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


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


WORST_CASE(?, O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 158 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, 413 ms]
(18) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   f(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x)))))))))

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(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b


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

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


The TRS R consists of the following rules:

   f(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x)))))))))

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b

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^1).


The TRS R consists of the following rules:

   f(a, f(a, f(b, f(a, f(a, f(b, f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x)))))))))

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b

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^1).


The TRS R consists of the following rules:

   f(a, c_f(a, c_f(b, c_f(a, c_f(a, c_f(b, c_f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x)))))))))

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b
   f(x0, x1) -> c_f(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^1).


The TRS R consists of the following rules:

   f(a, c_f(a, c_f(b, c_f(a, c_f(a, c_f(b, c_f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x)))))))))

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b
   f(x0, x1) -> c_f(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^1).


The TRS R consists of the following rules:

   f(a, c_f(a, c_f(b, c_f(a, c_f(a, c_f(b, c_f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))))) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   f(x0, x1) -> c_f(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:

   f(a, c_f(a, c_f(b, c_f(a, c_f(a, c_f(b, c_f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))))) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   f(x0, x1) -> c_f(x0, x1) [0]

The TRS has the following type information:
f :: a:b:c_f:cons_f -> a:b:c_f:cons_f -> a:b:c_f:cons_f
a :: a:b:c_f:cons_f
c_f :: a:b:c_f:cons_f -> a:b:c_f:cons_f -> a:b:c_f:cons_f
b :: a:b:c_f:cons_f
encArg :: a:b:c_f:cons_f -> a:b:c_f:cons_f
cons_f :: a:b:c_f:cons_f -> a:b:c_f:cons_f -> a:b:c_f:cons_f
encode_f :: a:b:c_f:cons_f -> a:b:c_f:cons_f -> a:b:c_f:cons_f
encode_a :: a:b:c_f:cons_f
encode_b :: a:b:c_f:cons_f

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_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_a, null_encode_b, null_f

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

(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:

   f(a, c_f(a, c_f(b, c_f(a, c_f(a, c_f(b, c_f(a, x))))))) -> f(a, f(b, f(a, f(a, f(b, f(a, f(a, f(a, f(b, x))))))))) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   f(x0, x1) -> c_f(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
a :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
c_f :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
b :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
encArg :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
cons_f :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
encode_f :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f -> a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
encode_a :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
encode_b :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
null_encArg :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
null_encode_f :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
null_encode_a :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
null_encode_b :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f
null_f :: a:b:c_f:cons_f:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_f

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:

   a => 0
   b => 1
   null_encArg => 0
   null_encode_f => 0
   null_encode_a => 0
   null_encode_b => 0
   null_f => 0

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

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

   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 0 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 1 }-> f(0, f(1, f(0, f(0, f(1, f(0, f(0, f(0, f(1, x))))))))) :|: z' = 1 + 0 + (1 + 1 + (1 + 0 + (1 + 0 + (1 + 1 + (1 + 0 + x))))), x >= 0, z = 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1

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(V1, V),0,[f(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(Out)],[]).
eq(start(V1, V),0,[fun2(Out)],[]).
eq(f(V1, V, Out),1,[f(1, V2, Ret11111111),f(0, Ret11111111, Ret1111111),f(0, Ret1111111, Ret111111),f(0, Ret111111, Ret11111),f(1, Ret11111, Ret1111),f(0, Ret1111, Ret111),f(0, Ret111, Ret11),f(1, Ret11, Ret1),f(0, Ret1, Ret)],[Out = Ret,V = 8 + V2,V2 >= 0,V1 = 0]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[],[Out = 1,V1 = 1]).
eq(encArg(V1, Out),0,[encArg(V4, Ret0),encArg(V3, Ret12),f(Ret0, Ret12, Ret2)],[Out = Ret2,V4 >= 0,V1 = 1 + V3 + V4,V3 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V5, Ret01),encArg(V6, Ret13),f(Ret01, Ret13, Ret3)],[Out = Ret3,V5 >= 0,V6 >= 0,V1 = V5,V = V6]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(Out),0,[],[Out = 1]).
eq(f(V1, V, Out),0,[],[Out = 1 + V7 + V8,V1 = V8,V8 >= 0,V7 >= 0,V = V7]).
eq(encArg(V1, Out),0,[],[Out = 0,V9 >= 0,V1 = V9]).
eq(fun(V1, V, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V1 = V11,V = V10]).
eq(fun2(Out),0,[],[Out = 0]).
eq(f(V1, V, Out),0,[],[Out = 0,V13 >= 0,V12 >= 0,V1 = V13,V = V12]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(Out),[],[Out]).
input_output_vars(fun2(Out),[],[Out]).


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

#### Computed strongly connected components 
0. recursive [multiple] : [f/3]
1. recursive [non_tail,multiple] : [encArg/2]
2. non_recursive  : [fun/3]
3. non_recursive  : [fun1/1]
4. non_recursive  : [fun2/1]
5. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into f/3
1. SCC is partially evaluated into encArg/2
2. SCC is partially evaluated into fun/3
3. SCC is completely evaluated into other SCCs
4. SCC is partially evaluated into fun2/1
5. SCC is partially evaluated into start/2

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

### Specialization of cost equations f/3 
* CE 7 is refined into CE [16] 
* CE 8 is refined into CE [17] 
* CE 6 is refined into CE [18] 


### Cost equations --> "Loop" of f/3 
* CEs [18] --> Loop 11 
* CEs [16] --> Loop 12 
* CEs [17] --> Loop 13 

### Ranking functions of CR f(V1,V,Out) 

#### Partial ranking functions of CR f(V1,V,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V/8-7/8 depends on loops [11:2,11:3,11:4,11:5,11:6,11:7,11:8,11:9] 
  - RF of loop [11:1,11:5,11:8]:
    -V1+1


### Specialization of cost equations encArg/2 
* CE 9 is refined into CE [19] 
* CE 10 is refined into CE [20] 
* CE 11 is refined into CE [21,22,23] 


### Cost equations --> "Loop" of encArg/2 
* CEs [23] --> Loop 14 
* CEs [22] --> Loop 15 
* CEs [21] --> Loop 16 
* CEs [19] --> Loop 17 
* CEs [20] --> Loop 18 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [14,15,16]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [14,15,16]:
  - RF of loop [14:1,14:2,15:1,15:2,16:1,16:2]:
    V1


### Specialization of cost equations fun/3 
* CE 12 is refined into CE [24,25,26,27,28,29,30,31,32,33] 
* CE 13 is refined into CE [34] 


### Cost equations --> "Loop" of fun/3 
* CEs [28] --> Loop 19 
* CEs [25,30] --> Loop 20 
* CEs [33] --> Loop 21 
* CEs [24,27,29,32,34] --> Loop 22 
* CEs [26,31] --> Loop 23 

### Ranking functions of CR fun(V1,V,Out) 

#### Partial ranking functions of CR fun(V1,V,Out) 


### Specialization of cost equations fun2/1 
* CE 14 is refined into CE [35] 
* CE 15 is refined into CE [36] 


### Cost equations --> "Loop" of fun2/1 
* CEs [35] --> Loop 24 
* CEs [36] --> Loop 25 

### Ranking functions of CR fun2(Out) 

#### Partial ranking functions of CR fun2(Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [37,38,39] 
* CE 2 is refined into CE [40,41] 
* CE 3 is refined into CE [42,43,44,45,46] 
* CE 4 is refined into CE [47] 
* CE 5 is refined into CE [48,49] 


### Cost equations --> "Loop" of start/2 
* CEs [37,38,40,41,42,43,44,45,47,48,49] --> Loop 26 
* CEs [39,46] --> Loop 27 

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

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


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

#### Cost of chains of f(V1,V,Out):
* Chain [multiple([11],[[],[13],[12]])]...: 1*it(11)+0
  Such that:it(11) =< -3*V1+1

  with precondition: [V1=0,V>=8] 

* Chain [13]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

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


#### Cost of chains of encArg(V1,Out):
* Chain [18]: 0
  with precondition: [V1=1,Out=1] 

* Chain [17]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([14,15,16],[[18],[17]])]: 1*s(3)+0
  Such that:aux(3) =< V1
s(3) =< aux(3)

  with precondition: [V1>=1] 


#### Cost of chains of fun(V1,V,Out):
* Chain [23]...: 1*s(7)+2*s(9)+2*s(10)+0
  Such that:s(6) =< V1
aux(5) =< 1
aux(6) =< V
s(10) =< aux(5)
s(9) =< aux(6)
s(7) =< s(6)

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

* Chain [22]: 2*s(15)+2*s(17)+0
  Such that:aux(7) =< V1
aux(8) =< V
s(17) =< aux(8)
s(15) =< aux(7)

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

* Chain [21]: 0
  with precondition: [Out=1,V1>=0,V>=0] 

* Chain [20]: 1*s(23)+2*s(25)+0
  Such that:s(22) =< V1
aux(9) =< V
s(25) =< aux(9)
s(23) =< s(22)

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

* Chain [19]: 1*s(29)+0
  Such that:s(28) =< V1
s(29) =< s(28)

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


#### Cost of chains of fun2(Out):
* Chain [25]: 0
  with precondition: [Out=0] 

* Chain [24]: 0
  with precondition: [Out=1] 


#### Cost of chains of start(V1,V):
* Chain [27]...: 3*s(30)+2*s(35)+1*s(36)+0
  Such that:s(31) =< V1
s(33) =< V
aux(10) =< 1
s(30) =< aux(10)
s(35) =< s(33)
s(36) =< s(31)

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

* Chain [26]: 5*s(38)+4*s(41)+0
  Such that:aux(11) =< V1
aux(12) =< V
s(38) =< aux(11)
s(41) =< aux(12)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [27]... with precondition: [V1>=0,V>=1] 
    - Upper bound: V1+2*V+3 
    - Complexity: n 
* Chain [26] with precondition: [] 
    - Upper bound: nat(V)*4+nat(V1)*5 
    - Complexity: n 

### Maximum cost of start(V1,V): nat(V)*2+nat(V1)+max([3,nat(V)*2+nat(V1)*4]) 
Asymptotic class: n 
* Total analysis performed in 338 ms.


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

(18)
BOUNDS(1, n^1)