/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^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^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), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 157 ms]
    (12) BOUNDS(1, n^1)
(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), 3478 ms]
    (20) proven lower bound
    (21) LowerBoundPropagationProof [FINISHED, 0 ms]
    (22) BOUNDS(n^1, INF)


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(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:

   a__f(X, X) -> a__f(a, b)
   a__b -> a
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2) -> f(X1, X2)
   a__b -> b

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

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

As the TRS is a non-duplicating overlay system, we have rc = irc.
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(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:

   a__f(X, X) -> a__f(a, b)
   a__b -> a
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2) -> f(X1, X2)
   a__b -> b

S is empty.
Rewrite Strategy: INNERMOST
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(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:

   a__f(X, X) -> a__f(a, b) [1]
   a__b -> a [1]
   mark(f(X1, X2)) -> a__f(mark(X1), X2) [1]
   mark(b) -> a__b [1]
   mark(a) -> a [1]
   a__f(X1, X2) -> f(X1, X2) [1]
   a__b -> b [1]

Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

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

   a__f(X, X) -> a__f(a, b) [1]
   a__b -> a [1]
   mark(f(X1, X2)) -> a__f(mark(X1), X2) [1]
   mark(b) -> a__b [1]
   mark(a) -> a [1]
   a__f(X1, X2) -> f(X1, X2) [1]
   a__b -> b [1]

The TRS has the following type information:
a__f :: a:b:f -> a:b:f -> a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f -> a:b:f
f :: a:b:f -> a:b:f -> a:b:f

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: none

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(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:

   a__f(X, X) -> a__f(a, b) [1]
   a__b -> a [1]
   mark(f(X1, X2)) -> a__f(mark(X1), X2) [1]
   mark(b) -> a__b [1]
   mark(a) -> a [1]
   a__f(X1, X2) -> f(X1, X2) [1]
   a__b -> b [1]

The TRS has the following type information:
a__f :: a:b:f -> a:b:f -> a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f -> a:b:f
f :: a:b:f -> a:b:f -> a:b:f

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

   a => 0
   b => 1

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

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

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z') -{ 1 }-> a__f(0, 1) :|: z' = X, X >= 0, z = X
   a__f(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   mark(z) -{ 1 }-> a__f(mark(X1), X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 1 }-> a__b :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 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(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(Out)],[]).
eq(start(V1, V),0,[mark(V1, Out)],[V1 >= 0]).
eq(fun(V1, V, Out),1,[fun(0, 1, Ret)],[Out = Ret,V = X3,X3 >= 0,V1 = X3]).
eq(fun1(Out),1,[],[Out = 0]).
eq(mark(V1, Out),1,[mark(X11, Ret0),fun(Ret0, X21, Ret1)],[Out = Ret1,X11 >= 0,X21 >= 0,V1 = 1 + X11 + X21]).
eq(mark(V1, Out),1,[fun1(Ret2)],[Out = Ret2,V1 = 1]).
eq(mark(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(fun(V1, V, Out),1,[],[Out = 1 + X12 + X22,X12 >= 0,X22 >= 0,V1 = X12,V = X22]).
eq(fun1(Out),1,[],[Out = 1]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(Out),[],[Out]).
input_output_vars(mark(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [fun/3]
1. non_recursive  : [fun1/1]
2. recursive [non_tail] : [mark/2]
3. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into fun1/1
2. SCC is partially evaluated into mark/2
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations fun/3 
* CE 5 is refined into CE [11] 
* CE 4 is refined into CE [12] 


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

### Ranking functions of CR fun(V1,V,Out) 

#### Partial ranking functions of CR fun(V1,V,Out) 


### Specialization of cost equations fun1/1 
* CE 7 is refined into CE [13] 
* CE 6 is refined into CE [14] 


### Cost equations --> "Loop" of fun1/1 
* CEs [13] --> Loop 11 
* CEs [14] --> Loop 12 

### Ranking functions of CR fun1(Out) 

#### Partial ranking functions of CR fun1(Out) 


### Specialization of cost equations mark/2 
* CE 9 is refined into CE [15,16] 
* CE 10 is refined into CE [17] 
* CE 8 is refined into CE [18,19] 


### Cost equations --> "Loop" of mark/2 
* CEs [19] --> Loop 13 
* CEs [18] --> Loop 14 
* CEs [16] --> Loop 15 
* CEs [15] --> Loop 16 
* CEs [17] --> Loop 17 

### Ranking functions of CR mark(V1,Out) 
* RF of phase [13,14]: [V1]

#### Partial ranking functions of CR mark(V1,Out) 
* Partial RF of phase [13,14]:
  - RF of loop [13:1,14:1]:
    V1


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


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

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

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


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

#### Cost of chains of fun(V1,V,Out):
* Chain [10]: 1
  with precondition: [V+V1+1=Out,V1>=0,V>=0] 

* Chain [9,10]: 2
  with precondition: [Out=2,V1=V,V1>=0] 


#### Cost of chains of fun1(Out):
* Chain [12]: 1
  with precondition: [Out=0] 

* Chain [11]: 1
  with precondition: [Out=1] 


#### Cost of chains of mark(V1,Out):
* Chain [[13,14],17]: 5*it(13)+1
  Such that:aux(3) =< V1
it(13) =< aux(3)

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

* Chain [[13,14],16]: 5*it(13)+2
  Such that:aux(4) =< V1
it(13) =< aux(4)

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

* Chain [[13,14],15]: 5*it(13)+2
  Such that:aux(5) =< V1
it(13) =< aux(5)

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

* Chain [17]: 1
  with precondition: [V1=0,Out=0] 

* Chain [16]: 2
  with precondition: [V1=1,Out=0] 

* Chain [15]: 2
  with precondition: [V1=1,Out=1] 


#### Cost of chains of start(V1,V):
* Chain [18]: 15*s(8)+2
  Such that:s(7) =< V1
s(8) =< s(7)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [18] with precondition: [] 
    - Upper bound: nat(V1)*15+2 
    - Complexity: n 

### Maximum cost of start(V1,V): nat(V1)*15+2 
Asymptotic class: n 
* Total analysis performed in 97 ms.


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(12)
BOUNDS(1, n^1)

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(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:

   a__f(X, X) -> a__f(a, b)
   a__b -> a
   mark(f(X1, X2)) -> a__f(mark(X1), X2)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2) -> f(X1, X2)
   a__b -> b

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

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

(16)
Obligation:
TRS:
Rules:
a__f(X, X) -> a__f(a, b)
a__b -> a
mark(f(X1, X2)) -> a__f(mark(X1), X2)
mark(b) -> a__b
mark(a) -> a
a__f(X1, X2) -> f(X1, X2)
a__b -> b

Types:
a__f :: a:b:f -> a:b:f -> a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f -> a:b:f
f :: a:b:f -> a:b:f -> a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat -> a:b:f

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

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

They will be analysed ascendingly in the following order:
a__f < mark

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(18)
Obligation:
TRS:
Rules:
a__f(X, X) -> a__f(a, b)
a__b -> a
mark(f(X1, X2)) -> a__f(mark(X1), X2)
mark(b) -> a__b
mark(a) -> a
a__f(X1, X2) -> f(X1, X2)
a__b -> b

Types:
a__f :: a:b:f -> a:b:f -> a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f -> a:b:f
f :: a:b:f -> a:b:f -> a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat -> a:b:f


Generator Equations:
gen_a:b:f2_0(0) <=> a
gen_a:b:f2_0(+(x, 1)) <=> f(gen_a:b:f2_0(x), a)


The following defined symbols remain to be analysed:
a__f, mark

They will be analysed ascendingly in the following order:
a__f < mark

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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_a:b:f2_0(+(1, n22_0))) -> *3_0, rt in Omega(n22_0)

Induction Base:
mark(gen_a:b:f2_0(+(1, 0)))

Induction Step:
mark(gen_a:b:f2_0(+(1, +(n22_0, 1)))) ->_R^Omega(1)
a__f(mark(gen_a:b:f2_0(+(1, n22_0))), a) ->_IH
a__f(*3_0, a)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(20)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
a__f(X, X) -> a__f(a, b)
a__b -> a
mark(f(X1, X2)) -> a__f(mark(X1), X2)
mark(b) -> a__b
mark(a) -> a
a__f(X1, X2) -> f(X1, X2)
a__b -> b

Types:
a__f :: a:b:f -> a:b:f -> a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f -> a:b:f
f :: a:b:f -> a:b:f -> a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat -> a:b:f


Generator Equations:
gen_a:b:f2_0(0) <=> a
gen_a:b:f2_0(+(x, 1)) <=> f(gen_a:b:f2_0(x), a)


The following defined symbols remain to be analysed:
mark
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(21) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(22)
BOUNDS(n^1, INF)