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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 147 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, 654 ms]
    (16) BOUNDS(1, n^2)
(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(18) TRS for Loop Detection
    (19) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   implies(not(x), y) -> or(x, y)
   implies(not(x), or(y, z)) -> implies(y, or(x, z))
   implies(x, or(y, z)) -> or(y, implies(x, 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(not(x_1)) -> not(encArg(x_1))
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   implies(not(x), y) -> or(x, y)
   implies(not(x), or(y, z)) -> implies(y, or(x, z))
   implies(x, or(y, z)) -> or(y, implies(x, z))

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

   encArg(not(x_1)) -> not(encArg(x_1))
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(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(n^1, n^2).


The TRS R consists of the following rules:

   implies(not(x), y) -> or(x, y)
   implies(not(x), or(y, z)) -> implies(y, or(x, z))
   implies(x, or(y, z)) -> or(y, implies(x, z))

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

   encArg(not(x_1)) -> not(encArg(x_1))
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

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^2).


The TRS R consists of the following rules:

   implies(not(x), y) -> or(x, y)
   implies(not(x), or(y, z)) -> implies(y, or(x, z))
   implies(x, or(y, z)) -> or(y, implies(x, z))

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

   encArg(not(x_1)) -> not(encArg(x_1))
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

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^2).


The TRS R consists of the following rules:

   implies(not(x), y) -> or(x, y) [1]
   implies(not(x), or(y, z)) -> implies(y, or(x, z)) [1]
   implies(x, or(y, z)) -> or(y, implies(x, z)) [1]
   encArg(not(x_1)) -> not(encArg(x_1)) [0]
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [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:

   implies(not(x), y) -> or(x, y) [1]
   implies(not(x), or(y, z)) -> implies(y, or(x, z)) [1]
   implies(x, or(y, z)) -> or(y, implies(x, z)) [1]
   encArg(not(x_1)) -> not(encArg(x_1)) [0]
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0]

The TRS has the following type information:
implies :: not:or:cons_implies -> not:or:cons_implies -> not:or:cons_implies
not :: not:or:cons_implies -> not:or:cons_implies
or :: not:or:cons_implies -> not:or:cons_implies -> not:or:cons_implies
encArg :: not:or:cons_implies -> not:or:cons_implies
cons_implies :: not:or:cons_implies -> not:or:cons_implies -> not:or:cons_implies
encode_implies :: not:or:cons_implies -> not:or:cons_implies -> not:or:cons_implies
encode_not :: not:or:cons_implies -> not:or:cons_implies
encode_or :: not:or:cons_implies -> not:or:cons_implies -> not:or:cons_implies

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_implies(v0, v1) -> null_encode_implies [0]
   encode_not(v0) -> null_encode_not [0]
   encode_or(v0, v1) -> null_encode_or [0]
   implies(v0, v1) -> null_implies [0]

And the following fresh constants:    null_encArg, null_encode_implies, null_encode_not, null_encode_or, null_implies

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

(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:

   implies(not(x), y) -> or(x, y) [1]
   implies(not(x), or(y, z)) -> implies(y, or(x, z)) [1]
   implies(x, or(y, z)) -> or(y, implies(x, z)) [1]
   encArg(not(x_1)) -> not(encArg(x_1)) [0]
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2)) [0]
   encode_not(x_1) -> not(encArg(x_1)) [0]
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2)) [0]
   encArg(v0) -> null_encArg [0]
   encode_implies(v0, v1) -> null_encode_implies [0]
   encode_not(v0) -> null_encode_not [0]
   encode_or(v0, v1) -> null_encode_or [0]
   implies(v0, v1) -> null_implies [0]

The TRS has the following type information:
implies :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
not :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
or :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
encArg :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
cons_implies :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
encode_implies :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
encode_not :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
encode_or :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies -> not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
null_encArg :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
null_encode_implies :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
null_encode_not :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
null_encode_or :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies
null_implies :: not:or:cons_implies:null_encArg:null_encode_implies:null_encode_not:null_encode_or:null_implies

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:

   null_encArg => 0
   null_encode_implies => 0
   null_encode_not => 0
   null_encode_or => 0
   null_implies => 0

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

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

   encArg(z') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encArg(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = 1 + x_1
   encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2
   encode_implies(z', z'') -{ 0 }-> implies(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   encode_implies(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_not(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_not(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = x_1
   encode_or(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   encode_or(z', z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2
   implies(z', z'') -{ 1 }-> implies(y, 1 + x + z) :|: z' = 1 + x, z >= 0, x >= 0, y >= 0, z'' = 1 + y + z
   implies(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   implies(z', z'') -{ 1 }-> 1 + x + y :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   implies(z', z'') -{ 1 }-> 1 + y + implies(x, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z

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(V, V1),0,[implies(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[encArg(V, Out)],[V >= 0]).
eq(start(V, V1),0,[fun(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[fun1(V, Out)],[V >= 0]).
eq(start(V, V1),0,[fun2(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(implies(V, V1, Out),1,[],[Out = 1 + V2 + V3,V = 1 + V3,V1 = V2,V3 >= 0,V2 >= 0]).
eq(implies(V, V1, Out),1,[implies(V5, 1 + V4 + V6, Ret)],[Out = Ret,V = 1 + V4,V6 >= 0,V4 >= 0,V5 >= 0,V1 = 1 + V5 + V6]).
eq(implies(V, V1, Out),1,[implies(V8, V9, Ret1)],[Out = 1 + Ret1 + V7,V9 >= 0,V = V8,V8 >= 0,V7 >= 0,V1 = 1 + V7 + V9]).
eq(encArg(V, Out),0,[encArg(V10, Ret11)],[Out = 1 + Ret11,V10 >= 0,V = 1 + V10]).
eq(encArg(V, Out),0,[encArg(V11, Ret01),encArg(V12, Ret12)],[Out = 1 + Ret01 + Ret12,V11 >= 0,V12 >= 0,V = 1 + V11 + V12]).
eq(encArg(V, Out),0,[encArg(V13, Ret0),encArg(V14, Ret13),implies(Ret0, Ret13, Ret2)],[Out = Ret2,V13 >= 0,V14 >= 0,V = 1 + V13 + V14]).
eq(fun(V, V1, Out),0,[encArg(V16, Ret02),encArg(V15, Ret14),implies(Ret02, Ret14, Ret3)],[Out = Ret3,V16 >= 0,V = V16,V15 >= 0,V1 = V15]).
eq(fun1(V, Out),0,[encArg(V17, Ret15)],[Out = 1 + Ret15,V17 >= 0,V = V17]).
eq(fun2(V, V1, Out),0,[encArg(V19, Ret011),encArg(V18, Ret16)],[Out = 1 + Ret011 + Ret16,V19 >= 0,V = V19,V18 >= 0,V1 = V18]).
eq(encArg(V, Out),0,[],[Out = 0,V20 >= 0,V = V20]).
eq(fun(V, V1, Out),0,[],[Out = 0,V22 >= 0,V21 >= 0,V1 = V21,V = V22]).
eq(fun1(V, Out),0,[],[Out = 0,V23 >= 0,V = V23]).
eq(fun2(V, V1, Out),0,[],[Out = 0,V24 >= 0,V25 >= 0,V1 = V25,V = V24]).
eq(implies(V, V1, Out),0,[],[Out = 0,V26 >= 0,V27 >= 0,V1 = V27,V = V26]).
input_output_vars(implies(V,V1,Out),[V,V1],[Out]).
input_output_vars(encArg(V,Out),[V],[Out]).
input_output_vars(fun(V,V1,Out),[V,V1],[Out]).
input_output_vars(fun1(V,Out),[V],[Out]).
input_output_vars(fun2(V,V1,Out),[V,V1],[Out]).


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into implies/3
1. SCC is partially evaluated into encArg/2
2. SCC is partially evaluated into fun/3
3. SCC is partially evaluated into fun1/2
4. SCC is partially evaluated into fun2/3
5. SCC is partially evaluated into start/2

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

### Specialization of cost equations implies/3 
* CE 6 is refined into CE [20] 
* CE 9 is refined into CE [21] 
* CE 7 is refined into CE [22] 
* CE 8 is refined into CE [23] 


### Cost equations --> "Loop" of implies/3 
* CEs [22] --> Loop 12 
* CEs [23] --> Loop 13 
* CEs [20] --> Loop 14 
* CEs [21] --> Loop 15 

### Ranking functions of CR implies(V,V1,Out) 
* RF of phase [12,13]: [V+V1]

#### Partial ranking functions of CR implies(V,V1,Out) 
* Partial RF of phase [12,13]:
  - RF of loop [12:1]:
    V+V1-1
  - RF of loop [13:1]:
    V1 depends on loops [12:1] 


### Specialization of cost equations encArg/2 
* CE 13 is refined into CE [24] 
* CE 11 is refined into CE [25] 
* CE 12 is refined into CE [26,27,28] 
* CE 10 is refined into CE [29] 


### Cost equations --> "Loop" of encArg/2 
* CEs [29] --> Loop 16 
* CEs [28] --> Loop 17 
* CEs [25] --> Loop 18 
* CEs [27] --> Loop 19 
* CEs [26] --> Loop 20 
* CEs [24] --> Loop 21 

### Ranking functions of CR encArg(V,Out) 
* RF of phase [16,17,18,19,20]: [V]

#### Partial ranking functions of CR encArg(V,Out) 
* Partial RF of phase [16,17,18,19,20]:
  - RF of loop [16:1,17:1,17:2,18:1,18:2,19:1,19:2,20:1,20:2]:
    V


### Specialization of cost equations fun/3 
* CE 14 is refined into CE [30,31,32,33,34,35,36,37] 
* CE 15 is refined into CE [38] 


### Cost equations --> "Loop" of fun/3 
* CEs [34] --> Loop 22 
* CEs [36,37] --> Loop 23 
* CEs [32] --> Loop 24 
* CEs [30,31,33,35,38] --> Loop 25 

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

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


### Specialization of cost equations fun1/2 
* CE 16 is refined into CE [39,40] 
* CE 17 is refined into CE [41] 


### Cost equations --> "Loop" of fun1/2 
* CEs [40] --> Loop 26 
* CEs [39] --> Loop 27 
* CEs [41] --> Loop 28 

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

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


### Specialization of cost equations fun2/3 
* CE 18 is refined into CE [42,43,44,45] 
* CE 19 is refined into CE [46] 


### Cost equations --> "Loop" of fun2/3 
* CEs [45] --> Loop 29 
* CEs [44] --> Loop 30 
* CEs [43] --> Loop 31 
* CEs [42] --> Loop 32 
* CEs [46] --> Loop 33 

### Ranking functions of CR fun2(V,V1,Out) 

#### Partial ranking functions of CR fun2(V,V1,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [47,48,49] 
* CE 2 is refined into CE [50,51] 
* CE 3 is refined into CE [52,53,54,55] 
* CE 4 is refined into CE [56,57,58] 
* CE 5 is refined into CE [59,60,61,62,63] 


### Cost equations --> "Loop" of start/2 
* CEs [47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63] --> Loop 34 

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

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


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

#### Cost of chains of implies(V,V1,Out):
* Chain [[12,13],15]: 1*it(12)+1*it(13)+0
  Such that:aux(9) =< V+V1
aux(10) =< V1
it(12) =< aux(9)
it(13) =< aux(9)
aux(1) =< it(12)*aux(9)
it(13) =< aux(1)+aux(10)

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

* Chain [[12,13],14]: 1*it(12)+1*it(13)+1
  Such that:aux(11) =< V+V1
aux(12) =< V1
it(12) =< aux(11)
it(13) =< aux(11)
aux(1) =< it(12)*aux(11)
it(13) =< aux(1)+aux(12)

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

* Chain [15]: 0
  with precondition: [Out=0,V>=0,V1>=0] 

* Chain [14]: 1
  with precondition: [V+V1=Out,V>=1,V1>=0] 


#### Cost of chains of encArg(V,Out):
* Chain [21]: 0
  with precondition: [Out=0,V>=0] 

* Chain [multiple([16,17,18,19,20],[[21]])]: 2*it(17)+2*s(21)+2*s(22)+0
  Such that:aux(22) =< V
aux(23) =< 2/3*V
it(17) =< aux(22)
it(17) =< aux(23)
aux(17) =< aux(22)-1
aux(16) =< aux(22)
s(23) =< it(17)*aux(17)
s(25) =< it(17)*aux(16)
s(21) =< s(25)
s(22) =< s(25)
s(24) =< s(21)*aux(22)
s(22) =< s(24)+s(23)

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


#### Cost of chains of fun(V,V1,Out):
* Chain [25]: 4*s(28)+4*s(33)+4*s(34)+4*s(38)+4*s(43)+4*s(44)+0
  Such that:aux(24) =< V
aux(25) =< 2/3*V
aux(26) =< V1
aux(27) =< 2/3*V1
s(38) =< aux(24)
s(38) =< aux(25)
s(39) =< aux(24)-1
s(40) =< aux(24)
s(41) =< s(38)*s(39)
s(42) =< s(38)*s(40)
s(43) =< s(42)
s(44) =< s(42)
s(45) =< s(43)*aux(24)
s(44) =< s(45)+s(41)
s(28) =< aux(26)
s(28) =< aux(27)
s(29) =< aux(26)-1
s(30) =< aux(26)
s(31) =< s(28)*s(29)
s(32) =< s(28)*s(30)
s(33) =< s(32)
s(34) =< s(32)
s(35) =< s(33)*aux(26)
s(34) =< s(35)+s(31)

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

* Chain [24]: 2*s(68)+2*s(73)+2*s(74)+2*s(78)+2*s(79)+1
  Such that:s(67) =< 2/3*V1
aux(28) =< V1
s(78) =< aux(28)
s(79) =< aux(28)
s(80) =< s(78)*aux(28)
s(79) =< s(80)+aux(28)
s(68) =< aux(28)
s(68) =< s(67)
s(69) =< aux(28)-1
s(70) =< aux(28)
s(71) =< s(68)*s(69)
s(72) =< s(68)*s(70)
s(73) =< s(72)
s(74) =< s(72)
s(75) =< s(73)*aux(28)
s(74) =< s(75)+s(71)

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

* Chain [23]: 4*s(83)+4*s(88)+4*s(89)+4*s(93)+4*s(98)+4*s(99)+2*s(123)+2*s(124)+1
  Such that:s(121) =< V+V1
aux(30) =< V
aux(31) =< 2/3*V
aux(32) =< V1
aux(33) =< 2/3*V1
s(93) =< aux(32)
s(93) =< aux(33)
s(94) =< aux(32)-1
s(95) =< aux(32)
s(96) =< s(93)*s(94)
s(97) =< s(93)*s(95)
s(98) =< s(97)
s(99) =< s(97)
s(100) =< s(98)*aux(32)
s(99) =< s(100)+s(96)
s(83) =< aux(30)
s(83) =< aux(31)
s(84) =< aux(30)-1
s(85) =< aux(30)
s(86) =< s(83)*s(84)
s(87) =< s(83)*s(85)
s(88) =< s(87)
s(89) =< s(87)
s(90) =< s(88)*aux(30)
s(89) =< s(90)+s(86)
s(123) =< s(121)
s(124) =< s(121)
s(125) =< s(123)*s(121)
s(124) =< s(125)+aux(32)

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

* Chain [22]: 2*s(128)+2*s(133)+2*s(134)+1
  Such that:s(126) =< V
s(127) =< 2/3*V
s(128) =< s(126)
s(128) =< s(127)
s(129) =< s(126)-1
s(130) =< s(126)
s(131) =< s(128)*s(129)
s(132) =< s(128)*s(130)
s(133) =< s(132)
s(134) =< s(132)
s(135) =< s(133)*s(126)
s(134) =< s(135)+s(131)

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


#### Cost of chains of fun1(V,Out):
* Chain [28]: 0
  with precondition: [Out=0,V>=0] 

* Chain [27]: 0
  with precondition: [Out=1,V>=0] 

* Chain [26]: 2*s(138)+2*s(143)+2*s(144)+0
  Such that:s(136) =< V
s(137) =< 2/3*V
s(138) =< s(136)
s(138) =< s(137)
s(139) =< s(136)-1
s(140) =< s(136)
s(141) =< s(138)*s(139)
s(142) =< s(138)*s(140)
s(143) =< s(142)
s(144) =< s(142)
s(145) =< s(143)*s(136)
s(144) =< s(145)+s(141)

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


#### Cost of chains of fun2(V,V1,Out):
* Chain [33]: 0
  with precondition: [Out=0,V>=0,V1>=0] 

* Chain [32]: 0
  with precondition: [Out=1,V>=0,V1>=0] 

* Chain [31]: 2*s(148)+2*s(153)+2*s(154)+0
  Such that:s(146) =< V1
s(147) =< 2/3*V1
s(148) =< s(146)
s(148) =< s(147)
s(149) =< s(146)-1
s(150) =< s(146)
s(151) =< s(148)*s(149)
s(152) =< s(148)*s(150)
s(153) =< s(152)
s(154) =< s(152)
s(155) =< s(153)*s(146)
s(154) =< s(155)+s(151)

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

* Chain [30]: 2*s(158)+2*s(163)+2*s(164)+0
  Such that:s(156) =< V
s(157) =< 2/3*V
s(158) =< s(156)
s(158) =< s(157)
s(159) =< s(156)-1
s(160) =< s(156)
s(161) =< s(158)*s(159)
s(162) =< s(158)*s(160)
s(163) =< s(162)
s(164) =< s(162)
s(165) =< s(163)*s(156)
s(164) =< s(165)+s(161)

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

* Chain [29]: 2*s(168)+2*s(173)+2*s(174)+2*s(178)+2*s(183)+2*s(184)+0
  Such that:s(166) =< V
s(167) =< 2/3*V
s(176) =< V1
s(177) =< 2/3*V1
s(178) =< s(176)
s(178) =< s(177)
s(179) =< s(176)-1
s(180) =< s(176)
s(181) =< s(178)*s(179)
s(182) =< s(178)*s(180)
s(183) =< s(182)
s(184) =< s(182)
s(185) =< s(183)*s(176)
s(184) =< s(185)+s(181)
s(168) =< s(166)
s(168) =< s(167)
s(169) =< s(166)-1
s(170) =< s(166)
s(171) =< s(168)*s(169)
s(172) =< s(168)*s(170)
s(173) =< s(172)
s(174) =< s(172)
s(175) =< s(173)*s(166)
s(174) =< s(175)+s(171)

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


#### Cost of chains of start(V,V1):
* Chain [34]: 4*s(188)+4*s(189)+18*s(193)+18*s(198)+18*s(199)+14*s(213)+14*s(218)+14*s(219)+2*s(223)+2*s(224)+1
  Such that:aux(34) =< V
aux(35) =< V+V1
aux(36) =< 2/3*V
aux(37) =< V1
aux(38) =< 2/3*V1
s(193) =< aux(34)
s(193) =< aux(36)
s(194) =< aux(34)-1
s(195) =< aux(34)
s(196) =< s(193)*s(194)
s(197) =< s(193)*s(195)
s(198) =< s(197)
s(199) =< s(197)
s(200) =< s(198)*aux(34)
s(199) =< s(200)+s(196)
s(213) =< aux(37)
s(213) =< aux(38)
s(214) =< aux(37)-1
s(215) =< aux(37)
s(216) =< s(213)*s(214)
s(217) =< s(213)*s(215)
s(218) =< s(217)
s(219) =< s(217)
s(220) =< s(218)*aux(37)
s(219) =< s(220)+s(216)
s(188) =< aux(35)
s(189) =< aux(35)
s(190) =< s(188)*aux(35)
s(189) =< s(190)+aux(37)
s(223) =< aux(37)
s(224) =< aux(37)
s(225) =< s(223)*aux(37)
s(224) =< s(225)+aux(37)

  with precondition: [V>=0] 


Closed-form bounds of start(V,V1): 
-------------------------------------
* Chain [34] with precondition: [V>=0] 
    - Upper bound: 18*V+1+36*V*V+nat(V1)*18+nat(V1)*28*nat(V1)+nat(V+V1)*8 
    - Complexity: n^2 

### Maximum cost of start(V,V1): 18*V+1+36*V*V+nat(V1)*18+nat(V1)*28*nat(V1)+nat(V+V1)*8 
Asymptotic class: n^2 
* Total analysis performed in 534 ms.


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

(16)
BOUNDS(1, n^2)

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

(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   implies(not(x), y) -> or(x, y)
   implies(not(x), or(y, z)) -> implies(y, or(x, z))
   implies(x, or(y, z)) -> or(y, implies(x, z))

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

   encArg(not(x_1)) -> not(encArg(x_1))
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

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

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

The rewrite sequence

implies(x, or(y, z)) ->^+ or(y, implies(x, z))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [z / or(y, z)].

The result substitution is [ ].




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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   implies(not(x), y) -> or(x, y)
   implies(not(x), or(y, z)) -> implies(y, or(x, z))
   implies(x, or(y, z)) -> or(y, implies(x, z))

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

   encArg(not(x_1)) -> not(encArg(x_1))
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   implies(not(x), y) -> or(x, y)
   implies(not(x), or(y, z)) -> implies(y, or(x, z))
   implies(x, or(y, z)) -> or(y, implies(x, z))

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

   encArg(not(x_1)) -> not(encArg(x_1))
   encArg(or(x_1, x_2)) -> or(encArg(x_1), encArg(x_2))
   encArg(cons_implies(x_1, x_2)) -> implies(encArg(x_1), encArg(x_2))
   encode_implies(x_1, x_2) -> implies(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_or(x_1, x_2) -> or(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL