/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 4 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 129 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), 1885 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 (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)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, 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)) -> +(sum(x), s(x)) [1]
   sum1(0) -> 0 [1]
   sum1(s(x)) -> s(+(sum1(x), +(x, 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)) -> +(sum(x), s(x)) [1]
   sum1(0) -> 0 [1]
   sum1(s(x)) -> s(+(sum1(x), +(x, 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:+ -> 0:s:+
sum1 :: 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]
   sum1(v0) -> null_sum1 [0]

And the following fresh constants:    null_sum, null_sum1

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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)) -> +(sum(x), s(x)) [1]
   sum1(0) -> 0 [1]
   sum1(s(x)) -> s(+(sum1(x), +(x, x))) [1]
   sum(v0) -> null_sum [0]
   sum1(v0) -> null_sum1 [0]

The TRS has the following type information:
sum :: 0:s:+:null_sum:null_sum1 -> 0:s:+:null_sum:null_sum1
0 :: 0:s:+:null_sum:null_sum1
s :: 0:s:+:null_sum:null_sum1 -> 0:s:+:null_sum:null_sum1
+ :: 0:s:+:null_sum:null_sum1 -> 0:s:+:null_sum:null_sum1 -> 0:s:+:null_sum:null_sum1
sum1 :: 0:s:+:null_sum:null_sum1 -> 0:s:+:null_sum:null_sum1
null_sum :: 0:s:+:null_sum:null_sum1
null_sum1 :: 0:s:+:null_sum:null_sum1

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
   null_sum1 => 0

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

   sum(z) -{ 1 }-> 0 :|: z = 0
   sum(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   sum(z) -{ 1 }-> 1 + sum(x) + (1 + x) :|: x >= 0, z = 1 + x
   sum1(z) -{ 1 }-> 0 :|: z = 0
   sum1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   sum1(z) -{ 1 }-> 1 + (1 + sum1(x) + (1 + x + 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,[sum1(V, Out)],[V >= 0]).
eq(sum(V, Out),1,[],[Out = 0,V = 0]).
eq(sum(V, Out),1,[sum(V1, Ret01)],[Out = 2 + Ret01 + V1,V1 >= 0,V = 1 + V1]).
eq(sum1(V, Out),1,[],[Out = 0,V = 0]).
eq(sum1(V, Out),1,[sum1(V2, Ret101)],[Out = 3 + Ret101 + 2*V2,V2 >= 0,V = 1 + V2]).
eq(sum(V, Out),0,[],[Out = 0,V3 >= 0,V = V3]).
eq(sum1(V, Out),0,[],[Out = 0,V4 >= 0,V = V4]).
input_output_vars(sum(V,Out),[V],[Out]).
input_output_vars(sum1(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [sum/2]
1. recursive  : [sum1/2]
2. non_recursive  : [start/1]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into sum/2
1. SCC is partially evaluated into sum1/2
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 [9] 
* CE 5 is refined into CE [10] 
* CE 4 is refined into CE [11] 


### Cost equations --> "Loop" of sum/2 
* CEs [11] --> Loop 6 
* CEs [9,10] --> Loop 7 

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

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


### Specialization of cost equations sum1/2 
* CE 6 is refined into CE [12] 
* CE 8 is refined into CE [13] 
* CE 7 is refined into CE [14] 


### Cost equations --> "Loop" of sum1/2 
* CEs [14] --> Loop 8 
* CEs [12,13] --> Loop 9 

### Ranking functions of CR sum1(V,Out) 
* RF of phase [8]: [V]

#### Partial ranking functions of CR sum1(V,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1]:
    V


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


### Cost equations --> "Loop" of start/1 
* CEs [15,16,17,18] --> Loop 10 

### Ranking functions of CR start(V) 

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


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

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

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

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


#### Cost of chains of sum1(V,Out):
* Chain [[8],9]: 1*it(8)+1
  Such that:it(8) =< V

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

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


#### Cost of chains of start(V):
* Chain [10]: 2*s(1)+1
  Such that:aux(1) =< V
s(1) =< aux(1)

  with precondition: [V>=0] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [10] 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 57 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)) -> +'(sum(x), s(x))
   sum1(0') -> 0'
   sum1(s(x)) -> s(+'(sum1(x), +'(x, x)))

S is empty.
Rewrite Strategy: INNERMOST
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(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
+'/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)) -> +'(sum(x))
   sum1(0') -> 0'
   sum1(s(x)) -> s(+'(sum1(x)))

S is empty.
Rewrite Strategy: INNERMOST
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(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x)))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat -> 0':s:+'

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(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sum, sum1
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(18)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x)))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat -> 0':s:+'


Generator Equations:
gen_0':s:+'2_0(0) <=> 0'
gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))


The following defined symbols remain to be analysed:
sum, sum1
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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_0':s:+'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
sum(gen_0':s:+'2_0(+(1, 0)))

Induction Step:
sum(gen_0':s:+'2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
+'(sum(gen_0':s:+'2_0(+(1, n4_0)))) ->_IH
+'(*3_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)
Complex Obligation (BEST)

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(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x)))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat -> 0':s:+'


Generator Equations:
gen_0':s:+'2_0(0) <=> 0'
gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))


The following defined symbols remain to be analysed:
sum, sum1
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(22) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(23)
BOUNDS(n^1, INF)

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(24)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x))
sum1(0') -> 0'
sum1(s(x)) -> s(+'(sum1(x)))

Types:
sum :: 0':s:+' -> 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+'
sum1 :: 0':s:+' -> 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat -> 0':s:+'


Lemmas:
sum(gen_0':s:+'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_0':s:+'2_0(0) <=> 0'
gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x))


The following defined symbols remain to be analysed:
sum1