/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 2 ms]
(2) CpxTRS
    (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, 503 ms]
    (12) BOUNDS(1, n^3)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (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), 293 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), 87 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
            (30) BEST
                (31) proven lower bound
                    (32) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (33) BOUNDS(n^2, INF)
                (34) typed CpxTrs
                    (35) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
                    (36) BEST
                        (37) proven lower bound
                            (38) LowerBoundPropagationProof [FINISHED, 0 ms]
                            (39) BOUNDS(n^3, INF)
                        (40) typed CpxTrs
                            (41) RewriteLemmaProof [LOWER BOUND(ID), 21 ms]
                            (42) typed CpxTrs
                            (43) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
                            (44) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   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)))
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(x, leaf) -> false
   less_leaves(leaf, cons(w, z)) -> true
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
quot(s(x), s([]))


The defined contexts are:
quot([], s(x1))
shuffle([])
less_leaves([], x1)
less_leaves(x0, [])
app([], add(x1, nil))
minus([], x1)
app([], x1)
reverse([])
concat([], x1)
concat(x0, [])
app(x0, add([], nil))
app(x0, [])


[] just represents basic- or constructor-terms in the following defined contexts:
quot([], s(x1))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   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)))
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(x, leaf) -> false
   less_leaves(leaf, cons(w, z)) -> true
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

S is empty.
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^3).


The TRS R consists of the following rules:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   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]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(x, leaf) -> false [1]
   less_leaves(leaf, cons(w, z)) -> true [1]
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z)) [1]

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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   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]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(x, leaf) -> false [1]
   less_leaves(leaf, cons(w, z)) -> true [1]
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z)) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
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:

   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]

And the following fresh constants:    null_minus, null_quot, 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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   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]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(x, leaf) -> false [1]
   less_leaves(leaf, cons(w, z)) -> true [1]
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z)) [1]
   minus(v0, v1) -> null_minus [0]
   quot(v0, v1) -> null_quot [0]

The TRS has the following type information:
minus :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
0 :: 0:s:null_minus:null_quot
s :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
quot :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
null_minus :: 0:s:null_minus:null_quot
null_quot :: 0:s:null_minus:null_quot
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:

   0 => 0
   nil => 0
   leaf => 0
   false => 0
   true => 1
   null_minus => 0
   null_quot => 0
   const => 0

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

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

   app(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0
   concat(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, y) :|: v >= 0, z' = 1 + u + v, z'' = y, y >= 0, u >= 0
   less_leaves(z', z'') -{ 1 }-> less_leaves(concat(u, v), concat(w, z)) :|: v >= 0, z >= 0, z' = 1 + u + v, z'' = 1 + w + z, w >= 0, u >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
   minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   minus(z', z'') -{ 1 }-> minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   quot(z', z'') -{ 1 }-> 0 :|: y >= 0, z'' = 1 + y, z' = 0
   quot(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   quot(z', z'') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   reverse(z') -{ 1 }-> app(reverse(x), 1 + n + 0) :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 1 + n + shuffle(reverse(x)) :|: n >= 0, z' = 1 + n + x, x >= 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(V, V1),0,[minus(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[quot(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[app(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[reverse(V, Out)],[V >= 0]).
eq(start(V, V1),0,[shuffle(V, Out)],[V >= 0]).
eq(start(V, V1),0,[concat(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[fun(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(minus(V, V1, Out),1,[],[Out = V2,V1 = 0,V = V2,V2 >= 0]).
eq(minus(V, V1, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V3,V3 >= 0,V4 >= 0,V1 = 1 + V4]).
eq(quot(V, V1, Out),1,[],[Out = 0,V5 >= 0,V1 = 1 + V5,V = 0]).
eq(quot(V, V1, Out),1,[minus(V7, V6, Ret10),quot(Ret10, 1 + V6, Ret1)],[Out = 1 + Ret1,V = 1 + V7,V7 >= 0,V6 >= 0,V1 = 1 + V6]).
eq(app(V, V1, Out),1,[],[Out = V8,V1 = V8,V8 >= 0,V = 0]).
eq(app(V, V1, Out),1,[app(V9, V10, Ret11)],[Out = 1 + Ret11 + V11,V11 >= 0,V1 = V10,V = 1 + V11 + V9,V9 >= 0,V10 >= 0]).
eq(reverse(V, Out),1,[],[Out = 0,V = 0]).
eq(reverse(V, Out),1,[reverse(V12, Ret0),app(Ret0, 1 + V13 + 0, Ret2)],[Out = Ret2,V13 >= 0,V = 1 + V12 + V13,V12 >= 0]).
eq(shuffle(V, Out),1,[],[Out = 0,V = 0]).
eq(shuffle(V, Out),1,[reverse(V14, Ret101),shuffle(Ret101, Ret12)],[Out = 1 + Ret12 + V15,V15 >= 0,V = 1 + V14 + V15,V14 >= 0]).
eq(concat(V, V1, Out),1,[],[Out = V16,V1 = V16,V16 >= 0,V = 0]).
eq(concat(V, V1, Out),1,[concat(V18, V19, Ret13)],[Out = 1 + Ret13 + V17,V18 >= 0,V = 1 + V17 + V18,V1 = V19,V19 >= 0,V17 >= 0]).
eq(fun(V, V1, Out),1,[],[Out = 0,V1 = 0,V = V20,V20 >= 0]).
eq(fun(V, V1, Out),1,[],[Out = 1,V22 >= 0,V1 = 1 + V21 + V22,V21 >= 0,V = 0]).
eq(fun(V, V1, Out),1,[concat(V23, V24, Ret01),concat(V25, V26, Ret14),fun(Ret01, Ret14, Ret3)],[Out = Ret3,V24 >= 0,V26 >= 0,V = 1 + V23 + V24,V1 = 1 + V25 + V26,V25 >= 0,V23 >= 0]).
eq(minus(V, V1, Out),0,[],[Out = 0,V28 >= 0,V27 >= 0,V1 = V27,V = V28]).
eq(quot(V, V1, Out),0,[],[Out = 0,V30 >= 0,V29 >= 0,V1 = V29,V = V30]).
input_output_vars(minus(V,V1,Out),[V,V1],[Out]).
input_output_vars(quot(V,V1,Out),[V,V1],[Out]).
input_output_vars(app(V,V1,Out),[V,V1],[Out]).
input_output_vars(reverse(V,Out),[V],[Out]).
input_output_vars(shuffle(V,Out),[V],[Out]).
input_output_vars(concat(V,V1,Out),[V,V1],[Out]).
input_output_vars(fun(V,V1,Out),[V,V1],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [app/3]
1. recursive  : [concat/3]
2. recursive  : [fun/3]
3. recursive  : [minus/3]
4. recursive  : [quot/3]
5. recursive [non_tail] : [reverse/2]
6. recursive  : [shuffle/2]
7. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into app/3
1. SCC is partially evaluated into concat/3
2. SCC is partially evaluated into fun/3
3. SCC is partially evaluated into minus/3
4. SCC is partially evaluated into quot/3
5. SCC is partially evaluated into reverse/2
6. SCC is partially evaluated into shuffle/2
7. SCC is partially evaluated into start/2

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

### Specialization of cost equations app/3 
* CE 15 is refined into CE [25] 
* CE 14 is refined into CE [26] 


### Cost equations --> "Loop" of app/3 
* CEs [26] --> Loop 18 
* CEs [25] --> Loop 19 

### Ranking functions of CR app(V,V1,Out) 
* RF of phase [19]: [V]

#### Partial ranking functions of CR app(V,V1,Out) 
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V


### Specialization of cost equations concat/3 
* CE 21 is refined into CE [27] 
* CE 20 is refined into CE [28] 


### Cost equations --> "Loop" of concat/3 
* CEs [28] --> Loop 20 
* CEs [27] --> Loop 21 

### Ranking functions of CR concat(V,V1,Out) 
* RF of phase [21]: [V]

#### Partial ranking functions of CR concat(V,V1,Out) 
* Partial RF of phase [21]:
  - RF of loop [21:1]:
    V


### Specialization of cost equations fun/3 
* CE 24 is refined into CE [29,30,31,32] 
* CE 22 is refined into CE [33] 
* CE 23 is refined into CE [34] 


### Cost equations --> "Loop" of fun/3 
* CEs [33] --> Loop 22 
* CEs [34] --> Loop 23 
* CEs [29,30,31,32] --> Loop 24 

### Ranking functions of CR fun(V,V1,Out) 
* RF of phase [24]: [V,V1]

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


### Specialization of cost equations minus/3 
* CE 10 is refined into CE [35] 
* CE 8 is refined into CE [36] 
* CE 9 is refined into CE [37] 


### Cost equations --> "Loop" of minus/3 
* CEs [37] --> Loop 25 
* CEs [35] --> Loop 26 
* CEs [36] --> Loop 27 

### Ranking functions of CR minus(V,V1,Out) 
* RF of phase [25]: [V,V1]

#### Partial ranking functions of CR minus(V,V1,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V
    V1


### Specialization of cost equations quot/3 
* CE 11 is refined into CE [38] 
* CE 13 is refined into CE [39] 
* CE 12 is refined into CE [40,41,42] 


### Cost equations --> "Loop" of quot/3 
* CEs [42] --> Loop 28 
* CEs [41] --> Loop 29 
* CEs [40] --> Loop 30 
* CEs [38,39] --> Loop 31 

### Ranking functions of CR quot(V,V1,Out) 
* RF of phase [28]: [V-1,V-V1+1]
* RF of phase [30]: [V]

#### Partial ranking functions of CR quot(V,V1,Out) 
* Partial RF of phase [28]:
  - RF of loop [28:1]:
    V-1
    V-V1+1
* Partial RF of phase [30]:
  - RF of loop [30:1]:
    V


### Specialization of cost equations reverse/2 
* CE 17 is refined into CE [43,44] 
* CE 16 is refined into CE [45] 


### Cost equations --> "Loop" of reverse/2 
* CEs [45] --> Loop 32 
* CEs [44] --> Loop 33 
* CEs [43] --> Loop 34 

### Ranking functions of CR reverse(V,Out) 
* RF of phase [33]: [V]

#### Partial ranking functions of CR reverse(V,Out) 
* Partial RF of phase [33]:
  - RF of loop [33:1]:
    V


### Specialization of cost equations shuffle/2 
* CE 19 is refined into CE [46,47] 
* CE 18 is refined into CE [48] 


### Cost equations --> "Loop" of shuffle/2 
* CEs [48] --> Loop 35 
* CEs [47] --> Loop 36 
* CEs [46] --> Loop 37 

### Ranking functions of CR shuffle(V,Out) 
* RF of phase [36]: [V-1]

#### Partial ranking functions of CR shuffle(V,Out) 
* Partial RF of phase [36]:
  - RF of loop [36:1]:
    V-1


### Specialization of cost equations start/2 
* CE 1 is refined into CE [49,50,51] 
* CE 2 is refined into CE [52,53,54,55,56] 
* CE 3 is refined into CE [57,58] 
* CE 4 is refined into CE [59,60] 
* CE 5 is refined into CE [61,62] 
* CE 6 is refined into CE [63,64] 
* CE 7 is refined into CE [65,66,67,68] 


### Cost equations --> "Loop" of start/2 
* CEs [52,60,62] --> Loop 38 
* CEs [49,50,51,53,54,55,56,58,64,66,67,68] --> Loop 39 
* CEs [57,59,61,63,65] --> Loop 40 

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

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


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

#### Cost of chains of app(V,V1,Out):
* Chain [[19],18]: 1*it(19)+1
  Such that:it(19) =< -V1+Out

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

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


#### Cost of chains of concat(V,V1,Out):
* Chain [[21],20]: 1*it(21)+1
  Such that:it(21) =< -V1+Out

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

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


#### Cost of chains of fun(V,V1,Out):
* Chain [[24],23]: 3*it(24)+2*s(9)+2*s(10)+1
  Such that:aux(6) =< V1
aux(7) =< V
it(24) =< aux(7)
it(24) =< aux(6)
s(11) =< it(24)*aux(6)
s(12) =< it(24)*aux(7)
s(10) =< s(12)
s(9) =< s(11)

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

* Chain [[24],22]: 3*it(24)+2*s(9)+2*s(10)+1
  Such that:aux(5) =< V
aux(8) =< V1
it(24) =< aux(8)
it(24) =< aux(5)
s(11) =< it(24)*aux(8)
s(12) =< it(24)*aux(5)
s(10) =< s(12)
s(9) =< s(11)

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

* Chain [23]: 1
  with precondition: [V=0,Out=1,V1>=1] 

* Chain [22]: 1
  with precondition: [V1=0,Out=0,V>=0] 


#### Cost of chains of minus(V,V1,Out):
* Chain [[25],27]: 1*it(25)+1
  Such that:it(25) =< V1

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

* Chain [[25],26]: 1*it(25)+0
  Such that:it(25) =< V1

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

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

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


#### Cost of chains of quot(V,V1,Out):
* Chain [[30],31]: 2*it(30)+1
  Such that:it(30) =< Out

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

* Chain [[30],29,31]: 2*it(30)+1*s(14)+2
  Such that:s(14) =< 1
it(30) =< Out

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

* Chain [[28],31]: 2*it(28)+1*s(17)+1
  Such that:it(28) =< V-V1+1
aux(11) =< V
it(28) =< aux(11)
s(17) =< aux(11)

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

* Chain [[28],29,31]: 2*it(28)+1*s(14)+1*s(17)+2
  Such that:it(28) =< V-V1+1
s(14) =< V1
aux(12) =< V
it(28) =< aux(12)
s(17) =< aux(12)

  with precondition: [V1>=2,Out>=2,V+3>=2*Out+V1] 

* Chain [31]: 1
  with precondition: [Out=0,V>=0,V1>=0] 

* Chain [29,31]: 1*s(14)+2
  Such that:s(14) =< V1

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


#### Cost of chains of reverse(V,Out):
* Chain [[33],34,32]: 2*it(33)+1*s(23)+3
  Such that:aux(16) =< Out
it(33) =< aux(16)
s(23) =< it(33)*aux(16)

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

* Chain [34,32]: 3
  with precondition: [V=Out,V>=1] 

* Chain [32]: 1
  with precondition: [V=0,Out=0] 


#### Cost of chains of shuffle(V,Out):
* Chain [[36],37,35]: 4*it(36)+2*s(33)+1*s(34)+3
  Such that:aux(19) =< Out
it(36) =< aux(19)
aux(17) =< aux(19)
s(35) =< it(36)*aux(17)
s(33) =< s(35)
s(34) =< s(33)*aux(19)

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

* Chain [37,35]: 3
  with precondition: [V=Out,V>=1] 

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


#### Cost of chains of start(V,V1):
* Chain [40]: 1
  with precondition: [V=0] 

* Chain [39]: 4*s(42)+4*s(45)+4*s(47)+6*s(56)+4*s(59)+4*s(60)+2
  Such that:aux(20) =< V
aux(21) =< V-V1+1
aux(22) =< V1
s(47) =< aux(20)
s(45) =< aux(21)
s(42) =< aux(22)
s(45) =< aux(20)
s(56) =< aux(22)
s(56) =< aux(20)
s(57) =< s(56)*aux(22)
s(58) =< s(56)*aux(20)
s(59) =< s(58)
s(60) =< s(57)

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

* Chain [38]: 1*s(68)+10*s(70)+1*s(73)+2*s(78)+1*s(79)+3
  Such that:s(68) =< 1
aux(23) =< V
s(70) =< aux(23)
s(73) =< s(70)*aux(23)
s(76) =< aux(23)
s(77) =< s(70)*s(76)
s(78) =< s(77)
s(79) =< s(78)*aux(23)

  with precondition: [V>=1] 


Closed-form bounds of start(V,V1): 
-------------------------------------
* Chain [40] with precondition: [V=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [39] with precondition: [V>=0,V1>=0] 
    - Upper bound: 4*V+2+4*V*V1+10*V1+4*V1*V1+nat(V-V1+1)*4 
    - Complexity: n^2 
* Chain [38] with precondition: [V>=1] 
    - Upper bound: 10*V+4+3*V*V+V*V*V 
    - Complexity: n^3 

### Maximum cost of start(V,V1): 4*V+1+max([6*V+2+3*V*V+V*V*V,4*V*nat(V1)+nat(V1)*10+nat(V1)*4*nat(V1)+nat(V-V1+1)*4])+1 
Asymptotic class: n^3 
* Total analysis performed in 421 ms.


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

(12)
BOUNDS(1, n^3)

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

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

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


The TRS R consists of the following rules:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   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)))
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(x, leaf) -> false
   less_leaves(leaf, cons(w, z)) -> true
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

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

(16)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   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)))
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(x, leaf) -> false
   less_leaves(leaf, cons(w, z)) -> true
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

(18)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
minus < quot
app < reverse
reverse < shuffle
concat < less_leaves

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

(20)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
minus, quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
minus < quot
app < reverse
reverse < shuffle
concat < less_leaves

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)

Induction Base:
minus(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
gen_0':s5_0(0)

Induction Step:
minus(gen_0':s5_0(+(n9_0, 1)), gen_0':s5_0(+(n9_0, 1))) ->_R^Omega(1)
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) ->_IH
gen_0':s5_0(0)

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

(22)
Complex Obligation (BEST)

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

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

TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
minus, quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
minus < quot
app < reverse
reverse < shuffle
concat < less_leaves

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


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

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

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)

Induction Base:
app(gen_nil:add6_0(0), gen_nil:add6_0(b)) ->_R^Omega(1)
gen_nil:add6_0(b)

Induction Step:
app(gen_nil:add6_0(+(n589_0, 1)), gen_nil:add6_0(b)) ->_R^Omega(1)
add(app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b))) ->_IH
add(gen_nil:add6_0(+(b, c590_0)))

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

(28)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


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

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

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)

Induction Base:
reverse(gen_nil:add6_0(0)) ->_R^Omega(1)
nil

Induction Step:
reverse(gen_nil:add6_0(+(n1508_0, 1))) ->_R^Omega(1)
app(reverse(gen_nil:add6_0(n1508_0)), add(nil)) ->_IH
app(gen_nil:add6_0(c1509_0), add(nil)) ->_L^Omega(1 + n1508_0)
gen_nil:add6_0(+(n1508_0, +(0, 1)))

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

(30)
Complex Obligation (BEST)

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

(31)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


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

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

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

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

(33)
BOUNDS(n^2, INF)

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

(34)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
concat < less_leaves

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuffle(gen_nil:add6_0(n1776_0)) -> gen_nil:add6_0(n1776_0), rt in Omega(1 + n1776_0 + n1776_0^2 + n1776_0^3)

Induction Base:
shuffle(gen_nil:add6_0(0)) ->_R^Omega(1)
nil

Induction Step:
shuffle(gen_nil:add6_0(+(n1776_0, 1))) ->_R^Omega(1)
add(shuffle(reverse(gen_nil:add6_0(n1776_0)))) ->_L^Omega(1 + n1776_0 + n1776_0^2)
add(shuffle(gen_nil:add6_0(n1776_0))) ->_IH
add(gen_nil:add6_0(c1777_0))

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

(36)
Complex Obligation (BEST)

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

(37)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
concat < less_leaves

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

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

(39)
BOUNDS(n^3, INF)

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

(40)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)
shuffle(gen_nil:add6_0(n1776_0)) -> gen_nil:add6_0(n1776_0), rt in Omega(1 + n1776_0 + n1776_0^2 + n1776_0^3)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
concat, less_leaves

They will be analysed ascendingly in the following order:
concat < less_leaves

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
concat(gen_leaf:cons7_0(n1977_0), gen_leaf:cons7_0(b)) -> gen_leaf:cons7_0(+(n1977_0, b)), rt in Omega(1 + n1977_0)

Induction Base:
concat(gen_leaf:cons7_0(0), gen_leaf:cons7_0(b)) ->_R^Omega(1)
gen_leaf:cons7_0(b)

Induction Step:
concat(gen_leaf:cons7_0(+(n1977_0, 1)), gen_leaf:cons7_0(b)) ->_R^Omega(1)
cons(leaf, concat(gen_leaf:cons7_0(n1977_0), gen_leaf:cons7_0(b))) ->_IH
cons(leaf, gen_leaf:cons7_0(+(b, c1978_0)))

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

(42)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
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)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
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
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)
shuffle(gen_nil:add6_0(n1776_0)) -> gen_nil:add6_0(n1776_0), rt in Omega(1 + n1776_0 + n1776_0^2 + n1776_0^3)
concat(gen_leaf:cons7_0(n1977_0), gen_leaf:cons7_0(b)) -> gen_leaf:cons7_0(+(n1977_0, b)), rt in Omega(1 + n1977_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
less_leaves(gen_leaf:cons7_0(n3068_0), gen_leaf:cons7_0(n3068_0)) -> false, rt in Omega(1 + n3068_0)

Induction Base:
less_leaves(gen_leaf:cons7_0(0), gen_leaf:cons7_0(0)) ->_R^Omega(1)
false

Induction Step:
less_leaves(gen_leaf:cons7_0(+(n3068_0, 1)), gen_leaf:cons7_0(+(n3068_0, 1))) ->_R^Omega(1)
less_leaves(concat(leaf, gen_leaf:cons7_0(n3068_0)), concat(leaf, gen_leaf:cons7_0(n3068_0))) ->_L^Omega(1)
less_leaves(gen_leaf:cons7_0(+(0, n3068_0)), concat(leaf, gen_leaf:cons7_0(n3068_0))) ->_L^Omega(1)
less_leaves(gen_leaf:cons7_0(n3068_0), gen_leaf:cons7_0(+(0, n3068_0))) ->_IH
false

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

(44)
BOUNDS(1, INF)