/export/starexec/sandbox2/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 177 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [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), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 537 ms]
    (12) BOUNDS(1, n^2)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRelTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 269 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 18 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 23 ms]
            (28) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
   overlap(Nil, ys) -> False
   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs))
   member(x, Nil) -> False
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs, ys) -> overlap(xs, ys)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys)
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
   overlap[Ite][True][Ite](True, xs, ys) -> True
   member[Ite][True][Ite](True, x, xs) -> True

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

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

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


The TRS R consists of the following rules:

   overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
   overlap(Nil, ys) -> False
   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs))
   member(x, Nil) -> False
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs, ys) -> overlap(xs, ys)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0, S(y)) -> False
   !EQ(S(x), 0) -> False
   !EQ(0, 0) -> True
   overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys)
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
   overlap[Ite][True][Ite](True, xs, ys) -> True
   member[Ite][True][Ite](True, x, xs) -> True

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

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

(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:

   overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1]
   overlap(Nil, ys) -> False [1]
   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs)) [1]
   member(x, Nil) -> False [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs, ys) -> overlap(xs, ys) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) [0]
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0]
   overlap[Ite][True][Ite](True, xs, ys) -> True [0]
   member[Ite][True][Ite](True, x, xs) -> True [0]

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:

   overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1]
   overlap(Nil, ys) -> False [1]
   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs)) [1]
   member(x, Nil) -> False [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs, ys) -> overlap(xs, ys) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) [0]
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0]
   overlap[Ite][True][Ite](True, xs, ys) -> True [0]
   member[Ite][True][Ite](True, x, xs) -> True [0]

The TRS has the following type information:
overlap :: Cons:Nil -> Cons:Nil -> False:True
Cons :: S:0 -> Cons:Nil -> Cons:Nil
overlap[Ite][True][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> False:True
member :: S:0 -> Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
member[Ite][True][Ite] :: False:True -> S:0 -> Cons:Nil -> False:True
!EQ :: S:0 -> S:0 -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
S :: S:0 -> S:0
0 :: S:0

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:

   !EQ(v0, v1) -> null_!EQ [0]
   overlap[Ite][True][Ite](v0, v1, v2) -> null_overlap[Ite][True][Ite] [0]
   member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0]

And the following fresh constants:    null_!EQ, null_overlap[Ite][True][Ite], null_member[Ite][True][Ite]

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

(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:

   overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys) [1]
   overlap(Nil, ys) -> False [1]
   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs)) [1]
   member(x, Nil) -> False [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   goal(xs, ys) -> overlap(xs, ys) [1]
   !EQ(S(x), S(y)) -> !EQ(x, y) [0]
   !EQ(0, S(y)) -> False [0]
   !EQ(S(x), 0) -> False [0]
   !EQ(0, 0) -> True [0]
   overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys) [0]
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs) [0]
   overlap[Ite][True][Ite](True, xs, ys) -> True [0]
   member[Ite][True][Ite](True, x, xs) -> True [0]
   !EQ(v0, v1) -> null_!EQ [0]
   overlap[Ite][True][Ite](v0, v1, v2) -> null_overlap[Ite][True][Ite] [0]
   member[Ite][True][Ite](v0, v1, v2) -> null_member[Ite][True][Ite] [0]

The TRS has the following type information:
overlap :: Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
Cons :: S:0 -> Cons:Nil -> Cons:Nil
overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] -> Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
member :: S:0 -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
Nil :: Cons:Nil
False :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite] -> S:0 -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
!EQ :: S:0 -> S:0 -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
notEmpty :: Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
True :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
goal :: Cons:Nil -> Cons:Nil -> False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
S :: S:0 -> S:0
0 :: S:0
null_!EQ :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
null_overlap[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]
null_member[Ite][True][Ite] :: False:True:null_!EQ:null_overlap[Ite][True][Ite]:null_member[Ite][True][Ite]

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:

   Nil => 0
   False => 1
   True => 2
   0 => 0
   null_!EQ => 0
   null_overlap[Ite][True][Ite] => 0
   null_member[Ite][True][Ite] => 0

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

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

   !EQ(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   !EQ(z, z') -{ 0 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   !EQ(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   !EQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   !EQ(z, z') -{ 0 }-> !EQ(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   goal(z, z') -{ 1 }-> overlap(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0
   member(z, z') -{ 1 }-> member[Ite][True][Ite](!EQ(x, x'), x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   member(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   member[Ite][True][Ite](z, z', z'') -{ 0 }-> member(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
   member[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, xs >= 0, z' = x, x >= 0, z'' = xs
   member[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 0
   overlap(z, z') -{ 1 }-> overlap[Ite][True][Ite](member(x, ys), 1 + x + xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
   overlap(z, z') -{ 1 }-> 1 :|: z' = ys, ys >= 0, z = 0
   overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> overlap(xs, ys) :|: xs >= 0, z' = 1 + x + xs, z = 1, ys >= 0, x >= 0, z'' = ys
   overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 2 :|: z = 2, xs >= 0, ys >= 0, z'' = ys, z' = xs
   overlap[Ite][True][Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0

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, V18),0,[overlap(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V18),0,[member(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V18),0,[notEmpty(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V18),0,[goal(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V18),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V18),0,[fun(V1, V, V18, Out)],[V1 >= 0,V >= 0,V18 >= 0]).
eq(start(V1, V, V18),0,[fun1(V1, V, V18, Out)],[V1 >= 0,V >= 0,V18 >= 0]).
eq(overlap(V1, V, Out),1,[member(V4, V3, Ret0),fun(Ret0, 1 + V4 + V2, V3, Ret)],[Out = Ret,V1 = 1 + V2 + V4,V2 >= 0,V = V3,V3 >= 0,V4 >= 0]).
eq(overlap(V1, V, Out),1,[],[Out = 1,V = V5,V5 >= 0,V1 = 0]).
eq(member(V1, V, Out),1,[fun2(V6, V8, Ret01),fun1(Ret01, V8, 1 + V6 + V7, Ret1)],[Out = Ret1,V7 >= 0,V = 1 + V6 + V7,V8 >= 0,V6 >= 0,V1 = V8]).
eq(member(V1, V, Out),1,[],[Out = 1,V9 >= 0,V1 = V9,V = 0]).
eq(notEmpty(V1, Out),1,[],[Out = 2,V1 = 1 + V10 + V11,V11 >= 0,V10 >= 0]).
eq(notEmpty(V1, Out),1,[],[Out = 1,V1 = 0]).
eq(goal(V1, V, Out),1,[overlap(V12, V13, Ret2)],[Out = Ret2,V12 >= 0,V1 = V12,V = V13,V13 >= 0]).
eq(fun2(V1, V, Out),0,[fun2(V14, V15, Ret3)],[Out = Ret3,V = 1 + V15,V14 >= 0,V15 >= 0,V1 = 1 + V14]).
eq(fun2(V1, V, Out),0,[],[Out = 1,V = 1 + V16,V16 >= 0,V1 = 0]).
eq(fun2(V1, V, Out),0,[],[Out = 1,V17 >= 0,V1 = 1 + V17,V = 0]).
eq(fun2(V1, V, Out),0,[],[Out = 2,V1 = 0,V = 0]).
eq(fun(V1, V, V18, Out),0,[overlap(V19, V20, Ret4)],[Out = Ret4,V19 >= 0,V = 1 + V19 + V21,V1 = 1,V20 >= 0,V21 >= 0,V18 = V20]).
eq(fun1(V1, V, V18, Out),0,[member(V24, V23, Ret5)],[Out = Ret5,V = V24,V23 >= 0,V1 = 1,V24 >= 0,V22 >= 0,V18 = 1 + V22 + V23]).
eq(fun(V1, V, V18, Out),0,[],[Out = 2,V1 = 2,V26 >= 0,V25 >= 0,V18 = V25,V = V26]).
eq(fun1(V1, V, V18, Out),0,[],[Out = 2,V1 = 2,V28 >= 0,V = V27,V27 >= 0,V18 = V28]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V30 >= 0,V29 >= 0,V1 = V30,V = V29]).
eq(fun(V1, V, V18, Out),0,[],[Out = 0,V32 >= 0,V18 = V33,V31 >= 0,V1 = V32,V = V31,V33 >= 0]).
eq(fun1(V1, V, V18, Out),0,[],[Out = 0,V36 >= 0,V18 = V34,V35 >= 0,V1 = V36,V = V35,V34 >= 0]).
input_output_vars(overlap(V1,V,Out),[V1,V],[Out]).
input_output_vars(member(V1,V,Out),[V1,V],[Out]).
input_output_vars(notEmpty(V1,Out),[V1],[Out]).
input_output_vars(goal(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,V18,Out),[V1,V,V18],[Out]).
input_output_vars(fun1(V1,V,V18,Out),[V1,V,V18],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [fun2/3]
1. recursive  : [fun1/4,member/3]
2. recursive  : [fun/4,overlap/3]
3. non_recursive  : [goal/3]
4. non_recursive  : [notEmpty/2]
5. non_recursive  : [start/3]

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

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

### Specialization of cost equations fun2/3 
* CE 23 is refined into CE [24] 
* CE 21 is refined into CE [25] 
* CE 20 is refined into CE [26] 
* CE 22 is refined into CE [27] 
* CE 19 is refined into CE [28] 


### Cost equations --> "Loop" of fun2/3 
* CEs [28] --> Loop 18 
* CEs [24] --> Loop 19 
* CEs [25] --> Loop 20 
* CEs [26] --> Loop 21 
* CEs [27] --> Loop 22 

### Ranking functions of CR fun2(V1,V,Out) 
* RF of phase [18]: [V,V1]

#### Partial ranking functions of CR fun2(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V
    V1


### Specialization of cost equations member/3 
* CE 14 is refined into CE [29,30] 
* CE 13 is refined into CE [31,32,33,34,35,36,37] 
* CE 16 is refined into CE [38] 
* CE 15 is refined into CE [39,40,41,42] 


### Cost equations --> "Loop" of member/3 
* CEs [39,41,42] --> Loop 23 
* CEs [40] --> Loop 24 
* CEs [30] --> Loop 25 
* CEs [38] --> Loop 26 
* CEs [29] --> Loop 27 
* CEs [31,32,33,34,35,36,37] --> Loop 28 

### Ranking functions of CR member(V1,V,Out) 
* RF of phase [23]: [V]
* RF of phase [24]: [V-1]

#### Partial ranking functions of CR member(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V-1


### Specialization of cost equations overlap/3 
* CE 10 is refined into CE [43,44] 
* CE 9 is refined into CE [45,46,47,48,49,50] 
* CE 12 is refined into CE [51] 
* CE 11 is refined into CE [52,53,54] 


### Cost equations --> "Loop" of overlap/3 
* CEs [52] --> Loop 29 
* CEs [54] --> Loop 30 
* CEs [53] --> Loop 31 
* CEs [43,44] --> Loop 32 
* CEs [45,46,47,49,50] --> Loop 33 
* CEs [48] --> Loop 34 
* CEs [51] --> Loop 35 

### Ranking functions of CR overlap(V1,V,Out) 
* RF of phase [29,30]: [V1,V1+V-2]
* RF of phase [31]: [V1]

#### Partial ranking functions of CR overlap(V1,V,Out) 
* Partial RF of phase [29,30]:
  - RF of loop [29:1]:
    V1
  - RF of loop [30:1]:
    V1-1
* Partial RF of phase [31]:
  - RF of loop [31:1]:
    V1


### Specialization of cost equations notEmpty/2 
* CE 17 is refined into CE [55] 
* CE 18 is refined into CE [56] 


### Cost equations --> "Loop" of notEmpty/2 
* CEs [55] --> Loop 36 
* CEs [56] --> Loop 37 

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

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


### Specialization of cost equations start/3 
* CE 2 is refined into CE [57,58,59,60,61,62] 
* CE 1 is refined into CE [63] 
* CE 3 is refined into CE [64,65,66,67,68,69] 
* CE 4 is refined into CE [70,71,72,73,74,75] 
* CE 5 is refined into CE [76,77,78,79,80,81] 
* CE 6 is refined into CE [82,83] 
* CE 7 is refined into CE [84,85,86,87,88,89] 
* CE 8 is refined into CE [90,91,92,93,94,95,96] 


### Cost equations --> "Loop" of start/3 
* CEs [96] --> Loop 38 
* CEs [71,72,79,85,86,92] --> Loop 39 
* CEs [57,58,59,83] --> Loop 40 
* CEs [60,61,62,63,64,65,66,67,68,69,73,74,75,76,80,81,87,88,89,93,94,95] --> Loop 41 
* CEs [70,77,78,82,84,90,91] --> Loop 42 

### Ranking functions of CR start(V1,V,V18) 

#### Partial ranking functions of CR start(V1,V,V18) 


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

#### Cost of chains of fun2(V1,V,Out):
* Chain [[18],22]: 0
  with precondition: [Out=2,V1=V,V1>=1] 

* Chain [[18],21]: 0
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [[18],20]: 0
  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [[18],19]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [22]: 0
  with precondition: [V1=0,V=0,Out=2] 

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

* Chain [20]: 0
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [19]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of member(V1,V,Out):
* Chain [[24],28]: 1*it(24)+1
  Such that:it(24) =< V

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

* Chain [[24],27]: 1*it(24)+1
  Such that:it(24) =< V

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

* Chain [[24],26]: 1*it(24)+1
  Such that:it(24) =< V

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

* Chain [[23],28]: 1*it(23)+1
  Such that:it(23) =< V

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

* Chain [[23],26]: 1*it(23)+1
  Such that:it(23) =< V

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

* Chain [[23],25]: 1*it(23)+1
  Such that:it(23) =< -V1+V

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

* Chain [28]: 1
  with precondition: [Out=0,V1>=0,V>=1] 

* Chain [27]: 1
  with precondition: [V1=0,Out=2,V>=1] 

* Chain [26]: 1
  with precondition: [V=0,Out=1,V1>=0] 

* Chain [25]: 1
  with precondition: [Out=2,V1>=1,V>=V1+1] 


#### Cost of chains of overlap(V1,V,Out):
* Chain [[31],35]: 2*it(31)+1
  Such that:it(31) =< V1

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

* Chain [[31],34]: 2*it(31)+2
  Such that:it(31) =< V1

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

* Chain [[29,30],35]: 4*it(29)+1*s(9)+1*s(10)+1
  Such that:aux(5) =< V1+V
aux(2) =< V
aux(8) =< V1
it(29) =< aux(8)
it(29) =< aux(5)
aux(3) =< aux(2)
s(9) =< it(29)*aux(2)
s(10) =< it(29)*aux(3)

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

* Chain [[29,30],33]: 4*it(29)+1*s(9)+1*s(10)+6*s(12)+2
  Such that:aux(5) =< V1+V
aux(10) =< V1
aux(11) =< V
s(12) =< aux(11)
it(29) =< aux(10)
it(29) =< aux(5)
aux(3) =< aux(11)
s(9) =< it(29)*aux(11)
s(10) =< it(29)*aux(3)

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

* Chain [[29,30],32]: 4*it(29)+1*s(9)+1*s(10)+2*s(17)+2
  Such that:aux(5) =< V1+V
aux(13) =< V1
aux(14) =< V
s(17) =< aux(14)
it(29) =< aux(13)
it(29) =< aux(5)
aux(3) =< aux(14)
s(9) =< it(29)*aux(14)
s(10) =< it(29)*aux(3)

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

* Chain [35]: 1
  with precondition: [V1=0,Out=1,V>=0] 

* Chain [34]: 2
  with precondition: [V=0,Out=0,V1>=1] 

* Chain [33]: 6*s(12)+2
  Such that:aux(9) =< V
s(12) =< aux(9)

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

* Chain [32]: 2*s(17)+2
  Such that:aux(12) =< V
s(17) =< aux(12)

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


#### Cost of chains of notEmpty(V1,Out):
* Chain [37]: 1
  with precondition: [V1=0,Out=1] 

* Chain [36]: 1
  with precondition: [Out=2,V1>=1] 


#### Cost of chains of start(V1,V,V18):
* Chain [42]: 2*s(40)+2
  Such that:aux(17) =< V
s(40) =< aux(17)

  with precondition: [V1=0] 

* Chain [41]: 21*s(45)+12*s(46)+3*s(48)+3*s(49)+1*s(70)+35*s(74)+24*s(75)+6*s(77)+6*s(78)+1*s(97)+3
  Such that:s(97) =< -V1+V
s(70) =< -V+V18
aux(18) =< V1
aux(19) =< V1+V
aux(20) =< V
aux(21) =< V+V18
aux(22) =< V18
s(74) =< aux(20)
s(45) =< aux(22)
s(75) =< aux(18)
s(75) =< aux(19)
s(76) =< aux(20)
s(77) =< s(75)*aux(20)
s(78) =< s(75)*s(76)
s(46) =< aux(20)
s(46) =< aux(21)
s(47) =< aux(22)
s(48) =< s(46)*aux(22)
s(49) =< s(46)*s(47)

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

* Chain [40]: 4*s(121)+2
  Such that:aux(23) =< V
s(121) =< aux(23)

  with precondition: [V1>=1] 

* Chain [39]: 8*s(123)+3
  Such that:aux(24) =< V1
s(123) =< aux(24)

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

* Chain [38]: 0
  with precondition: [V1=V,V1>=1] 


Closed-form bounds of start(V1,V,V18): 
-------------------------------------
* Chain [42] with precondition: [V1=0] 
    - Upper bound: nat(V)*2+2 
    - Complexity: n 
* Chain [41] with precondition: [V1>=0,V>=0] 
    - Upper bound: 24*V1+3+12*V1*V+47*V+6*V*nat(V18)+nat(V18)*21+nat(-V1+V)+nat(-V+V18) 
    - Complexity: n^2 
* Chain [40] with precondition: [V1>=1] 
    - Upper bound: nat(V)*4+2 
    - Complexity: n 
* Chain [39] with precondition: [V=0,V1>=0] 
    - Upper bound: 8*V1+3 
    - Complexity: n 
* Chain [38] with precondition: [V1=V,V1>=1] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V18): max([8*V1+3,24*V1+1+12*V1*nat(V)+nat(V)*43+nat(V)*6*nat(V18)+nat(V18)*21+nat(-V1+V)+nat(-V+V18)+nat(V)*2+(nat(V)*2+2)]) 
Asymptotic class: n^2 
* Total analysis performed in 495 ms.


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

(12)
BOUNDS(1, n^2)

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

(13) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
   overlap(Nil, ys) -> False
   member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs))
   member(x, Nil) -> False
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs, ys) -> overlap(xs, ys)

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

   !EQ(S(x), S(y)) -> !EQ(x, y)
   !EQ(0', S(y)) -> False
   !EQ(S(x), 0') -> False
   !EQ(0', 0') -> True
   overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys)
   member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
   overlap[Ite][True][Ite](True, xs, ys) -> True
   member[Ite][True][Ite](True, x, xs) -> True

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

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

(16)
Obligation:
Innermost TRS:
Rules:
overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
overlap(Nil, ys) -> False
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs, ys) -> overlap(xs, ys)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys)
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
overlap[Ite][True][Ite](True, xs, ys) -> True
member[Ite][True][Ite](True, x, xs) -> True

Types:
overlap :: Cons:Nil -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
overlap[Ite][True][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> False:True
member :: S:0' -> Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
overlap, member, !EQ

They will be analysed ascendingly in the following order:
member < overlap
!EQ < member

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

(18)
Obligation:
Innermost TRS:
Rules:
overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
overlap(Nil, ys) -> False
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs, ys) -> overlap(xs, ys)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys)
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
overlap[Ite][True][Ite](True, xs, ys) -> True
member[Ite][True][Ite](True, x, xs) -> True

Types:
overlap :: Cons:Nil -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
overlap[Ite][True][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> False:True
member :: S:0' -> Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
!EQ, overlap, member

They will be analysed ascendingly in the following order:
member < overlap
!EQ < member

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> False, rt in Omega(0)

Induction Base:
!EQ(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) ->_R^Omega(0)
False

Induction Step:
!EQ(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) ->_R^Omega(0)
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) ->_IH
False

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
overlap(Nil, ys) -> False
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs, ys) -> overlap(xs, ys)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys)
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
overlap[Ite][True][Ite](True, xs, ys) -> True
member[Ite][True][Ite](True, x, xs) -> True

Types:
overlap :: Cons:Nil -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
overlap[Ite][True][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> False:True
member :: S:0' -> Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> False, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
member, overlap

They will be analysed ascendingly in the following order:
member < overlap

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(n286_0)) -> False, rt in Omega(1 + n286_0)

Induction Base:
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
False

Induction Step:
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(n286_0, 1))) ->_R^Omega(1)
member[Ite][True][Ite](!EQ(0', gen_S:0'5_0(1)), gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(n286_0))) ->_L^Omega(0)
member[Ite][True][Ite](False, gen_S:0'5_0(1), Cons(0', gen_Cons:Nil4_0(n286_0))) ->_R^Omega(0)
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(n286_0)) ->_IH
False

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Complex Obligation (BEST)

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

(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
overlap(Nil, ys) -> False
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs, ys) -> overlap(xs, ys)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys)
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
overlap[Ite][True][Ite](True, xs, ys) -> True
member[Ite][True][Ite](True, x, xs) -> True

Types:
overlap :: Cons:Nil -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
overlap[Ite][True][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> False:True
member :: S:0' -> Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> False, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
member, overlap

They will be analysed ascendingly in the following order:
member < overlap

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
overlap(Cons(x, xs), ys) -> overlap[Ite][True][Ite](member(x, ys), Cons(x, xs), ys)
overlap(Nil, ys) -> False
member(x', Cons(x, xs)) -> member[Ite][True][Ite](!EQ(x, x'), x', Cons(x, xs))
member(x, Nil) -> False
notEmpty(Cons(x, xs)) -> True
notEmpty(Nil) -> False
goal(xs, ys) -> overlap(xs, ys)
!EQ(S(x), S(y)) -> !EQ(x, y)
!EQ(0', S(y)) -> False
!EQ(S(x), 0') -> False
!EQ(0', 0') -> True
overlap[Ite][True][Ite](False, Cons(x, xs), ys) -> overlap(xs, ys)
member[Ite][True][Ite](False, x', Cons(x, xs)) -> member(x', xs)
overlap[Ite][True][Ite](True, xs, ys) -> True
member[Ite][True][Ite](True, x, xs) -> True

Types:
overlap :: Cons:Nil -> Cons:Nil -> False:True
Cons :: S:0' -> Cons:Nil -> Cons:Nil
overlap[Ite][True][Ite] :: False:True -> Cons:Nil -> Cons:Nil -> False:True
member :: S:0' -> Cons:Nil -> False:True
Nil :: Cons:Nil
False :: False:True
member[Ite][True][Ite] :: False:True -> S:0' -> Cons:Nil -> False:True
!EQ :: S:0' -> S:0' -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
S :: S:0' -> S:0'
0' :: S:0'
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
hole_S:0'3_0 :: S:0'
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
!EQ(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> False, rt in Omega(0)
member(gen_S:0'5_0(1), gen_Cons:Nil4_0(n286_0)) -> False, rt in Omega(1 + n286_0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
overlap
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
overlap(gen_Cons:Nil4_0(n633_0), gen_Cons:Nil4_0(0)) -> False, rt in Omega(1 + n633_0)

Induction Base:
overlap(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
False

Induction Step:
overlap(gen_Cons:Nil4_0(+(n633_0, 1)), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
overlap[Ite][True][Ite](member(0', gen_Cons:Nil4_0(0)), Cons(0', gen_Cons:Nil4_0(n633_0)), gen_Cons:Nil4_0(0)) ->_R^Omega(1)
overlap[Ite][True][Ite](False, Cons(0', gen_Cons:Nil4_0(n633_0)), gen_Cons:Nil4_0(0)) ->_R^Omega(0)
overlap(gen_Cons:Nil4_0(n633_0), gen_Cons:Nil4_0(0)) ->_IH
False

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(28)
BOUNDS(1, INF)