/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 6 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 1519 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) -> if3(gt(x, 0), x, y)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   if3(true, x, y) -> gcd(x, minus(y, x))
   if3(false, x, y) -> y
   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) -> if3(gt(x, 0), x, y) [1]
   if2(true, x, y) -> gcd(minus(x, y), y) [1]
   if2(false, x, y) -> x [1]
   if3(true, x, y) -> gcd(x, minus(y, x)) [1]
   if3(false, x, y) -> y [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) -> if3(gt(x, 0), x, y) [1]
   if2(true, x, y) -> gcd(minus(x, y), y) [1]
   if2(false, x, y) -> x [1]
   if3(true, x, y) -> gcd(x, minus(y, x)) [1]
   if3(false, x, y) -> y [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
if3 :: 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]
   if3(v0, v1, v2) -> null_if3 [0]

And the following fresh constants:    null_minus, null_gt, null_ge, null_if, null_if1, null_if2, null_if3

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

(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) -> if3(gt(x, 0), x, y) [1]
   if2(true, x, y) -> gcd(minus(x, y), y) [1]
   if2(false, x, y) -> x [1]
   if3(true, x, y) -> gcd(x, minus(y, x)) [1]
   if3(false, x, y) -> y [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]
   if3(v0, v1, v2) -> null_if3 [0]

The TRS has the following type information:
minus :: s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3
s :: s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3
if :: true:false:null_gt:null_ge -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3
gt :: s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> 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:null_if3
gcd :: s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3
if1 :: true:false:null_gt:null_ge -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3
ge :: s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> true:false:null_gt:null_ge
if2 :: true:false:null_gt:null_ge -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3
if3 :: true:false:null_gt:null_ge -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3 -> s:0:null_minus:null_if:null_if1:null_if2:null_if3
null_minus :: s:0:null_minus:null_if:null_if1:null_if2:null_if3
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_if3
null_if1 :: s:0:null_minus:null_if:null_if1:null_if2:null_if3
null_if2 :: s:0:null_minus:null_if:null_if1:null_if2:null_if3
null_if3 :: s:0:null_minus:null_if:null_if1:null_if2:null_if3

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
   null_if3 => 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 }-> if3(gt(x, 0), x, y) :|: z' = x, z'' = y, z = 1, 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'') -{ 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
   if3(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if3(z, z', z'') -{ 1 }-> gcd(x, minus(y, x)) :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0
   if3(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,[if3(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,[gt(V14, 0, Ret03),if3(Ret03, V14, V13, Ret4)],[Out = Ret4,V = V14,V5 = V13,V1 = 1,V14 >= 0,V13 >= 0]).
eq(if2(V1, V, V5, Out),1,[minus(V16, V15, Ret04),gcd(Ret04, 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(if3(V1, V, V5, Out),1,[minus(V19, V20, Ret11),gcd(V20, Ret11, Ret6)],[Out = Ret6,V1 = 2,V = V20,V5 = V19,V20 >= 0,V19 >= 0]).
eq(if3(V1, V, V5, Out),1,[],[Out = V22,V = V21,V5 = V22,V1 = 1,V21 >= 0,V22 >= 0]).
eq(gt(V1, V, Out),1,[],[Out = 1,V23 >= 0,V1 = 0,V = V23]).
eq(gt(V1, V, Out),1,[],[Out = 2,V24 >= 0,V1 = 1 + V24,V = 0]).
eq(gt(V1, V, Out),1,[gt(V26, V25, Ret7)],[Out = Ret7,V = 1 + V25,V26 >= 0,V25 >= 0,V1 = 1 + V26]).
eq(ge(V1, V, Out),1,[],[Out = 2,V27 >= 0,V1 = V27,V = 0]).
eq(ge(V1, V, Out),1,[],[Out = 1,V = 1 + V28,V28 >= 0,V1 = 0]).
eq(ge(V1, V, Out),1,[ge(V30, V29, Ret8)],[Out = Ret8,V = 1 + V29,V30 >= 0,V29 >= 0,V1 = 1 + V30]).
eq(minus(V1, V, Out),0,[],[Out = 0,V32 >= 0,V31 >= 0,V1 = V32,V = V31]).
eq(gt(V1, V, Out),0,[],[Out = 0,V34 >= 0,V33 >= 0,V1 = V34,V = V33]).
eq(ge(V1, V, Out),0,[],[Out = 0,V36 >= 0,V35 >= 0,V1 = V36,V = V35]).
eq(if(V1, V, V5, Out),0,[],[Out = 0,V37 >= 0,V5 = V39,V38 >= 0,V1 = V37,V = V38,V39 >= 0]).
eq(if1(V1, V, V5, Out),0,[],[Out = 0,V41 >= 0,V5 = V42,V40 >= 0,V1 = V41,V = V40,V42 >= 0]).
eq(if2(V1, V, V5, Out),0,[],[Out = 0,V44 >= 0,V5 = V45,V43 >= 0,V1 = V44,V = V43,V45 >= 0]).
eq(if3(V1, V, V5, Out),0,[],[Out = 0,V47 >= 0,V5 = V48,V46 >= 0,V1 = V47,V = V46,V48 >= 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(if3(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,if3/4]
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 34 is refined into CE [35] 
* CE 31 is refined into CE [36] 
* CE 32 is refined into CE [37] 
* CE 33 is refined into CE [38] 


### Cost equations --> "Loop" of ge/3 
* CEs [38] --> Loop 18 
* CEs [35] --> Loop 19 
* CEs [36] --> Loop 20 
* CEs [37] --> Loop 21 

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

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


### Specialization of cost equations gt/3 
* CE 19 is refined into CE [39] 
* CE 17 is refined into CE [40] 
* CE 16 is refined into CE [41] 
* CE 18 is refined into CE [42] 


### Cost equations --> "Loop" of gt/3 
* CEs [42] --> Loop 22 
* CEs [39] --> Loop 23 
* CEs [40] --> Loop 24 
* CEs [41] --> Loop 25 

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

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


### Specialization of cost equations minus/3 
* CE 20 is refined into CE [43,44,45,46] 
* CE 21 is refined into CE [47] 
* CE 23 is refined into CE [48] 
* CE 22 is refined into CE [49,50] 


### Cost equations --> "Loop" of minus/3 
* CEs [50] --> Loop 26 
* CEs [49] --> Loop 27 
* CEs [43,44,45,46,47,48] --> Loop 28 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [26]: [V1-1,V1-V]
* RF of phase [27]: [V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [26]:
  - RF of loop [26:1]:
    V1-1
    V1-V
* Partial RF of phase [27]:
  - RF of loop [27:1]:
    V1


### Specialization of cost equations gcd/3 
* CE 25 is refined into CE [51] 
* CE 28 is refined into CE [52] 
* CE 24 is refined into CE [53,54,55,56] 
* CE 27 is refined into CE [57,58,59,60] 
* CE 30 is refined into CE [61,62,63,64,65] 
* CE 29 is refined into CE [66,67] 
* CE 26 is refined into CE [68,69] 


### Cost equations --> "Loop" of gcd/3 
* CEs [67] --> Loop 29 
* CEs [69] --> Loop 30 
* CEs [68] --> Loop 31 
* CEs [66] --> Loop 32 
* CEs [52] --> Loop 33 
* CEs [57,58,62] --> Loop 34 
* CEs [51] --> Loop 35 
* CEs [53,54,55,56,59,60,61,63,64,65] --> Loop 36 

### Ranking functions of CR gcd(V1,V,Out) 
* RF of phase [29,30]: [V1+V-2]

#### Partial ranking functions of CR gcd(V1,V,Out) 
* Partial RF of phase [29,30]:
  - RF of loop [29:1]:
    V1-1
    V1-V depends on loops [30:1] 
  - RF of loop [30:1]:
    V-1
    -V1+V depends on loops [29:1] 


### Specialization of cost equations start/3 
* CE 6 is refined into CE [70,71,72,73,74,75,76,77] 
* CE 7 is refined into CE [78,79,80] 
* CE 8 is refined into CE [81] 
* CE 9 is refined into CE [82,83,84,85,86,87] 
* CE 10 is refined into CE [88,89,90,91,92,93,94,95,96] 
* CE 11 is refined into CE [97,98,99] 
* CE 1 is refined into CE [100,101,102] 
* CE 2 is refined into CE [103] 
* CE 3 is refined into CE [104] 
* CE 4 is refined into CE [105] 
* CE 5 is refined into CE [106,107,108,109,110,111] 
* CE 12 is refined into CE [112,113,114] 
* CE 13 is refined into CE [115,116,117,118,119,120] 
* CE 14 is refined into CE [121,122,123,124,125] 
* CE 15 is refined into CE [126,127,128,129,130] 


### Cost equations --> "Loop" of start/3 
* CEs [112,117,122,127] --> Loop 37 
* CEs [78,81,88,89,92,97] --> Loop 38 
* CEs [70,71,72,73,74,75,76,77,79,80,82,83,84,85,86,87,90,91,93,94,95,96,98,99] --> Loop 39 
* CEs [100,101,102,104,105,106,107,108,109,110,111] --> Loop 40 
* CEs [103,113,114,115,116,118,119,120,121,123,124,125,126,128,129,130] --> Loop 41 

### 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 [[18],21]: 1*it(18)+1
  Such that:it(18) =< V1

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

* Chain [[18],20]: 1*it(18)+1
  Such that:it(18) =< V

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

* Chain [[18],19]: 1*it(18)+0
  Such that:it(18) =< V

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

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

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

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


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

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

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

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

* Chain [24]: 1
  with precondition: [V=0,Out=2,V1>=1] 

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


#### Cost of chains of minus(V1,V,Out):
* Chain [[27],28]: 3*it(27)+2*s(4)+3
  Such that:aux(1) =< V1-Out
it(27) =< Out
s(4) =< aux(1)

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

* Chain [[26],28]: 3*it(26)+2*s(3)+2*s(4)+1*s(9)+3
  Such that:aux(1) =< V1-Out
it(26) =< Out
aux(4) =< V
s(4) =< aux(1)
s(3) =< aux(4)
s(9) =< it(26)*aux(4)

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

* Chain [28]: 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 [[29,30],36]: 8*it(29)+8*it(30)+7*s(11)+5*s(46)+3*s(47)+1*s(49)+3*s(52)+3*s(53)+2*s(54)+1*s(55)+4
  Such that:aux(13) =< -V1+V
aux(10) =< V1-V
aux(21) =< V1
aux(22) =< V1+V
aux(23) =< V
s(11) =< aux(22)
it(29) =< aux(21)
it(29) =< aux(22)
it(30) =< aux(22)
s(57) =< aux(22)
it(30) =< aux(23)
s(57) =< aux(23)
it(30) =< aux(22)+aux(13)
it(29) =< aux(22)+aux(10)
s(53) =< it(30)*aux(23)
s(47) =< it(29)*aux(21)
s(52) =< s(57)
s(54) =< aux(23)
s(55) =< s(53)*aux(21)
s(46) =< aux(21)
s(49) =< s(47)*aux(23)

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

* Chain [[29,30],32,36]: 8*it(29)+8*it(30)+9*s(14)+5*s(46)+3*s(47)+1*s(49)+3*s(52)+3*s(53)+2*s(54)+1*s(55)+12
  Such that:aux(13) =< -V1+V
aux(10) =< V1-V
aux(26) =< V1
aux(27) =< V1+V
aux(28) =< V
s(14) =< aux(27)
it(29) =< aux(26)
it(29) =< aux(27)
it(30) =< aux(27)
s(57) =< aux(27)
it(30) =< aux(28)
s(57) =< aux(28)
it(30) =< aux(27)+aux(13)
it(29) =< aux(27)+aux(10)
s(53) =< it(30)*aux(28)
s(47) =< it(29)*aux(26)
s(52) =< s(57)
s(54) =< aux(28)
s(55) =< s(53)*aux(26)
s(46) =< aux(26)
s(49) =< s(47)*aux(28)

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

* Chain [[29,30],32,35]: 8*it(29)+8*it(30)+7*s(46)+3*s(47)+1*s(49)+3*s(52)+3*s(53)+2*s(54)+1*s(55)+3*s(58)+13
  Such that:aux(13) =< -V1+V
aux(10) =< V1-V
aux(16) =< V1+V
aux(17) =< V1+V-Out
aux(19) =< V
aux(20) =< V-Out
aux(24) =< Out
aux(29) =< V1
s(58) =< aux(24)
s(46) =< aux(29)
it(29) =< aux(29)
aux(9) =< aux(16)
it(29) =< aux(16)
it(30) =< aux(16)
s(57) =< aux(16)
aux(9) =< aux(17)
it(29) =< aux(17)
it(30) =< aux(17)
s(57) =< aux(17)
it(30) =< aux(19)
s(56) =< aux(19)
s(57) =< aux(19)
it(30) =< aux(20)
s(56) =< aux(20)
s(57) =< aux(20)
it(30) =< aux(9)+aux(13)
it(29) =< aux(9)+aux(10)
s(53) =< it(30)*aux(19)
s(47) =< it(29)*aux(29)
s(52) =< s(57)
s(54) =< s(56)
s(55) =< s(53)*aux(29)
s(49) =< s(47)*aux(19)

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

* Chain [[29,30],31,36]: 8*it(29)+8*it(30)+8*s(11)+5*s(46)+3*s(47)+1*s(49)+3*s(52)+3*s(53)+2*s(54)+1*s(55)+12
  Such that:aux(13) =< -V1+V
aux(10) =< V1-V
aux(32) =< V1
aux(33) =< V1+V
aux(34) =< V
s(11) =< aux(33)
it(29) =< aux(32)
it(29) =< aux(33)
it(30) =< aux(33)
s(57) =< aux(33)
it(30) =< aux(34)
s(57) =< aux(34)
it(30) =< aux(33)+aux(13)
it(29) =< aux(33)+aux(10)
s(53) =< it(30)*aux(34)
s(47) =< it(29)*aux(32)
s(52) =< s(57)
s(54) =< aux(34)
s(55) =< s(53)*aux(32)
s(46) =< aux(32)
s(49) =< s(47)*aux(34)

  with precondition: [Out=0,V1>=1,V>=2,V1+2*V>=7] 

* Chain [[29,30],31,34]: 8*it(29)+8*it(30)+5*s(46)+3*s(47)+1*s(49)+3*s(52)+3*s(53)+2*s(54)+1*s(55)+5*s(63)+12
  Such that:aux(13) =< -V1+V
aux(10) =< V1-V
aux(35) =< V1
aux(36) =< V1+V
aux(37) =< V
s(63) =< aux(36)
it(29) =< aux(35)
it(29) =< aux(36)
it(30) =< aux(36)
s(57) =< aux(36)
it(30) =< aux(37)
s(57) =< aux(37)
it(30) =< aux(36)+aux(13)
it(29) =< aux(36)+aux(10)
s(53) =< it(30)*aux(37)
s(47) =< it(29)*aux(35)
s(52) =< s(57)
s(54) =< aux(37)
s(55) =< s(53)*aux(35)
s(46) =< aux(35)
s(49) =< s(47)*aux(37)

  with precondition: [Out=0,V1>=1,V>=2,V1+2*V>=7] 

* Chain [[29,30],31,33]: 8*it(29)+8*it(30)+5*s(46)+3*s(47)+1*s(49)+3*s(52)+3*s(53)+4*s(54)+1*s(55)+3*s(63)+13
  Such that:aux(13) =< -V1+V
aux(15) =< V1
aux(10) =< V1-V
aux(16) =< V1+V
aux(17) =< V1+V-Out
aux(18) =< V1-Out
aux(30) =< Out
aux(38) =< V
s(63) =< aux(30)
s(54) =< aux(38)
it(29) =< aux(15)
s(50) =< aux(15)
aux(9) =< aux(16)
it(29) =< aux(16)
it(30) =< aux(16)
s(57) =< aux(16)
aux(9) =< aux(17)
it(29) =< aux(17)
it(30) =< aux(17)
s(57) =< aux(17)
it(29) =< aux(18)
s(50) =< aux(18)
it(30) =< aux(38)
s(57) =< aux(38)
it(30) =< aux(9)+aux(13)
it(29) =< aux(9)+aux(10)
s(53) =< it(30)*aux(38)
s(47) =< it(29)*aux(15)
s(52) =< s(57)
s(55) =< s(53)*aux(15)
s(46) =< s(50)
s(49) =< s(47)*aux(38)

  with precondition: [Out>=1,V1>=Out,V>=Out+1,V1+2*V>=4*Out+3,V+V1>=3*Out+1] 

* Chain [36]: 3*s(11)+4*s(14)+4
  Such that:aux(5) =< V1
aux(6) =< V
s(11) =< aux(5)
s(14) =< aux(6)

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

* Chain [35]: 5
  with precondition: [V1=0,V=Out,V>=1] 

* Chain [34]: 4
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [33]: 5
  with precondition: [V=0,V1=Out,V1>=0] 

* Chain [32,36]: 7*s(14)+2*s(61)+12
  Such that:s(59) =< V1
aux(25) =< V
s(14) =< aux(25)
s(61) =< s(59)

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

* Chain [32,35]: 3*s(58)+2*s(61)+13
  Such that:s(59) =< V1
aux(24) =< Out
s(58) =< aux(24)
s(61) =< s(59)

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

* Chain [31,36]: 6*s(11)+2*s(66)+12
  Such that:s(64) =< V
aux(31) =< V1
s(11) =< aux(31)
s(66) =< s(64)

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

* Chain [31,34]: 3*s(63)+2*s(66)+12
  Such that:aux(30) =< V1
s(64) =< V
s(63) =< aux(30)
s(66) =< s(64)

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

* Chain [31,33]: 3*s(63)+2*s(66)+13
  Such that:s(64) =< V
aux(30) =< Out
s(63) =< aux(30)
s(66) =< s(64)

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


#### Cost of chains of start(V1,V,V5):
* Chain [41]: 66*s(193)+42*s(194)+1*s(200)+29*s(206)+48*s(207)+48*s(208)+18*s(210)+18*s(211)+18*s(212)+6*s(214)+6*s(216)+13
  Such that:aux(55) =< -V1+V
aux(56) =< V1
aux(57) =< V1-V
aux(58) =< V1+V
aux(59) =< V
s(193) =< aux(56)
s(194) =< aux(59)
s(206) =< aux(58)
s(207) =< aux(56)
s(207) =< aux(58)
s(208) =< aux(58)
s(209) =< aux(58)
s(208) =< aux(59)
s(209) =< aux(59)
s(208) =< aux(58)+aux(55)
s(207) =< aux(58)+aux(57)
s(210) =< s(208)*aux(59)
s(211) =< s(207)*aux(56)
s(212) =< s(209)
s(214) =< s(210)*aux(56)
s(216) =< s(211)*aux(59)
s(200) =< s(193)*aux(59)

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

* Chain [40]: 56*s(267)+126*s(268)+32*s(275)+12*s(279)+4*s(294)+37*s(300)+48*s(301)+48*s(302)+18*s(304)+18*s(305)+18*s(306)+6*s(308)+6*s(310)+19
  Such that:aux(70) =< -2*V+V5
aux(71) =< V
aux(72) =< V+V5
aux(73) =< V5
s(267) =< aux(73)
s(300) =< aux(72)
s(301) =< aux(71)
s(301) =< aux(72)
s(302) =< aux(72)
s(303) =< aux(72)
s(302) =< aux(73)
s(303) =< aux(73)
s(302) =< aux(72)+aux(70)
s(301) =< aux(72)+aux(71)
s(304) =< s(302)*aux(73)
s(305) =< s(301)*aux(71)
s(306) =< s(303)
s(308) =< s(304)*aux(71)
s(268) =< aux(71)
s(310) =< s(305)*aux(73)
s(294) =< s(267)*aux(71)
s(275) =< aux(71)
s(275) =< aux(71)+aux(71)
s(279) =< s(275)*aux(71)

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

* Chain [39]: 354*s(371)+96*s(383)+36*s(385)+295*s(395)+32*s(402)+12*s(406)+4*s(421)+95*s(427)+48*s(428)+48*s(429)+18*s(431)+18*s(432)+54*s(433)+6*s(435)+6*s(437)+9*s(527)+96*s(534)+96*s(535)+36*s(537)+36*s(538)+12*s(541)+12*s(543)+19
  Such that:aux(108) =< -2*V+V5
aux(109) =< V
aux(110) =< V-2*V5
aux(111) =< V+V5
aux(112) =< V5
s(371) =< aux(112)
s(427) =< aux(111)
s(428) =< aux(109)
s(428) =< aux(111)
s(429) =< aux(111)
s(430) =< aux(111)
s(429) =< aux(112)
s(430) =< aux(112)
s(429) =< aux(111)+aux(108)
s(428) =< aux(111)+aux(109)
s(431) =< s(429)*aux(112)
s(432) =< s(428)*aux(109)
s(433) =< s(430)
s(435) =< s(431)*aux(109)
s(395) =< aux(109)
s(437) =< s(432)*aux(112)
s(421) =< s(371)*aux(109)
s(527) =< s(395)*aux(112)
s(383) =< aux(112)
s(383) =< aux(112)+aux(112)
s(385) =< s(383)*aux(112)
s(534) =< aux(109)
s(534) =< aux(111)
s(535) =< aux(111)
s(535) =< aux(112)
s(535) =< aux(111)+aux(112)
s(534) =< aux(111)+aux(110)
s(537) =< s(535)*aux(112)
s(538) =< s(534)*aux(109)
s(541) =< s(537)*aux(109)
s(543) =< s(538)*aux(112)
s(402) =< aux(109)
s(402) =< aux(109)+aux(109)
s(406) =< s(402)*aux(109)

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

* Chain [38]: 80*s(719)+32*s(727)+12*s(731)+16
  Such that:aux(117) =< V
s(719) =< aux(117)
s(727) =< aux(117)
s(727) =< aux(117)+aux(117)
s(731) =< s(727)*aux(117)

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

* Chain [37]: 5*s(748)+5
  Such that:aux(118) =< V1
s(748) =< aux(118)

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


Closed-form bounds of start(V1,V,V5): 
-------------------------------------
* Chain [41] with precondition: [V1>=0,V>=0] 
    - Upper bound: 114*V1+13+18*V1*V1+6*V1*V1*V+V*V1+(V1+V)*(6*V1*V)+42*V+(V1+V)*(18*V)+(95*V1+95*V) 
    - Complexity: n^3 
* Chain [40] with precondition: [V1=1,V>=0,V5>=0] 
    - Upper bound: 206*V+19+30*V*V+6*V*V*V5+4*V*V5+(V+V5)*(6*V*V5)+56*V5+(V+V5)*(18*V5)+(103*V+103*V5) 
    - Complexity: n^3 
* Chain [39] with precondition: [V1=2,V>=0,V5>=0] 
    - Upper bound: 471*V+19+66*V*V+18*V*V*V5+13*V*V5+(V+V5)*(18*V*V5)+450*V5+36*V5*V5+(V+V5)*(54*V5)+(293*V+293*V5) 
    - Complexity: n^3 
* Chain [38] with precondition: [V1=2,V5=0,V>=0] 
    - Upper bound: 112*V+16+12*V*V 
    - Complexity: n^2 
* Chain [37] with precondition: [V=0,V1>=0] 
    - Upper bound: 5*V1+5 
    - Complexity: n 

### Maximum cost of start(V1,V,V5): max([5*V1,42*V+8+max([18*V1*V1+114*V1+6*V1*V1*V+V*V1+(V1+V)*(6*V1*V)+(V1+V)*(18*V)+(95*V1+95*V),36*V*V+265*V+12*V*V*nat(V5)+9*V*nat(V5)+12*V*nat(V5)*nat(V+V5)+nat(V5)*394+nat(V5)*36*nat(V5)+nat(V5)*36*nat(V+V5)+nat(V+V5)*190+(94*V+3+18*V*V+6*V*V*nat(V5)+4*V*nat(V5)+6*V*nat(V5)*nat(V+V5)+nat(V5)*56+nat(V5)*18*nat(V+V5)+nat(V+V5)*103)+(70*V+3+12*V*V)])])+5 
Asymptotic class: n^3 
* Total analysis performed in 1296 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) -> if3(gt(x, 0), x, y)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   if3(true, x, y) -> gcd(x, minus(y, x))
   if3(false, x, y) -> y
   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) -> if3(gt(x, 0), x, y)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   if3(true, x, y) -> gcd(x, minus(y, x))
   if3(false, x, y) -> y
   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
----------------------------------------

(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^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) -> if3(gt(x, 0), x, y)
   if2(true, x, y) -> gcd(minus(x, y), y)
   if2(false, x, y) -> x
   if3(true, x, y) -> gcd(x, minus(y, x))
   if3(false, x, y) -> y
   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