WORST_CASE(Omega(n^3),O(n^3))
proof of input_QNbWTH78Kb.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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, 509 ms]
    (10) BOUNDS(1, n^3)
(11) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
    (13) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRelTRS
    (15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 341 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), 93 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 72 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 27 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 98 ms]
            (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 225 ms]
            (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 36 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 873 ms]
            (42) BEST
                (43) proven lower bound
                    (44) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (45) BOUNDS(n^2, INF)
                (46) typed CpxTrs
                    (47) RewriteLemmaProof [LOWER BOUND(ID), 962 ms]
                    (48) BEST
                        (49) proven lower bound
                            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
                            (51) BOUNDS(n^3, INF)
                        (52) typed CpxTrs
                            (53) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
                            (54) typed CpxTrs
                            (55) RewriteLemmaProof [LOWER BOUND(ID), 35 ms]
                            (56) BOUNDS(1, INF)


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

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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:

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

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

(4)
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
----------------------------------------

(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:

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

And the following fresh constants:    null_minus, null_quot, 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:

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

(7) 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

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

(8)
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.

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

(9) 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 418 ms.


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

(10)
BOUNDS(1, n^3)

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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   reverse(nil) -> nil
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
   shuffle(nil) -> nil
   shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
   concat(leaf, z0) -> z0
   concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
   less_leaves(z0, leaf) -> false
   less_leaves(leaf, cons(z0, z1)) -> true
   less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))
Tuples:
   MINUS(z0, 0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c2
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   APP(nil, z0) -> c4
   APP(add(z0, z1), z2) -> c5(APP(z1, z2))
   REVERSE(nil) -> c6
   REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(nil) -> c8
   SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
   CONCAT(leaf, z0) -> c10
   CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
   LESS_LEAVES(z0, leaf) -> c12
   LESS_LEAVES(leaf, cons(z0, z1)) -> c13
   LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
   LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
S tuples:
   MINUS(z0, 0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c2
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   APP(nil, z0) -> c4
   APP(add(z0, z1), z2) -> c5(APP(z1, z2))
   REVERSE(nil) -> c6
   REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(nil) -> c8
   SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
   CONCAT(leaf, z0) -> c10
   CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
   LESS_LEAVES(z0, leaf) -> c12
   LESS_LEAVES(leaf, cons(z0, z1)) -> c13
   LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
   LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
K tuples:none
Defined Rule Symbols:   minus_2, quot_2, app_2, reverse_1, shuffle_1, concat_2, less_leaves_2

Defined Pair Symbols:   MINUS_2, QUOT_2, APP_2, REVERSE_1, SHUFFLE_1, CONCAT_2, LESS_LEAVES_2

Compound Symbols:   c, c1_1, c2, c3_2, c4, c5_1, c6, c7_2, c8, c9_2, c10, c11_1, c12, c13, c14_2, c15_2


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

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

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


The TRS R consists of the following rules:

   MINUS(z0, 0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(0, s(z0)) -> c2
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   APP(nil, z0) -> c4
   APP(add(z0, z1), z2) -> c5(APP(z1, z2))
   REVERSE(nil) -> c6
   REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(nil) -> c8
   SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
   CONCAT(leaf, z0) -> c10
   CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
   LESS_LEAVES(z0, leaf) -> c12
   LESS_LEAVES(leaf, cons(z0, z1)) -> c13
   LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
   LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))

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

   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0, s(z0)) -> 0
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   reverse(nil) -> nil
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
   shuffle(nil) -> nil
   shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
   concat(leaf, z0) -> z0
   concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
   less_leaves(z0, leaf) -> false
   less_leaves(leaf, cons(z0, z1)) -> true
   less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

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

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

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


The TRS R consists of the following rules:

   MINUS(z0, 0') -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   QUOT(0', s(z0)) -> c2
   QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
   APP(nil, z0) -> c4
   APP(add(z0, z1), z2) -> c5(APP(z1, z2))
   REVERSE(nil) -> c6
   REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(nil) -> c8
   SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
   CONCAT(leaf, z0) -> c10
   CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
   LESS_LEAVES(z0, leaf) -> c12
   LESS_LEAVES(leaf, cons(z0, z1)) -> c13
   LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
   LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))

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

   minus(z0, 0') -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   quot(0', s(z0)) -> 0'
   quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   reverse(nil) -> nil
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
   shuffle(nil) -> nil
   shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
   concat(leaf, z0) -> z0
   concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
   less_leaves(z0, leaf) -> false
   less_leaves(leaf, cons(z0, z1)) -> true
   less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, QUOT, minus, APP, REVERSE, reverse, SHUFFLE, CONCAT, LESS_LEAVES, concat, quot, app, shuffle, less_leaves

They will be analysed ascendingly in the following order:
MINUS < QUOT
minus < QUOT
minus < quot
APP < REVERSE
reverse < REVERSE
REVERSE < SHUFFLE
reverse < SHUFFLE
app < reverse
reverse < shuffle
CONCAT < LESS_LEAVES
concat < LESS_LEAVES
concat < less_leaves

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

(20)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


The following defined symbols remain to be analysed:
MINUS, QUOT, minus, APP, REVERSE, reverse, SHUFFLE, CONCAT, LESS_LEAVES, concat, quot, app, shuffle, less_leaves

They will be analysed ascendingly in the following order:
MINUS < QUOT
minus < QUOT
minus < quot
APP < REVERSE
reverse < REVERSE
REVERSE < SHUFFLE
reverse < SHUFFLE
app < reverse
reverse < shuffle
CONCAT < LESS_LEAVES
concat < LESS_LEAVES
concat < less_leaves

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)

Induction Base:
MINUS(gen_0':s14_16(0), gen_0':s14_16(0)) ->_R^Omega(1)
c

Induction Step:
MINUS(gen_0':s14_16(+(n24_16, 1)), gen_0':s14_16(+(n24_16, 1))) ->_R^Omega(1)
c1(MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16))) ->_IH
c1(gen_c:c113_16(c25_16))

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:

Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


The following defined symbols remain to be analysed:
MINUS, QUOT, minus, APP, REVERSE, reverse, SHUFFLE, CONCAT, LESS_LEAVES, concat, quot, app, shuffle, less_leaves

They will be analysed ascendingly in the following order:
MINUS < QUOT
minus < QUOT
minus < quot
APP < REVERSE
reverse < REVERSE
REVERSE < SHUFFLE
reverse < SHUFFLE
app < reverse
reverse < shuffle
CONCAT < LESS_LEAVES
concat < LESS_LEAVES
concat < less_leaves

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


The following defined symbols remain to be analysed:
minus, QUOT, APP, REVERSE, reverse, SHUFFLE, CONCAT, LESS_LEAVES, concat, quot, app, shuffle, less_leaves

They will be analysed ascendingly in the following order:
minus < QUOT
minus < quot
APP < REVERSE
reverse < REVERSE
REVERSE < SHUFFLE
reverse < SHUFFLE
app < reverse
reverse < shuffle
CONCAT < LESS_LEAVES
concat < LESS_LEAVES
concat < less_leaves

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)

Induction Base:
minus(gen_0':s14_16(0), gen_0':s14_16(0)) ->_R^Omega(0)
gen_0':s14_16(0)

Induction Step:
minus(gen_0':s14_16(+(n546_16, 1)), gen_0':s14_16(+(n546_16, 1))) ->_R^Omega(0)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) ->_IH
gen_0':s14_16(0)

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

(28)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


The following defined symbols remain to be analysed:
QUOT, APP, REVERSE, reverse, SHUFFLE, CONCAT, LESS_LEAVES, concat, quot, app, shuffle, less_leaves

They will be analysed ascendingly in the following order:
APP < REVERSE
reverse < REVERSE
REVERSE < SHUFFLE
reverse < SHUFFLE
app < reverse
reverse < shuffle
CONCAT < LESS_LEAVES
concat < LESS_LEAVES
concat < less_leaves

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)

Induction Base:
APP(gen_nil:add17_16(0), gen_nil:add17_16(b)) ->_R^Omega(1)
c4

Induction Step:
APP(gen_nil:add17_16(+(n3100_16, 1)), gen_nil:add17_16(b)) ->_R^Omega(1)
c5(APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b))) ->_IH
c5(gen_c4:c516_16(c3101_16))

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:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)

Induction Base:
CONCAT(gen_leaf:cons21_16(0), gen_leaf:cons21_16(b)) ->_R^Omega(1)
c10

Induction Step:
CONCAT(gen_leaf:cons21_16(+(n4018_16, 1)), gen_leaf:cons21_16(b)) ->_R^Omega(1)
c11(CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b))) ->_IH
c11(gen_c10:c1120_16(c4019_16))

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

(32)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

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

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)

Induction Base:
concat(gen_leaf:cons21_16(0), gen_leaf:cons21_16(b)) ->_R^Omega(0)
gen_leaf:cons21_16(b)

Induction Step:
concat(gen_leaf:cons21_16(+(n5061_16, 1)), gen_leaf:cons21_16(b)) ->_R^Omega(0)
cons(leaf, concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b))) ->_IH
cons(leaf, gen_leaf:cons21_16(+(b, c5062_16)))

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

(34)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


The following defined symbols remain to be analysed:
LESS_LEAVES, REVERSE, reverse, SHUFFLE, quot, app, shuffle, less_leaves

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

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)

Induction Base:
LESS_LEAVES(gen_leaf:cons21_16(0), gen_leaf:cons21_16(0)) ->_R^Omega(1)
c12

Induction Step:
LESS_LEAVES(gen_leaf:cons21_16(+(n7770_16, 1)), gen_leaf:cons21_16(+(n7770_16, 1))) ->_R^Omega(1)
c14(LESS_LEAVES(concat(leaf, gen_leaf:cons21_16(n7770_16)), concat(leaf, gen_leaf:cons21_16(n7770_16))), CONCAT(leaf, gen_leaf:cons21_16(n7770_16))) ->_L^Omega(0)
c14(LESS_LEAVES(gen_leaf:cons21_16(+(0, n7770_16)), concat(leaf, gen_leaf:cons21_16(n7770_16))), CONCAT(leaf, gen_leaf:cons21_16(n7770_16))) ->_L^Omega(0)
c14(LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(+(0, n7770_16))), CONCAT(leaf, gen_leaf:cons21_16(n7770_16))) ->_IH
c14(gen_c12:c13:c14:c1522_16(c7771_16), CONCAT(leaf, gen_leaf:cons21_16(n7770_16))) ->_L^Omega(1)
c14(gen_c12:c13:c14:c1522_16(n7770_16), gen_c10:c1120_16(0))

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

(36)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

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

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b)) -> gen_nil:add17_16(+(n13572_16, b)), rt in Omega(0)

Induction Base:
app(gen_nil:add17_16(0), gen_nil:add17_16(b)) ->_R^Omega(0)
gen_nil:add17_16(b)

Induction Step:
app(gen_nil:add17_16(+(n13572_16, 1)), gen_nil:add17_16(b)) ->_R^Omega(0)
add(hole_a6_16, app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b))) ->_IH
add(hole_a6_16, gen_nil:add17_16(+(b, c13573_16)))

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

(38)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)
app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b)) -> gen_nil:add17_16(+(n13572_16, b)), rt in Omega(0)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

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

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
reverse(gen_nil:add17_16(n15949_16)) -> gen_nil:add17_16(n15949_16), rt in Omega(0)

Induction Base:
reverse(gen_nil:add17_16(0)) ->_R^Omega(0)
nil

Induction Step:
reverse(gen_nil:add17_16(+(n15949_16, 1))) ->_R^Omega(0)
app(reverse(gen_nil:add17_16(n15949_16)), add(hole_a6_16, nil)) ->_IH
app(gen_nil:add17_16(c15950_16), add(hole_a6_16, nil)) ->_L^Omega(0)
gen_nil:add17_16(+(n15949_16, +(0, 1)))

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

(40)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)
app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b)) -> gen_nil:add17_16(+(n13572_16, b)), rt in Omega(0)
reverse(gen_nil:add17_16(n15949_16)) -> gen_nil:add17_16(n15949_16), rt in Omega(0)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

They will be analysed ascendingly in the following order:
REVERSE < SHUFFLE

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
REVERSE(gen_nil:add17_16(n16480_16)) -> *23_16, rt in Omega(n16480_16 + n16480_16^2)

Induction Base:
REVERSE(gen_nil:add17_16(0))

Induction Step:
REVERSE(gen_nil:add17_16(+(n16480_16, 1))) ->_R^Omega(1)
c7(APP(reverse(gen_nil:add17_16(n16480_16)), add(hole_a6_16, nil)), REVERSE(gen_nil:add17_16(n16480_16))) ->_L^Omega(0)
c7(APP(gen_nil:add17_16(n16480_16), add(hole_a6_16, nil)), REVERSE(gen_nil:add17_16(n16480_16))) ->_L^Omega(1 + n16480_16)
c7(gen_c4:c516_16(n16480_16), REVERSE(gen_nil:add17_16(n16480_16))) ->_IH
c7(gen_c4:c516_16(n16480_16), *23_16)

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

(42)
Complex Obligation (BEST)

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

(43)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)
app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b)) -> gen_nil:add17_16(+(n13572_16, b)), rt in Omega(0)
reverse(gen_nil:add17_16(n15949_16)) -> gen_nil:add17_16(n15949_16), rt in Omega(0)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

They will be analysed ascendingly in the following order:
REVERSE < SHUFFLE

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

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

(45)
BOUNDS(n^2, INF)

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

(46)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)
app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b)) -> gen_nil:add17_16(+(n13572_16, b)), rt in Omega(0)
reverse(gen_nil:add17_16(n15949_16)) -> gen_nil:add17_16(n15949_16), rt in Omega(0)
REVERSE(gen_nil:add17_16(n16480_16)) -> *23_16, rt in Omega(n16480_16 + n16480_16^2)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SHUFFLE(gen_nil:add17_16(n23880_16)) -> *23_16, rt in Omega(n23880_16 + n23880_16^2 + n23880_16^3)

Induction Base:
SHUFFLE(gen_nil:add17_16(0))

Induction Step:
SHUFFLE(gen_nil:add17_16(+(n23880_16, 1))) ->_R^Omega(1)
c9(SHUFFLE(reverse(gen_nil:add17_16(n23880_16))), REVERSE(gen_nil:add17_16(n23880_16))) ->_L^Omega(0)
c9(SHUFFLE(gen_nil:add17_16(n23880_16)), REVERSE(gen_nil:add17_16(n23880_16))) ->_IH
c9(*23_16, REVERSE(gen_nil:add17_16(n23880_16))) ->_L^Omega(n23880_16 + n23880_16^2)
c9(*23_16, *23_16)

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

(48)
Complex Obligation (BEST)

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

(49)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)
app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b)) -> gen_nil:add17_16(+(n13572_16, b)), rt in Omega(0)
reverse(gen_nil:add17_16(n15949_16)) -> gen_nil:add17_16(n15949_16), rt in Omega(0)
REVERSE(gen_nil:add17_16(n16480_16)) -> *23_16, rt in Omega(n16480_16 + n16480_16^2)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

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

(51)
BOUNDS(n^3, INF)

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

(52)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)
app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b)) -> gen_nil:add17_16(+(n13572_16, b)), rt in Omega(0)
reverse(gen_nil:add17_16(n15949_16)) -> gen_nil:add17_16(n15949_16), rt in Omega(0)
REVERSE(gen_nil:add17_16(n16480_16)) -> *23_16, rt in Omega(n16480_16 + n16480_16^2)
SHUFFLE(gen_nil:add17_16(n23880_16)) -> *23_16, rt in Omega(n23880_16 + n23880_16^2 + n23880_16^3)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuffle(gen_nil:add17_16(n38561_16)) -> gen_nil:add17_16(n38561_16), rt in Omega(0)

Induction Base:
shuffle(gen_nil:add17_16(0)) ->_R^Omega(0)
nil

Induction Step:
shuffle(gen_nil:add17_16(+(n38561_16, 1))) ->_R^Omega(0)
add(hole_a6_16, shuffle(reverse(gen_nil:add17_16(n38561_16)))) ->_L^Omega(0)
add(hole_a6_16, shuffle(gen_nil:add17_16(n38561_16))) ->_IH
add(hole_a6_16, gen_nil:add17_16(c38562_16))

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

(54)
Obligation:
Innermost TRS:
Rules:
MINUS(z0, 0') -> c
MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
QUOT(0', s(z0)) -> c2
QUOT(s(z0), s(z1)) -> c3(QUOT(minus(z0, z1), s(z1)), MINUS(z0, z1))
APP(nil, z0) -> c4
APP(add(z0, z1), z2) -> c5(APP(z1, z2))
REVERSE(nil) -> c6
REVERSE(add(z0, z1)) -> c7(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(nil) -> c8
SHUFFLE(add(z0, z1)) -> c9(SHUFFLE(reverse(z1)), REVERSE(z1))
CONCAT(leaf, z0) -> c10
CONCAT(cons(z0, z1), z2) -> c11(CONCAT(z1, z2))
LESS_LEAVES(z0, leaf) -> c12
LESS_LEAVES(leaf, cons(z0, z1)) -> c13
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c14(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z0, z1))
LESS_LEAVES(cons(z0, z1), cons(z2, z3)) -> c15(LESS_LEAVES(concat(z0, z1), concat(z2, z3)), CONCAT(z2, z3))
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
quot(0', s(z0)) -> 0'
quot(s(z0), s(z1)) -> s(quot(minus(z0, z1), s(z1)))
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(nil) -> nil
shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, leaf) -> false
less_leaves(leaf, cons(z0, z1)) -> true
less_leaves(cons(z0, z1), cons(z2, z3)) -> less_leaves(concat(z0, z1), concat(z2, z3))

Types:
MINUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
QUOT :: 0':s -> 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c:c1 -> c2:c3
minus :: 0':s -> 0':s -> 0':s
APP :: nil:add -> nil:add -> c4:c5
nil :: nil:add
c4 :: c4:c5
add :: a -> nil:add -> nil:add
c5 :: c4:c5 -> c4:c5
REVERSE :: nil:add -> c6:c7
c6 :: c6:c7
c7 :: c4:c5 -> c6:c7 -> c6:c7
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c6:c7 -> c8:c9
CONCAT :: leaf:cons -> leaf:cons -> c10:c11
leaf :: leaf:cons
c10 :: c10:c11
cons :: leaf:cons -> leaf:cons -> leaf:cons
c11 :: c10:c11 -> c10:c11
LESS_LEAVES :: leaf:cons -> leaf:cons -> c12:c13:c14:c15
c12 :: c12:c13:c14:c15
c13 :: c12:c13:c14:c15
c14 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
concat :: leaf:cons -> leaf:cons -> leaf:cons
c15 :: c12:c13:c14:c15 -> c10:c11 -> c12:c13:c14:c15
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
shuffle :: nil:add -> nil:add
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_c:c11_16 :: c:c1
hole_0':s2_16 :: 0':s
hole_c2:c33_16 :: c2:c3
hole_c4:c54_16 :: c4:c5
hole_nil:add5_16 :: nil:add
hole_a6_16 :: a
hole_c6:c77_16 :: c6:c7
hole_c8:c98_16 :: c8:c9
hole_c10:c119_16 :: c10:c11
hole_leaf:cons10_16 :: leaf:cons
hole_c12:c13:c14:c1511_16 :: c12:c13:c14:c15
hole_false:true12_16 :: false:true
gen_c:c113_16 :: Nat -> c:c1
gen_0':s14_16 :: Nat -> 0':s
gen_c2:c315_16 :: Nat -> c2:c3
gen_c4:c516_16 :: Nat -> c4:c5
gen_nil:add17_16 :: Nat -> nil:add
gen_c6:c718_16 :: Nat -> c6:c7
gen_c8:c919_16 :: Nat -> c8:c9
gen_c10:c1120_16 :: Nat -> c10:c11
gen_leaf:cons21_16 :: Nat -> leaf:cons
gen_c12:c13:c14:c1522_16 :: Nat -> c12:c13:c14:c15


Lemmas:
MINUS(gen_0':s14_16(n24_16), gen_0':s14_16(n24_16)) -> gen_c:c113_16(n24_16), rt in Omega(1 + n24_16)
minus(gen_0':s14_16(n546_16), gen_0':s14_16(n546_16)) -> gen_0':s14_16(0), rt in Omega(0)
APP(gen_nil:add17_16(n3100_16), gen_nil:add17_16(b)) -> gen_c4:c516_16(n3100_16), rt in Omega(1 + n3100_16)
CONCAT(gen_leaf:cons21_16(n4018_16), gen_leaf:cons21_16(b)) -> gen_c10:c1120_16(n4018_16), rt in Omega(1 + n4018_16)
concat(gen_leaf:cons21_16(n5061_16), gen_leaf:cons21_16(b)) -> gen_leaf:cons21_16(+(n5061_16, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons21_16(n7770_16), gen_leaf:cons21_16(n7770_16)) -> gen_c12:c13:c14:c1522_16(n7770_16), rt in Omega(1 + n7770_16)
app(gen_nil:add17_16(n13572_16), gen_nil:add17_16(b)) -> gen_nil:add17_16(+(n13572_16, b)), rt in Omega(0)
reverse(gen_nil:add17_16(n15949_16)) -> gen_nil:add17_16(n15949_16), rt in Omega(0)
REVERSE(gen_nil:add17_16(n16480_16)) -> *23_16, rt in Omega(n16480_16 + n16480_16^2)
SHUFFLE(gen_nil:add17_16(n23880_16)) -> *23_16, rt in Omega(n23880_16 + n23880_16^2 + n23880_16^3)
shuffle(gen_nil:add17_16(n38561_16)) -> gen_nil:add17_16(n38561_16), rt in Omega(0)


Generator Equations:
gen_c:c113_16(0) <=> c
gen_c:c113_16(+(x, 1)) <=> c1(gen_c:c113_16(x))
gen_0':s14_16(0) <=> 0'
gen_0':s14_16(+(x, 1)) <=> s(gen_0':s14_16(x))
gen_c2:c315_16(0) <=> c2
gen_c2:c315_16(+(x, 1)) <=> c3(gen_c2:c315_16(x), c)
gen_c4:c516_16(0) <=> c4
gen_c4:c516_16(+(x, 1)) <=> c5(gen_c4:c516_16(x))
gen_nil:add17_16(0) <=> nil
gen_nil:add17_16(+(x, 1)) <=> add(hole_a6_16, gen_nil:add17_16(x))
gen_c6:c718_16(0) <=> c6
gen_c6:c718_16(+(x, 1)) <=> c7(c4, gen_c6:c718_16(x))
gen_c8:c919_16(0) <=> c8
gen_c8:c919_16(+(x, 1)) <=> c9(gen_c8:c919_16(x), c6)
gen_c10:c1120_16(0) <=> c10
gen_c10:c1120_16(+(x, 1)) <=> c11(gen_c10:c1120_16(x))
gen_leaf:cons21_16(0) <=> leaf
gen_leaf:cons21_16(+(x, 1)) <=> cons(leaf, gen_leaf:cons21_16(x))
gen_c12:c13:c14:c1522_16(0) <=> c12
gen_c12:c13:c14:c1522_16(+(x, 1)) <=> c14(gen_c12:c13:c14:c1522_16(x), c10)


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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
less_leaves(gen_leaf:cons21_16(n39135_16), gen_leaf:cons21_16(n39135_16)) -> false, rt in Omega(0)

Induction Base:
less_leaves(gen_leaf:cons21_16(0), gen_leaf:cons21_16(0)) ->_R^Omega(0)
false

Induction Step:
less_leaves(gen_leaf:cons21_16(+(n39135_16, 1)), gen_leaf:cons21_16(+(n39135_16, 1))) ->_R^Omega(0)
less_leaves(concat(leaf, gen_leaf:cons21_16(n39135_16)), concat(leaf, gen_leaf:cons21_16(n39135_16))) ->_L^Omega(0)
less_leaves(gen_leaf:cons21_16(+(0, n39135_16)), concat(leaf, gen_leaf:cons21_16(n39135_16))) ->_L^Omega(0)
less_leaves(gen_leaf:cons21_16(n39135_16), gen_leaf:cons21_16(+(0, n39135_16))) ->_IH
false

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

(56)
BOUNDS(1, INF)