WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 42 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, 244 ms]
    (12) BOUNDS(1, n^1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRelTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 247 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), 3 ms]
            (28) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs)
   lte(Cons(x, xs), Nil) -> False
   even(Cons(x, Nil)) -> False
   even(Cons(x', Cons(x, xs))) -> even(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lte(Nil, y) -> True
   even(Nil) -> True
   goal(x, y) -> and(lte(x, y), even(x))

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> 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^1).


The TRS R consists of the following rules:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs)
   lte(Cons(x, xs), Nil) -> False
   even(Cons(x, Nil)) -> False
   even(Cons(x', Cons(x, xs))) -> even(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lte(Nil, y) -> True
   even(Nil) -> True
   goal(x, y) -> and(lte(x, y), even(x))

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> 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^1).


The TRS R consists of the following rules:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1]
   lte(Cons(x, xs), Nil) -> False [1]
   even(Cons(x, Nil)) -> False [1]
   even(Cons(x', Cons(x, xs))) -> even(xs) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   lte(Nil, y) -> True [1]
   even(Nil) -> True [1]
   goal(x, y) -> and(lte(x, y), even(x)) [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> 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:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1]
   lte(Cons(x, xs), Nil) -> False [1]
   even(Cons(x, Nil)) -> False [1]
   even(Cons(x', Cons(x, xs))) -> even(xs) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   lte(Nil, y) -> True [1]
   even(Nil) -> True [1]
   goal(x, y) -> and(lte(x, y), even(x)) [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]

The TRS has the following type information:
lte :: Cons:Nil -> Cons:Nil -> False:True
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
and :: False:True -> False:True -> False:True

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:

   and(v0, v1) -> null_and [0]

And the following fresh constants:    null_and, const

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

(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:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs) [1]
   lte(Cons(x, xs), Nil) -> False [1]
   even(Cons(x, Nil)) -> False [1]
   even(Cons(x', Cons(x, xs))) -> even(xs) [1]
   notEmpty(Cons(x, xs)) -> True [1]
   notEmpty(Nil) -> False [1]
   lte(Nil, y) -> True [1]
   even(Nil) -> True [1]
   goal(x, y) -> and(lte(x, y), even(x)) [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
lte :: Cons:Nil -> Cons:Nil -> False:True:null_and
Cons :: a -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True:null_and
even :: Cons:Nil -> False:True:null_and
notEmpty :: Cons:Nil -> False:True:null_and
True :: False:True:null_and
goal :: Cons:Nil -> Cons:Nil -> False:True:null_and
and :: False:True:null_and -> False:True:null_and -> False:True:null_and
null_and :: False:True:null_and
const :: a

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
   null_and => 0
   const => 0

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

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

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   even(z) -{ 1 }-> even(xs) :|: xs >= 0, x' >= 0, x >= 0, z = 1 + x' + (1 + x + xs)
   even(z) -{ 1 }-> 2 :|: z = 0
   even(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x + 0
   goal(z, z') -{ 1 }-> and(lte(x, y), even(x)) :|: x >= 0, y >= 0, z = x, z' = y
   lte(z, z') -{ 1 }-> lte(xs', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   lte(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y
   lte(z, z') -{ 1 }-> 1 :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   notEmpty(z) -{ 1 }-> 2 :|: z = 1 + x + xs, xs >= 0, x >= 0
   notEmpty(z) -{ 1 }-> 1 :|: z = 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(V2, V),0,[lte(V2, V, Out)],[V2 >= 0,V >= 0]).
eq(start(V2, V),0,[even(V2, Out)],[V2 >= 0]).
eq(start(V2, V),0,[notEmpty(V2, Out)],[V2 >= 0]).
eq(start(V2, V),0,[goal(V2, V, Out)],[V2 >= 0,V >= 0]).
eq(start(V2, V),0,[and(V2, V, Out)],[V2 >= 0,V >= 0]).
eq(lte(V2, V, Out),1,[lte(V1, V3, Ret)],[Out = Ret,V3 >= 0,V = 1 + V3 + V5,V4 >= 0,V1 >= 0,V5 >= 0,V2 = 1 + V1 + V4]).
eq(lte(V2, V, Out),1,[],[Out = 1,V2 = 1 + V6 + V7,V7 >= 0,V6 >= 0,V = 0]).
eq(even(V2, Out),1,[],[Out = 1,V8 >= 0,V2 = 1 + V8]).
eq(even(V2, Out),1,[even(V10, Ret1)],[Out = Ret1,V10 >= 0,V11 >= 0,V9 >= 0,V2 = 2 + V10 + V11 + V9]).
eq(notEmpty(V2, Out),1,[],[Out = 2,V2 = 1 + V12 + V13,V13 >= 0,V12 >= 0]).
eq(notEmpty(V2, Out),1,[],[Out = 1,V2 = 0]).
eq(lte(V2, V, Out),1,[],[Out = 2,V14 >= 0,V2 = 0,V = V14]).
eq(even(V2, Out),1,[],[Out = 2,V2 = 0]).
eq(goal(V2, V, Out),1,[lte(V15, V16, Ret0),even(V15, Ret11),and(Ret0, Ret11, Ret2)],[Out = Ret2,V15 >= 0,V16 >= 0,V2 = V15,V = V16]).
eq(and(V2, V, Out),0,[],[Out = 1,V2 = 1,V = 1]).
eq(and(V2, V, Out),0,[],[Out = 1,V2 = 2,V = 1]).
eq(and(V2, V, Out),0,[],[Out = 1,V = 2,V2 = 1]).
eq(and(V2, V, Out),0,[],[Out = 2,V2 = 2,V = 2]).
eq(and(V2, V, Out),0,[],[Out = 0,V18 >= 0,V17 >= 0,V2 = V18,V = V17]).
input_output_vars(lte(V2,V,Out),[V2,V],[Out]).
input_output_vars(even(V2,Out),[V2],[Out]).
input_output_vars(notEmpty(V2,Out),[V2],[Out]).
input_output_vars(goal(V2,V,Out),[V2,V],[Out]).
input_output_vars(and(V2,V,Out),[V2,V],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [and/3]
1. recursive  : [even/2]
2. recursive  : [lte/3]
3. non_recursive  : [goal/3]
4. non_recursive  : [notEmpty/2]
5. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into and/3
1. SCC is partially evaluated into even/2
2. SCC is partially evaluated into lte/3
3. SCC is partially evaluated into goal/3
4. SCC is partially evaluated into notEmpty/2
5. SCC is partially evaluated into start/2

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

### Specialization of cost equations and/3 
* CE 19 is refined into CE [20] 
* CE 18 is refined into CE [21] 
* CE 16 is refined into CE [22] 
* CE 17 is refined into CE [23] 
* CE 15 is refined into CE [24] 


### Cost equations --> "Loop" of and/3 
* CEs [20] --> Loop 16 
* CEs [21] --> Loop 17 
* CEs [22] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [24] --> Loop 20 

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

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


### Specialization of cost equations even/2 
* CE 9 is refined into CE [25] 
* CE 11 is refined into CE [26] 
* CE 10 is refined into CE [27] 


### Cost equations --> "Loop" of even/2 
* CEs [27] --> Loop 21 
* CEs [25] --> Loop 22 
* CEs [26] --> Loop 23 

### Ranking functions of CR even(V2,Out) 
* RF of phase [21]: [V2-1]

#### Partial ranking functions of CR even(V2,Out) 
* Partial RF of phase [21]:
  - RF of loop [21:1]:
    V2-1


### Specialization of cost equations lte/3 
* CE 7 is refined into CE [28] 
* CE 8 is refined into CE [29] 
* CE 6 is refined into CE [30] 


### Cost equations --> "Loop" of lte/3 
* CEs [30] --> Loop 24 
* CEs [28] --> Loop 25 
* CEs [29] --> Loop 26 

### Ranking functions of CR lte(V2,V,Out) 
* RF of phase [24]: [V,V2]

#### Partial ranking functions of CR lte(V2,V,Out) 
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V
    V2


### Specialization of cost equations goal/3 
* CE 14 is refined into CE [31,32,33,34,35,36,37,38,39,40,41,42,43,44] 


### Cost equations --> "Loop" of goal/3 
* CEs [43] --> Loop 27 
* CEs [37,39,41] --> Loop 28 
* CEs [38,40,42,44] --> Loop 29 
* CEs [33,35] --> Loop 30 
* CEs [34,36] --> Loop 31 
* CEs [31] --> Loop 32 
* CEs [32] --> Loop 33 

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

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


### Specialization of cost equations notEmpty/2 
* CE 12 is refined into CE [45] 
* CE 13 is refined into CE [46] 


### Cost equations --> "Loop" of notEmpty/2 
* CEs [45] --> Loop 34 
* CEs [46] --> Loop 35 

### Ranking functions of CR notEmpty(V2,Out) 

#### Partial ranking functions of CR notEmpty(V2,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [47,48,49,50] 
* CE 2 is refined into CE [51,52,53] 
* CE 3 is refined into CE [54,55] 
* CE 4 is refined into CE [56,57,58,59,60,61,62] 
* CE 5 is refined into CE [63,64,65,66,67] 


### Cost equations --> "Loop" of start/2 
* CEs [48,58,59] --> Loop 36 
* CEs [66] --> Loop 37 
* CEs [65] --> Loop 38 
* CEs [52,53,55,64] --> Loop 39 
* CEs [49,50,60,61,62,63,67] --> Loop 40 
* CEs [47,51,54,56,57] --> Loop 41 

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

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


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

#### Cost of chains of and(V2,V,Out):
* Chain [20]: 0
  with precondition: [V2=1,V=1,Out=1] 

* Chain [19]: 0
  with precondition: [V2=1,V=2,Out=1] 

* Chain [18]: 0
  with precondition: [V2=2,V=1,Out=1] 

* Chain [17]: 0
  with precondition: [V2=2,V=2,Out=2] 

* Chain [16]: 0
  with precondition: [Out=0,V2>=0,V>=0] 


#### Cost of chains of even(V2,Out):
* Chain [[21],23]: 1*it(21)+1
  Such that:it(21) =< V2

  with precondition: [Out=2,V2>=2] 

* Chain [[21],22]: 1*it(21)+1
  Such that:it(21) =< V2

  with precondition: [Out=1,V2>=3] 

* Chain [23]: 1
  with precondition: [V2=0,Out=2] 

* Chain [22]: 1
  with precondition: [Out=1,V2>=1] 


#### Cost of chains of lte(V2,V,Out):
* Chain [[24],26]: 1*it(24)+1
  Such that:it(24) =< V

  with precondition: [Out=2,V2>=1,V>=1] 

* Chain [[24],25]: 1*it(24)+1
  Such that:it(24) =< V

  with precondition: [Out=1,V2>=2,V>=1] 

* Chain [26]: 1
  with precondition: [V2=0,Out=2,V>=0] 

* Chain [25]: 1
  with precondition: [V=0,Out=1,V2>=1] 


#### Cost of chains of goal(V2,V,Out):
* Chain [33]: 3
  with precondition: [V2=0,Out=0,V>=0] 

* Chain [32]: 3
  with precondition: [V2=0,Out=2,V>=0] 

* Chain [31]: 2*s(2)+3
  Such that:aux(1) =< V2
s(2) =< aux(1)

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

* Chain [30]: 2*s(4)+3
  Such that:aux(2) =< V2
s(4) =< aux(2)

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

* Chain [29]: 4*s(6)+4*s(7)+3
  Such that:aux(3) =< V2
aux(4) =< V
s(7) =< aux(3)
s(6) =< aux(4)

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

* Chain [28]: 3*s(14)+3*s(15)+3
  Such that:aux(5) =< V2
aux(6) =< V
s(15) =< aux(5)
s(14) =< aux(6)

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

* Chain [27]: 1*s(20)+1*s(21)+3
  Such that:s(21) =< V2
s(20) =< V

  with precondition: [Out=2,V2>=2,V>=1] 


#### Cost of chains of notEmpty(V2,Out):
* Chain [35]: 1
  with precondition: [V2=0,Out=1] 

* Chain [34]: 1
  with precondition: [Out=2,V2>=1] 


#### Cost of chains of start(V2,V):
* Chain [41]: 3
  with precondition: [V2=0] 

* Chain [40]: 10*s(22)+8*s(26)+3
  Such that:aux(7) =< V2
aux(8) =< V
s(26) =< aux(7)
s(22) =< aux(8)

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

* Chain [39]: 2*s(34)+1
  Such that:aux(9) =< V2
s(34) =< aux(9)

  with precondition: [V2>=1] 

* Chain [38]: 0
  with precondition: [V2=2,V=1] 

* Chain [37]: 0
  with precondition: [V2=2,V=2] 

* Chain [36]: 4*s(37)+3
  Such that:aux(10) =< V2
s(37) =< aux(10)

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


Closed-form bounds of start(V2,V): 
-------------------------------------
* Chain [41] with precondition: [V2=0] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [40] with precondition: [V2>=0,V>=0] 
    - Upper bound: 8*V2+10*V+3 
    - Complexity: n 
* Chain [39] with precondition: [V2>=1] 
    - Upper bound: 2*V2+1 
    - Complexity: n 
* Chain [38] with precondition: [V2=2,V=1] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [37] with precondition: [V2=2,V=2] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [36] with precondition: [V=0,V2>=1] 
    - Upper bound: 4*V2+3 
    - Complexity: n 

### Maximum cost of start(V2,V): max([2,nat(V)*10+4*V2+(2*V2+2)+2*V2])+1 
Asymptotic class: n 
* Total analysis performed in 231 ms.


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

(12)
BOUNDS(1, n^1)

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

(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:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs)
   lte(Cons(x, xs), Nil) -> False
   even(Cons(x, Nil)) -> False
   even(Cons(x', Cons(x, xs))) -> even(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lte(Nil, y) -> True
   even(Nil) -> True
   goal(x, y) -> and(lte(x, y), even(x))

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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

(16)
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:

   lte(Cons(xs'), Cons(xs)) -> lte(xs', xs)
   lte(Cons(xs), Nil) -> False
   even(Cons(Nil)) -> False
   even(Cons(Cons(xs))) -> even(xs)
   notEmpty(Cons(xs)) -> True
   notEmpty(Nil) -> False
   lte(Nil, y) -> True
   even(Nil) -> True
   goal(x, y) -> and(lte(x, y), even(x))

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

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

(18)
Obligation:
Innermost TRS:
Rules:
lte(Cons(xs'), Cons(xs)) -> lte(xs', xs)
lte(Cons(xs), Nil) -> False
even(Cons(Nil)) -> False
even(Cons(Cons(xs))) -> even(xs)
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
lte(Nil, y) -> True
even(Nil) -> True
goal(x, y) -> and(lte(x, y), even(x))
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True

Types:
lte :: Cons:Nil -> Cons:Nil -> False:True
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
and :: False:True -> False:True -> False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
lte, even
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
lte(Cons(xs'), Cons(xs)) -> lte(xs', xs)
lte(Cons(xs), Nil) -> False
even(Cons(Nil)) -> False
even(Cons(Cons(xs))) -> even(xs)
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
lte(Nil, y) -> True
even(Nil) -> True
goal(x, y) -> and(lte(x, y), even(x))
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True

Types:
lte :: Cons:Nil -> Cons:Nil -> False:True
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
and :: False:True -> False:True -> False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
lte, even
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lte(gen_Cons:Nil3_0(+(1, n5_0)), gen_Cons:Nil3_0(n5_0)) -> False, rt in Omega(1 + n5_0)

Induction Base:
lte(gen_Cons:Nil3_0(+(1, 0)), gen_Cons:Nil3_0(0)) ->_R^Omega(1)
False

Induction Step:
lte(gen_Cons:Nil3_0(+(1, +(n5_0, 1))), gen_Cons:Nil3_0(+(n5_0, 1))) ->_R^Omega(1)
lte(gen_Cons:Nil3_0(+(1, n5_0)), gen_Cons:Nil3_0(n5_0)) ->_IH
False

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(22)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
lte(Cons(xs'), Cons(xs)) -> lte(xs', xs)
lte(Cons(xs), Nil) -> False
even(Cons(Nil)) -> False
even(Cons(Cons(xs))) -> even(xs)
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
lte(Nil, y) -> True
even(Nil) -> True
goal(x, y) -> and(lte(x, y), even(x))
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True

Types:
lte :: Cons:Nil -> Cons:Nil -> False:True
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
and :: False:True -> False:True -> False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


The following defined symbols remain to be analysed:
lte, even
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(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(25)
BOUNDS(n^1, INF)

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(26)
Obligation:
Innermost TRS:
Rules:
lte(Cons(xs'), Cons(xs)) -> lte(xs', xs)
lte(Cons(xs), Nil) -> False
even(Cons(Nil)) -> False
even(Cons(Cons(xs))) -> even(xs)
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
lte(Nil, y) -> True
even(Nil) -> True
goal(x, y) -> and(lte(x, y), even(x))
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True

Types:
lte :: Cons:Nil -> Cons:Nil -> False:True
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
False :: False:True
even :: Cons:Nil -> False:True
notEmpty :: Cons:Nil -> False:True
True :: False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
and :: False:True -> False:True -> False:True
hole_False:True1_0 :: False:True
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Lemmas:
lte(gen_Cons:Nil3_0(+(1, n5_0)), gen_Cons:Nil3_0(n5_0)) -> False, rt in Omega(1 + n5_0)


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_Cons:Nil3_0(+(1, *(2, n234_0)))) -> False, rt in Omega(1 + n234_0)

Induction Base:
even(gen_Cons:Nil3_0(+(1, *(2, 0)))) ->_R^Omega(1)
False

Induction Step:
even(gen_Cons:Nil3_0(+(1, *(2, +(n234_0, 1))))) ->_R^Omega(1)
even(gen_Cons:Nil3_0(+(1, *(2, n234_0)))) ->_IH
False

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(28)
BOUNDS(1, INF)