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


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


WORST_CASE(Omega(n^3), O(n^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 186 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), 302 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), 28 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:

   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(nil) -> nil
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

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:

   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [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:

   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1]

The TRS has the following type information:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: a -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add

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:

   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1]

The TRS has the following type information:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: a -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
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:

   app(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   app(z, z') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, x >= 0, y >= 0, z = 1 + n + x, z' = y
   reverse(z) -{ 1 }-> app(reverse(x), 1 + n + 0) :|: n >= 0, x >= 0, z = 1 + n + x
   reverse(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 0 :|: z = 0
   shuffle(z) -{ 1 }-> 1 + n + shuffle(reverse(x)) :|: n >= 0, x >= 0, z = 1 + n + x

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


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

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

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into app/3
1. SCC is partially evaluated into reverse/2
2. SCC is partially evaluated into shuffle/2
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations app/3 
* CE 5 is refined into CE [10] 
* CE 4 is refined into CE [11] 


### Cost equations --> "Loop" of app/3 
* CEs [11] --> Loop 8 
* CEs [10] --> Loop 9 

### Ranking functions of CR app(V1,V,Out) 
* RF of phase [9]: [V1]

#### Partial ranking functions of CR app(V1,V,Out) 
* Partial RF of phase [9]:
  - RF of loop [9:1]:
    V1


### Specialization of cost equations reverse/2 
* CE 7 is refined into CE [12,13] 
* CE 6 is refined into CE [14] 


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

### Ranking functions of CR reverse(V1,Out) 
* RF of phase [11]: [V1]

#### Partial ranking functions of CR reverse(V1,Out) 
* Partial RF of phase [11]:
  - RF of loop [11:1]:
    V1


### Specialization of cost equations shuffle/2 
* CE 9 is refined into CE [15,16] 
* CE 8 is refined into CE [17] 


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

### Ranking functions of CR shuffle(V1,Out) 
* RF of phase [14]: [V1-1]

#### Partial ranking functions of CR shuffle(V1,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V1-1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [18,19] 
* CE 2 is refined into CE [20,21] 
* CE 3 is refined into CE [22,23] 


### Cost equations --> "Loop" of start/2 
* CEs [19,21,23] --> Loop 16 
* CEs [18,20,22] --> Loop 17 

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

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


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

#### Cost of chains of app(V1,V,Out):
* Chain [[9],8]: 1*it(9)+1
  Such that:it(9) =< -V+Out

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

* Chain [8]: 1
  with precondition: [V1=0,V=Out,V>=0] 


#### Cost of chains of reverse(V1,Out):
* Chain [[11],12,10]: 2*it(11)+1*s(3)+3
  Such that:aux(3) =< Out
it(11) =< aux(3)
s(3) =< it(11)*aux(3)

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

* Chain [12,10]: 3
  with precondition: [V1=Out,V1>=1] 

* Chain [10]: 1
  with precondition: [V1=0,Out=0] 


#### Cost of chains of shuffle(V1,Out):
* Chain [[14],15,13]: 4*it(14)+2*s(13)+1*s(14)+3
  Such that:aux(6) =< Out
it(14) =< aux(6)
aux(4) =< aux(6)
s(15) =< it(14)*aux(4)
s(13) =< s(15)
s(14) =< s(13)*aux(6)

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

* Chain [15,13]: 3
  with precondition: [V1=Out,V1>=1] 

* Chain [13]: 1
  with precondition: [V1=0,Out=0] 


#### Cost of chains of start(V1,V):
* Chain [17]: 1
  with precondition: [V1=0] 

* Chain [16]: 7*s(22)+1*s(25)+2*s(30)+1*s(31)+3
  Such that:aux(7) =< V1
s(22) =< aux(7)
s(25) =< s(22)*aux(7)
s(28) =< aux(7)
s(29) =< s(22)*s(28)
s(30) =< s(29)
s(31) =< s(30)*aux(7)

  with precondition: [V1>=1] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [17] with precondition: [V1=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [16] with precondition: [V1>=1] 
    - Upper bound: 7*V1+3+3*V1*V1+V1*V1*V1 
    - Complexity: n^3 

### Maximum cost of start(V1,V): 7*V1+2+3*V1*V1+V1*V1*V1+1 
Asymptotic class: n^3 
* Total analysis performed in 128 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:

   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(nil) -> nil
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

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

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

   app(nil, y) -> y
   app(add(x), y) -> add(app(x, y))
   reverse(nil) -> nil
   reverse(add(x)) -> app(reverse(x), add(nil))
   shuffle(nil) -> nil
   shuffle(add(x)) -> add(shuffle(reverse(x)))

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

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

(16)
Obligation:
Innermost TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add

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

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

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

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

(18)
Obligation:
Innermost TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
app(gen_nil:add2_0(0), gen_nil:add2_0(b)) ->_R^Omega(1)
gen_nil:add2_0(b)

Induction Step:
app(gen_nil:add2_0(+(n4_0, 1)), gen_nil:add2_0(b)) ->_R^Omega(1)
add(app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b))) ->_IH
add(gen_nil:add2_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:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


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

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

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_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_nil:add2_0(n487_0)) -> gen_nil:add2_0(n487_0), rt in Omega(1 + n487_0 + n487_0^2)

Induction Base:
reverse(gen_nil:add2_0(0)) ->_R^Omega(1)
nil

Induction Step:
reverse(gen_nil:add2_0(+(n487_0, 1))) ->_R^Omega(1)
app(reverse(gen_nil:add2_0(n487_0)), add(nil)) ->_IH
app(gen_nil:add2_0(c488_0), add(nil)) ->_L^Omega(1 + n487_0)
gen_nil:add2_0(+(n487_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:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_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:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
reverse(gen_nil:add2_0(n487_0)) -> gen_nil:add2_0(n487_0), rt in Omega(1 + n487_0 + n487_0^2)


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuffle(gen_nil:add2_0(n701_0)) -> gen_nil:add2_0(n701_0), rt in Omega(1 + n701_0 + n701_0^2 + n701_0^3)

Induction Base:
shuffle(gen_nil:add2_0(0)) ->_R^Omega(1)
nil

Induction Step:
shuffle(gen_nil:add2_0(+(n701_0, 1))) ->_R^Omega(1)
add(shuffle(reverse(gen_nil:add2_0(n701_0)))) ->_L^Omega(1 + n701_0 + n701_0^2)
add(shuffle(gen_nil:add2_0(n701_0))) ->_IH
add(gen_nil:add2_0(c702_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:
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))

Types:
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
hole_nil:add1_0 :: nil:add
gen_nil:add2_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add2_0(n4_0), gen_nil:add2_0(b)) -> gen_nil:add2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
reverse(gen_nil:add2_0(n487_0)) -> gen_nil:add2_0(n487_0), rt in Omega(1 + n487_0 + n487_0^2)


Generator Equations:
gen_nil:add2_0(0) <=> nil
gen_nil:add2_0(+(x, 1)) <=> add(gen_nil:add2_0(x))


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

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

(34)
BOUNDS(n^3, INF)