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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(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, 137 ms]
    (10) BOUNDS(1, n^2)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) SlicingProof [LOWER BOUND(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), 280 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), 1705 ms]
            (26) proven lower bound
            (27) LowerBoundPropagationProof [FINISHED, 0 ms]
            (28) BOUNDS(n^2, INF)


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

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


The TRS R consists of the following rules:

   fac(s(x)) -> *(fac(p(s(x))), s(x))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))

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

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

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


The TRS R consists of the following rules:

   fac(s(x)) -> *(fac(p(s(x))), s(x)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   fac(s(x)) -> *(fac(p(s(x))), s(x)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]

The TRS has the following type information:
fac :: s:0 -> *
s :: s:0 -> s:0
* :: * -> s:0 -> *
p :: s:0 -> s:0
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:

   fac(v0) -> null_fac [0]
   p(v0) -> null_p [0]

And the following fresh constants:    null_fac, null_p

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

(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:

   fac(s(x)) -> *(fac(p(s(x))), s(x)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   fac(v0) -> null_fac [0]
   p(v0) -> null_p [0]

The TRS has the following type information:
fac :: s:0:null_p -> *:null_fac
s :: s:0:null_p -> s:0:null_p
* :: *:null_fac -> s:0:null_p -> *:null_fac
p :: s:0:null_p -> s:0:null_p
0 :: s:0:null_p
null_fac :: *:null_fac
null_p :: s:0:null_p

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

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

   0 => 0
   null_fac => 0
   null_p => 0

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

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

   fac(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   fac(z) -{ 1 }-> 1 + fac(p(1 + x)) + (1 + x) :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x)

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(V),0,[fac(V, Out)],[V >= 0]).
eq(start(V),0,[p(V, Out)],[V >= 0]).
eq(fac(V, Out),1,[p(1 + V1, Ret010),fac(Ret010, Ret01)],[Out = 2 + Ret01 + V1,V1 >= 0,V = 1 + V1]).
eq(p(V, Out),1,[],[Out = 0,V = 1]).
eq(p(V, Out),1,[p(1 + V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V = 2 + V2]).
eq(fac(V, Out),0,[],[Out = 0,V3 >= 0,V = V3]).
eq(p(V, Out),0,[],[Out = 0,V4 >= 0,V = V4]).
input_output_vars(fac(V,Out),[V],[Out]).
input_output_vars(p(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [p/2]
1. recursive  : [fac/2]
2. non_recursive  : [start/1]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into p/2
1. SCC is partially evaluated into fac/2
2. SCC is partially evaluated into start/1

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

### Specialization of cost equations p/2 
* CE 5 is refined into CE [8] 
* CE 7 is refined into CE [9] 
* CE 6 is refined into CE [10] 


### Cost equations --> "Loop" of p/2 
* CEs [10] --> Loop 6 
* CEs [8,9] --> Loop 7 

### Ranking functions of CR p(V,Out) 
* RF of phase [6]: [V-1]

#### Partial ranking functions of CR p(V,Out) 
* Partial RF of phase [6]:
  - RF of loop [6:1]:
    V-1


### Specialization of cost equations fac/2 
* CE 4 is refined into CE [11] 
* CE 3 is refined into CE [12,13] 


### Cost equations --> "Loop" of fac/2 
* CEs [13] --> Loop 8 
* CEs [12] --> Loop 9 
* CEs [11] --> Loop 10 

### Ranking functions of CR fac(V,Out) 
* RF of phase [8]: [V-1]

#### Partial ranking functions of CR fac(V,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1]:
    V-1


### Specialization of cost equations start/1 
* CE 1 is refined into CE [14,15,16] 
* CE 2 is refined into CE [17,18] 


### Cost equations --> "Loop" of start/1 
* CEs [14,15,16,17,18] --> Loop 11 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of p(V,Out):
* Chain [[6],7]: 1*it(6)+1
  Such that:it(6) =< Out

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

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


#### Cost of chains of fac(V,Out):
* Chain [[8],10]: 2*it(8)+1*s(3)+0
  Such that:aux(3) =< V
it(8) =< aux(3)
s(3) =< it(8)*aux(3)

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

* Chain [[8],9,10]: 2*it(8)+1*s(3)+2
  Such that:aux(4) =< V
it(8) =< aux(4)
s(3) =< it(8)*aux(4)

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

* Chain [10]: 0
  with precondition: [Out=0,V>=0] 

* Chain [9,10]: 2
  with precondition: [V+1=Out,V>=1] 


#### Cost of chains of start(V):
* Chain [11]: 5*s(11)+2*s(12)+2
  Such that:aux(6) =< V
s(11) =< aux(6)
s(12) =< s(11)*aux(6)

  with precondition: [V>=0] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [11] with precondition: [V>=0] 
    - Upper bound: 5*V+2+2*V*V 
    - Complexity: n^2 

### Maximum cost of start(V): 5*V+2+2*V*V 
Asymptotic class: n^2 
* Total analysis performed in 77 ms.


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(10)
BOUNDS(1, n^2)

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   fac(s(x)) -> *'(fac(p(s(x))), s(x))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))

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

(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
*'/1

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   fac(s(x)) -> *'(fac(p(s(x))))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))

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

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

(16)
Obligation:
Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
fac, p

They will be analysed ascendingly in the following order:
p < fac

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

(18)
Obligation:
Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_*'3_0(0) <=> hole_*'1_0
gen_*'3_0(+(x, 1)) <=> *'(gen_*'3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
p, fac

They will be analysed ascendingly in the following order:
p < fac

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_s:0'4_0(+(1, n6_0))) -> gen_s:0'4_0(n6_0), rt in Omega(1 + n6_0)

Induction Base:
p(gen_s:0'4_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
p(gen_s:0'4_0(+(1, +(n6_0, 1)))) ->_R^Omega(1)
s(p(s(gen_s:0'4_0(n6_0)))) ->_IH
s(gen_s:0'4_0(c7_0))

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:

Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_*'3_0(0) <=> hole_*'1_0
gen_*'3_0(+(x, 1)) <=> *'(gen_*'3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
p, fac

They will be analysed ascendingly in the following order:
p < fac

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
p(gen_s:0'4_0(+(1, n6_0))) -> gen_s:0'4_0(n6_0), rt in Omega(1 + n6_0)


Generator Equations:
gen_*'3_0(0) <=> hole_*'1_0
gen_*'3_0(+(x, 1)) <=> *'(gen_*'3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fac(gen_s:0'4_0(+(1, n221_0))) -> *5_0, rt in Omega(n221_0 + n221_0^2)

Induction Base:
fac(gen_s:0'4_0(+(1, 0)))

Induction Step:
fac(gen_s:0'4_0(+(1, +(n221_0, 1)))) ->_R^Omega(1)
*'(fac(p(s(gen_s:0'4_0(+(1, n221_0)))))) ->_L^Omega(2 + n221_0)
*'(fac(gen_s:0'4_0(+(1, n221_0)))) ->_IH
*'(*5_0)

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(26)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
fac(s(x)) -> *'(fac(p(s(x))))
p(s(0')) -> 0'
p(s(s(x))) -> s(p(s(x)))

Types:
fac :: s:0' -> *'
s :: s:0' -> s:0'
*' :: *' -> *'
p :: s:0' -> s:0'
0' :: s:0'
hole_*'1_0 :: *'
hole_s:0'2_0 :: s:0'
gen_*'3_0 :: Nat -> *'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
p(gen_s:0'4_0(+(1, n6_0))) -> gen_s:0'4_0(n6_0), rt in Omega(1 + n6_0)


Generator Equations:
gen_*'3_0(0) <=> hole_*'1_0
gen_*'3_0(+(x, 1)) <=> *'(gen_*'3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


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

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

(28)
BOUNDS(n^2, INF)