WORST_CASE(Omega(n^1),O(n^1))
proof of input_ObdkSMjP8O.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 171 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [UPPER BOUND(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, 307 ms]
    (12) BOUNDS(1, n^1)
(13) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
    (15) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 404 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 104 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 62 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 37 ms]
            (34) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

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

(2)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (UPPER BOUND(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 232 ms.


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

(12)
BOUNDS(1, n^1)

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

(13) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
   lte(Cons(z0, z1), Nil) -> False
   lte(Nil, z0) -> True
   even(Cons(z0, Nil)) -> False
   even(Cons(z0, Cons(z1, z2))) -> even(z2)
   even(Nil) -> True
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0, z1) -> and(lte(z0, z1), even(z0))
Tuples:
   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   LTE(Cons(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
   LTE(Cons(z0, z1), Nil) -> c5
   LTE(Nil, z0) -> c6
   EVEN(Cons(z0, Nil)) -> c7
   EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
   EVEN(Nil) -> c9
   NOTEMPTY(Cons(z0, z1)) -> c10
   NOTEMPTY(Nil) -> c11
   GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
   GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))
S tuples:
   LTE(Cons(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
   LTE(Cons(z0, z1), Nil) -> c5
   LTE(Nil, z0) -> c6
   EVEN(Cons(z0, Nil)) -> c7
   EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
   EVEN(Nil) -> c9
   NOTEMPTY(Cons(z0, z1)) -> c10
   NOTEMPTY(Nil) -> c11
   GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
   GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))
K tuples:none
Defined Rule Symbols:   lte_2, even_1, notEmpty_1, goal_2, and_2

Defined Pair Symbols:   AND_2, LTE_2, EVEN_1, NOTEMPTY_1, GOAL_2

Compound Symbols:   c, c1, c2, c3, c4_1, c5, c6, c7, c8_1, c9, c10, c11, c12_2, c13_2


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

(15) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(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(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
   LTE(Cons(z0, z1), Nil) -> c5
   LTE(Nil, z0) -> c6
   EVEN(Cons(z0, Nil)) -> c7
   EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
   EVEN(Nil) -> c9
   NOTEMPTY(Cons(z0, z1)) -> c10
   NOTEMPTY(Nil) -> c11
   GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
   GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
   lte(Cons(z0, z1), Nil) -> False
   lte(Nil, z0) -> True
   even(Cons(z0, Nil)) -> False
   even(Cons(z0, Cons(z1, z2))) -> even(z2)
   even(Nil) -> True
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0, z1) -> and(lte(z0, z1), even(z0))

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

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

(18)
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(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
   LTE(Cons(z0, z1), Nil) -> c5
   LTE(Nil, z0) -> c6
   EVEN(Cons(z0, Nil)) -> c7
   EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
   EVEN(Nil) -> c9
   NOTEMPTY(Cons(z0, z1)) -> c10
   NOTEMPTY(Nil) -> c11
   GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
   GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
   lte(Cons(z0, z1), Nil) -> False
   lte(Nil, z0) -> True
   even(Cons(z0, Nil)) -> False
   even(Cons(z0, Cons(z1, z2))) -> even(z2)
   even(Nil) -> True
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0, z1) -> and(lte(z0, z1), even(z0))

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

(19) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
LTE(Cons(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
LTE(Cons(z0, z1), Nil) -> c5
LTE(Nil, z0) -> c6
EVEN(Cons(z0, Nil)) -> c7
EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
EVEN(Nil) -> c9
NOTEMPTY(Cons(z0, z1)) -> c10
NOTEMPTY(Nil) -> c11
GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
lte(Cons(z0, z1), Nil) -> False
lte(Nil, z0) -> True
even(Cons(z0, Nil)) -> False
even(Cons(z0, Cons(z1, z2))) -> even(z2)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0, z1) -> and(lte(z0, z1), even(z0))

Types:
LTE :: Cons:Nil -> Cons:Nil -> c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c4 :: c4:c5:c6 -> c4:c5:c6
Nil :: Cons:Nil
c5 :: c4:c5:c6
c6 :: c4:c5:c6
EVEN :: Cons:Nil -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9 -> c7:c8:c9
c9 :: c7:c8:c9
NOTEMPTY :: Cons:Nil -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
GOAL :: Cons:Nil -> Cons:Nil -> c12:c13
c12 :: c:c1:c2:c3 -> c4:c5:c6 -> c12:c13
AND :: False:True -> False:True -> c:c1:c2:c3
lte :: Cons:Nil -> Cons:Nil -> False:True
even :: Cons:Nil -> False:True
c13 :: c:c1:c2:c3 -> c7:c8:c9 -> c12:c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
notEmpty :: Cons:Nil -> False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
hole_c4:c5:c61_14 :: c4:c5:c6
hole_Cons:Nil2_14 :: Cons:Nil
hole_a3_14 :: a
hole_c7:c8:c94_14 :: c7:c8:c9
hole_c10:c115_14 :: c10:c11
hole_c12:c136_14 :: c12:c13
hole_c:c1:c2:c37_14 :: c:c1:c2:c3
hole_False:True8_14 :: False:True
gen_c4:c5:c69_14 :: Nat -> c4:c5:c6
gen_Cons:Nil10_14 :: Nat -> Cons:Nil
gen_c7:c8:c911_14 :: Nat -> c7:c8:c9

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

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

(22)
Obligation:
Innermost TRS:
Rules:
LTE(Cons(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
LTE(Cons(z0, z1), Nil) -> c5
LTE(Nil, z0) -> c6
EVEN(Cons(z0, Nil)) -> c7
EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
EVEN(Nil) -> c9
NOTEMPTY(Cons(z0, z1)) -> c10
NOTEMPTY(Nil) -> c11
GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
lte(Cons(z0, z1), Nil) -> False
lte(Nil, z0) -> True
even(Cons(z0, Nil)) -> False
even(Cons(z0, Cons(z1, z2))) -> even(z2)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0, z1) -> and(lte(z0, z1), even(z0))

Types:
LTE :: Cons:Nil -> Cons:Nil -> c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c4 :: c4:c5:c6 -> c4:c5:c6
Nil :: Cons:Nil
c5 :: c4:c5:c6
c6 :: c4:c5:c6
EVEN :: Cons:Nil -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9 -> c7:c8:c9
c9 :: c7:c8:c9
NOTEMPTY :: Cons:Nil -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
GOAL :: Cons:Nil -> Cons:Nil -> c12:c13
c12 :: c:c1:c2:c3 -> c4:c5:c6 -> c12:c13
AND :: False:True -> False:True -> c:c1:c2:c3
lte :: Cons:Nil -> Cons:Nil -> False:True
even :: Cons:Nil -> False:True
c13 :: c:c1:c2:c3 -> c7:c8:c9 -> c12:c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
notEmpty :: Cons:Nil -> False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
hole_c4:c5:c61_14 :: c4:c5:c6
hole_Cons:Nil2_14 :: Cons:Nil
hole_a3_14 :: a
hole_c7:c8:c94_14 :: c7:c8:c9
hole_c10:c115_14 :: c10:c11
hole_c12:c136_14 :: c12:c13
hole_c:c1:c2:c37_14 :: c:c1:c2:c3
hole_False:True8_14 :: False:True
gen_c4:c5:c69_14 :: Nat -> c4:c5:c6
gen_Cons:Nil10_14 :: Nat -> Cons:Nil
gen_c7:c8:c911_14 :: Nat -> c7:c8:c9


Generator Equations:
gen_c4:c5:c69_14(0) <=> c5
gen_c4:c5:c69_14(+(x, 1)) <=> c4(gen_c4:c5:c69_14(x))
gen_Cons:Nil10_14(0) <=> Nil
gen_Cons:Nil10_14(+(x, 1)) <=> Cons(hole_a3_14, gen_Cons:Nil10_14(x))
gen_c7:c8:c911_14(0) <=> c7
gen_c7:c8:c911_14(+(x, 1)) <=> c8(gen_c7:c8:c911_14(x))


The following defined symbols remain to be analysed:
LTE, EVEN, lte, even
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LTE(gen_Cons:Nil10_14(+(1, n13_14)), gen_Cons:Nil10_14(n13_14)) -> gen_c4:c5:c69_14(n13_14), rt in Omega(1 + n13_14)

Induction Base:
LTE(gen_Cons:Nil10_14(+(1, 0)), gen_Cons:Nil10_14(0)) ->_R^Omega(1)
c5

Induction Step:
LTE(gen_Cons:Nil10_14(+(1, +(n13_14, 1))), gen_Cons:Nil10_14(+(n13_14, 1))) ->_R^Omega(1)
c4(LTE(gen_Cons:Nil10_14(+(1, n13_14)), gen_Cons:Nil10_14(n13_14))) ->_IH
c4(gen_c4:c5:c69_14(c14_14))

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LTE(Cons(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
LTE(Cons(z0, z1), Nil) -> c5
LTE(Nil, z0) -> c6
EVEN(Cons(z0, Nil)) -> c7
EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
EVEN(Nil) -> c9
NOTEMPTY(Cons(z0, z1)) -> c10
NOTEMPTY(Nil) -> c11
GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
lte(Cons(z0, z1), Nil) -> False
lte(Nil, z0) -> True
even(Cons(z0, Nil)) -> False
even(Cons(z0, Cons(z1, z2))) -> even(z2)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0, z1) -> and(lte(z0, z1), even(z0))

Types:
LTE :: Cons:Nil -> Cons:Nil -> c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c4 :: c4:c5:c6 -> c4:c5:c6
Nil :: Cons:Nil
c5 :: c4:c5:c6
c6 :: c4:c5:c6
EVEN :: Cons:Nil -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9 -> c7:c8:c9
c9 :: c7:c8:c9
NOTEMPTY :: Cons:Nil -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
GOAL :: Cons:Nil -> Cons:Nil -> c12:c13
c12 :: c:c1:c2:c3 -> c4:c5:c6 -> c12:c13
AND :: False:True -> False:True -> c:c1:c2:c3
lte :: Cons:Nil -> Cons:Nil -> False:True
even :: Cons:Nil -> False:True
c13 :: c:c1:c2:c3 -> c7:c8:c9 -> c12:c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
notEmpty :: Cons:Nil -> False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
hole_c4:c5:c61_14 :: c4:c5:c6
hole_Cons:Nil2_14 :: Cons:Nil
hole_a3_14 :: a
hole_c7:c8:c94_14 :: c7:c8:c9
hole_c10:c115_14 :: c10:c11
hole_c12:c136_14 :: c12:c13
hole_c:c1:c2:c37_14 :: c:c1:c2:c3
hole_False:True8_14 :: False:True
gen_c4:c5:c69_14 :: Nat -> c4:c5:c6
gen_Cons:Nil10_14 :: Nat -> Cons:Nil
gen_c7:c8:c911_14 :: Nat -> c7:c8:c9


Generator Equations:
gen_c4:c5:c69_14(0) <=> c5
gen_c4:c5:c69_14(+(x, 1)) <=> c4(gen_c4:c5:c69_14(x))
gen_Cons:Nil10_14(0) <=> Nil
gen_Cons:Nil10_14(+(x, 1)) <=> Cons(hole_a3_14, gen_Cons:Nil10_14(x))
gen_c7:c8:c911_14(0) <=> c7
gen_c7:c8:c911_14(+(x, 1)) <=> c8(gen_c7:c8:c911_14(x))


The following defined symbols remain to be analysed:
LTE, EVEN, lte, even
----------------------------------------

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
LTE(Cons(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
LTE(Cons(z0, z1), Nil) -> c5
LTE(Nil, z0) -> c6
EVEN(Cons(z0, Nil)) -> c7
EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
EVEN(Nil) -> c9
NOTEMPTY(Cons(z0, z1)) -> c10
NOTEMPTY(Nil) -> c11
GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
lte(Cons(z0, z1), Nil) -> False
lte(Nil, z0) -> True
even(Cons(z0, Nil)) -> False
even(Cons(z0, Cons(z1, z2))) -> even(z2)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0, z1) -> and(lte(z0, z1), even(z0))

Types:
LTE :: Cons:Nil -> Cons:Nil -> c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c4 :: c4:c5:c6 -> c4:c5:c6
Nil :: Cons:Nil
c5 :: c4:c5:c6
c6 :: c4:c5:c6
EVEN :: Cons:Nil -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9 -> c7:c8:c9
c9 :: c7:c8:c9
NOTEMPTY :: Cons:Nil -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
GOAL :: Cons:Nil -> Cons:Nil -> c12:c13
c12 :: c:c1:c2:c3 -> c4:c5:c6 -> c12:c13
AND :: False:True -> False:True -> c:c1:c2:c3
lte :: Cons:Nil -> Cons:Nil -> False:True
even :: Cons:Nil -> False:True
c13 :: c:c1:c2:c3 -> c7:c8:c9 -> c12:c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
notEmpty :: Cons:Nil -> False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
hole_c4:c5:c61_14 :: c4:c5:c6
hole_Cons:Nil2_14 :: Cons:Nil
hole_a3_14 :: a
hole_c7:c8:c94_14 :: c7:c8:c9
hole_c10:c115_14 :: c10:c11
hole_c12:c136_14 :: c12:c13
hole_c:c1:c2:c37_14 :: c:c1:c2:c3
hole_False:True8_14 :: False:True
gen_c4:c5:c69_14 :: Nat -> c4:c5:c6
gen_Cons:Nil10_14 :: Nat -> Cons:Nil
gen_c7:c8:c911_14 :: Nat -> c7:c8:c9


Lemmas:
LTE(gen_Cons:Nil10_14(+(1, n13_14)), gen_Cons:Nil10_14(n13_14)) -> gen_c4:c5:c69_14(n13_14), rt in Omega(1 + n13_14)


Generator Equations:
gen_c4:c5:c69_14(0) <=> c5
gen_c4:c5:c69_14(+(x, 1)) <=> c4(gen_c4:c5:c69_14(x))
gen_Cons:Nil10_14(0) <=> Nil
gen_Cons:Nil10_14(+(x, 1)) <=> Cons(hole_a3_14, gen_Cons:Nil10_14(x))
gen_c7:c8:c911_14(0) <=> c7
gen_c7:c8:c911_14(+(x, 1)) <=> c8(gen_c7:c8:c911_14(x))


The following defined symbols remain to be analysed:
EVEN, lte, even
----------------------------------------

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EVEN(gen_Cons:Nil10_14(+(1, *(2, n596_14)))) -> gen_c7:c8:c911_14(n596_14), rt in Omega(1 + n596_14)

Induction Base:
EVEN(gen_Cons:Nil10_14(+(1, *(2, 0)))) ->_R^Omega(1)
c7

Induction Step:
EVEN(gen_Cons:Nil10_14(+(1, *(2, +(n596_14, 1))))) ->_R^Omega(1)
c8(EVEN(gen_Cons:Nil10_14(+(1, *(2, n596_14))))) ->_IH
c8(gen_c7:c8:c911_14(c597_14))

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

(30)
Obligation:
Innermost TRS:
Rules:
LTE(Cons(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
LTE(Cons(z0, z1), Nil) -> c5
LTE(Nil, z0) -> c6
EVEN(Cons(z0, Nil)) -> c7
EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
EVEN(Nil) -> c9
NOTEMPTY(Cons(z0, z1)) -> c10
NOTEMPTY(Nil) -> c11
GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
lte(Cons(z0, z1), Nil) -> False
lte(Nil, z0) -> True
even(Cons(z0, Nil)) -> False
even(Cons(z0, Cons(z1, z2))) -> even(z2)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0, z1) -> and(lte(z0, z1), even(z0))

Types:
LTE :: Cons:Nil -> Cons:Nil -> c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c4 :: c4:c5:c6 -> c4:c5:c6
Nil :: Cons:Nil
c5 :: c4:c5:c6
c6 :: c4:c5:c6
EVEN :: Cons:Nil -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9 -> c7:c8:c9
c9 :: c7:c8:c9
NOTEMPTY :: Cons:Nil -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
GOAL :: Cons:Nil -> Cons:Nil -> c12:c13
c12 :: c:c1:c2:c3 -> c4:c5:c6 -> c12:c13
AND :: False:True -> False:True -> c:c1:c2:c3
lte :: Cons:Nil -> Cons:Nil -> False:True
even :: Cons:Nil -> False:True
c13 :: c:c1:c2:c3 -> c7:c8:c9 -> c12:c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
notEmpty :: Cons:Nil -> False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
hole_c4:c5:c61_14 :: c4:c5:c6
hole_Cons:Nil2_14 :: Cons:Nil
hole_a3_14 :: a
hole_c7:c8:c94_14 :: c7:c8:c9
hole_c10:c115_14 :: c10:c11
hole_c12:c136_14 :: c12:c13
hole_c:c1:c2:c37_14 :: c:c1:c2:c3
hole_False:True8_14 :: False:True
gen_c4:c5:c69_14 :: Nat -> c4:c5:c6
gen_Cons:Nil10_14 :: Nat -> Cons:Nil
gen_c7:c8:c911_14 :: Nat -> c7:c8:c9


Lemmas:
LTE(gen_Cons:Nil10_14(+(1, n13_14)), gen_Cons:Nil10_14(n13_14)) -> gen_c4:c5:c69_14(n13_14), rt in Omega(1 + n13_14)
EVEN(gen_Cons:Nil10_14(+(1, *(2, n596_14)))) -> gen_c7:c8:c911_14(n596_14), rt in Omega(1 + n596_14)


Generator Equations:
gen_c4:c5:c69_14(0) <=> c5
gen_c4:c5:c69_14(+(x, 1)) <=> c4(gen_c4:c5:c69_14(x))
gen_Cons:Nil10_14(0) <=> Nil
gen_Cons:Nil10_14(+(x, 1)) <=> Cons(hole_a3_14, gen_Cons:Nil10_14(x))
gen_c7:c8:c911_14(0) <=> c7
gen_c7:c8:c911_14(+(x, 1)) <=> c8(gen_c7:c8:c911_14(x))


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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lte(gen_Cons:Nil10_14(+(1, n1108_14)), gen_Cons:Nil10_14(n1108_14)) -> False, rt in Omega(0)

Induction Base:
lte(gen_Cons:Nil10_14(+(1, 0)), gen_Cons:Nil10_14(0)) ->_R^Omega(0)
False

Induction Step:
lte(gen_Cons:Nil10_14(+(1, +(n1108_14, 1))), gen_Cons:Nil10_14(+(n1108_14, 1))) ->_R^Omega(0)
lte(gen_Cons:Nil10_14(+(1, n1108_14)), gen_Cons:Nil10_14(n1108_14)) ->_IH
False

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

(32)
Obligation:
Innermost TRS:
Rules:
LTE(Cons(z0, z1), Cons(z2, z3)) -> c4(LTE(z1, z3))
LTE(Cons(z0, z1), Nil) -> c5
LTE(Nil, z0) -> c6
EVEN(Cons(z0, Nil)) -> c7
EVEN(Cons(z0, Cons(z1, z2))) -> c8(EVEN(z2))
EVEN(Nil) -> c9
NOTEMPTY(Cons(z0, z1)) -> c10
NOTEMPTY(Nil) -> c11
GOAL(z0, z1) -> c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1))
GOAL(z0, z1) -> c13(AND(lte(z0, z1), even(z0)), EVEN(z0))
AND(False, False) -> c
AND(True, False) -> c1
AND(False, True) -> c2
AND(True, True) -> c3
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True
lte(Cons(z0, z1), Cons(z2, z3)) -> lte(z1, z3)
lte(Cons(z0, z1), Nil) -> False
lte(Nil, z0) -> True
even(Cons(z0, Nil)) -> False
even(Cons(z0, Cons(z1, z2))) -> even(z2)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0, z1) -> and(lte(z0, z1), even(z0))

Types:
LTE :: Cons:Nil -> Cons:Nil -> c4:c5:c6
Cons :: a -> Cons:Nil -> Cons:Nil
c4 :: c4:c5:c6 -> c4:c5:c6
Nil :: Cons:Nil
c5 :: c4:c5:c6
c6 :: c4:c5:c6
EVEN :: Cons:Nil -> c7:c8:c9
c7 :: c7:c8:c9
c8 :: c7:c8:c9 -> c7:c8:c9
c9 :: c7:c8:c9
NOTEMPTY :: Cons:Nil -> c10:c11
c10 :: c10:c11
c11 :: c10:c11
GOAL :: Cons:Nil -> Cons:Nil -> c12:c13
c12 :: c:c1:c2:c3 -> c4:c5:c6 -> c12:c13
AND :: False:True -> False:True -> c:c1:c2:c3
lte :: Cons:Nil -> Cons:Nil -> False:True
even :: Cons:Nil -> False:True
c13 :: c:c1:c2:c3 -> c7:c8:c9 -> c12:c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
notEmpty :: Cons:Nil -> False:True
goal :: Cons:Nil -> Cons:Nil -> False:True
hole_c4:c5:c61_14 :: c4:c5:c6
hole_Cons:Nil2_14 :: Cons:Nil
hole_a3_14 :: a
hole_c7:c8:c94_14 :: c7:c8:c9
hole_c10:c115_14 :: c10:c11
hole_c12:c136_14 :: c12:c13
hole_c:c1:c2:c37_14 :: c:c1:c2:c3
hole_False:True8_14 :: False:True
gen_c4:c5:c69_14 :: Nat -> c4:c5:c6
gen_Cons:Nil10_14 :: Nat -> Cons:Nil
gen_c7:c8:c911_14 :: Nat -> c7:c8:c9


Lemmas:
LTE(gen_Cons:Nil10_14(+(1, n13_14)), gen_Cons:Nil10_14(n13_14)) -> gen_c4:c5:c69_14(n13_14), rt in Omega(1 + n13_14)
EVEN(gen_Cons:Nil10_14(+(1, *(2, n596_14)))) -> gen_c7:c8:c911_14(n596_14), rt in Omega(1 + n596_14)
lte(gen_Cons:Nil10_14(+(1, n1108_14)), gen_Cons:Nil10_14(n1108_14)) -> False, rt in Omega(0)


Generator Equations:
gen_c4:c5:c69_14(0) <=> c5
gen_c4:c5:c69_14(+(x, 1)) <=> c4(gen_c4:c5:c69_14(x))
gen_Cons:Nil10_14(0) <=> Nil
gen_Cons:Nil10_14(+(x, 1)) <=> Cons(hole_a3_14, gen_Cons:Nil10_14(x))
gen_c7:c8:c911_14(0) <=> c7
gen_c7:c8:c911_14(+(x, 1)) <=> c8(gen_c7:c8:c911_14(x))


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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_Cons:Nil10_14(+(1, *(2, n1475_14)))) -> False, rt in Omega(0)

Induction Base:
even(gen_Cons:Nil10_14(+(1, *(2, 0)))) ->_R^Omega(0)
False

Induction Step:
even(gen_Cons:Nil10_14(+(1, *(2, +(n1475_14, 1))))) ->_R^Omega(0)
even(gen_Cons:Nil10_14(+(1, *(2, n1475_14)))) ->_IH
False

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

(34)
BOUNDS(1, INF)