/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) CpxWeightedTrsRenamingProof [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), 3 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 461 ms]
    (12) BOUNDS(1, n^2)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
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:

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

Rewrite Strategy: INNERMOST
----------------------------------------

(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus

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

(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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), y) -> f(p(minus(s(x), y)), p(minus(y, s(x)))) [1]
   f(x, s(y)) -> f(p(minus(x, s(y))), p(minus(s(y), x))) [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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), y) -> f(p(minus(s(x), y)), p(minus(y, s(x)))) [1]
   f(x, s(y)) -> f(p(minus(x, s(y))), p(minus(s(y), x))) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
p :: 0:s -> 0:s
f :: 0:s -> 0:s -> 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:

   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_minus, null_p, 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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), y) -> f(p(minus(s(x), y)), p(minus(y, s(x)))) [1]
   f(x, s(y)) -> f(p(minus(x, s(y))), p(minus(s(y), x))) [1]
   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
minus :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p -> 0:s:null_minus:null_p
0 :: 0:s:null_minus:null_p
s :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p
p :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p
f :: 0:s:null_minus:null_p -> 0:s:null_minus:null_p -> null_f
null_minus :: 0:s:null_minus:null_p
null_p :: 0:s:null_minus:null_p
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:

   0 => 0
   null_minus => 0
   null_p => 0
   null_f => 0

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

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

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

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,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[p(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[f(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
eq(minus(V1, V, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = 1 + V3]).
eq(p(V1, Out),1,[],[Out = V5,V5 >= 0,V1 = 1 + V5]).
eq(f(V1, V, Out),1,[minus(1 + V6, V7, Ret00),p(Ret00, Ret0),minus(V7, 1 + V6, Ret10),p(Ret10, Ret1),f(Ret0, Ret1, Ret2)],[Out = Ret2,V6 >= 0,V7 >= 0,V1 = 1 + V6,V = V7]).
eq(f(V1, V, Out),1,[minus(V8, 1 + V9, Ret001),p(Ret001, Ret01),minus(1 + V9, V8, Ret101),p(Ret101, Ret11),f(Ret01, Ret11, Ret3)],[Out = Ret3,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = V8]).
eq(minus(V1, V, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V1 = V11,V = V10]).
eq(p(V1, Out),0,[],[Out = 0,V12 >= 0,V1 = V12]).
eq(f(V1, V, Out),0,[],[Out = 0,V14 >= 0,V13 >= 0,V1 = V14,V = V13]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(p(V1,Out),[V1],[Out]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).


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

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

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

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

### Specialization of cost equations minus/3 
* CE 6 is refined into CE [12] 
* CE 4 is refined into CE [13] 
* CE 5 is refined into CE [14] 


### Cost equations --> "Loop" of minus/3 
* CEs [14] --> Loop 10 
* CEs [12] --> Loop 11 
* CEs [13] --> Loop 12 

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

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


### Specialization of cost equations p/2 
* CE 7 is refined into CE [15] 
* CE 8 is refined into CE [16] 


### Cost equations --> "Loop" of p/2 
* CEs [15] --> Loop 13 
* CEs [16] --> Loop 14 

### Ranking functions of CR p(V1,Out) 

#### Partial ranking functions of CR p(V1,Out) 


### Specialization of cost equations f/3 
* CE 11 is refined into CE [17] 
* CE 9 is refined into CE [18,19,20,21,22,23,24,25] 
* CE 10 is refined into CE [26,27,28,29,30,31,32,33] 


### Cost equations --> "Loop" of f/3 
* CEs [25,33] --> Loop 15 
* CEs [22,30] --> Loop 16 
* CEs [24,32] --> Loop 17 
* CEs [19] --> Loop 18 
* CEs [18,20] --> Loop 19 
* CEs [27] --> Loop 20 
* CEs [21,23,26,28,29,31] --> Loop 21 
* CEs [17] --> Loop 22 

### Ranking functions of CR f(V1,V,Out) 
* RF of phase [18]: [V1]
* RF of phase [20]: [V]

#### Partial ranking functions of CR f(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V1
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V


### Specialization of cost equations start/2 
* CE 1 is refined into CE [34,35,36] 
* CE 2 is refined into CE [37,38] 
* CE 3 is refined into CE [39,40,41] 


### Cost equations --> "Loop" of start/2 
* CEs [41] --> Loop 23 
* CEs [34,35,36,37,38,39,40] --> Loop 24 

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

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


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

#### Cost of chains of minus(V1,V,Out):
* Chain [[10],12]: 1*it(10)+1
  Such that:it(10) =< V

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

* Chain [[10],11]: 1*it(10)+0
  Such that:it(10) =< V

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

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

* Chain [11]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of p(V1,Out):
* Chain [14]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [13]: 1
  with precondition: [V1=Out+1,V1>=1] 


#### Cost of chains of f(V1,V,Out):
* Chain [[20],22]: 3*it(20)+1*s(4)+0
  Such that:aux(3) =< V
it(20) =< aux(3)
s(4) =< it(20)*aux(3)

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

* Chain [[20],21,22]: 9*it(20)+1*s(4)+2
  Such that:aux(6) =< V
it(20) =< aux(6)
s(4) =< it(20)*aux(6)

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

* Chain [[18],22]: 3*it(18)+1*s(18)+0
  Such that:aux(9) =< V1
it(18) =< aux(9)
s(18) =< it(18)*aux(9)

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

* Chain [[18],19,22]: 5*it(18)+1*s(18)+2
  Such that:aux(11) =< V1
it(18) =< aux(11)
s(18) =< it(18)*aux(11)

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

* Chain [22]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [21,22]: 6*s(5)+5*s(6)+2
  Such that:aux(4) =< V1
aux(5) =< V
s(6) =< aux(4)
s(5) =< aux(5)

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

* Chain [19,22]: 2*s(19)+1*s(20)+2
  Such that:aux(10) =< V1
s(20) =< V
s(19) =< aux(10)

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

* Chain [17,22]: 4*s(22)+3
  Such that:aux(14) =< V1
s(22) =< aux(14)

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

* Chain [16,[20],22]: 3*it(20)+1*s(4)+2*s(26)+2*s(27)+3
  Such that:aux(3) =< -V1+V
aux(16) =< V1+1
aux(15) =< V
it(20) =< aux(3)
s(4) =< it(20)*aux(3)
s(26) =< aux(15)
s(27) =< aux(16)

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

* Chain [16,[20],21,22]: 9*it(20)+1*s(4)+2*s(26)+2*s(27)+5
  Such that:aux(6) =< -V1+V
aux(16) =< V1+1
aux(15) =< V
it(20) =< aux(6)
s(4) =< it(20)*aux(6)
s(26) =< aux(15)
s(27) =< aux(16)

  with precondition: [Out=0,V1>=1,V>=V1+3] 

* Chain [16,22]: 2*s(26)+2*s(27)+3
  Such that:aux(16) =< V1+1
aux(15) =< V
s(26) =< aux(15)
s(27) =< aux(16)

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

* Chain [16,21,22]: 6*s(5)+2*s(26)+2*s(27)+5
  Such that:aux(5) =< -V1+V
aux(16) =< V1+1
aux(15) =< V
s(5) =< aux(5)
s(26) =< aux(15)
s(27) =< aux(16)

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

* Chain [15,[18],22]: 3*it(18)+1*s(18)+2*s(30)+2*s(31)+3
  Such that:aux(18) =< V1
aux(9) =< V1-V
aux(17) =< V
it(18) =< aux(9)
s(18) =< it(18)*aux(9)
s(30) =< aux(17)
s(31) =< aux(18)

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

* Chain [15,[18],19,22]: 5*it(18)+1*s(18)+2*s(30)+2*s(31)+5
  Such that:aux(18) =< V1
aux(11) =< V1-V
aux(17) =< V
it(18) =< aux(11)
s(18) =< it(18)*aux(11)
s(30) =< aux(17)
s(31) =< aux(18)

  with precondition: [Out=0,V>=1,V1>=V+3] 

* Chain [15,22]: 2*s(30)+2*s(31)+3
  Such that:aux(18) =< V1
aux(17) =< V
s(30) =< aux(17)
s(31) =< aux(18)

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

* Chain [15,19,22]: 2*s(19)+2*s(30)+2*s(31)+5
  Such that:aux(18) =< V1
aux(10) =< V1-V
aux(17) =< V
s(19) =< aux(10)
s(30) =< aux(17)
s(31) =< aux(18)

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


#### Cost of chains of start(V1,V):
* Chain [24]: 37*s(101)+18*s(109)+8*s(110)+2*s(111)+10*s(112)+23*s(113)+2*s(114)+2*s(115)+2*s(118)+5
  Such that:s(103) =< -V1+V
s(105) =< V1+1
s(106) =< V1-V
aux(25) =< V1
aux(26) =< V
s(101) =< aux(26)
s(109) =< s(103)
s(110) =< s(105)
s(111) =< s(109)*s(103)
s(112) =< s(106)
s(113) =< aux(25)
s(114) =< s(112)*s(106)
s(115) =< s(101)*aux(26)
s(118) =< s(113)*aux(25)

  with precondition: [V1>=0] 

* Chain [23]: 4*s(120)+3
  Such that:s(119) =< V
s(120) =< s(119)

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


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [24] with precondition: [V1>=0] 
    - Upper bound: 23*V1+5+2*V1*V1+nat(V)*37+nat(V)*2*nat(V)+(8*V1+8)+nat(-V1+V)*18+nat(-V1+V)*2*nat(-V1+V)+nat(V1-V)*10+nat(V1-V)*2*nat(V1-V) 
    - Complexity: n^2 
* Chain [23] with precondition: [V1=V,V1>=1] 
    - Upper bound: 4*V+3 
    - Complexity: n 

### Maximum cost of start(V1,V): 23*V1+2+2*V1*V1+nat(V)*33+nat(V)*2*nat(V)+(8*V1+8)+nat(-V1+V)*18+nat(-V1+V)*2*nat(-V1+V)+nat(V1-V)*10+nat(V1-V)*2*nat(V1-V)+(nat(V)*4+3) 
Asymptotic class: n^2 
* Total analysis performed in 367 ms.


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

(12)
BOUNDS(1, n^2)

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

(15) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

-(s(x), s(y)) ->^+ -(x, y)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x / s(x), y / s(y)].

The result substitution is [ ].




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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

S is empty.
Rewrite Strategy: INNERMOST