/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 168 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedTrs
(11) CompletionProof [UPPER BOUND(ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(14) CpxRNTS
(15) CompleteCoflocoProof [FINISHED, 222 ms]
(16) 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(x, x) -> f(a, b)
   b -> c

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


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

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

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

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

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

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

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

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

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

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

(6)
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(x, x) -> f(a, b)
   b -> c

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

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

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

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

(8)
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(x, x) -> f(a, b) [1]
   b -> c [1]
   encArg(a) -> a [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [0]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, x) -> f(a, b) [1]
   b -> c [1]
   encArg(a) -> a [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [0]

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

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

(11) 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]
   encode_c -> null_encode_c [0]
   f(v0, v1) -> null_f [0]

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

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

(12)
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(x, x) -> f(a, b) [1]
   b -> c [1]
   encArg(a) -> a [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [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]
   encode_c -> null_encode_c [0]
   f(v0, v1) -> null_f [0]

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

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

(13) 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
   c => 1
   cons_b => 2
   null_encArg => 0
   null_encode_f => 0
   null_encode_a => 0
   null_encode_b => 0
   null_encode_c => 0
   null_f => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   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 }-> b :|: z = 2
   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 }-> b :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 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, b) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(15) 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,[b(Out)],[]).
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(start(V1, V),0,[fun3(Out)],[]).
eq(f(V1, V, Out),1,[b(Ret1),f(0, Ret1, Ret)],[Out = Ret,V = V2,V2 >= 0,V1 = V2]).
eq(b(Out),1,[],[Out = 1]).
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, Ret11),f(Ret0, Ret11, Ret2)],[Out = Ret2,V4 >= 0,V1 = 1 + V3 + V4,V3 >= 0]).
eq(encArg(V1, Out),0,[b(Ret3)],[Out = Ret3,V1 = 2]).
eq(fun(V1, V, Out),0,[encArg(V5, Ret01),encArg(V6, Ret12),f(Ret01, Ret12, Ret4)],[Out = Ret4,V5 >= 0,V6 >= 0,V1 = V5,V = V6]).
eq(fun1(Out),0,[],[Out = 0]).
eq(fun2(Out),0,[b(Ret5)],[Out = Ret5]).
eq(fun3(Out),0,[],[Out = 1]).
eq(encArg(V1, Out),0,[],[Out = 0,V7 >= 0,V1 = V7]).
eq(fun(V1, V, Out),0,[],[Out = 0,V9 >= 0,V8 >= 0,V1 = V9,V = V8]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(Out),0,[],[Out = 0]).
eq(f(V1, V, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V1 = V11,V = V10]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(b(Out),[],[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]).
input_output_vars(fun3(Out),[],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [b/1]
1. recursive  : [f/3]
2. recursive [non_tail,multiple] : [encArg/2]
3. non_recursive  : [fun/3]
4. non_recursive  : [fun1/1]
5. non_recursive  : [fun2/1]
6. non_recursive  : [fun3/1]
7. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into f/3
2. SCC is partially evaluated into encArg/2
3. SCC is partially evaluated into fun/3
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into fun2/1
6. SCC is partially evaluated into fun3/1
7. SCC is partially evaluated into start/2

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

### Specialization of cost equations f/3 
* CE 9 is refined into CE [20] 
* CE 8 is refined into CE [21] 


### Cost equations --> "Loop" of f/3 
* CEs [21] --> Loop 13 
* CEs [20] --> Loop 14 

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

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


### Specialization of cost equations encArg/2 
* CE 10 is refined into CE [22] 
* CE 13 is refined into CE [23] 
* CE 11 is refined into CE [24] 
* CE 12 is refined into CE [25] 


### Cost equations --> "Loop" of encArg/2 
* CEs [25] --> Loop 15 
* CEs [22] --> Loop 16 
* CEs [23] --> Loop 17 
* CEs [24] --> Loop 18 

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

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


### Specialization of cost equations fun/3 
* CE 14 is refined into CE [26,27,28,29,30,31,32,33,34] 
* CE 15 is refined into CE [35] 


### Cost equations --> "Loop" of fun/3 
* CEs [32] --> Loop 19 
* CEs [30] --> Loop 20 
* CEs [29,31] --> Loop 21 
* CEs [27,33] --> Loop 22 
* CEs [26,28,34,35] --> 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 16 is refined into CE [36] 
* CE 17 is refined into CE [37] 


### Cost equations --> "Loop" of fun2/1 
* CEs [36] --> Loop 24 
* CEs [37] --> Loop 25 

### Ranking functions of CR fun2(Out) 

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


### Specialization of cost equations fun3/1 
* CE 18 is refined into CE [38] 
* CE 19 is refined into CE [39] 


### Cost equations --> "Loop" of fun3/1 
* CEs [38] --> Loop 26 
* CEs [39] --> Loop 27 

### Ranking functions of CR fun3(Out) 

#### Partial ranking functions of CR fun3(Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [40] 
* CE 2 is refined into CE [41] 
* CE 3 is refined into CE [42,43,44] 
* CE 4 is refined into CE [45,46,47] 
* CE 5 is refined into CE [48] 
* CE 6 is refined into CE [49,50] 
* CE 7 is refined into CE [51,52] 


### Cost equations --> "Loop" of start/2 
* CEs [40,41,42,43,44,45,46,47,48,49,50,51,52] --> Loop 28 

### 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 [14]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [13,14]: 2
  with precondition: [Out=0,V1=V,V1>=0] 


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

* Chain [17]: 1
  with precondition: [V1=2,Out=1] 

* Chain [16]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([15],[[18],[17],[16]])]: 2*it(15)+1*it([17])+0
  Such that:it([17]) =< V1/3+1/3
aux(1) =< V1
aux(2) =< V1+1
it(15) =< aux(1)
it([17]) =< aux(1)
it([17]) =< aux(2)

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


#### Cost of chains of fun(V1,V,Out):
* Chain [23]: 2*s(7)+4*s(8)+1*s(11)+2*s(12)+2
  Such that:s(9) =< V1
s(10) =< V1+1
s(11) =< V1/3+1/3
aux(3) =< V
aux(4) =< V+1
aux(5) =< V/3+1/3
s(7) =< aux(5)
s(8) =< aux(3)
s(7) =< aux(3)
s(7) =< aux(4)
s(12) =< s(9)
s(11) =< s(9)
s(11) =< s(10)

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

* Chain [22]: 1*s(19)+2*s(20)+3
  Such that:s(17) =< V1
s(18) =< V1+1
s(19) =< V1/3+1/3
s(20) =< s(17)
s(19) =< s(17)
s(19) =< s(18)

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

* Chain [21]: 1*s(23)+2*s(24)+3
  Such that:s(21) =< V
s(22) =< V+1
s(23) =< V/3+1/3
s(24) =< s(21)
s(23) =< s(21)
s(23) =< s(22)

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

* Chain [20]: 4
  with precondition: [V1=2,V=2,Out=0] 

* Chain [19]: 1*s(27)+2*s(28)+2
  Such that:s(25) =< V1
s(26) =< V1+1
s(27) =< V1/3+1/3
s(28) =< s(25)
s(27) =< s(25)
s(27) =< s(26)

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


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

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


#### Cost of chains of fun3(Out):
* Chain [27]: 0
  with precondition: [Out=0] 

* Chain [26]: 0
  with precondition: [Out=1] 


#### Cost of chains of start(V1,V):
* Chain [28]: 4*s(44)+8*s(45)+3*s(52)+6*s(53)+4
  Such that:s(49) =< V
s(50) =< V+1
s(51) =< V/3+1/3
aux(9) =< V1
aux(10) =< V1+1
aux(11) =< V1/3+1/3
s(44) =< aux(11)
s(45) =< aux(9)
s(44) =< aux(9)
s(44) =< aux(10)
s(52) =< s(51)
s(53) =< s(49)
s(52) =< s(49)
s(52) =< s(50)

  with precondition: [] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [28] with precondition: [] 
    - Upper bound: nat(V1)*8+4+nat(V)*6+nat(V1/3+1/3)*4+nat(V/3+1/3)*3 
    - Complexity: n 

### Maximum cost of start(V1,V): nat(V1)*8+4+nat(V)*6+nat(V1/3+1/3)*4+nat(V/3+1/3)*3 
Asymptotic class: n 
* Total analysis performed in 167 ms.


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

(16)
BOUNDS(1, n^1)