/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(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), 1 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 188 ms]
    (10) BOUNDS(1, n^1)
(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), 275 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   digits -> d(0)
   d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x)
   if(true, x) -> cons(x, d(s(x)))
   if(false, x) -> nil
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(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^1).


The TRS R consists of the following rules:

   digits -> d(0) [1]
   d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) [1]
   if(true, x) -> cons(x, d(s(x))) [1]
   if(false, x) -> nil [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(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:

   digits -> d(0) [1]
   d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) [1]
   if(true, x) -> cons(x, d(s(x))) [1]
   if(false, x) -> nil [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]

The TRS has the following type information:
digits :: cons:nil
d :: 0:s -> cons:nil
0 :: 0:s
if :: true:false -> 0:s -> cons:nil
le :: 0:s -> 0:s -> true:false
s :: 0:s -> 0:s
true :: true:false
cons :: 0:s -> cons:nil -> cons:nil
false :: true:false
nil :: cons:nil

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:
none

And the following fresh constants: none

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

(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:

   digits -> d(0) [1]
   d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) [1]
   if(true, x) -> cons(x, d(s(x))) [1]
   if(false, x) -> nil [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]

The TRS has the following type information:
digits :: cons:nil
d :: 0:s -> cons:nil
0 :: 0:s
if :: true:false -> 0:s -> cons:nil
le :: 0:s -> 0:s -> true:false
s :: 0:s -> 0:s
true :: true:false
cons :: 0:s -> cons:nil -> cons:nil
false :: true:false
nil :: cons:nil

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
   true => 1
   false => 0
   nil => 0

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

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

   d(z) -{ 1 }-> if(le(x, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), x) :|: x >= 0, z = x
   digits -{ 1 }-> d(0) :|: 
   if(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   if(z, z') -{ 1 }-> 1 + x + d(1 + x) :|: z' = x, z = 1, x >= 0
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0

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, V2),0,[digits(Out)],[]).
eq(start(V, V2),0,[d(V, Out)],[V >= 0]).
eq(start(V, V2),0,[if(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[le(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(digits(Out),1,[d(0, Ret)],[Out = Ret]).
eq(d(V, Out),1,[le(V1, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), Ret0),if(Ret0, V1, Ret1)],[Out = Ret1,V1 >= 0,V = V1]).
eq(if(V, V2, Out),1,[d(1 + V3, Ret11)],[Out = 1 + Ret11 + V3,V2 = V3,V = 1,V3 >= 0]).
eq(if(V, V2, Out),1,[],[Out = 0,V2 = V4,V4 >= 0,V = 0]).
eq(le(V, V2, Out),1,[],[Out = 1,V5 >= 0,V = 0,V2 = V5]).
eq(le(V, V2, Out),1,[],[Out = 0,V6 >= 0,V = 1 + V6,V2 = 0]).
eq(le(V, V2, Out),1,[le(V7, V8, Ret2)],[Out = Ret2,V2 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]).
input_output_vars(digits(Out),[],[Out]).
input_output_vars(d(V,Out),[V],[Out]).
input_output_vars(if(V,V2,Out),[V,V2],[Out]).
input_output_vars(le(V,V2,Out),[V,V2],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [le/3]
1. recursive  : [d/2,if/3]
2. non_recursive  : [digits/1]
3. non_recursive  : [start/2]

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

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

### Specialization of cost equations le/3 
* CE 10 is refined into CE [11] 
* CE 9 is refined into CE [12] 
* CE 8 is refined into CE [13] 


### Cost equations --> "Loop" of le/3 
* CEs [12] --> Loop 7 
* CEs [13] --> Loop 8 
* CEs [11] --> Loop 9 

### Ranking functions of CR le(V,V2,Out) 
* RF of phase [9]: [V,V2]

#### Partial ranking functions of CR le(V,V2,Out) 
* Partial RF of phase [9]:
  - RF of loop [9:1]:
    V
    V2


### Specialization of cost equations d/2 
* CE 7 is refined into CE [14,15] 
* CE 6 is refined into CE [16] 


### Cost equations --> "Loop" of d/2 
* CEs [16] --> Loop 10 
* CEs [15] --> Loop 11 
* CEs [14] --> Loop 12 

### Ranking functions of CR d(V,Out) 
* RF of phase [11]: [-V+10]

#### Partial ranking functions of CR d(V,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    -V+10


### Specialization of cost equations start/2 
* CE 1 is refined into CE [17] 
* CE 2 is refined into CE [18,19] 
* CE 3 is refined into CE [20] 
* CE 4 is refined into CE [21,22,23] 
* CE 5 is refined into CE [24,25,26,27] 


### Cost equations --> "Loop" of start/2 
* CEs [17,18,19,20,21,22,23,24,25,26,27] --> Loop 13 

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

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


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

#### Cost of chains of le(V,V2,Out):
* Chain [[9],8]: 1*it(9)+1
  Such that:it(9) =< V

  with precondition: [Out=1,V>=1,V2>=V] 

* Chain [[9],7]: 1*it(9)+1
  Such that:it(9) =< V2

  with precondition: [Out=0,V2>=1,V>=V2+1] 

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

* Chain [7]: 1
  with precondition: [V2=0,Out=0,V>=1] 


#### Cost of chains of d(V,Out):
* Chain [[11],10]: 3*it(11)+1*s(1)+1*s(4)+3
  Such that:s(1) =< 9
s(4) =< -9*V+90
it(11) =< -V+10

  with precondition: [9>=V,V>=1] 

* Chain [12,[11],10]: 4*it(11)+1*s(4)+6
  Such that:s(4) =< 81
aux(1) =< 9
it(11) =< aux(1)

  with precondition: [V=0] 

* Chain [10]: 1*s(1)+3
  Such that:s(1) =< 9

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


#### Cost of chains of start(V,V2):
* Chain [13]: 12*s(5)+1*s(7)+3*s(8)+2*s(9)+1*s(17)+3*s(18)+1*s(19)+1*s(20)+7
  Such that:s(17) =< -9*V+90
s(18) =< -V+10
s(20) =< V
s(7) =< -9*V2+81
s(8) =< -V2+9
s(19) =< V2
aux(2) =< 9
aux(3) =< 81
s(5) =< aux(2)
s(9) =< aux(3)

  with precondition: [] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [13] with precondition: [] 
    - Upper bound: nat(V)+277+nat(V2)+nat(-V+10)*3+nat(-V2+9)*3+nat(-9*V+90)+nat(-9*V2+81) 
    - Complexity: n 

### Maximum cost of start(V,V2): nat(V)+277+nat(V2)+nat(-V+10)*3+nat(-V2+9)*3+nat(-9*V+90)+nat(-9*V2+81) 
Asymptotic class: n 
* Total analysis performed in 123 ms.


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

(10)
BOUNDS(1, n^1)

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

(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^1, INF).


The TRS R consists of the following rules:

   digits -> d(0')
   d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0')))))))))), x)
   if(true, x) -> cons(x, d(s(x)))
   if(false, x) -> nil
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)

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

(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/0

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

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


The TRS R consists of the following rules:

   digits -> d(0')
   d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0')))))))))), x)
   if(true, x) -> cons(d(s(x)))
   if(false, x) -> nil
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)

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

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

(16)
Obligation:
Innermost TRS:
Rules:
digits -> d(0')
d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0')))))))))), x)
if(true, x) -> cons(d(s(x)))
if(false, x) -> nil
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)

Types:
digits :: cons:nil
d :: 0':s -> cons:nil
0' :: 0':s
if :: true:false -> 0':s -> cons:nil
le :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
true :: true:false
cons :: cons:nil -> cons:nil
false :: true:false
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s

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

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

They will be analysed ascendingly in the following order:
le < d

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

(18)
Obligation:
Innermost TRS:
Rules:
digits -> d(0')
d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0')))))))))), x)
if(true, x) -> cons(d(s(x)))
if(false, x) -> nil
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)

Types:
digits :: cons:nil
d :: 0':s -> cons:nil
0' :: 0':s
if :: true:false -> 0':s -> cons:nil
le :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
true :: true:false
cons :: cons:nil -> cons:nil
false :: true:false
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
le, d

They will be analysed ascendingly in the following order:
le < d

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) -> true, rt in Omega(1 + n7_0)

Induction Base:
le(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s5_0(+(n7_0, 1)), gen_0':s5_0(+(n7_0, 1))) ->_R^Omega(1)
le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_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:

Innermost TRS:
Rules:
digits -> d(0')
d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0')))))))))), x)
if(true, x) -> cons(d(s(x)))
if(false, x) -> nil
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)

Types:
digits :: cons:nil
d :: 0':s -> cons:nil
0' :: 0':s
if :: true:false -> 0':s -> cons:nil
le :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
true :: true:false
cons :: cons:nil -> cons:nil
false :: true:false
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
le, d

They will be analysed ascendingly in the following order:
le < d

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
digits -> d(0')
d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0')))))))))), x)
if(true, x) -> cons(d(s(x)))
if(false, x) -> nil
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)

Types:
digits :: cons:nil
d :: 0':s -> cons:nil
0' :: 0':s
if :: true:false -> 0':s -> cons:nil
le :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
true :: true:false
cons :: cons:nil -> cons:nil
false :: true:false
nil :: cons:nil
hole_cons:nil1_0 :: cons:nil
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) -> true, rt in Omega(1 + n7_0)


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
d