/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 200 ms]
    (14) BOUNDS(1, n^1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 6178 ms]
    (22) proven lower bound
    (23) LowerBoundPropagationProof [FINISHED, 0 ms]
    (24) 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:

   f(a) -> b
   f(c) -> d
   f(g(x, y)) -> g(f(x), f(y))
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
   g(x, x) -> h(e, x)

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of g: f, g
The following defined symbols can occur below the 1th argument of g: f, g

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(g(x, y)) -> g(f(x), f(y))

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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(a) -> b
   f(c) -> d
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
   g(x, x) -> h(e, x)

S is empty.
Rewrite Strategy: FULL
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(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

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


The defined contexts are:
g(h(x0, []), h(x2, x3))
g(h(x0, x1), h(x2, []))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
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(4)
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:

   f(a) -> b
   f(c) -> d
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
   g(x, x) -> h(e, x)

S is empty.
Rewrite Strategy: INNERMOST
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(5) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(6)
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:

   f(a) -> b [1]
   f(c) -> d [1]
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y))) [1]
   g(x, x) -> h(e, x) [1]

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(a) -> b [1]
   f(c) -> d [1]
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y))) [1]
   g(x, x) -> h(e, x) [1]

The TRS has the following type information:
f :: a:b:c:d:h:e -> a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
h :: a:b:c:d:h:e -> a:b:c:d:h:e -> a:b:c:d:h:e
g :: a:b:c:d:h:e -> a:b:c:d:h:e -> a:b:c:d:h:e
e :: a:b:c:d:h:e

Rewrite Strategy: INNERMOST
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(9) 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:

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

And the following fresh constants:    null_f, null_g

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(10)
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(a) -> b [1]
   f(c) -> d [1]
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y))) [1]
   g(x, x) -> h(e, x) [1]
   f(v0) -> null_f [0]
   g(v0, v1) -> null_g [0]

The TRS has the following type information:
f :: a:b:c:d:h:e:null_f:null_g -> a:b:c:d:h:e:null_f:null_g
a :: a:b:c:d:h:e:null_f:null_g
b :: a:b:c:d:h:e:null_f:null_g
c :: a:b:c:d:h:e:null_f:null_g
d :: a:b:c:d:h:e:null_f:null_g
h :: a:b:c:d:h:e:null_f:null_g -> a:b:c:d:h:e:null_f:null_g -> a:b:c:d:h:e:null_f:null_g
g :: a:b:c:d:h:e:null_f:null_g -> a:b:c:d:h:e:null_f:null_g -> a:b:c:d:h:e:null_f:null_g
e :: a:b:c:d:h:e:null_f:null_g
null_f :: a:b:c:d:h:e:null_f:null_g
null_g :: a:b:c:d:h:e:null_f:null_g

Rewrite Strategy: INNERMOST
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(11) 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
   c => 2
   d => 3
   e => 4
   null_f => 0
   null_g => 0

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(12)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 1 }-> g(1 + y + f(x), 1 + x + f(y)) :|: z = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 1 }-> 3 :|: z = 2
   f(z) -{ 1 }-> 1 :|: z = 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z, z') -{ 1 }-> 1 + 4 + x :|: z' = x, x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

eq(start(V, V3),0,[f(V, Out)],[V >= 0]).
eq(start(V, V3),0,[g(V, V3, Out)],[V >= 0,V3 >= 0]).
eq(f(V, Out),1,[],[Out = 1,V = 0]).
eq(f(V, Out),1,[],[Out = 3,V = 2]).
eq(f(V, Out),1,[f(V2, Ret01),f(V1, Ret11),g(1 + V1 + Ret01, 1 + V2 + Ret11, Ret)],[Out = Ret,V = 1 + V1 + V2,V2 >= 0,V1 >= 0]).
eq(g(V, V3, Out),1,[],[Out = 5 + V4,V3 = V4,V4 >= 0,V = V4]).
eq(f(V, Out),0,[],[Out = 0,V5 >= 0,V = V5]).
eq(g(V, V3, Out),0,[],[Out = 0,V7 >= 0,V6 >= 0,V = V7,V3 = V6]).
input_output_vars(f(V,Out),[V],[Out]).
input_output_vars(g(V,V3,Out),[V,V3],[Out]).


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

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

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

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

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


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

### Ranking functions of CR g(V,V3,Out) 

#### Partial ranking functions of CR g(V,V3,Out) 


### Specialization of cost equations f/2 
* CE 6 is refined into CE [11] 
* CE 4 is refined into CE [12] 
* CE 3 is refined into CE [13] 
* CE 5 is refined into CE [14,15] 


### Cost equations --> "Loop" of f/2 
* CEs [15] --> Loop 10 
* CEs [14] --> Loop 11 
* CEs [11] --> Loop 12 
* CEs [12] --> Loop 13 
* CEs [13] --> Loop 14 

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

#### Partial ranking functions of CR f(V,Out) 
* Partial RF of phase [10,11]:
  - RF of loop [10:1,10:2,11:1,11:2]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [16,17,18] 
* CE 2 is refined into CE [19,20] 


### Cost equations --> "Loop" of start/2 
* CEs [20] --> Loop 15 
* CEs [16,17,18,19] --> Loop 16 

### Ranking functions of CR start(V,V3) 

#### Partial ranking functions of CR start(V,V3) 


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

#### Cost of chains of g(V,V3,Out):
* Chain [9]: 0
  with precondition: [Out=0,V>=0,V3>=0] 

* Chain [8]: 1
  with precondition: [V=V3,V+5=Out,V>=0] 


#### Cost of chains of f(V,Out):
* Chain [14]: 1
  with precondition: [V=0,Out=1] 

* Chain [13]: 1
  with precondition: [V=2,Out=3] 

* Chain [12]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([10,11],[[14],[13],[12]])]: 2*it(10)+1*it(11)+1*it([13])+1*it([14])+0
  Such that:it([13]) =< V/3+1/3
aux(1) =< V+1
aux(2) =< 10/9*V+1/9
aux(3) =< 12/11*V+1/11
it([13]) =< aux(1)
it([14]) =< aux(1)
it(10) =< aux(2)
it(11) =< aux(2)
it([13]) =< aux(2)
it(11) =< aux(3)
it([13]) =< aux(3)

  with precondition: [V>=1,Out>=0,7*V+7>=2*Out] 


#### Cost of chains of start(V,V3):
* Chain [16]: 1*s(9)+1*s(12)+2*s(13)+1*s(14)+1
  Such that:s(8) =< V+1
s(9) =< V/3+1/3
s(10) =< 10/9*V+1/9
s(11) =< 12/11*V+1/11
s(9) =< s(8)
s(12) =< s(8)
s(13) =< s(10)
s(14) =< s(10)
s(9) =< s(10)
s(14) =< s(11)
s(9) =< s(11)

  with precondition: [V>=0] 

* Chain [15]: 1
  with precondition: [V=V3,V>=0] 


Closed-form bounds of start(V,V3): 
-------------------------------------
* Chain [16] with precondition: [V>=0] 
    - Upper bound: 14/3*V+8/3 
    - Complexity: n 
* Chain [15] with precondition: [V=V3,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V,V3): 14/3*V+8/3 
Asymptotic class: n 
* Total analysis performed in 128 ms.


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(14)
BOUNDS(1, n^1)

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(15) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(16)
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(a) -> b
   f(c) -> d
   f(g(x, y)) -> g(f(x), f(y))
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
   g(x, x) -> h(e, x)

S is empty.
Rewrite Strategy: FULL
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(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(18)
Obligation:
TRS:
Rules:
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)

Types:
f :: a:b:c:d:h:e -> a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e -> a:b:c:d:h:e -> a:b:c:d:h:e
h :: a:b:c:d:h:e -> a:b:c:d:h:e -> a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat -> a:b:c:d:h:e

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(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f
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(20)
Obligation:
TRS:
Rules:
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)

Types:
f :: a:b:c:d:h:e -> a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e -> a:b:c:d:h:e -> a:b:c:d:h:e
h :: a:b:c:d:h:e -> a:b:c:d:h:e -> a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat -> a:b:c:d:h:e


Generator Equations:
gen_a:b:c:d:h:e2_0(0) <=> a
gen_a:b:c:d:h:e2_0(+(x, 1)) <=> h(a, gen_a:b:c:d:h:e2_0(x))


The following defined symbols remain to be analysed:
f
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_a:b:c:d:h:e2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
f(gen_a:b:c:d:h:e2_0(0))

Induction Step:
f(gen_a:b:c:d:h:e2_0(+(n4_0, 1))) ->_R^Omega(1)
g(h(gen_a:b:c:d:h:e2_0(n4_0), f(a)), h(a, f(gen_a:b:c:d:h:e2_0(n4_0)))) ->_R^Omega(1)
g(h(gen_a:b:c:d:h:e2_0(n4_0), b), h(a, f(gen_a:b:c:d:h:e2_0(n4_0)))) ->_IH
g(h(gen_a:b:c:d:h:e2_0(n4_0), b), h(a, *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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(22)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)

Types:
f :: a:b:c:d:h:e -> a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e -> a:b:c:d:h:e -> a:b:c:d:h:e
h :: a:b:c:d:h:e -> a:b:c:d:h:e -> a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat -> a:b:c:d:h:e


Generator Equations:
gen_a:b:c:d:h:e2_0(0) <=> a
gen_a:b:c:d:h:e2_0(+(x, 1)) <=> h(a, gen_a:b:c:d:h:e2_0(x))


The following defined symbols remain to be analysed:
f
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(23) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(24)
BOUNDS(n^1, INF)