/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^2))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 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), 1 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 247 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), 249 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs


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(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:

   f(t, x, y) -> f(g(x, y), x, s(y))
   g(s(x), 0) -> t
   g(s(x), s(y)) -> g(x, y)

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

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

The duplicating contexts are:
f(t, [], y)
f(t, x, [])


The defined contexts are:
f([], x1, s(x2))


[] just represents basic- or constructor-terms in the following defined contexts:
f([], x1, s(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:

   f(t, x, y) -> f(g(x, y), x, s(y))
   g(s(x), 0) -> t
   g(s(x), s(y)) -> g(x, 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:

   f(t, x, y) -> f(g(x, y), x, s(y)) [1]
   g(s(x), 0) -> t [1]
   g(s(x), s(y)) -> g(x, 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:

   f(t, x, y) -> f(g(x, y), x, s(y)) [1]
   g(s(x), 0) -> t [1]
   g(s(x), s(y)) -> g(x, y) [1]

The TRS has the following type information:
f :: t -> s:0 -> s:0 -> f
t :: t
g :: s:0 -> s:0 -> t
s :: s:0 -> s:0
0 :: s:0

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:

   g(v0, v1) -> null_g [0]
   f(v0, v1, v2) -> null_f [0]

And the following fresh constants:    null_g, null_f

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

(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:

   f(t, x, y) -> f(g(x, y), x, s(y)) [1]
   g(s(x), 0) -> t [1]
   g(s(x), s(y)) -> g(x, y) [1]
   g(v0, v1) -> null_g [0]
   f(v0, v1, v2) -> null_f [0]

The TRS has the following type information:
f :: t:null_g -> s:0 -> s:0 -> null_f
t :: t:null_g
g :: s:0 -> s:0 -> t:null_g
s :: s:0 -> s:0
0 :: s:0
null_g :: t:null_g
null_f :: null_f

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:

   t => 1
   0 => 0
   null_g => 0
   null_f => 0

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

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

   f(z, z', z'') -{ 1 }-> f(g(x, y), x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   g(z, z') -{ 1 }-> g(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   g(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

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(V1, V, V2),0,[f(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(start(V1, V, V2),0,[g(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(f(V1, V, V2, Out),1,[g(V4, V3, Ret0),f(Ret0, V4, 1 + V3, Ret)],[Out = Ret,V = V4,V2 = V3,V1 = 1,V4 >= 0,V3 >= 0]).
eq(g(V1, V, Out),1,[],[Out = 1,V5 >= 0,V1 = 1 + V5,V = 0]).
eq(g(V1, V, Out),1,[g(V6, V7, Ret1)],[Out = Ret1,V = 1 + V7,V6 >= 0,V7 >= 0,V1 = 1 + V6]).
eq(g(V1, V, Out),0,[],[Out = 0,V9 >= 0,V8 >= 0,V1 = V9,V = V8]).
eq(f(V1, V, V2, Out),0,[],[Out = 0,V11 >= 0,V2 = V12,V10 >= 0,V1 = V11,V = V10,V12 >= 0]).
input_output_vars(f(V1,V,V2,Out),[V1,V,V2],[Out]).
input_output_vars(g(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [g/3]
1. recursive  : [f/4]
2. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into g/3
1. SCC is partially evaluated into f/4
2. SCC is partially evaluated into start/3

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

### Specialization of cost equations g/3 
* CE 7 is refined into CE [8] 
* CE 5 is refined into CE [9] 
* CE 6 is refined into CE [10] 


### Cost equations --> "Loop" of g/3 
* CEs [10] --> Loop 7 
* CEs [8] --> Loop 8 
* CEs [9] --> Loop 9 

### Ranking functions of CR g(V1,V,Out) 
* RF of phase [7]: [V,V1]

#### Partial ranking functions of CR g(V1,V,Out) 
* Partial RF of phase [7]:
  - RF of loop [7:1]:
    V
    V1


### Specialization of cost equations f/4 
* CE 4 is refined into CE [11] 
* CE 3 is refined into CE [12,13,14] 


### Cost equations --> "Loop" of f/4 
* CEs [14] --> Loop 10 
* CEs [13] --> Loop 11 
* CEs [12] --> Loop 12 
* CEs [11] --> Loop 13 

### Ranking functions of CR f(V1,V,V2,Out) 
* RF of phase [10]: [V-V2]

#### Partial ranking functions of CR f(V1,V,V2,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V-V2


### Specialization of cost equations start/3 
* CE 1 is refined into CE [15] 
* CE 2 is refined into CE [16,17,18] 


### Cost equations --> "Loop" of start/3 
* CEs [15,16,17,18] --> Loop 14 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of g(V1,V,Out):
* Chain [[7],9]: 1*it(7)+1
  Such that:it(7) =< V

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

* Chain [[7],8]: 1*it(7)+0
  Such that:it(7) =< V

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

* Chain [9]: 1
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [8]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of f(V1,V,V2,Out):
* Chain [[10],13]: 2*it(10)+1*s(4)+0
  Such that:aux(1) =< V
it(10) =< V-V2
s(4) =< it(10)*aux(1)

  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 

* Chain [[10],11,13]: 2*it(10)+1*s(4)+1*s(5)+1
  Such that:aux(1) =< V
s(5) =< V+1
it(10) =< V-V2
s(4) =< it(10)*aux(1)

  with precondition: [V1=1,Out=0,V2>=1,V>=V2+1] 

* Chain [13]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 

* Chain [12,[10],13]: 2*it(10)+1*s(4)+2
  Such that:aux(2) =< V
it(10) =< aux(2)
s(4) =< it(10)*aux(2)

  with precondition: [V1=1,V2=0,Out=0,V>=2] 

* Chain [12,[10],11,13]: 2*it(10)+1*s(4)+1*s(5)+3
  Such that:s(5) =< V+1
aux(3) =< V
it(10) =< aux(3)
s(4) =< it(10)*aux(3)

  with precondition: [V1=1,V2=0,Out=0,V>=2] 

* Chain [12,13]: 2
  with precondition: [V1=1,V2=0,Out=0,V>=1] 

* Chain [12,11,13]: 1*s(5)+3
  Such that:s(5) =< 2

  with precondition: [V1=1,V2=0,Out=0,V>=1] 

* Chain [11,13]: 1*s(5)+1
  Such that:s(5) =< V2+1

  with precondition: [V1=1,Out=0,V>=0,V2>=0] 


#### Cost of chains of start(V1,V,V2):
* Chain [14]: 1*s(22)+1*s(23)+2*s(27)+4*s(28)+6*s(29)+2*s(30)+2*s(31)+3
  Such that:s(22) =< 2
s(25) =< V+1
s(26) =< V-V2
s(23) =< V2+1
aux(7) =< V
s(29) =< aux(7)
s(27) =< s(25)
s(28) =< s(26)
s(30) =< s(29)*aux(7)
s(31) =< s(28)*aux(7)

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


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [14] with precondition: [V1>=0,V>=0] 
    - Upper bound: 6*V+5+2*V*V+2*V*nat(V-V2)+(2*V+2)+nat(V2+1)+nat(V-V2)*4 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V2): 6*V+5+2*V*V+2*V*nat(V-V2)+(2*V+2)+nat(V2+1)+nat(V-V2)*4 
Asymptotic class: n^2 
* Total analysis performed in 171 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:

   f(t, x, y) -> f(g(x, y), x, s(y))
   g(s(x), 0') -> t
   g(s(x), s(y)) -> g(x, y)

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

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

(16)
Obligation:
TRS:
Rules:
f(t, x, y) -> f(g(x, y), x, s(y))
g(s(x), 0') -> t
g(s(x), s(y)) -> g(x, y)

Types:
f :: t -> s:0' -> s:0' -> f
t :: t
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'

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

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

They will be analysed ascendingly in the following order:
g < f

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

(18)
Obligation:
TRS:
Rules:
f(t, x, y) -> f(g(x, y), x, s(y))
g(s(x), 0') -> t
g(s(x), s(y)) -> g(x, y)

Types:
f :: t -> s:0' -> s:0' -> f
t :: t
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
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:
g, f

They will be analysed ascendingly in the following order:
g < f

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) -> t, rt in Omega(1 + n6_0)

Induction Base:
g(gen_s:0'4_0(+(1, 0)), gen_s:0'4_0(0)) ->_R^Omega(1)
t

Induction Step:
g(gen_s:0'4_0(+(1, +(n6_0, 1))), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) ->_IH
t

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:
f(t, x, y) -> f(g(x, y), x, s(y))
g(s(x), 0') -> t
g(s(x), s(y)) -> g(x, y)

Types:
f :: t -> s:0' -> s:0' -> f
t :: t
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
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:
g, f

They will be analysed ascendingly in the following order:
g < f

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
f(t, x, y) -> f(g(x, y), x, s(y))
g(s(x), 0') -> t
g(s(x), s(y)) -> g(x, y)

Types:
f :: t -> s:0' -> s:0' -> f
t :: t
g :: s:0' -> s:0' -> t
s :: s:0' -> s:0'
0' :: s:0'
hole_f1_0 :: f
hole_t2_0 :: t
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
g(gen_s:0'4_0(+(1, n6_0)), gen_s:0'4_0(n6_0)) -> t, rt in Omega(1 + n6_0)


Generator Equations:
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:
f