/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^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 15 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 806 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 272 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 60 ms]
            (26) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   gt(s(x), 0) -> true
   gt(0, y) -> false
   gt(s(x), s(y)) -> gt(x, y)
   divides(x, y) -> div(x, y, y)
   div(0, 0, z) -> true
   div(0, s(x), z) -> false
   div(s(x), 0, s(z)) -> div(s(x), s(z), s(z))
   div(s(x), s(y), z) -> div(x, y, z)
   prime(x) -> test(x, s(s(0)))
   test(x, y) -> if1(gt(x, y), x, y)
   if1(true, x, y) -> if2(divides(x, y), x, y)
   if1(false, x, y) -> true
   if2(true, x, y) -> false
   if2(false, x, y) -> test(x, s(y))

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

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

The duplicating contexts are:
divides(x, [])
div(s(x), 0, s([]))
test([], y)
test(x, [])
if1(true, [], y)
if1(true, x, [])


The defined contexts are:
if2([], x1, x2)
if1([], x1, x2)


[] just represents basic- or constructor-terms in the following defined contexts:
if2([], x1, x2)
if1([], x1, x2)


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   gt(s(x), 0) -> true
   gt(0, y) -> false
   gt(s(x), s(y)) -> gt(x, y)
   divides(x, y) -> div(x, y, y)
   div(0, 0, z) -> true
   div(0, s(x), z) -> false
   div(s(x), 0, s(z)) -> div(s(x), s(z), s(z))
   div(s(x), s(y), z) -> div(x, y, z)
   prime(x) -> test(x, s(s(0)))
   test(x, y) -> if1(gt(x, y), x, y)
   if1(true, x, y) -> if2(divides(x, y), x, y)
   if1(false, x, y) -> true
   if2(true, x, y) -> false
   if2(false, x, y) -> test(x, s(y))

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   gt(s(x), 0) -> true [1]
   gt(0, y) -> false [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   divides(x, y) -> div(x, y, y) [1]
   div(0, 0, z) -> true [1]
   div(0, s(x), z) -> false [1]
   div(s(x), 0, s(z)) -> div(s(x), s(z), s(z)) [1]
   div(s(x), s(y), z) -> div(x, y, z) [1]
   prime(x) -> test(x, s(s(0))) [1]
   test(x, y) -> if1(gt(x, y), x, y) [1]
   if1(true, x, y) -> if2(divides(x, y), x, y) [1]
   if1(false, x, y) -> true [1]
   if2(true, x, y) -> false [1]
   if2(false, x, y) -> test(x, s(y)) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   gt(s(x), 0) -> true [1]
   gt(0, y) -> false [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   divides(x, y) -> div(x, y, y) [1]
   div(0, 0, z) -> true [1]
   div(0, s(x), z) -> false [1]
   div(s(x), 0, s(z)) -> div(s(x), s(z), s(z)) [1]
   div(s(x), s(y), z) -> div(x, y, z) [1]
   prime(x) -> test(x, s(s(0))) [1]
   test(x, y) -> if1(gt(x, y), x, y) [1]
   if1(true, x, y) -> if2(divides(x, y), x, y) [1]
   if1(false, x, y) -> true [1]
   if2(true, x, y) -> false [1]
   if2(false, x, y) -> test(x, s(y)) [1]

The TRS has the following type information:
gt :: s:0 -> s:0 -> true:false
s :: s:0 -> s:0
0 :: s:0
true :: true:false
false :: true:false
divides :: s:0 -> s:0 -> true:false
div :: s:0 -> s:0 -> s:0 -> true:false
prime :: s:0 -> true:false
test :: s:0 -> s:0 -> true:false
if1 :: true:false -> s:0 -> s:0 -> true:false
if2 :: true:false -> s:0 -> s:0 -> true:false

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

(7) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   div(v0, v1, v2) -> null_div [0]
   if1(v0, v1, v2) -> null_if1 [0]
   if2(v0, v1, v2) -> null_if2 [0]

And the following fresh constants:    null_div, null_if1, null_if2

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

(8)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   gt(s(x), 0) -> true [1]
   gt(0, y) -> false [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   divides(x, y) -> div(x, y, y) [1]
   div(0, 0, z) -> true [1]
   div(0, s(x), z) -> false [1]
   div(s(x), 0, s(z)) -> div(s(x), s(z), s(z)) [1]
   div(s(x), s(y), z) -> div(x, y, z) [1]
   prime(x) -> test(x, s(s(0))) [1]
   test(x, y) -> if1(gt(x, y), x, y) [1]
   if1(true, x, y) -> if2(divides(x, y), x, y) [1]
   if1(false, x, y) -> true [1]
   if2(true, x, y) -> false [1]
   if2(false, x, y) -> test(x, s(y)) [1]
   div(v0, v1, v2) -> null_div [0]
   if1(v0, v1, v2) -> null_if1 [0]
   if2(v0, v1, v2) -> null_if2 [0]

The TRS has the following type information:
gt :: s:0 -> s:0 -> true:false:null_div:null_if1:null_if2
s :: s:0 -> s:0
0 :: s:0
true :: true:false:null_div:null_if1:null_if2
false :: true:false:null_div:null_if1:null_if2
divides :: s:0 -> s:0 -> true:false:null_div:null_if1:null_if2
div :: s:0 -> s:0 -> s:0 -> true:false:null_div:null_if1:null_if2
prime :: s:0 -> true:false:null_div:null_if1:null_if2
test :: s:0 -> s:0 -> true:false:null_div:null_if1:null_if2
if1 :: true:false:null_div:null_if1:null_if2 -> s:0 -> s:0 -> true:false:null_div:null_if1:null_if2
if2 :: true:false:null_div:null_if1:null_if2 -> s:0 -> s:0 -> true:false:null_div:null_if1:null_if2
null_div :: true:false:null_div:null_if1:null_if2
null_if1 :: true:false:null_div:null_if1:null_if2
null_if2 :: true:false:null_div:null_if1:null_if2

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

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 2
   false => 1
   null_div => 0
   null_if1 => 0
   null_if2 => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'', z1) -{ 1 }-> div(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
   div(z', z'', z1) -{ 1 }-> div(1 + x, 1 + z, 1 + z) :|: z'' = 0, z' = 1 + x, z >= 0, x >= 0, z1 = 1 + z
   div(z', z'', z1) -{ 1 }-> 2 :|: z'' = 0, z1 = z, z >= 0, z' = 0
   div(z', z'', z1) -{ 1 }-> 1 :|: z1 = z, z >= 0, x >= 0, z'' = 1 + x, z' = 0
   div(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   divides(z', z'') -{ 1 }-> div(x, y, y) :|: z' = x, z'' = y, x >= 0, y >= 0
   gt(z', z'') -{ 1 }-> gt(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   gt(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 1 + x, x >= 0
   gt(z', z'') -{ 1 }-> 1 :|: z'' = y, y >= 0, z' = 0
   if1(z', z'', z1) -{ 1 }-> if2(divides(x, y), x, y) :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   if1(z', z'', z1) -{ 1 }-> 2 :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   if1(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if2(z', z'', z1) -{ 1 }-> test(x, 1 + y) :|: z1 = y, x >= 0, y >= 0, z'' = x, z' = 1
   if2(z', z'', z1) -{ 1 }-> 1 :|: z1 = y, z' = 2, x >= 0, y >= 0, z'' = x
   if2(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   prime(z') -{ 1 }-> test(x, 1 + (1 + 0)) :|: z' = x, x >= 0
   test(z', z'') -{ 1 }-> if1(gt(x, y), x, y) :|: z' = x, z'' = y, x >= 0, y >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V1, V9),0,[gt(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V9),0,[divides(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V9),0,[div(V, V1, V9, Out)],[V >= 0,V1 >= 0,V9 >= 0]).
eq(start(V, V1, V9),0,[prime(V, Out)],[V >= 0]).
eq(start(V, V1, V9),0,[test(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V9),0,[if1(V, V1, V9, Out)],[V >= 0,V1 >= 0,V9 >= 0]).
eq(start(V, V1, V9),0,[if2(V, V1, V9, Out)],[V >= 0,V1 >= 0,V9 >= 0]).
eq(gt(V, V1, Out),1,[],[Out = 2,V1 = 0,V = 1 + V2,V2 >= 0]).
eq(gt(V, V1, Out),1,[],[Out = 1,V1 = V3,V3 >= 0,V = 0]).
eq(gt(V, V1, Out),1,[gt(V4, V5, Ret)],[Out = Ret,V = 1 + V4,V4 >= 0,V5 >= 0,V1 = 1 + V5]).
eq(divides(V, V1, Out),1,[div(V7, V6, V6, Ret1)],[Out = Ret1,V = V7,V1 = V6,V7 >= 0,V6 >= 0]).
eq(div(V, V1, V9, Out),1,[],[Out = 2,V1 = 0,V9 = V8,V8 >= 0,V = 0]).
eq(div(V, V1, V9, Out),1,[],[Out = 1,V9 = V11,V11 >= 0,V10 >= 0,V1 = 1 + V10,V = 0]).
eq(div(V, V1, V9, Out),1,[div(1 + V12, 1 + V13, 1 + V13, Ret2)],[Out = Ret2,V1 = 0,V = 1 + V12,V13 >= 0,V12 >= 0,V9 = 1 + V13]).
eq(div(V, V1, V9, Out),1,[div(V16, V14, V15, Ret3)],[Out = Ret3,V = 1 + V16,V9 = V15,V15 >= 0,V16 >= 0,V14 >= 0,V1 = 1 + V14]).
eq(prime(V, Out),1,[test(V17, 1 + (1 + 0), Ret4)],[Out = Ret4,V = V17,V17 >= 0]).
eq(test(V, V1, Out),1,[gt(V19, V18, Ret0),if1(Ret0, V19, V18, Ret5)],[Out = Ret5,V = V19,V1 = V18,V19 >= 0,V18 >= 0]).
eq(if1(V, V1, V9, Out),1,[divides(V21, V20, Ret01),if2(Ret01, V21, V20, Ret6)],[Out = Ret6,V9 = V20,V = 2,V21 >= 0,V20 >= 0,V1 = V21]).
eq(if1(V, V1, V9, Out),1,[],[Out = 2,V9 = V22,V23 >= 0,V22 >= 0,V1 = V23,V = 1]).
eq(if2(V, V1, V9, Out),1,[],[Out = 1,V9 = V24,V = 2,V25 >= 0,V24 >= 0,V1 = V25]).
eq(if2(V, V1, V9, Out),1,[test(V27, 1 + V26, Ret7)],[Out = Ret7,V9 = V26,V27 >= 0,V26 >= 0,V1 = V27,V = 1]).
eq(div(V, V1, V9, Out),0,[],[Out = 0,V29 >= 0,V9 = V30,V28 >= 0,V1 = V28,V30 >= 0,V = V29]).
eq(if1(V, V1, V9, Out),0,[],[Out = 0,V33 >= 0,V9 = V31,V32 >= 0,V1 = V32,V31 >= 0,V = V33]).
eq(if2(V, V1, V9, Out),0,[],[Out = 0,V36 >= 0,V9 = V34,V35 >= 0,V1 = V35,V34 >= 0,V = V36]).
input_output_vars(gt(V,V1,Out),[V,V1],[Out]).
input_output_vars(divides(V,V1,Out),[V,V1],[Out]).
input_output_vars(div(V,V1,V9,Out),[V,V1,V9],[Out]).
input_output_vars(prime(V,Out),[V],[Out]).
input_output_vars(test(V,V1,Out),[V,V1],[Out]).
input_output_vars(if1(V,V1,V9,Out),[V,V1,V9],[Out]).
input_output_vars(if2(V,V1,V9,Out),[V,V1,V9],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [(div)/4]
1. non_recursive  : [divides/3]
2. recursive  : [gt/3]
3. recursive  : [if1/4,if2/4,test/3]
4. non_recursive  : [prime/2]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into (div)/4
1. SCC is completely evaluated into other SCCs
2. SCC is partially evaluated into gt/3
3. SCC is partially evaluated into test/3
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations (div)/4 
* CE 17 is refined into CE [26] 
* CE 14 is refined into CE [27] 
* CE 13 is refined into CE [28] 
* CE 16 is refined into CE [29] 
* CE 15 is refined into CE [30] 


### Cost equations --> "Loop" of (div)/4 
* CEs [29] --> Loop 15 
* CEs [30] --> Loop 16 
* CEs [26] --> Loop 17 
* CEs [27] --> Loop 18 
* CEs [28] --> Loop 19 

### Ranking functions of CR div(V,V1,V9,Out) 

#### Partial ranking functions of CR div(V,V1,V9,Out) 
* Partial RF of phase [15,16]:
  - RF of loop [15:1]:
    V
    V1 depends on loops [16:1] 
  - RF of loop [16:1]:
    -V1+1 depends on loops [15:1] 


### Specialization of cost equations gt/3 
* CE 25 is refined into CE [31] 
* CE 23 is refined into CE [32] 
* CE 24 is refined into CE [33] 


### Cost equations --> "Loop" of gt/3 
* CEs [32] --> Loop 20 
* CEs [33] --> Loop 21 
* CEs [31] --> Loop 22 

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

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


### Specialization of cost equations test/3 
* CE 22 is refined into CE [34,35] 
* CE 20 is refined into CE [36] 
* CE 18 is refined into CE [37,38,39,40] 
* CE 21 is refined into CE [41,42,43,44] 
* CE 19 is refined into CE [45] 


### Cost equations --> "Loop" of test/3 
* CEs [45] --> Loop 23 
* CEs [35] --> Loop 24 
* CEs [36] --> Loop 25 
* CEs [38,39,40,44] --> Loop 26 
* CEs [43] --> Loop 27 
* CEs [37,42] --> Loop 28 
* CEs [34] --> Loop 29 
* CEs [41] --> Loop 30 

### Ranking functions of CR test(V,V1,Out) 
* RF of phase [23]: [V-V1]

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [46,47,48,49,50] 
* CE 3 is refined into CE [51,52,53,54,55,56,57] 
* CE 5 is refined into CE [58,59] 
* CE 6 is refined into CE [60] 
* CE 2 is refined into CE [61] 
* CE 4 is refined into CE [62,63,64,65,66,67,68] 
* CE 7 is refined into CE [69] 
* CE 8 is refined into CE [70,71,72,73] 
* CE 9 is refined into CE [74,75,76,77,78] 
* CE 10 is refined into CE [79,80,81,82,83] 
* CE 11 is refined into CE [84,85,86,87,88,89,90] 
* CE 12 is refined into CE [91,92,93,94,95,96,97,98] 


### Cost equations --> "Loop" of start/3 
* CEs [87,88,90] --> Loop 31 
* CEs [47,51,52,71,93] --> Loop 32 
* CEs [46,48,49,50,53,54,55,56,57,58,59,60,61,72,73,76,77,78,81,82,83,94,95,96,97,98] --> Loop 33 
* CEs [62,63,64,65,66,67,68,69,86,89] --> Loop 34 
* CEs [70,74,75,79,80,84,85,91,92] --> Loop 35 

### Ranking functions of CR start(V,V1,V9) 

#### Partial ranking functions of CR start(V,V1,V9) 


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

#### Cost of chains of div(V,V1,V9,Out):
* Chain [[15,16],19]: 1*it(15)+1*it(16)+1
  Such that:it(15) =< V
aux(6) =< -V1
aux(5) =< -V1+1
it(16) =< it(15)+aux(6)
it(16) =< it(15)+aux(5)

  with precondition: [Out=2,V>=1,V1>=0,V9>=0,V>=V1,V1+V9>=1] 

* Chain [[15,16],18]: 1*it(15)+1*it(16)+1
  Such that:it(15) =< V
aux(5) =< -V1+1
it(16) =< it(15)+aux(5)

  with precondition: [Out=1,V>=1,V1>=0,V9>=0,V1+V9>=1] 

* Chain [[15,16],17]: 1*it(15)+1*it(16)+0
  Such that:it(15) =< V
aux(5) =< -V1+1
it(16) =< it(15)+aux(5)

  with precondition: [Out=0,V>=1,V1>=0,V9>=0,V1+V9>=1] 

* Chain [19]: 1
  with precondition: [V=0,V1=0,Out=2,V9>=0] 

* Chain [18]: 1
  with precondition: [V=0,Out=1,V1>=1,V9>=0] 

* Chain [17]: 0
  with precondition: [Out=0,V>=0,V1>=0,V9>=0] 


#### Cost of chains of gt(V,V1,Out):
* Chain [[22],21]: 1*it(22)+1
  Such that:it(22) =< V

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

* Chain [[22],20]: 1*it(22)+1
  Such that:it(22) =< V1

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

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

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


#### Cost of chains of test(V,V1,Out):
* Chain [[23],27]: 6*it(23)+1*s(4)+2*s(13)+1*s(15)+2
  Such that:s(16) =< -V1+1
aux(14) =< V
aux(15) =< V-V1
s(4) =< aux(14)
it(23) =< aux(15)
s(16) =< aux(15)
s(13) =< it(23)*aux(14)
s(15) =< s(13)+s(16)

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

* Chain [[23],26]: 6*it(23)+2*s(13)+1*s(15)+7*s(17)+3*s(20)+5
  Such that:s(16) =< -V1+1
aux(19) =< V
aux(20) =< V-V1
aux(21) =< -V1
it(23) =< aux(20)
s(16) =< aux(20)
s(17) =< aux(19)
s(20) =< s(17)+aux(21)
s(13) =< it(23)*aux(19)
s(15) =< s(13)+s(16)

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

* Chain [[23],25]: 6*it(23)+2*s(13)+1*s(15)+2*s(31)+1*s(35)+6
  Such that:s(16) =< -V1+1
aux(22) =< V
aux(23) =< V-V1
aux(24) =< -V1
s(31) =< aux(22)
it(23) =< aux(23)
s(16) =< aux(23)
s(35) =< s(31)+aux(24)
s(13) =< it(23)*aux(22)
s(15) =< s(13)+s(16)

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

* Chain [[23],24]: 6*it(23)+2*s(13)+1*s(15)+1*s(36)+3
  Such that:s(16) =< -V1+1
aux(25) =< V
aux(26) =< V-V1
s(36) =< aux(25)
it(23) =< aux(26)
s(16) =< aux(26)
s(13) =< it(23)*aux(25)
s(15) =< s(13)+s(16)

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

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

* Chain [29]: 3
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [28]: 1*s(37)+1*s(39)+4
  Such that:s(38) =< 1
s(37) =< V
s(39) =< s(37)+s(38)

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

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

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

* Chain [26]: 4*s(17)+3*s(18)+2*s(20)+1*s(29)+5
  Such that:s(27) =< -V1
aux(16) =< V
aux(17) =< -V1+1
aux(18) =< V1
s(18) =< aux(16)
s(17) =< aux(18)
s(20) =< s(18)+aux(17)
s(29) =< s(18)+s(27)
s(29) =< s(18)+aux(17)

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

* Chain [25]: 1*s(31)+1*s(32)+1*s(35)+6
  Such that:s(32) =< V
s(33) =< -V1
s(34) =< -V1+1
s(31) =< V1
s(35) =< s(32)+s(33)
s(35) =< s(32)+s(34)

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

* Chain [24]: 1*s(36)+3
  Such that:s(36) =< V

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


#### Cost of chains of start(V,V1,V9):
* Chain [35]: 4
  with precondition: [V=0] 

* Chain [34]: 17*s(78)+5*s(86)+8*s(87)+24*s(89)+8*s(91)+4*s(92)+2*s(113)+7
  Such that:aux(36) =< V
aux(37) =< V1
aux(38) =< V1-V9
aux(39) =< -V9
aux(40) =< V9+1
s(113) =< aux(36)
s(78) =< aux(37)
s(84) =< aux(39)
s(86) =< aux(40)
s(87) =< s(78)+aux(39)
s(89) =< aux(38)
s(84) =< aux(38)
s(91) =< s(89)*aux(37)
s(92) =< s(91)+s(84)

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

* Chain [33]: 32*s(115)+7*s(117)+2*s(124)+5*s(139)+8*s(140)+24*s(142)+8*s(144)+4*s(145)+24*s(179)+6*s(183)+4*s(190)+24*s(212)+4*s(213)+8*s(214)+4*s(215)+10
  Such that:aux(48) =< V
aux(49) =< V-V1
aux(50) =< -V1
aux(51) =< -V1+1
aux(52) =< V1
aux(53) =< V1-V9
aux(54) =< -V9
aux(55) =< -V9+1
aux(56) =< V9+1
s(179) =< aux(48)
s(207) =< aux(51)
s(115) =< aux(52)
s(137) =< aux(54)
s(139) =< aux(56)
s(183) =< s(179)+aux(51)
s(190) =< s(179)+aux(50)
s(190) =< s(179)+aux(51)
s(212) =< aux(49)
s(207) =< aux(49)
s(213) =< s(179)+aux(50)
s(214) =< s(212)*aux(48)
s(215) =< s(214)+s(207)
s(117) =< s(115)+aux(55)
s(124) =< s(115)+aux(54)
s(124) =< s(115)+aux(55)
s(140) =< s(115)+aux(54)
s(142) =< aux(53)
s(137) =< aux(53)
s(144) =< s(142)*aux(52)
s(145) =< s(144)+s(137)

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

* Chain [32]: 1*s(237)+1*s(238)+7
  Such that:s(236) =< 1
s(237) =< V
s(238) =< s(237)+s(236)

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

* Chain [31]: 39*s(245)+5*s(246)+8*s(247)+8*s(251)+4*s(252)+7
  Such that:aux(62) =< 2
aux(63) =< V
s(246) =< aux(62)
s(245) =< aux(63)
s(247) =< s(245)
s(251) =< s(245)*aux(63)
s(252) =< s(251)

  with precondition: [V>=3] 


Closed-form bounds of start(V,V1,V9): 
-------------------------------------
* Chain [35] with precondition: [V=0] 
    - Upper bound: 4 
    - Complexity: constant 
* Chain [34] with precondition: [2>=V,V>=1] 
    - Upper bound: 2*V+7+nat(V1)*25+nat(V1)*12*nat(V1-V9)+nat(-V9)*12+nat(V9+1)*5+nat(V1-V9)*24 
    - Complexity: n^2 
* Chain [33] with precondition: [V>=0,V1>=0] 
    - Upper bound: 38*V+10+12*V*nat(V-V1)+49*V1+12*V1*nat(V1-V9)+nat(-V9)*14+nat(V9+1)*5+nat(-V1+1)*10+nat(-V9+1)*7+nat(V-V1)*24+nat(V1-V9)*24 
    - Complexity: n^2 
* Chain [32] with precondition: [V1=0,V>=1] 
    - Upper bound: 2*V+8 
    - Complexity: n 
* Chain [31] with precondition: [V>=3] 
    - Upper bound: 47*V+17+12*V*V 
    - Complexity: n^2 

### Maximum cost of start(V,V1,V9): 2*V+3+max([max([1,nat(V1)*12*nat(V1-V9)+nat(V1)*25+nat(-V9)*12+nat(V9+1)*5+nat(V1-V9)*24]),36*V+3+max([9*V+7+12*V*V,12*V*nat(V-V1)+nat(V1)*49+nat(V1)*12*nat(V1-V9)+nat(-V9)*14+nat(V9+1)*5+nat(-V1+1)*10+nat(-V9+1)*7+nat(V-V1)*24+nat(V1-V9)*24])])+4 
Asymptotic class: n^2 
* Total analysis performed in 677 ms.


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

(12)
BOUNDS(1, n^2)

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

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

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


The TRS R consists of the following rules:

   gt(s(x), 0') -> true
   gt(0', y) -> false
   gt(s(x), s(y)) -> gt(x, y)
   divides(x, y) -> div(x, y, y)
   div(0', 0', z) -> true
   div(0', s(x), z) -> false
   div(s(x), 0', s(z)) -> div(s(x), s(z), s(z))
   div(s(x), s(y), z) -> div(x, y, z)
   prime(x) -> test(x, s(s(0')))
   test(x, y) -> if1(gt(x, y), x, y)
   if1(true, x, y) -> if2(divides(x, y), x, y)
   if1(false, x, y) -> true
   if2(true, x, y) -> false
   if2(false, x, y) -> test(x, s(y))

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

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

(16)
Obligation:
TRS:
Rules:
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)
divides(x, y) -> div(x, y, y)
div(0', 0', z) -> true
div(0', s(x), z) -> false
div(s(x), 0', s(z)) -> div(s(x), s(z), s(z))
div(s(x), s(y), z) -> div(x, y, z)
prime(x) -> test(x, s(s(0')))
test(x, y) -> if1(gt(x, y), x, y)
if1(true, x, y) -> if2(divides(x, y), x, y)
if1(false, x, y) -> true
if2(true, x, y) -> false
if2(false, x, y) -> test(x, s(y))

Types:
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
test :: s:0' -> s:0' -> true:false
if1 :: true:false -> s:0' -> s:0' -> true:false
if2 :: true:false -> s:0' -> s:0' -> true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat -> s:0'

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
gt, div, test

They will be analysed ascendingly in the following order:
gt < test

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

(18)
Obligation:
TRS:
Rules:
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)
divides(x, y) -> div(x, y, y)
div(0', 0', z) -> true
div(0', s(x), z) -> false
div(s(x), 0', s(z)) -> div(s(x), s(z), s(z))
div(s(x), s(y), z) -> div(x, y, z)
prime(x) -> test(x, s(s(0')))
test(x, y) -> if1(gt(x, y), x, y)
if1(true, x, y) -> if2(divides(x, y), x, y)
if1(false, x, y) -> true
if2(true, x, y) -> false
if2(false, x, y) -> test(x, s(y))

Types:
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
test :: s:0' -> s:0' -> true:false
if1 :: true:false -> s:0' -> s:0' -> true:false
if2 :: true:false -> s:0' -> s:0' -> true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
gt, div, test

They will be analysed ascendingly in the following order:
gt < test

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'3_0(+(1, n5_0)), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
gt(gen_s:0'3_0(+(1, 0)), gen_s:0'3_0(0)) ->_R^Omega(1)
true

Induction Step:
gt(gen_s:0'3_0(+(1, +(n5_0, 1))), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
gt(gen_s:0'3_0(+(1, n5_0)), gen_s:0'3_0(n5_0)) ->_IH
true

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

(20)
Complex Obligation (BEST)

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

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

TRS:
Rules:
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)
divides(x, y) -> div(x, y, y)
div(0', 0', z) -> true
div(0', s(x), z) -> false
div(s(x), 0', s(z)) -> div(s(x), s(z), s(z))
div(s(x), s(y), z) -> div(x, y, z)
prime(x) -> test(x, s(s(0')))
test(x, y) -> if1(gt(x, y), x, y)
if1(true, x, y) -> if2(divides(x, y), x, y)
if1(false, x, y) -> true
if2(true, x, y) -> false
if2(false, x, y) -> test(x, s(y))

Types:
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
test :: s:0' -> s:0' -> true:false
if1 :: true:false -> s:0' -> s:0' -> true:false
if2 :: true:false -> s:0' -> s:0' -> true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
gt, div, test

They will be analysed ascendingly in the following order:
gt < test

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)
divides(x, y) -> div(x, y, y)
div(0', 0', z) -> true
div(0', s(x), z) -> false
div(s(x), 0', s(z)) -> div(s(x), s(z), s(z))
div(s(x), s(y), z) -> div(x, y, z)
prime(x) -> test(x, s(s(0')))
test(x, y) -> if1(gt(x, y), x, y)
if1(true, x, y) -> if2(divides(x, y), x, y)
if1(false, x, y) -> true
if2(true, x, y) -> false
if2(false, x, y) -> test(x, s(y))

Types:
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
test :: s:0' -> s:0' -> true:false
if1 :: true:false -> s:0' -> s:0' -> true:false
if2 :: true:false -> s:0' -> s:0' -> true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'3_0(+(1, n5_0)), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
div, test
----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
div(gen_s:0'3_0(n276_0), gen_s:0'3_0(n276_0), gen_s:0'3_0(c)) -> true, rt in Omega(1 + n276_0)

Induction Base:
div(gen_s:0'3_0(0), gen_s:0'3_0(0), gen_s:0'3_0(c)) ->_R^Omega(1)
true

Induction Step:
div(gen_s:0'3_0(+(n276_0, 1)), gen_s:0'3_0(+(n276_0, 1)), gen_s:0'3_0(c)) ->_R^Omega(1)
div(gen_s:0'3_0(n276_0), gen_s:0'3_0(n276_0), gen_s:0'3_0(c)) ->_IH
true

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

(26)
Obligation:
TRS:
Rules:
gt(s(x), 0') -> true
gt(0', y) -> false
gt(s(x), s(y)) -> gt(x, y)
divides(x, y) -> div(x, y, y)
div(0', 0', z) -> true
div(0', s(x), z) -> false
div(s(x), 0', s(z)) -> div(s(x), s(z), s(z))
div(s(x), s(y), z) -> div(x, y, z)
prime(x) -> test(x, s(s(0')))
test(x, y) -> if1(gt(x, y), x, y)
if1(true, x, y) -> if2(divides(x, y), x, y)
if1(false, x, y) -> true
if2(true, x, y) -> false
if2(false, x, y) -> test(x, s(y))

Types:
gt :: s:0' -> s:0' -> true:false
s :: s:0' -> s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0' -> true:false
prime :: s:0' -> true:false
test :: s:0' -> s:0' -> true:false
if1 :: true:false -> s:0' -> s:0' -> true:false
if2 :: true:false -> s:0' -> s:0' -> true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'3_0(+(1, n5_0)), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
div(gen_s:0'3_0(n276_0), gen_s:0'3_0(n276_0), gen_s:0'3_0(c)) -> true, rt in Omega(1 + n276_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
test