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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 34 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), 746 ms]
    (20) proven lower bound
    (21) LowerBoundPropagationProof [FINISHED, 0 ms]
    (22) BOUNDS(n^1, INF)


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(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:

   sum(0) -> 0
   sum(s(x)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(x))

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(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:

   sum(0) -> 0 [1]
   sum(s(x)) -> +(sqr(s(x)), sum(x)) [1]
   sqr(x) -> *(x, x) [1]
   sum(s(x)) -> +(*(s(x), s(x)), sum(x)) [1]

Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   sum(0) -> 0 [1]
   sum(s(x)) -> +(sqr(s(x)), sum(x)) [1]
   sqr(x) -> *(x, x) [1]
   sum(s(x)) -> +(*(s(x), s(x)), sum(x)) [1]

The TRS has the following type information:
sum :: 0:s:+ -> 0:s:+
0 :: 0:s:+
s :: 0:s:+ -> 0:s:+
+ :: * -> 0:s:+ -> 0:s:+
sqr :: 0:s:+ -> *
* :: 0:s:+ -> 0:s:+ -> *

Rewrite Strategy: INNERMOST
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(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:

   sum(v0) -> null_sum [0]

And the following fresh constants:    null_sum, const

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(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:

   sum(0) -> 0 [1]
   sum(s(x)) -> +(sqr(s(x)), sum(x)) [1]
   sqr(x) -> *(x, x) [1]
   sum(s(x)) -> +(*(s(x), s(x)), sum(x)) [1]
   sum(v0) -> null_sum [0]

The TRS has the following type information:
sum :: 0:s:+:null_sum -> 0:s:+:null_sum
0 :: 0:s:+:null_sum
s :: 0:s:+:null_sum -> 0:s:+:null_sum
+ :: * -> 0:s:+:null_sum -> 0:s:+:null_sum
sqr :: 0:s:+:null_sum -> *
* :: 0:s:+:null_sum -> 0:s:+:null_sum -> *
null_sum :: 0:s:+:null_sum
const :: *

Rewrite Strategy: INNERMOST
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(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_sum => 0
   const => 0

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

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

Only complete derivations are relevant for the runtime complexity.

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(9) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V),0,[sum(V, Out)],[V >= 0]).
eq(start(V),0,[sqr(V, Out)],[V >= 0]).
eq(sum(V, Out),1,[],[Out = 0,V = 0]).
eq(sum(V, Out),1,[sqr(1 + V1, Ret01),sum(V1, Ret1)],[Out = 1 + Ret01 + Ret1,V1 >= 0,V = 1 + V1]).
eq(sqr(V, Out),1,[],[Out = 1 + 2*V2,V2 >= 0,V = V2]).
eq(sum(V, Out),1,[sum(V3, Ret11)],[Out = 4 + Ret11 + 2*V3,V3 >= 0,V = 1 + V3]).
eq(sum(V, Out),0,[],[Out = 0,V4 >= 0,V = V4]).
input_output_vars(sum(V,Out),[V],[Out]).
input_output_vars(sqr(V,Out),[V],[Out]).


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

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

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

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

### Specialization of cost equations sum/2 
* CE 3 is refined into CE [7] 
* CE 6 is refined into CE [8] 
* CE 4 is refined into CE [9] 
* CE 5 is refined into CE [10] 


### Cost equations --> "Loop" of sum/2 
* CEs [9,10] --> Loop 4 
* CEs [7,8] --> Loop 5 

### Ranking functions of CR sum(V,Out) 
* RF of phase [4]: [V]

#### Partial ranking functions of CR sum(V,Out) 
* Partial RF of phase [4]:
  - RF of loop [4:1]:
    V


### Specialization of cost equations start/1 
* CE 1 is refined into CE [11,12] 
* CE 2 is refined into CE [13] 


### Cost equations --> "Loop" of start/1 
* CEs [11,12,13] --> Loop 6 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of sum(V,Out):
* Chain [[4],5]: 2*it(4)+1
  Such that:it(4) =< V

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

* Chain [5]: 1
  with precondition: [Out=0,V>=0] 


#### Cost of chains of start(V):
* Chain [6]: 2*s(1)+1
  Such that:s(1) =< V

  with precondition: [V>=0] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [6] with precondition: [V>=0] 
    - Upper bound: 2*V+1 
    - Complexity: n 

### Maximum cost of start(V): 2*V+1 
Asymptotic class: n 
* Total analysis performed in 36 ms.


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(10)
BOUNDS(1, n^1)

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

   sum(0') -> 0'
   sum(s(x)) -> +'(sqr(s(x)), sum(x))
   sqr(x) -> *'(x, x)
   sum(s(x)) -> +'(*'(s(x), s(x)), sum(x))

S is empty.
Rewrite Strategy: INNERMOST
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(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
sqr/0
*'/0
*'/1

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(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:

   sum(0') -> 0'
   sum(s(x)) -> +'(sqr, sum(x))
   sqr -> *'
   sum(s(x)) -> +'(*', sum(x))

S is empty.
Rewrite Strategy: INNERMOST
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(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(16)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sqr, sum(x))
sqr -> *'
sum(s(x)) -> +'(*', sum(x))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: *'
*' :: *'
hole_0':s:+'1_0 :: 0':s:+'
hole_*'2_0 :: *'
gen_0':s:+'3_0 :: Nat -> 0':s:+'

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(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sum
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(18)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sqr, sum(x))
sqr -> *'
sum(s(x)) -> +'(*', sum(x))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: *'
*' :: *'
hole_0':s:+'1_0 :: 0':s:+'
hole_*'2_0 :: *'
gen_0':s:+'3_0 :: Nat -> 0':s:+'


Generator Equations:
gen_0':s:+'3_0(0) <=> 0'
gen_0':s:+'3_0(+(x, 1)) <=> s(gen_0':s:+'3_0(x))


The following defined symbols remain to be analysed:
sum
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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_0':s:+'3_0(n5_0)) -> *4_0, rt in Omega(n5_0)

Induction Base:
sum(gen_0':s:+'3_0(0))

Induction Step:
sum(gen_0':s:+'3_0(+(n5_0, 1))) ->_R^Omega(1)
+'(*', sum(gen_0':s:+'3_0(n5_0))) ->_IH
+'(*', *4_0)

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:

Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sqr, sum(x))
sqr -> *'
sum(s(x)) -> +'(*', sum(x))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: *' -> 0':s:+' -> 0':s:+'
sqr :: *'
*' :: *'
hole_0':s:+'1_0 :: 0':s:+'
hole_*'2_0 :: *'
gen_0':s:+'3_0 :: Nat -> 0':s:+'


Generator Equations:
gen_0':s:+'3_0(0) <=> 0'
gen_0':s:+'3_0(+(x, 1)) <=> s(gen_0':s:+'3_0(x))


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