(4) =< aux(1)
s(3) =< aux(2)

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


#### Cost of chains of gcd(V1,V,Out):
* Chain [[31,32],33,34,30]: 8*it(31)+3*it(32)+6*s(14)+3*s(36)+3*s(37)+2*s(38)+1*s(39)+1*s(42)+15
  Such that:aux(36) =< -V1+V
aux(12) =< V1-V
aux(25) =< -V1/2+V/2
aux(42) =< V1/2+V/2
aux(45) =< V1
aux(46) =< V1+V
aux(47) =< V1+2*V
aux(48) =< V
it(32) =< aux(48)
s(14) =< aux(47)
it(31) =< aux(47)
it(32) =< aux(47)
aux(24) =< aux(42)
aux(24) =< aux(46)
aux(23) =< aux(48)
aux(26) =< aux(46)+1
aux(27) =< aux(24)*2
it(32) =< aux(24)+aux(25)
aux(9) =< aux(27)+aux(36)
aux(9) =< it(32)*aux(23)
aux(11) =< aux(27)+aux(36)
aux(11) =< it(32)*aux(48)
s(42) =< it(32)*aux(26)
s(41) =< aux(9)+aux(45)
it(31) =< aux(11)+aux(12)
it(31) =< aux(9)+aux(45)
s(40) =< aux(9)+aux(45)
s(41) =< it(31)*aux(23)
s(37) =< it(31)*aux(46)
s(40) =< it(31)*aux(46)
s(36) =< s(41)
s(38) =< s(40)
s(39) =< s(37)*aux(48)

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

* Chain [[31,32],33,34,29]: 8*it(31)+3*it(32)+5*s(16)+3*s(36)+3*s(37)+2*s(38)+1*s(39)+1*s(42)+15
  Such that:aux(36) =< -V1+V
aux(12) =< V1-V
aux(25) =< -V1/2+V/2
aux(42) =< V1/2+V/2
aux(49) =< V1
aux(50) =< V1+V
aux(51) =< V1+2*V
aux(52) =< V
it(32) =< aux(52)
s(16) =< aux(51)
it(31) =< aux(51)
it(32) =< aux(51)
aux(24) =< aux(42)
aux(24) =< aux(50)
aux(23) =< aux(52)
aux(26) =< aux(50)+1
aux(27) =< aux(24)*2
it(32) =< aux(24)+aux(25)
aux(9) =< aux(27)+aux(36)
aux(9) =< it(32)*aux(23)
aux(11) =< aux(27)+aux(36)
aux(11) =< it(32)*aux(52)
s(42) =< it(32)*aux(26)
s(41) =< aux(9)+aux(49)
it(31) =< aux(11)+aux(12)
it(31) =< aux(9)+aux(49)
s(40) =< aux(9)+aux(49)
s(41) =< it(31)*aux(23)
s(37) =< it(31)*aux(50)
s(40) =< it(31)*aux(50)
s(36) =< s(41)
s(38) =< s(40)
s(39) =< s(37)*aux(52)

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

* Chain [[31,32],33,34,28]: 8*it(31)+3*it(32)+3*s(16)+2*s(19)+3*s(36)+3*s(37)+2*s(38)+1*s(39)+1*s(42)+16
  Such that:aux(36) =< -V1+V
aux(37) =< V1
aux(12) =< V1-V
aux(38) =< V1+V
aux(39) =< V1+2*V
aux(40) =< V1+2*V-2*Out
aux(41) =< V1-Out
aux(25) =< -V1/2+V/2
aux(42) =< V1/2+V/2
aux(44) =< V
it(32) =< V-Out
aux(6) =< Out
aux(53) =< V1+V-Out
s(16) =< aux(6)
s(19) =< aux(53)
it(31) =< aux(39)
it(32) =< aux(39)
it(31) =< aux(40)
it(32) =< aux(40)
aux(24) =< aux(42)
aux(24) =< aux(53)
it(32) =< aux(44)
aux(23) =< aux(44)
aux(26) =< aux(38)+1
aux(27) =< aux(24)*2
it(32) =< aux(24)+aux(25)
aux(9) =< aux(27)+aux(36)
aux(9) =< it(32)*aux(23)
aux(11) =< aux(27)+aux(36)
aux(11) =< it(32)*aux(44)
s(42) =< it(32)*aux(26)
s(41) =< aux(9)+aux(37)
it(31) =< aux(11)+aux(12)
it(31) =< aux(9)+aux(37)
s(41) =< aux(9)+aux(41)
s(40) =< aux(9)+aux(41)
s(40) =< aux(9)+aux(37)
it(31) =< aux(9)+aux(41)
s(41) =< it(31)*aux(23)
s(37) =< it(31)*aux(38)
s(40) =< it(31)*aux(38)
s(36) =< s(41)
s(38) =< s(40)
s(39) =< s(37)*aux(44)

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

* Chain [[31,32],33,30]: 8*it(31)+3*it(32)+9*s(10)+3*s(36)+3*s(37)+2*s(38)+1*s(39)+1*s(42)+12
  Such that:aux(36) =< -V1+V
aux(12) =< V1-V
aux(25) =< -V1/2+V/2
aux(42) =< V1/2+V/2
aux(55) =< V1
aux(56) =< V1+V
aux(57) =< V1+2*V
aux(58) =< V
it(32) =< aux(58)
s(10) =< aux(56)
it(31) =< aux(57)
it(32) =< aux(57)
aux(24) =< aux(42)
aux(24) =< aux(56)
aux(23) =< aux(58)
aux(26) =< aux(56)+1
aux(27) =< aux(24)*2
it(32) =< aux(24)+aux(25)
aux(9) =< aux(27)+aux(36)
aux(9) =< it(32)*aux(23)
aux(11) =< aux(27)+aux(36)
aux(11) =< it(32)*aux(58)
s(42) =< it(32)*aux(26)
s(41) =< aux(9)+aux(55)
it(31) =< aux(11)+aux(12)
it(31) =< aux(9)+aux(55)
s(40) =< aux(9)+aux(55)
s(41) =< it(31)*aux(23)
s(37) =< it(31)*aux(56)
s(40) =< it(31)*aux(56)
s(36) =< s(41)
s(38) =< s(40)
s(39) =< s(37)*aux(58)

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

* Chain [[31,32],30]: 8*it(31)+3*it(32)+5*s(10)+3*s(36)+3*s(37)+2*s(38)+1*s(39)+1*s(42)+4
  Such that:aux(36) =< -V1+V
aux(12) =< V1-V
aux(25) =< -V1/2+V/2
aux(42) =< V1/2+V/2
aux(59) =< V1
aux(60) =< V1+V
aux(61) =< V1+2*V
aux(62) =< V
s(10) =< aux(60)
it(32) =< aux(62)
it(31) =< aux(61)
it(32) =< aux(61)
aux(24) =< aux(42)
aux(24) =< aux(60)
aux(23) =< aux(62)
aux(26) =< aux(60)+1
aux(27) =< aux(24)*2
it(32) =< aux(24)+aux(25)
aux(9) =< aux(27)+aux(36)
aux(9) =< it(32)*aux(23)
aux(11) =< aux(27)+aux(36)
aux(11) =< it(32)*aux(62)
s(42) =< it(32)*aux(26)
s(41) =< aux(9)+aux(59)
it(31) =< aux(11)+aux(12)
it(31) =< aux(9)+aux(59)
s(40) =< aux(9)+aux(59)
s(41) =< it(31)*aux(23)
s(37) =< it(31)*aux(60)
s(40) =< it(31)*aux(60)
s(36) =< s(41)
s(38) =< s(40)
s(39) =< s(37)*aux(62)

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

* Chain [34,30]: 1*s(14)+7
  Such that:s(14) =< V

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

* Chain [34,29]: 7
  with precondition: [V1=0,Out=0,V>=1] 

* Chain [34,28]: 8
  with precondition: [V1=0,V=Out,V>=1] 

* Chain [33,34,30]: 4*s(14)+2*s(19)+15
  Such that:s(17) =< V1
aux(7) =< V
s(14) =< aux(7)
s(19) =< s(17)

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

* Chain [33,34,29]: 3*s(16)+2*s(19)+15
  Such that:s(17) =< V1
aux(6) =< V
s(16) =< aux(6)
s(19) =< s(17)

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

* Chain [33,34,28]: 3*s(16)+2*s(19)+16
  Such that:s(17) =< V1
aux(6) =< Out
s(16) =< aux(6)
s(19) =< s(17)

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

* Chain [33,30]: 7*s(10)+2*s(19)+12
  Such that:s(17) =< V1
aux(54) =< V
s(10) =< aux(54)
s(19) =< s(17)

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

* Chain [30]: 4*s(10)+1*s(14)+4
  Such that:s(14) =< V1
aux(5) =< V
s(10) =< aux(5)

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

* Chain [29]: 4
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [28]: 5
  with precondition: [V=0,V1=Out,V1>=0] 


#### Cost of chains of start(V1,V,V5):
* Chain [39]: 18*s(158)+30*s(159)+1*s(165)+19*s(176)+15*s(177)+40*s(178)+5*s(185)+15*s(188)+15*s(189)+10*s(190)+5*s(191)+11*s(192)+16
  Such that:aux(76) =< -V1+V
aux(77) =< V1
aux(78) =< V1-V
aux(79) =< V1+V
aux(80) =< V1+2*V
aux(81) =< -V1/2+V/2
aux(82) =< V1/2+V/2
aux(83) =< V
s(158) =< aux(77)
s(159) =< aux(83)
s(176) =< aux(79)
s(177) =< aux(83)
s(178) =< aux(80)
s(177) =< aux(80)
s(179) =< aux(82)
s(179) =< aux(79)
s(180) =< aux(83)
s(181) =< aux(79)+1
s(182) =< s(179)*2
s(177) =< s(179)+aux(81)
s(183) =< s(182)+aux(76)
s(183) =< s(177)*s(180)
s(184) =< s(182)+aux(76)
s(184) =< s(177)*aux(83)
s(185) =< s(177)*s(181)
s(186) =< s(183)+aux(77)
s(178) =< s(184)+aux(78)
s(178) =< s(183)+aux(77)
s(187) =< s(183)+aux(77)
s(186) =< s(178)*s(180)
s(188) =< s(178)*aux(79)
s(187) =< s(178)*aux(79)
s(189) =< s(186)
s(190) =< s(187)
s(191) =< s(188)*aux(83)
s(192) =< aux(80)
s(165) =< s(158)*aux(83)

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

* Chain [38]: 9*s(240)+22*s(241)+19*s(242)+15*s(243)+40*s(244)+5*s(251)+15*s(254)+15*s(255)+10*s(256)+5*s(257)+11*s(258)+17
  Such that:aux(88) =< -V+V5
aux(89) =< V
aux(90) =< V-V5
aux(91) =< V+V5
aux(92) =< 2*V+V5
aux(93) =< V/2-V5/2
aux(94) =< V/2+V5/2
aux(95) =< V5
s(240) =< aux(95)
s(241) =< aux(89)
s(242) =< aux(91)
s(243) =< aux(89)
s(244) =< aux(92)
s(243) =< aux(92)
s(245) =< aux(94)
s(245) =< aux(91)
s(246) =< aux(89)
s(247) =< aux(91)+1
s(248) =< s(245)*2
s(243) =< s(245)+aux(93)
s(249) =< s(248)+aux(90)
s(249) =< s(243)*s(246)
s(250) =< s(248)+aux(90)
s(250) =< s(243)*aux(89)
s(251) =< s(243)*s(247)
s(252) =< s(249)+aux(95)
s(244) =< s(250)+aux(88)
s(244) =< s(249)+aux(95)
s(253) =< s(249)+aux(95)
s(252) =< s(244)*s(246)
s(254) =< s(244)*aux(91)
s(253) =< s(244)*aux(91)
s(255) =< s(252)
s(256) =< s(253)
s(257) =< s(254)*aux(89)
s(258) =< aux(92)

  with precondition: [V1=1,V>=0,V5>=0] 

* Chain [37]: 9
  with precondition: [V1=1,V5=0,V>=1] 

* Chain [36]: 147*s(295)+134*s(296)+24*s(308)+64*s(309)+8*s(316)+24*s(319)+24*s(320)+16*s(321)+8*s(322)+22*s(323)+7*s(333)+42*s(344)+24*s(345)+64*s(346)+8*s(353)+24*s(356)+24*s(357)+16*s(358)+8*s(359)+22*s(360)+6*s(387)+16*s(392)+2*s(399)+6*s(402)+6*s(403)+4*s(404)+2*s(405)+12*s(431)+8*s(433)+22
  Such that:aux(124) =< V
aux(125) =< V-2*V5
aux(126) =< V+V5
aux(127) =< V+2*V5
aux(128) =< -V5
aux(129) =< V5
aux(130) =< 2*V5
aux(131) =< V5/2
s(295) =< aux(124)
s(296) =< aux(129)
s(333) =< s(295)*aux(129)
s(430) =< aux(124)
s(431) =< s(295)*aux(124)
s(430) =< s(295)*aux(124)
s(433) =< s(430)
s(308) =< aux(129)
s(309) =< aux(130)
s(308) =< aux(130)
s(310) =< aux(131)
s(310) =< aux(129)
s(311) =< aux(129)
s(312) =< aux(129)+1
s(313) =< s(310)*2
s(308) =< s(310)+aux(131)
s(314) =< s(313)+aux(129)
s(314) =< s(308)*s(311)
s(315) =< s(313)+aux(129)
s(315) =< s(308)*aux(129)
s(316) =< s(308)*s(312)
s(317) =< s(314)
s(309) =< s(315)+aux(128)
s(309) =< s(314)
s(318) =< s(314)
s(317) =< s(309)*s(311)
s(319) =< s(309)*aux(129)
s(318) =< s(309)*aux(129)
s(320) =< s(317)
s(321) =< s(318)
s(322) =< s(319)*aux(129)
s(323) =< aux(130)
s(344) =< aux(126)
s(345) =< aux(129)
s(346) =< aux(127)
s(345) =< aux(127)
s(349) =< aux(126)+1
s(350) =< aux(126)*2
s(345) =< aux(126)+aux(131)
s(351) =< s(350)+aux(129)
s(351) =< s(345)*s(311)
s(352) =< s(350)+aux(129)
s(352) =< s(345)*aux(129)
s(353) =< s(345)*s(349)
s(354) =< s(351)+aux(124)
s(346) =< s(352)+aux(125)
s(346) =< s(351)+aux(124)
s(355) =< s(351)+aux(124)
s(354) =< s(346)*s(311)
s(356) =< s(346)*aux(126)
s(355) =< s(346)*aux(126)
s(357) =< s(354)
s(358) =< s(355)
s(359) =< s(356)*aux(129)
s(360) =< aux(127)
s(387) =< aux(129)
s(392) =< aux(127)
s(387) =< aux(127)
s(387) =< aux(126)+aux(129)
s(397) =< s(350)+aux(129)
s(397) =< s(387)*s(311)
s(398) =< s(350)+aux(129)
s(398) =< s(387)*aux(129)
s(399) =< s(387)*s(349)
s(400) =< s(397)+aux(124)
s(392) =< s(398)+aux(125)
s(392) =< s(397)+aux(124)
s(401) =< s(397)+aux(124)
s(400) =< s(392)*s(311)
s(402) =< s(392)*aux(126)
s(401) =< s(392)*aux(126)
s(403) =< s(400)
s(404) =< s(401)
s(405) =< s(402)*aux(129)

  with precondition: [V1=2,V>=0,V5>=0] 

* Chain [35]: 5*s(577)+5
  Such that:aux(132) =< V1
s(577) =< aux(132)

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


Closed-form bounds of start(V1,V,V5): 
-------------------------------------
* Chain [39] with precondition: [V1>=0,V>=0] 
    - Upper bound: 43*V1+16+V*V1+50*V+(V1+V)*(5*V)+(V1+2*V)*((V1+V)*(5*V))+(19*V1+19*V)+(15*V1+15*V)*(V1+2*V)+(51*V1+102*V)+nat(-V1+V)*25+(25*V1+25*V) 
    - Complexity: n^3 
* Chain [38] with precondition: [V1=1,V>=0,V5>=0] 
    - Upper bound: 42*V+17+(V+V5)*(5*V)+(2*V+V5)*((V+V5)*(5*V))+34*V5+(19*V+19*V5)+(15*V+15*V5)*(2*V+V5)+(102*V+51*V5)+(25*V+25*V5)+nat(V-V5)*25 
    - Complexity: n^3 
* Chain [37] with precondition: [V1=1,V5=0,V>=1] 
    - Upper bound: 9 
    - Complexity: constant 
* Chain [36] with precondition: [V1=2,V>=0,V5>=0] 
    - Upper bound: 205*V+22+12*V*V+7*V*V5+296*V5+8*V5*V5+8*V5*V5*(2*V5)+24*V5*(2*V5)+(V+V5)*(10*V5)+(V+2*V5)*((V+V5)*(10*V5))+172*V5+(142*V+142*V5)+(30*V+30*V5)*(V+2*V5)+(102*V+204*V5)+40*V5 
    - Complexity: n^3 
* Chain [35] with precondition: [V=0,V1>=0] 
    - Upper bound: 5*V1+5 
    - Complexity: n 

### Maximum cost of start(V1,V,V5): max([max([5*V1,4]),42*V+11+max([8*V+max([43*V1+V*V1+(V1+V)*(5*V)+(V1+2*V)*((V1+V)*(5*V))+(19*V1+19*V)+(15*V1+15*V)*(V1+2*V)+(51*V1+102*V)+nat(-V1+V)*25+(25*V1+25*V),155*V+6+12*V*V+7*V*nat(V5)+nat(V5)*296+nat(V5)*8*nat(V5)+nat(V5)*8*nat(V5)*nat(2*V5)+nat(V5)*24*nat(2*V5)+nat(V5)*10*nat(V+V5)+nat(V5)*10*nat(V+V5)*nat(V+2*V5)+nat(2*V5)*86+nat(V+V5)*142+nat(V+V5)*30*nat(V+2*V5)+nat(V+2*V5)*102+nat(V5/2)*80]),5*V*nat(V+V5)+1+5*V*nat(V+V5)*nat(2*V+V5)+nat(V5)*34+nat(V+V5)*19+nat(V+V5)*15*nat(2*V+V5)+nat(2*V+V5)*51+nat(V/2+V5/2)*50+nat(V-V5)*25])])+5 
Asymptotic class: n^3 
* Total analysis performed in 1376 ms.


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

(10)
BOUNDS(1, n^3)

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

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

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   gcd(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0), x, y)
   if1(false, x, y) -> gcd(y, x)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)

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

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

The rewrite sequence

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

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

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

The result substitution is [ ].




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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   gcd(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0), x, y)
   if1(false, x, y) -> gcd(y, x)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(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(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   gcd(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0), x, y)
   if1(false, x, y) -> gcd(y, x)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)

S is empty.
Rewrite Strategy: INNERMOST