/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?, O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 172 ms]
(4) CpxRelTRS
(5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(6) CpxRelTRS
(7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
(9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxWeightedTrs
(11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedTrs
(13) CompletionProof [UPPER BOUND(ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) CompleteCoflocoProof [FINISHED, 329 ms]
(18) BOUNDS(1, n^2)


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

(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   .(.(x, y), z) -> .(x, .(y, z))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   .(.(x, y), z) -> .(x, .(y, z))

The (relative) TRS S consists of the following rules:

   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   .(.(x, y), z) -> .(x, .(y, z))

The (relative) TRS S consists of the following rules:

   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
----------------------------------------

(5) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   .(c_.(x, y), z) -> .(x, .(y, z))

The (relative) TRS S consists of the following rules:

   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   .(x0, x1) -> c_.(x0, x1)

Rewrite Strategy: FULL
----------------------------------------

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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   .(c_.(x, y), z) -> .(x, .(y, z))

The (relative) TRS S consists of the following rules:

   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   .(x0, x1) -> c_.(x0, x1)

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

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

(10)
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:

   .(c_.(x, y), z) -> .(x, .(y, z)) [1]
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   .(x0, x1) -> c_.(x0, x1) [0]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   .(c_.(x, y), z) -> .(x, .(y, z)) [1]
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   .(x0, x1) -> c_.(x0, x1) [0]

The TRS has the following type information:
. :: c_. -> c_. -> c_.
c_. :: c_. -> c_. -> c_.
encArg :: cons_. -> c_.
cons_. :: cons_. -> cons_. -> cons_.
encode_. :: cons_. -> cons_. -> c_.

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

(13) 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:

   encArg(v0) -> null_encArg [0]
   encode_.(v0, v1) -> null_encode_. [0]
   .(v0, v1) -> null_. [0]

And the following fresh constants:    null_encArg, null_encode_., null_., const

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

(14)
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:

   .(c_.(x, y), z) -> .(x, .(y, z)) [1]
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2)) [0]
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2)) [0]
   .(x0, x1) -> c_.(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_.(v0, v1) -> null_encode_. [0]
   .(v0, v1) -> null_. [0]

The TRS has the following type information:
. :: c_.:null_encArg:null_encode_.:null_. -> c_.:null_encArg:null_encode_.:null_. -> c_.:null_encArg:null_encode_.:null_.
c_. :: c_.:null_encArg:null_encode_.:null_. -> c_.:null_encArg:null_encode_.:null_. -> c_.:null_encArg:null_encode_.:null_.
encArg :: cons_. -> c_.:null_encArg:null_encode_.:null_.
cons_. :: cons_. -> cons_. -> cons_.
encode_. :: cons_. -> cons_. -> c_.:null_encArg:null_encode_.:null_.
null_encArg :: c_.:null_encArg:null_encode_.:null_.
null_encode_. :: c_.:null_encArg:null_encode_.:null_.
null_. :: c_.:null_encArg:null_encode_.:null_.
const :: cons_.

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

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

   null_encArg => 0
   null_encode_. => 0
   null_. => 0
   const => 0

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   .(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   .(z', z'') -{ 1 }-> .(x, .(y, z)) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
   .(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encArg(z') -{ 0 }-> .(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_.(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_.(z', z'') -{ 0 }-> .(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V2),0,[fun(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(start(V, V2),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V2),0,[fun1(V, V2, Out)],[V >= 0,V2 >= 0]).
eq(fun(V, V2, Out),1,[fun(V3, V4, Ret1),fun(V1, Ret1, Ret)],[Out = Ret,V2 = V4,V4 >= 0,V = 1 + V1 + V3,V1 >= 0,V3 >= 0]).
eq(encArg(V, Out),0,[encArg(V6, Ret0),encArg(V5, Ret11),fun(Ret0, Ret11, Ret2)],[Out = Ret2,V6 >= 0,V5 >= 0,V = 1 + V5 + V6]).
eq(fun1(V, V2, Out),0,[encArg(V7, Ret01),encArg(V8, Ret12),fun(Ret01, Ret12, Ret3)],[Out = Ret3,V7 >= 0,V = V7,V8 >= 0,V2 = V8]).
eq(fun(V, V2, Out),0,[],[Out = 1 + V10 + V9,V2 = V9,V10 >= 0,V9 >= 0,V = V10]).
eq(encArg(V, Out),0,[],[Out = 0,V11 >= 0,V = V11]).
eq(fun1(V, V2, Out),0,[],[Out = 0,V13 >= 0,V12 >= 0,V2 = V12,V = V13]).
eq(fun(V, V2, Out),0,[],[Out = 0,V15 >= 0,V14 >= 0,V2 = V14,V = V15]).
input_output_vars(fun(V,V2,Out),[V,V2],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun1(V,V2,Out),[V,V2],[Out]).


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

#### Computed strongly connected components 
0. recursive [multiple] : [fun/3]
1. recursive [non_tail,multiple] : [encArg/2]
2. non_recursive  : [fun1/3]
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 encArg/2
2. SCC is partially evaluated into fun1/3
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 6 is refined into CE [12] 
* CE 4 is refined into CE [13] 


### Cost equations --> "Loop" of fun/3 
* CEs [13] --> Loop 8 
* CEs [11] --> Loop 9 
* CEs [12] --> Loop 10 

### Ranking functions of CR fun(V,V2,Out) 
* RF of phase [8]: [V]

#### Partial ranking functions of CR fun(V,V2,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1,8:2]:
    V


### Specialization of cost equations encArg/2 
* CE 8 is refined into CE [14] 
* CE 7 is refined into CE [15,16,17] 


### Cost equations --> "Loop" of encArg/2 
* CEs [17] --> Loop 11 
* CEs [16] --> Loop 12 
* CEs [15] --> Loop 13 
* CEs [14] --> Loop 14 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [11,12,13]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [11,12,13]:
  - RF of loop [11:1,11:2,12:1,12:2,13:1,13:2]:
    V


### Specialization of cost equations fun1/3 
* CE 9 is refined into CE [18,19,20,21,22,23,24,25,26,27] 
* CE 10 is refined into CE [28] 


### Cost equations --> "Loop" of fun1/3 
* CEs [26,27] --> Loop 15 
* CEs [23,24] --> Loop 16 
* CEs [21] --> Loop 17 
* CEs [19] --> Loop 18 
* CEs [18,20,22,25,28] --> Loop 19 

### Ranking functions of CR fun1(V,V2,Out) 

#### Partial ranking functions of CR fun1(V,V2,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [29,30,31] 
* CE 2 is refined into CE [32,33] 
* CE 3 is refined into CE [34,35,36,37,38] 


### Cost equations --> "Loop" of start/2 
* CEs [29,30,31,32,33,34,35,36,37,38] --> Loop 20 

### Ranking functions of CR start(V,V2) 

#### Partial ranking functions of CR start(V,V2) 


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

#### Cost of chains of fun(V,V2,Out):
* Chain [10]: 0
  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [9]: 0
  with precondition: [V+V2+1=Out,V>=0,V2>=0] 

* Chain [multiple([8],[[10],[9]])]: 1*it(8)+0
  Such that:it(8) =< V

  with precondition: [V>=1,V2>=0,Out>=0,V+V2+1>=Out] 


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

* Chain [multiple([11,12,13],[[14]])]: 1*s(3)+0
  Such that:aux(3) =< V
aux(4) =< 2*V+1
it(11) =< aux(4)
s(3) =< it(11)*aux(3)

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


#### Cost of chains of fun1(V,V2,Out):
* Chain [19]: 2*s(7)+2*s(11)+0
  Such that:aux(5) =< V
aux(6) =< 2*V+1
aux(7) =< V2
aux(8) =< 2*V2+1
s(11) =< aux(6)*aux(5)
s(7) =< aux(8)*aux(7)

  with precondition: [Out=0,V>=0,V2>=0] 

* Chain [18]: 0
  with precondition: [Out=1,V>=0,V2>=0] 

* Chain [17]: 1*s(23)+0
  Such that:s(20) =< V2
s(21) =< 2*V2+1
s(23) =< s(21)*s(20)

  with precondition: [V>=0,V2>=1,Out>=1,V2+1>=Out] 

* Chain [16]: 2*s(27)+1*s(32)+0
  Such that:aux(10) =< V
aux(11) =< 2*V+1
s(27) =< aux(11)*aux(10)
s(32) =< aux(10)

  with precondition: [V>=1,V2>=0,Out>=0,V+1>=Out] 

* Chain [15]: 2*s(36)+2*s(40)+1*s(49)+0
  Such that:aux(13) =< V
aux(14) =< 2*V+1
aux(15) =< V2
aux(16) =< 2*V2+1
s(40) =< aux(16)*aux(15)
s(36) =< aux(14)*aux(13)
s(49) =< aux(13)

  with precondition: [V>=1,V2>=1,Out>=0,V+V2+1>=Out] 


#### Cost of chains of start(V,V2):
* Chain [20]: 3*s(50)+7*s(54)+5*s(60)+0
  Such that:aux(17) =< V
aux(18) =< 2*V+1
aux(19) =< V2
aux(20) =< 2*V2+1
s(50) =< aux(17)
s(54) =< aux(18)*aux(17)
s(60) =< aux(20)*aux(19)

  with precondition: [V>=0] 


Closed-form bounds of start(V,V2): 
-------------------------------------
* Chain [20] with precondition: [V>=0] 
    - Upper bound: (2*V+1)*(7*V)+3*V+nat(V2)*5*nat(2*V2+1) 
    - Complexity: n^2 

### Maximum cost of start(V,V2): (2*V+1)*(7*V)+3*V+nat(V2)*5*nat(2*V2+1) 
Asymptotic class: n^2 
* Total analysis performed in 260 ms.


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(18)
BOUNDS(1, n^2)