/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^3), O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 4 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 208 ms]
    (10) BOUNDS(1, n^3)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) SlicingProof [LOWER BOUND(ID), 0 ms]
    (14) CpxTRS
    (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), 278 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 16 ms]
            (26) BEST
                (27) proven lower bound
                    (28) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (29) BOUNDS(n^2, INF)
                (30) typed CpxTrs
                    (31) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
                    (32) proven lower bound
                    (33) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (34) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   shuffle(Cons(x, xs)) -> Cons(x, shuffle(reverse(xs)))
   reverse(Cons(x, xs)) -> append(reverse(xs), Cons(x, Nil))
   append(Cons(x, xs), ys) -> Cons(x, append(xs, ys))
   shuffle(Nil) -> Nil
   reverse(Nil) -> Nil
   append(Nil, ys) -> ys
   goal(xs) -> shuffle(xs)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   shuffle(Cons(x, xs)) -> Cons(x, shuffle(reverse(xs))) [1]
   reverse(Cons(x, xs)) -> append(reverse(xs), Cons(x, Nil)) [1]
   append(Cons(x, xs), ys) -> Cons(x, append(xs, ys)) [1]
   shuffle(Nil) -> Nil [1]
   reverse(Nil) -> Nil [1]
   append(Nil, ys) -> ys [1]
   goal(xs) -> shuffle(xs) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   shuffle(Cons(x, xs)) -> Cons(x, shuffle(reverse(xs))) [1]
   reverse(Cons(x, xs)) -> append(reverse(xs), Cons(x, Nil)) [1]
   append(Cons(x, xs), ys) -> Cons(x, append(xs, ys)) [1]
   shuffle(Nil) -> Nil [1]
   reverse(Nil) -> Nil [1]
   append(Nil, ys) -> ys [1]
   goal(xs) -> shuffle(xs) [1]

The TRS has the following type information:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil

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

(5) 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:
none

And the following fresh constants:    const

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

(6)
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:

   shuffle(Cons(x, xs)) -> Cons(x, shuffle(reverse(xs))) [1]
   reverse(Cons(x, xs)) -> append(reverse(xs), Cons(x, Nil)) [1]
   append(Cons(x, xs), ys) -> Cons(x, append(xs, ys)) [1]
   shuffle(Nil) -> Nil [1]
   reverse(Nil) -> Nil [1]
   append(Nil, ys) -> ys [1]
   goal(xs) -> shuffle(xs) [1]

The TRS has the following type information:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: a -> Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
const :: a

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

(7) 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
   const => 0

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

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   append(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0
   append(z, z') -{ 1 }-> 1 + x + append(xs, ys) :|: z = 1 + x + xs, xs >= 0, z' = ys, ys >= 0, x >= 0
   goal(z) -{ 1 }-> shuffle(xs) :|: xs >= 0, z = xs
   reverse(z) -{ 1 }-> append(reverse(xs), 1 + x + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 1 + x + shuffle(reverse(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0

Only complete derivations are relevant for the runtime complexity.

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

(9) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V, V5),0,[shuffle(V, Out)],[V >= 0]).
eq(start(V, V5),0,[reverse(V, Out)],[V >= 0]).
eq(start(V, V5),0,[append(V, V5, Out)],[V >= 0,V5 >= 0]).
eq(start(V, V5),0,[goal(V, Out)],[V >= 0]).
eq(shuffle(V, Out),1,[reverse(V1, Ret10),shuffle(Ret10, Ret1)],[Out = 1 + Ret1 + V2,V = 1 + V1 + V2,V1 >= 0,V2 >= 0]).
eq(reverse(V, Out),1,[reverse(V4, Ret0),append(Ret0, 1 + V3 + 0, Ret)],[Out = Ret,V = 1 + V3 + V4,V4 >= 0,V3 >= 0]).
eq(append(V, V5, Out),1,[append(V8, V7, Ret11)],[Out = 1 + Ret11 + V6,V = 1 + V6 + V8,V8 >= 0,V5 = V7,V7 >= 0,V6 >= 0]).
eq(shuffle(V, Out),1,[],[Out = 0,V = 0]).
eq(reverse(V, Out),1,[],[Out = 0,V = 0]).
eq(append(V, V5, Out),1,[],[Out = V9,V5 = V9,V9 >= 0,V = 0]).
eq(goal(V, Out),1,[shuffle(V10, Ret2)],[Out = Ret2,V10 >= 0,V = V10]).
input_output_vars(shuffle(V,Out),[V],[Out]).
input_output_vars(reverse(V,Out),[V],[Out]).
input_output_vars(append(V,V5,Out),[V,V5],[Out]).
input_output_vars(goal(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [append/3]
1. recursive [non_tail] : [reverse/2]
2. recursive  : [shuffle/2]
3. non_recursive  : [goal/2]
4. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into append/3
1. SCC is partially evaluated into reverse/2
2. SCC is partially evaluated into shuffle/2
3. SCC is completely evaluated into other SCCs
4. SCC is partially evaluated into start/2

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

### Specialization of cost equations append/3 
* CE 10 is refined into CE [11] 
* CE 9 is refined into CE [12] 


### Cost equations --> "Loop" of append/3 
* CEs [12] --> Loop 8 
* CEs [11] --> Loop 9 

### Ranking functions of CR append(V,V5,Out) 
* RF of phase [8]: [V]

#### Partial ranking functions of CR append(V,V5,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1]:
    V


### Specialization of cost equations reverse/2 
* CE 8 is refined into CE [13] 
* CE 7 is refined into CE [14,15] 


### Cost equations --> "Loop" of reverse/2 
* CEs [15] --> Loop 10 
* CEs [14] --> Loop 11 
* CEs [13] --> Loop 12 

### Ranking functions of CR reverse(V,Out) 
* RF of phase [10]: [V]

#### Partial ranking functions of CR reverse(V,Out) 
* Partial RF of phase [10]:
  - RF of loop [10:1]:
    V


### Specialization of cost equations shuffle/2 
* CE 6 is refined into CE [16] 
* CE 5 is refined into CE [17,18] 


### Cost equations --> "Loop" of shuffle/2 
* CEs [18] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [16] --> Loop 15 

### Ranking functions of CR shuffle(V,Out) 
* RF of phase [13]: [V-1]

#### Partial ranking functions of CR shuffle(V,Out) 
* Partial RF of phase [13]:
  - RF of loop [13:1]:
    V-1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [19,20] 
* CE 2 is refined into CE [21,22] 
* CE 3 is refined into CE [23,24] 
* CE 4 is refined into CE [25,26] 


### Cost equations --> "Loop" of start/2 
* CEs [20,22,24,26] --> Loop 16 
* CEs [19,21,23,25] --> Loop 17 

### Ranking functions of CR start(V,V5) 

#### Partial ranking functions of CR start(V,V5) 


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

#### Cost of chains of append(V,V5,Out):
* Chain [[8],9]: 1*it(8)+1
  Such that:it(8) =< -V5+Out

  with precondition: [V+V5=Out,V>=1,V5>=0] 

* Chain [9]: 1
  with precondition: [V=0,V5=Out,V5>=0] 


#### Cost of chains of reverse(V,Out):
* Chain [[10],11,12]: 2*it(10)+1*s(3)+3
  Such that:aux(3) =< Out
it(10) =< aux(3)
s(3) =< it(10)*aux(3)

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

* Chain [12]: 1
  with precondition: [V=0,Out=0] 

* Chain [11,12]: 3
  with precondition: [V=Out,V>=1] 


#### Cost of chains of shuffle(V,Out):
* Chain [[13],14,15]: 4*it(13)+2*s(13)+1*s(14)+3
  Such that:aux(6) =< Out
it(13) =< aux(6)
aux(4) =< aux(6)
s(15) =< it(13)*aux(4)
s(13) =< s(15)
s(14) =< s(13)*aux(6)

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

* Chain [15]: 1
  with precondition: [V=0,Out=0] 

* Chain [14,15]: 3
  with precondition: [V=Out,V>=1] 


#### Cost of chains of start(V,V5):
* Chain [17]: 2
  with precondition: [V=0] 

* Chain [16]: 11*s(23)+4*s(26)+2*s(27)+1*s(30)+4
  Such that:aux(7) =< V
s(23) =< aux(7)
s(24) =< aux(7)
s(25) =< s(23)*s(24)
s(26) =< s(25)
s(27) =< s(26)*aux(7)
s(30) =< s(23)*aux(7)

  with precondition: [V>=1] 


Closed-form bounds of start(V,V5): 
-------------------------------------
* Chain [17] with precondition: [V=0] 
    - Upper bound: 2 
    - Complexity: constant 
* Chain [16] with precondition: [V>=1] 
    - Upper bound: 11*V+4+5*V*V+2*V*V*V 
    - Complexity: n^3 

### Maximum cost of start(V,V5): 11*V+2+5*V*V+2*V*V*V+2 
Asymptotic class: n^3 
* Total analysis performed in 133 ms.


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

(10)
BOUNDS(1, n^3)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   shuffle(Cons(x, xs)) -> Cons(x, shuffle(reverse(xs)))
   reverse(Cons(x, xs)) -> append(reverse(xs), Cons(x, Nil))
   append(Cons(x, xs), ys) -> Cons(x, append(xs, ys))
   shuffle(Nil) -> Nil
   reverse(Nil) -> Nil
   append(Nil, ys) -> ys
   goal(xs) -> shuffle(xs)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   shuffle(Cons(xs)) -> Cons(shuffle(reverse(xs)))
   reverse(Cons(xs)) -> append(reverse(xs), Cons(Nil))
   append(Cons(xs), ys) -> Cons(append(xs, ys))
   shuffle(Nil) -> Nil
   reverse(Nil) -> Nil
   append(Nil, ys) -> ys
   goal(xs) -> shuffle(xs)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

(16)
Obligation:
Innermost TRS:
Rules:
shuffle(Cons(xs)) -> Cons(shuffle(reverse(xs)))
reverse(Cons(xs)) -> append(reverse(xs), Cons(Nil))
append(Cons(xs), ys) -> Cons(append(xs, ys))
shuffle(Nil) -> Nil
reverse(Nil) -> Nil
append(Nil, ys) -> ys
goal(xs) -> shuffle(xs)

Types:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
shuffle, reverse, append

They will be analysed ascendingly in the following order:
reverse < shuffle
append < reverse

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

(18)
Obligation:
Innermost TRS:
Rules:
shuffle(Cons(xs)) -> Cons(shuffle(reverse(xs)))
reverse(Cons(xs)) -> append(reverse(xs), Cons(Nil))
append(Cons(xs), ys) -> Cons(append(xs, ys))
shuffle(Nil) -> Nil
reverse(Nil) -> Nil
append(Nil, ys) -> ys
goal(xs) -> shuffle(xs)

Types:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
append, shuffle, reverse

They will be analysed ascendingly in the following order:
reverse < shuffle
append < reverse

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
append(gen_Cons:Nil2_0(0), gen_Cons:Nil2_0(b)) ->_R^Omega(1)
gen_Cons:Nil2_0(b)

Induction Step:
append(gen_Cons:Nil2_0(+(n4_0, 1)), gen_Cons:Nil2_0(b)) ->_R^Omega(1)
Cons(append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b))) ->_IH
Cons(gen_Cons:Nil2_0(+(b, c5_0)))

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

(20)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
shuffle(Cons(xs)) -> Cons(shuffle(reverse(xs)))
reverse(Cons(xs)) -> append(reverse(xs), Cons(Nil))
append(Cons(xs), ys) -> Cons(append(xs, ys))
shuffle(Nil) -> Nil
reverse(Nil) -> Nil
append(Nil, ys) -> ys
goal(xs) -> shuffle(xs)

Types:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
append, shuffle, reverse

They will be analysed ascendingly in the following order:
reverse < shuffle
append < reverse

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
shuffle(Cons(xs)) -> Cons(shuffle(reverse(xs)))
reverse(Cons(xs)) -> append(reverse(xs), Cons(Nil))
append(Cons(xs), ys) -> Cons(append(xs, ys))
shuffle(Nil) -> Nil
reverse(Nil) -> Nil
append(Nil, ys) -> ys
goal(xs) -> shuffle(xs)

Types:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
reverse, shuffle

They will be analysed ascendingly in the following order:
reverse < shuffle

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
reverse(gen_Cons:Nil2_0(n504_0)) -> gen_Cons:Nil2_0(n504_0), rt in Omega(1 + n504_0 + n504_0^2)

Induction Base:
reverse(gen_Cons:Nil2_0(0)) ->_R^Omega(1)
Nil

Induction Step:
reverse(gen_Cons:Nil2_0(+(n504_0, 1))) ->_R^Omega(1)
append(reverse(gen_Cons:Nil2_0(n504_0)), Cons(Nil)) ->_IH
append(gen_Cons:Nil2_0(c505_0), Cons(Nil)) ->_L^Omega(1 + n504_0)
gen_Cons:Nil2_0(+(n504_0, +(0, 1)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
shuffle(Cons(xs)) -> Cons(shuffle(reverse(xs)))
reverse(Cons(xs)) -> append(reverse(xs), Cons(Nil))
append(Cons(xs), ys) -> Cons(append(xs, ys))
shuffle(Nil) -> Nil
reverse(Nil) -> Nil
append(Nil, ys) -> ys
goal(xs) -> shuffle(xs)

Types:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
reverse, shuffle

They will be analysed ascendingly in the following order:
reverse < shuffle

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

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

(29)
BOUNDS(n^2, INF)

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

(30)
Obligation:
Innermost TRS:
Rules:
shuffle(Cons(xs)) -> Cons(shuffle(reverse(xs)))
reverse(Cons(xs)) -> append(reverse(xs), Cons(Nil))
append(Cons(xs), ys) -> Cons(append(xs, ys))
shuffle(Nil) -> Nil
reverse(Nil) -> Nil
append(Nil, ys) -> ys
goal(xs) -> shuffle(xs)

Types:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
reverse(gen_Cons:Nil2_0(n504_0)) -> gen_Cons:Nil2_0(n504_0), rt in Omega(1 + n504_0 + n504_0^2)


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuffle(gen_Cons:Nil2_0(n719_0)) -> gen_Cons:Nil2_0(n719_0), rt in Omega(1 + n719_0 + n719_0^2 + n719_0^3)

Induction Base:
shuffle(gen_Cons:Nil2_0(0)) ->_R^Omega(1)
Nil

Induction Step:
shuffle(gen_Cons:Nil2_0(+(n719_0, 1))) ->_R^Omega(1)
Cons(shuffle(reverse(gen_Cons:Nil2_0(n719_0)))) ->_L^Omega(1 + n719_0 + n719_0^2)
Cons(shuffle(gen_Cons:Nil2_0(n719_0))) ->_IH
Cons(gen_Cons:Nil2_0(c720_0))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(32)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
shuffle(Cons(xs)) -> Cons(shuffle(reverse(xs)))
reverse(Cons(xs)) -> append(reverse(xs), Cons(Nil))
append(Cons(xs), ys) -> Cons(append(xs, ys))
shuffle(Nil) -> Nil
reverse(Nil) -> Nil
append(Nil, ys) -> ys
goal(xs) -> shuffle(xs)

Types:
shuffle :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
reverse :: Cons:Nil -> Cons:Nil
append :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat -> Cons:Nil


Lemmas:
append(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) -> gen_Cons:Nil2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
reverse(gen_Cons:Nil2_0(n504_0)) -> gen_Cons:Nil2_0(n504_0), rt in Omega(1 + n504_0 + n504_0^2)


Generator Equations:
gen_Cons:Nil2_0(0) <=> Nil
gen_Cons:Nil2_0(+(x, 1)) <=> Cons(gen_Cons:Nil2_0(x))


The following defined symbols remain to be analysed:
shuffle
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(33) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(34)
BOUNDS(n^3, INF)