/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 152 ms]
    (12) BOUNDS(1, n^1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 3804 ms]
    (22) proven lower bound
    (23) LowerBoundPropagationProof [FINISHED, 0 ms]
    (24) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 0)

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

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

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   prime(0) -> false
   prime(s(0)) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0) -> false
   prime1(x, s(0)) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> =(rem(x, y), 0)

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

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

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


The TRS R consists of the following rules:

   prime(0) -> false [1]
   prime(s(0)) -> false [1]
   prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1]
   prime1(x, 0) -> false [1]
   prime1(x, s(0)) -> true [1]
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
   divp(x, y) -> =(rem(x, y), 0) [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:

   prime(0) -> false [1]
   prime(s(0)) -> false [1]
   prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1]
   prime1(x, 0) -> false [1]
   prime1(x, s(0)) -> true [1]
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
   divp(x, y) -> =(rem(x, y), 0) [1]

The TRS has the following type information:
prime :: 0:s -> false:true:and
0 :: 0:s
false :: false:true:and
s :: 0:s -> 0:s
prime1 :: 0:s -> 0:s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: = -> not
divp :: 0:s -> 0:s -> =
= :: rem -> 0:s -> =
rem :: 0:s -> 0:s -> rem

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:
none

And the following fresh constants:    const, const1, const2

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

(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:

   prime(0) -> false [1]
   prime(s(0)) -> false [1]
   prime(s(s(x))) -> prime1(s(s(x)), s(x)) [1]
   prime1(x, 0) -> false [1]
   prime1(x, s(0)) -> true [1]
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
   divp(x, y) -> =(rem(x, y), 0) [1]

The TRS has the following type information:
prime :: 0:s -> false:true:and
0 :: 0:s
false :: false:true:and
s :: 0:s -> 0:s
prime1 :: 0:s -> 0:s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: = -> not
divp :: 0:s -> 0:s -> =
= :: rem -> 0:s -> =
rem :: 0:s -> 0:s -> rem
const :: not
const1 :: =
const2 :: rem

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
   false => 0
   true => 1
   const => 0
   const1 => 0
   const2 => 0

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

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

   divp(z, z') -{ 1 }-> 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y
   prime(z) -{ 1 }-> prime1(1 + (1 + x), 1 + x) :|: x >= 0, z = 1 + (1 + x)
   prime(z) -{ 1 }-> 0 :|: z = 0
   prime(z) -{ 1 }-> 0 :|: z = 1 + 0
   prime1(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1 + 0, z = x
   prime1(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   prime1(z, z') -{ 1 }-> 1 + (1 + divp(1 + (1 + y), x)) + prime1(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x

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, V2),0,[prime(V, Out)],[V >= 0]).
eq(start(V, V2),0,[prime1(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[divp(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(prime(V, Out),1,[],[Out = 0,V = 0]).
eq(prime(V, Out),1,[],[Out = 0,V = 1]).
eq(prime(V, Out),1,[prime1(1 + (1 + V1), 1 + V1, Ret)],[Out = Ret,V1 >= 0,V = 2 + V1]).
eq(prime1(V, V2, Out),1,[],[Out = 0,V3 >= 0,V = V3,V2 = 0]).
eq(prime1(V, V2, Out),1,[],[Out = 1,V4 >= 0,V2 = 1,V = V4]).
eq(prime1(V, V2, Out),1,[divp(1 + (1 + V6), V5, Ret011),prime1(V5, 1 + V6, Ret1)],[Out = 2 + Ret011 + Ret1,V2 = 2 + V6,V5 >= 0,V6 >= 0,V = V5]).
eq(divp(V, V2, Out),1,[],[Out = 2 + V7 + V8,V7 >= 0,V8 >= 0,V = V7,V2 = V8]).
input_output_vars(prime(V,Out),[V],[Out]).
input_output_vars(prime1(V,V2,Out),[V,V2],[Out]).
input_output_vars(divp(V,V2,Out),[V,V2],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [divp/3]
1. recursive  : [prime1/3]
2. non_recursive  : [prime/2]
3. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into prime1/3
2. SCC is partially evaluated into prime/2
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations prime1/3 
* CE 9 is refined into CE [10] 
* CE 8 is refined into CE [11] 
* CE 7 is refined into CE [12] 


### Cost equations --> "Loop" of prime1/3 
* CEs [11] --> Loop 8 
* CEs [12] --> Loop 9 
* CEs [10] --> Loop 10 

### Ranking functions of CR prime1(V,V2,Out) 
* RF of phase [10]: [V2-1]

#### Partial ranking functions of CR prime1(V,V2,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V2-1


### Specialization of cost equations prime/2 
* CE 6 is refined into CE [13,14] 
* CE 5 is refined into CE [15] 
* CE 4 is refined into CE [16] 


### Cost equations --> "Loop" of prime/2 
* CEs [14] --> Loop 11 
* CEs [13] --> Loop 12 
* CEs [15] --> Loop 13 
* CEs [16] --> Loop 14 

### Ranking functions of CR prime(V,Out) 

#### Partial ranking functions of CR prime(V,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [17,18,19,20] 
* CE 2 is refined into CE [21,22,23] 
* CE 3 is refined into CE [24] 


### Cost equations --> "Loop" of start/2 
* CEs [20] --> Loop 15 
* CEs [22] --> Loop 16 
* CEs [21,23,24] --> Loop 17 
* CEs [19] --> Loop 18 
* CEs [18] --> Loop 19 
* CEs [17] --> Loop 20 

### Ranking functions of CR start(V,V2) 

#### Partial ranking functions of CR start(V,V2) 


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

#### Cost of chains of prime1(V,V2,Out):
* Chain [[10],8]: 2*it(10)+1
  Such that:it(10) =< V2

  with precondition: [V>=0,V2>=2,Out+7>=7*V2+V] 

* Chain [9]: 1
  with precondition: [V2=0,Out=0,V>=0] 

* Chain [8]: 1
  with precondition: [V2=1,Out=1,V>=0] 


#### Cost of chains of prime(V,Out):
* Chain [14]: 1
  with precondition: [V=0,Out=0] 

* Chain [13]: 1
  with precondition: [V=1,Out=0] 

* Chain [12]: 2
  with precondition: [V=2,Out=1] 

* Chain [11]: 2*s(1)+2
  Such that:s(1) =< V

  with precondition: [V>=3,Out+14>=8*V] 


#### Cost of chains of start(V,V2):
* Chain [20]: 1
  with precondition: [V=0] 

* Chain [19]: 1
  with precondition: [V=1] 

* Chain [18]: 2
  with precondition: [V=2] 

* Chain [17]: 2*s(2)+1
  Such that:s(2) =< V2

  with precondition: [V>=0,V2>=0] 

* Chain [16]: 1
  with precondition: [V2=1,V>=0] 

* Chain [15]: 2*s(3)+2
  Such that:s(3) =< V

  with precondition: [V>=3] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [20] with precondition: [V=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [19] with precondition: [V=1] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [18] with precondition: [V=2] 
    - Upper bound: 2 
    - Complexity: constant 
* Chain [17] with precondition: [V>=0,V2>=0] 
    - Upper bound: 2*V2+1 
    - Complexity: n 
* Chain [16] with precondition: [V2=1,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [15] with precondition: [V>=3] 
    - Upper bound: 2*V+2 
    - Complexity: n 

### Maximum cost of start(V,V2): max([2*V+1,nat(V2)*2])+1 
Asymptotic class: n 
* Total analysis performed in 79 ms.


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

(12)
BOUNDS(1, n^1)

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

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

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


The TRS R consists of the following rules:

   prime(0') -> false
   prime(s(0')) -> false
   prime(s(s(x))) -> prime1(s(s(x)), s(x))
   prime1(x, 0') -> false
   prime1(x, s(0')) -> true
   prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
   divp(x, y) -> ='(rem(x, y), 0')

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

(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
prime1/0
divp/0
divp/1
='/0
='/1
rem/0
rem/1

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

(16)
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:

   prime(0') -> false
   prime(s(0')) -> false
   prime(s(s(x))) -> prime1(s(x))
   prime1(0') -> false
   prime1(s(0')) -> true
   prime1(s(s(y))) -> and(not(divp), prime1(s(y)))
   divp -> ='

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

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

(18)
Obligation:
TRS:
Rules:
prime(0') -> false
prime(s(0')) -> false
prime(s(s(x))) -> prime1(s(x))
prime1(0') -> false
prime1(s(0')) -> true
prime1(s(s(y))) -> and(not(divp), prime1(s(y)))
divp -> ='

Types:
prime :: 0':s -> false:true:and
0' :: 0':s
false :: false:true:and
s :: 0':s -> 0':s
prime1 :: 0':s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: =' -> not
divp :: ='
=' :: ='
hole_false:true:and1_0 :: false:true:and
hole_0':s2_0 :: 0':s
hole_not3_0 :: not
hole_='4_0 :: ='
gen_false:true:and5_0 :: Nat -> false:true:and
gen_0':s6_0 :: Nat -> 0':s

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
prime1
----------------------------------------

(20)
Obligation:
TRS:
Rules:
prime(0') -> false
prime(s(0')) -> false
prime(s(s(x))) -> prime1(s(x))
prime1(0') -> false
prime1(s(0')) -> true
prime1(s(s(y))) -> and(not(divp), prime1(s(y)))
divp -> ='

Types:
prime :: 0':s -> false:true:and
0' :: 0':s
false :: false:true:and
s :: 0':s -> 0':s
prime1 :: 0':s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: =' -> not
divp :: ='
=' :: ='
hole_false:true:and1_0 :: false:true:and
hole_0':s2_0 :: 0':s
hole_not3_0 :: not
hole_='4_0 :: ='
gen_false:true:and5_0 :: Nat -> false:true:and
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_false:true:and5_0(0) <=> false
gen_false:true:and5_0(+(x, 1)) <=> and(not(='), gen_false:true:and5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
prime1
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
prime1(gen_0':s6_0(+(1, n8_0))) -> *7_0, rt in Omega(n8_0)

Induction Base:
prime1(gen_0':s6_0(+(1, 0)))

Induction Step:
prime1(gen_0':s6_0(+(1, +(n8_0, 1)))) ->_R^Omega(1)
and(not(divp), prime1(s(gen_0':s6_0(n8_0)))) ->_R^Omega(1)
and(not(='), prime1(s(gen_0':s6_0(n8_0)))) ->_IH
and(not(='), *7_0)

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

(22)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
prime(0') -> false
prime(s(0')) -> false
prime(s(s(x))) -> prime1(s(x))
prime1(0') -> false
prime1(s(0')) -> true
prime1(s(s(y))) -> and(not(divp), prime1(s(y)))
divp -> ='

Types:
prime :: 0':s -> false:true:and
0' :: 0':s
false :: false:true:and
s :: 0':s -> 0':s
prime1 :: 0':s -> false:true:and
true :: false:true:and
and :: not -> false:true:and -> false:true:and
not :: =' -> not
divp :: ='
=' :: ='
hole_false:true:and1_0 :: false:true:and
hole_0':s2_0 :: 0':s
hole_not3_0 :: not
hole_='4_0 :: ='
gen_false:true:and5_0 :: Nat -> false:true:and
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_false:true:and5_0(0) <=> false
gen_false:true:and5_0(+(x, 1)) <=> and(not(='), gen_false:true:and5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
prime1
----------------------------------------

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

(24)
BOUNDS(n^1, INF)