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


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 1 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, 1219 ms]
    (10) BOUNDS(1, n^2)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 308 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 52 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 70 ms]
            (28) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   if(true, c, l, ys, zs) -> nil
   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
   take(0, cons(x, xs), ys) -> x
   take(0, nil, cons(y, ys)) -> y
   take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

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

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

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


The TRS R consists of the following rules:

   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   length(nil) -> 0 [1]
   length(cons(x, y)) -> s(length(y)) [1]
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   if(true, c, l, ys, zs) -> nil [1]
   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs) [1]
   take(0, cons(x, xs), ys) -> x [1]
   take(0, nil, cons(y, ys)) -> y [1]
   take(s(c), cons(x, xs), ys) -> take(c, xs, ys) [1]
   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys) [1]
   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs)) [1]

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

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

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

   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   length(nil) -> 0 [1]
   length(cons(x, y)) -> s(length(y)) [1]
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   if(true, c, l, ys, zs) -> nil [1]
   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs) [1]
   take(0, cons(x, xs), ys) -> x [1]
   take(0, nil, cons(y, ys)) -> y [1]
   take(s(c), cons(x, xs), ys) -> take(c, xs, ys) [1]
   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys) [1]
   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs)) [1]

The TRS has the following type information:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0:s -> 0:s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0 :: 0:s
plus :: 0:s -> 0:s -> 0:s
length :: nil:cons:xs -> 0:s
s :: 0:s -> 0:s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0:s -> 0:s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
helpb :: 0:s -> 0:s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0:s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs

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:

   length(v0) -> null_length [0]
   take(v0, v1, v2) -> null_take [0]
   plus(v0, v1) -> null_plus [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]

And the following fresh constants:    null_length, null_take, null_plus, null_ge, null_if

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

(6)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   app(x, y) -> helpa(0, plus(length(x), length(y)), x, y) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   length(nil) -> 0 [1]
   length(cons(x, y)) -> s(length(y)) [1]
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   if(true, c, l, ys, zs) -> nil [1]
   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs) [1]
   take(0, cons(x, xs), ys) -> x [1]
   take(0, nil, cons(y, ys)) -> y [1]
   take(s(c), cons(x, xs), ys) -> take(c, xs, ys) [1]
   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys) [1]
   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs)) [1]
   length(v0) -> null_length [0]
   take(v0, v1, v2) -> null_take [0]
   plus(v0, v1) -> null_plus [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]

The TRS has the following type information:
app :: nil:cons:xs:null_if -> nil:cons:xs:null_if -> nil:cons:xs:null_if
helpa :: 0:s:null_length:null_plus -> 0:s:null_length:null_plus -> nil:cons:xs:null_if -> nil:cons:xs:null_if -> nil:cons:xs:null_if
0 :: 0:s:null_length:null_plus
plus :: 0:s:null_length:null_plus -> 0:s:null_length:null_plus -> 0:s:null_length:null_plus
length :: nil:cons:xs:null_if -> 0:s:null_length:null_plus
s :: 0:s:null_length:null_plus -> 0:s:null_length:null_plus
nil :: nil:cons:xs:null_if
cons :: null_take -> nil:cons:xs:null_if -> nil:cons:xs:null_if
if :: true:false:null_ge -> 0:s:null_length:null_plus -> 0:s:null_length:null_plus -> nil:cons:xs:null_if -> nil:cons:xs:null_if -> nil:cons:xs:null_if
ge :: 0:s:null_length:null_plus -> 0:s:null_length:null_plus -> true:false:null_ge
true :: true:false:null_ge
false :: true:false:null_ge
helpb :: 0:s:null_length:null_plus -> 0:s:null_length:null_plus -> nil:cons:xs:null_if -> nil:cons:xs:null_if -> nil:cons:xs:null_if
take :: 0:s:null_length:null_plus -> nil:cons:xs:null_if -> nil:cons:xs:null_if -> null_take
xs :: nil:cons:xs:null_if
null_length :: 0:s:null_length:null_plus
null_take :: null_take
null_plus :: 0:s:null_length:null_plus
null_ge :: true:false:null_ge
null_if :: nil:cons:xs:null_if

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
   true => 2
   false => 1
   xs => 1
   null_length => 0
   null_take => 0
   null_plus => 0
   null_ge => 0
   null_if => 0

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

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

   app(z, z') -{ 1 }-> helpa(0, plus(length(x), length(y)), x, y) :|: x >= 0, y >= 0, z = x, z' = y
   ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   ge(z, z') -{ 1 }-> 2 :|: x >= 0, z = x, z' = 0
   ge(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   helpa(z, z', z'', z1) -{ 1 }-> if(ge(c, l), c, l, ys, zs) :|: z' = l, c >= 0, ys >= 0, zs >= 0, z'' = ys, z = c, l >= 0, z1 = zs
   helpb(z, z', z'', z1) -{ 1 }-> 1 + take(c, ys, zs) + helpa(1 + c, l, ys, zs) :|: z' = l, c >= 0, ys >= 0, zs >= 0, z'' = ys, z = c, l >= 0, z1 = zs
   if(z, z', z'', z1, z2) -{ 1 }-> helpb(c, l, ys, zs) :|: z2 = zs, c >= 0, z = 1, ys >= 0, zs >= 0, l >= 0, z' = c, z'' = l, z1 = ys
   if(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z = 2, z2 = zs, c >= 0, ys >= 0, zs >= 0, l >= 0, z' = c, z'' = l, z1 = ys
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
   length(z) -{ 1 }-> 0 :|: z = 0
   length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   length(z) -{ 1 }-> 1 + length(y) :|: z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   take(z, z', z'') -{ 1 }-> x :|: ys >= 0, x >= 0, z'' = ys, z = 0, z' = 1 + x + 1
   take(z, z', z'') -{ 1 }-> y :|: z'' = 1 + y + ys, ys >= 0, y >= 0, z = 0, z' = 0
   take(z, z', z'') -{ 1 }-> take(c, 1, ys) :|: c >= 0, ys >= 0, x >= 0, z'' = ys, z = 1 + c, z' = 1 + x + 1
   take(z, z', z'') -{ 1 }-> take(c, 0, ys) :|: z'' = 1 + y + ys, c >= 0, ys >= 0, y >= 0, z = 1 + c, z' = 0
   take(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 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(V1, V, V9, V14, V22),0,[app(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V9, V14, V22),0,[plus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V9, V14, V22),0,[length(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V9, V14, V22),0,[helpa(V1, V, V9, V14, Out)],[V1 >= 0,V >= 0,V9 >= 0,V14 >= 0]).
eq(start(V1, V, V9, V14, V22),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V9, V14, V22),0,[if(V1, V, V9, V14, V22, Out)],[V1 >= 0,V >= 0,V9 >= 0,V14 >= 0,V22 >= 0]).
eq(start(V1, V, V9, V14, V22),0,[take(V1, V, V9, Out)],[V1 >= 0,V >= 0,V9 >= 0]).
eq(start(V1, V, V9, V14, V22),0,[helpb(V1, V, V9, V14, Out)],[V1 >= 0,V >= 0,V9 >= 0,V14 >= 0]).
eq(app(V1, V, Out),1,[length(V3, Ret10),length(V2, Ret11),plus(Ret10, Ret11, Ret1),helpa(0, Ret1, V3, V2, Ret)],[Out = Ret,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
eq(plus(V1, V, Out),1,[],[Out = V4,V4 >= 0,V1 = V4,V = 0]).
eq(plus(V1, V, Out),1,[plus(V5, V6, Ret12)],[Out = 1 + Ret12,V = 1 + V6,V5 >= 0,V6 >= 0,V1 = V5]).
eq(length(V1, Out),1,[],[Out = 0,V1 = 0]).
eq(length(V1, Out),1,[length(V8, Ret13)],[Out = 1 + Ret13,V1 = 1 + V7 + V8,V7 >= 0,V8 >= 0]).
eq(helpa(V1, V, V9, V14, Out),1,[ge(V10, V11, Ret0),if(Ret0, V10, V11, V12, V13, Ret2)],[Out = Ret2,V = V11,V10 >= 0,V12 >= 0,V13 >= 0,V9 = V12,V1 = V10,V11 >= 0,V14 = V13]).
eq(ge(V1, V, Out),1,[],[Out = 2,V15 >= 0,V1 = V15,V = 0]).
eq(ge(V1, V, Out),1,[],[Out = 1,V = 1 + V16,V16 >= 0,V1 = 0]).
eq(ge(V1, V, Out),1,[ge(V18, V17, Ret3)],[Out = Ret3,V = 1 + V17,V18 >= 0,V17 >= 0,V1 = 1 + V18]).
eq(if(V1, V, V9, V14, V22, Out),1,[],[Out = 0,V1 = 2,V22 = V21,V19 >= 0,V20 >= 0,V21 >= 0,V23 >= 0,V = V19,V9 = V23,V14 = V20]).
eq(if(V1, V, V9, V14, V22, Out),1,[helpb(V24, V27, V26, V25, Ret4)],[Out = Ret4,V22 = V25,V24 >= 0,V1 = 1,V26 >= 0,V25 >= 0,V27 >= 0,V = V24,V9 = V27,V14 = V26]).
eq(take(V1, V, V9, Out),1,[],[Out = V28,V29 >= 0,V28 >= 0,V9 = V29,V1 = 0,V = 2 + V28]).
eq(take(V1, V, V9, Out),1,[],[Out = V30,V9 = 1 + V30 + V31,V31 >= 0,V30 >= 0,V1 = 0,V = 0]).
eq(take(V1, V, V9, Out),1,[take(V32, 1, V34, Ret5)],[Out = Ret5,V32 >= 0,V34 >= 0,V33 >= 0,V9 = V34,V1 = 1 + V32,V = 2 + V33]).
eq(take(V1, V, V9, Out),1,[take(V37, 0, V36, Ret6)],[Out = Ret6,V9 = 1 + V35 + V36,V37 >= 0,V36 >= 0,V35 >= 0,V1 = 1 + V37,V = 0]).
eq(helpb(V1, V, V9, V14, Out),1,[take(V40, V38, V39, Ret01),helpa(1 + V40, V41, V38, V39, Ret14)],[Out = 1 + Ret01 + Ret14,V = V41,V40 >= 0,V38 >= 0,V39 >= 0,V9 = V38,V1 = V40,V41 >= 0,V14 = V39]).
eq(length(V1, Out),0,[],[Out = 0,V42 >= 0,V1 = V42]).
eq(take(V1, V, V9, Out),0,[],[Out = 0,V44 >= 0,V9 = V45,V43 >= 0,V1 = V44,V = V43,V45 >= 0]).
eq(plus(V1, V, Out),0,[],[Out = 0,V47 >= 0,V46 >= 0,V1 = V47,V = V46]).
eq(ge(V1, V, Out),0,[],[Out = 0,V48 >= 0,V49 >= 0,V1 = V48,V = V49]).
eq(if(V1, V, V9, V14, V22, Out),0,[],[Out = 0,V14 = V54,V50 >= 0,V53 >= 0,V9 = V51,V52 >= 0,V1 = V50,V = V52,V22 = V53,V51 >= 0,V54 >= 0]).
input_output_vars(app(V1,V,Out),[V1,V],[Out]).
input_output_vars(plus(V1,V,Out),[V1,V],[Out]).
input_output_vars(length(V1,Out),[V1],[Out]).
input_output_vars(helpa(V1,V,V9,V14,Out),[V1,V,V9,V14],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V9,V14,V22,Out),[V1,V,V9,V14,V22],[Out]).
input_output_vars(take(V1,V,V9,Out),[V1,V,V9],[Out]).
input_output_vars(helpb(V1,V,V9,V14,Out),[V1,V,V9,V14],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [ge/3]
1. recursive  : [take/4]
2. recursive  : [helpa/5,helpb/5,if/6]
3. recursive  : [length/2]
4. recursive  : [plus/3]
5. non_recursive  : [app/3]
6. non_recursive  : [start/5]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into ge/3
1. SCC is partially evaluated into take/4
2. SCC is partially evaluated into helpa/5
3. SCC is partially evaluated into length/2
4. SCC is partially evaluated into plus/3
5. SCC is partially evaluated into app/3
6. SCC is partially evaluated into start/5

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

### Specialization of cost equations ge/3 
* CE 29 is refined into CE [30] 
* CE 26 is refined into CE [31] 
* CE 27 is refined into CE [32] 
* CE 28 is refined into CE [33] 


### Cost equations --> "Loop" of ge/3 
* CEs [33] --> Loop 20 
* CEs [30] --> Loop 21 
* CEs [31] --> Loop 22 
* CEs [32] --> Loop 23 

### Ranking functions of CR ge(V1,V,Out) 
* RF of phase [20]: [V,V1]

#### Partial ranking functions of CR ge(V1,V,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V
    V1


### Specialization of cost equations take/4 
* CE 15 is refined into CE [34] 
* CE 11 is refined into CE [35] 
* CE 12 is refined into CE [36] 
* CE 13 is refined into CE [37] 
* CE 14 is refined into CE [38] 


### Cost equations --> "Loop" of take/4 
* CEs [37] --> Loop 24 
* CEs [38] --> Loop 25 
* CEs [34] --> Loop 26 
* CEs [35] --> Loop 27 
* CEs [36] --> Loop 28 

### Ranking functions of CR take(V1,V,V9,Out) 
* RF of phase [25]: [V1,V9]

#### Partial ranking functions of CR take(V1,V,V9,Out) 
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V1
    V9


### Specialization of cost equations helpa/5 
* CE 18 is refined into CE [39,40,41,42,43] 
* CE 16 is refined into CE [44,45,46,47,48] 
* CE 17 is refined into CE [49,50] 


### Cost equations --> "Loop" of helpa/5 
* CEs [45,49] --> Loop 29 
* CEs [44,46,47,48,50] --> Loop 30 
* CEs [42] --> Loop 31 
* CEs [43] --> Loop 32 
* CEs [40] --> Loop 33 
* CEs [41] --> Loop 34 
* CEs [39] --> Loop 35 

### Ranking functions of CR helpa(V1,V,V9,V14,Out) 
* RF of phase [31,32]: [-V1+V]

#### Partial ranking functions of CR helpa(V1,V,V9,V14,Out) 
* Partial RF of phase [31,32]:
  - RF of loop [31:1,32:1]:
    -V1+V
  - RF of loop [32:1]:
    -V1+V14


### Specialization of cost equations length/2 
* CE 23 is refined into CE [51] 
* CE 25 is refined into CE [52] 
* CE 24 is refined into CE [53] 


### Cost equations --> "Loop" of length/2 
* CEs [53] --> Loop 36 
* CEs [51,52] --> Loop 37 

### Ranking functions of CR length(V1,Out) 
* RF of phase [36]: [V1]

#### Partial ranking functions of CR length(V1,Out) 
* Partial RF of phase [36]:
  - RF of loop [36:1]:
    V1


### Specialization of cost equations plus/3 
* CE 22 is refined into CE [54] 
* CE 20 is refined into CE [55] 
* CE 21 is refined into CE [56] 


### Cost equations --> "Loop" of plus/3 
* CEs [56] --> Loop 38 
* CEs [54] --> Loop 39 
* CEs [55] --> Loop 40 

### Ranking functions of CR plus(V1,V,Out) 
* RF of phase [38]: [V]

#### Partial ranking functions of CR plus(V1,V,Out) 
* Partial RF of phase [38]:
  - RF of loop [38:1]:
    V


### Specialization of cost equations app/3 
* CE 19 is refined into CE [57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83] 


### Cost equations --> "Loop" of app/3 
* CEs [70] --> Loop 41 
* CEs [76] --> Loop 42 
* CEs [61,66,80] --> Loop 43 
* CEs [63,68,72,78,82] --> Loop 44 
* CEs [71] --> Loop 45 
* CEs [62,67,77,81] --> Loop 46 
* CEs [57,58,59,64,69,73,74,75,79,83] --> Loop 47 
* CEs [60,65] --> Loop 48 

### Ranking functions of CR app(V1,V,Out) 

#### Partial ranking functions of CR app(V1,V,Out) 


### Specialization of cost equations start/5 
* CE 2 is refined into CE [84] 
* CE 1 is refined into CE [85] 
* CE 3 is refined into CE [86,87,88,89,90,91,92,93] 
* CE 4 is refined into CE [94,95,96,97,98,99,100,101] 
* CE 5 is refined into CE [102,103,104,105,106,107,108,109] 
* CE 6 is refined into CE [110,111,112,113] 
* CE 7 is refined into CE [114,115] 
* CE 8 is refined into CE [116,117,118,119,120,121] 
* CE 9 is refined into CE [122,123,124,125,126] 
* CE 10 is refined into CE [127,128,129,130] 


### Cost equations --> "Loop" of start/5 
* CEs [100,101] --> Loop 49 
* CEs [84] --> Loop 50 
* CEs [92,93] --> Loop 51 
* CEs [88,89] --> Loop 52 
* CEs [86,87,90,91] --> Loop 53 
* CEs [102,117,118,122,128] --> Loop 54 
* CEs [96,97,119] --> Loop 55 
* CEs [94,95,116] --> Loop 56 
* CEs [85,98,99,103,104,105,106,107,108,109,110,111,112,113,114,115,120,121,123,124,125,126,127,129,130] --> Loop 57 

### Ranking functions of CR start(V1,V,V9,V14,V22) 

#### Partial ranking functions of CR start(V1,V,V9,V14,V22) 


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

#### Cost of chains of ge(V1,V,Out):
* Chain [[20],23]: 1*it(20)+1
  Such that:it(20) =< V1

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

* Chain [[20],22]: 1*it(20)+1
  Such that:it(20) =< V

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

* Chain [[20],21]: 1*it(20)+0
  Such that:it(20) =< V

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

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

* Chain [22]: 1
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [21]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of take(V1,V,V9,Out):
* Chain [[25],28]: 1*it(25)+1
  Such that:it(25) =< V1

  with precondition: [V=0,V1>=1,Out>=0,V9>=Out+V1+1] 

* Chain [[25],26]: 1*it(25)+0
  Such that:it(25) =< V9

  with precondition: [V=0,Out=0,V1>=1,V9>=1] 

* Chain [28]: 1
  with precondition: [V1=0,V=0,Out>=0,V9>=Out+1] 

* Chain [27]: 1
  with precondition: [V1=0,V=Out+2,V>=2,V9>=0] 

* Chain [26]: 0
  with precondition: [Out=0,V1>=0,V>=0,V9>=0] 

* Chain [24,26]: 1
  with precondition: [Out=0,V1>=1,V>=2,V9>=0] 


#### Cost of chains of helpa(V1,V,V9,V14,Out):
* Chain [[31,32],30]: 5*it(31)+5*it(32)+4*s(3)+1*s(15)+1*s(16)+2*s(17)+3
  Such that:it(32) =< -V1+V14
aux(4) =< V14
aux(8) =< -V1+V
aux(9) =< V
it(32) =< aux(8)
s(3) =< aux(9)
it(31) =< aux(8)
aux(5) =< aux(9)
s(16) =< it(31)*aux(4)
s(15) =< it(31)*aux(9)
s(18) =< it(32)*aux(5)
s(17) =< s(18)

  with precondition: [V1>=1,V9>=0,V14>=0,Out>=1,V>=V1+1] 

* Chain [35,[31,32],30]: 9*it(31)+5*it(32)+1*s(15)+1*s(16)+2*s(17)+8
  Such that:aux(10) =< V
aux(11) =< V14
it(32) =< aux(11)
it(32) =< aux(10)
it(31) =< aux(10)
aux(5) =< aux(10)
s(16) =< it(31)*aux(11)
s(15) =< it(31)*aux(10)
s(18) =< it(32)*aux(5)
s(17) =< s(18)

  with precondition: [V1=0,V9=0,V>=2,V14>=1,Out>=2] 

* Chain [35,30]: 3*s(3)+1*s(4)+8
  Such that:s(4) =< 1
aux(1) =< V
s(3) =< aux(1)

  with precondition: [V1=0,V9=0,V>=1,Out>=1,V14>=Out] 

* Chain [34,[31,32],30]: 9*it(31)+5*it(32)+1*s(15)+1*s(16)+2*s(17)+1*s(19)+8
  Such that:aux(12) =< V
aux(13) =< V14
it(32) =< aux(13)
s(19) =< aux(13)
it(32) =< aux(12)
it(31) =< aux(12)
aux(5) =< aux(12)
s(16) =< it(31)*aux(13)
s(15) =< it(31)*aux(12)
s(18) =< it(32)*aux(5)
s(17) =< s(18)

  with precondition: [V1=0,V>=2,V9>=0,V14>=0,Out>=2] 

* Chain [34,30]: 3*s(3)+1*s(4)+1*s(19)+8
  Such that:s(4) =< 1
aux(1) =< V
s(19) =< V14
s(3) =< aux(1)

  with precondition: [V1=0,Out=1,V>=1,V9>=0,V14>=0] 

* Chain [33,[31,32],30]: 9*it(31)+5*it(32)+1*s(15)+1*s(16)+2*s(17)+8
  Such that:aux(14) =< V
aux(15) =< V14
it(32) =< aux(15)
it(32) =< aux(14)
it(31) =< aux(14)
aux(5) =< aux(14)
s(16) =< it(31)*aux(15)
s(15) =< it(31)*aux(14)
s(18) =< it(32)*aux(5)
s(17) =< s(18)

  with precondition: [V1=0,V>=2,V9>=2,V14>=0,Out>=V9] 

* Chain [33,30]: 3*s(3)+1*s(4)+8
  Such that:s(4) =< 1
aux(1) =< V
s(3) =< aux(1)

  with precondition: [V1=0,V9=Out+1,V>=1,V9>=2,V14>=0] 

* Chain [30]: 3*s(3)+1*s(4)+3
  Such that:s(4) =< V1
aux(1) =< V
s(3) =< aux(1)

  with precondition: [Out=0,V1>=0,V>=0,V9>=0,V14>=0] 

* Chain [29]: 3
  with precondition: [V=0,Out=0,V1>=0,V9>=0,V14>=0] 


#### Cost of chains of length(V1,Out):
* Chain [[36],37]: 1*it(36)+1
  Such that:it(36) =< V1

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

* Chain [37]: 1
  with precondition: [Out=0,V1>=0] 


#### Cost of chains of plus(V1,V,Out):
* Chain [[38],40]: 1*it(38)+1
  Such that:it(38) =< V

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

* Chain [[38],39]: 1*it(38)+0
  Such that:it(38) =< Out

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

* Chain [40]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

* Chain [39]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of app(V1,V,Out):
* Chain [48]: 10*s(51)+2*s(53)+12
  Such that:aux(20) =< 1
aux(21) =< V
s(53) =< aux(20)
s(51) =< aux(21)

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

* Chain [47]: 19*s(67)+8*s(81)+3*s(99)+7
  Such that:s(98) =< V1+V
aux(32) =< V1
aux(33) =< V
s(81) =< aux(32)
s(67) =< aux(33)
s(99) =< s(98)

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

* Chain [46]: 21*s(106)+4*s(108)+2*s(118)+3*s(124)+12
  Such that:s(122) =< V1+V
aux(38) =< 1
aux(39) =< V1
aux(40) =< V
s(108) =< aux(38)
s(118) =< aux(39)
s(106) =< aux(40)
s(124) =< s(122)

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

* Chain [45]: 4*s(132)+1*s(133)+1*s(135)+12
  Such that:s(133) =< 1
s(135) =< V
aux(41) =< V1
s(132) =< aux(41)

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

* Chain [44]: 17*s(137)+5*s(139)+6*s(147)+3*s(156)+12
  Such that:s(155) =< V1+V
aux(47) =< 1
aux(48) =< V1
aux(49) =< V
s(139) =< aux(47)
s(147) =< aux(48)
s(137) =< aux(49)
s(156) =< s(155)

  with precondition: [V1=Out+1,V1>=2,V>=0] 

* Chain [43]: 135*s(163)+18*s(170)+18*s(173)+1*s(187)+12
  Such that:s(187) =< V1
aux(53) =< V
s(163) =< aux(53)
s(169) =< aux(53)
s(170) =< s(163)*aux(53)
s(172) =< s(163)*s(169)
s(173) =< s(172)

  with precondition: [V1>=0,V>=2,Out>=2] 

* Chain [42]: 1*s(200)+3*s(201)+15*s(205)+27*s(206)+3*s(208)+3*s(209)+6*s(211)+12
  Such that:s(200) =< V1
s(203) =< V1+V
aux(54) =< V
s(201) =< aux(54)
s(205) =< aux(54)
s(205) =< s(203)
s(206) =< s(203)
s(207) =< s(203)
s(208) =< s(206)*aux(54)
s(209) =< s(206)*s(203)
s(210) =< s(205)*s(207)
s(211) =< s(210)

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

* Chain [41]: 28*s(213)+15*s(216)+3*s(219)+3*s(220)+6*s(222)+1*s(223)+12
  Such that:s(215) =< V
aux(55) =< V1
s(213) =< aux(55)
s(216) =< s(215)
s(216) =< aux(55)
s(218) =< aux(55)
s(219) =< s(213)*s(215)
s(220) =< s(213)*aux(55)
s(221) =< s(216)*s(218)
s(222) =< s(221)
s(223) =< s(215)

  with precondition: [V1>=2,V>=0,Out>=2] 


#### Cost of chains of start(V1,V,V9,V14,V22):
* Chain [57]: 2*s(224)+1*s(225)+215*s(227)+10*s(229)+10*s(234)+2*s(236)+2*s(237)+4*s(239)+54*s(243)+36*s(245)+10*s(250)+18*s(270)+18*s(272)+15*s(277)+3*s(280)+3*s(281)+6*s(283)+15*s(287)+3*s(289)+3*s(290)+6*s(292)+1*s(314)+12
  Such that:s(225) =< V1+1
s(314) =< V9
aux(57) =< 1
aux(58) =< -V1+V
aux(59) =< -V1+V14
aux(60) =< V1
aux(61) =< V1+V
aux(62) =< V
aux(63) =< V14
s(250) =< aux(57)
s(229) =< aux(59)
s(243) =< aux(60)
s(227) =< aux(62)
s(224) =< aux(63)
s(245) =< aux(61)
s(229) =< aux(58)
s(234) =< aux(58)
s(235) =< aux(62)
s(236) =< s(234)*aux(63)
s(237) =< s(234)*aux(62)
s(238) =< s(229)*s(235)
s(239) =< s(238)
s(270) =< s(227)*aux(62)
s(271) =< s(227)*s(235)
s(272) =< s(271)
s(277) =< aux(62)
s(277) =< aux(61)
s(279) =< aux(61)
s(280) =< s(245)*aux(62)
s(281) =< s(245)*aux(61)
s(282) =< s(277)*s(279)
s(283) =< s(282)
s(287) =< aux(62)
s(287) =< aux(60)
s(288) =< aux(60)
s(289) =< s(243)*aux(62)
s(290) =< s(243)*aux(60)
s(291) =< s(287)*s(288)
s(292) =< s(291)

  with precondition: [V1>=0] 

* Chain [56]: 2*s(316)+15*s(318)+5*s(319)+1*s(326)+1*s(327)+2*s(329)+8
  Such that:aux(65) =< V14
aux(66) =< 1
aux(67) =< V
s(316) =< aux(66)
s(318) =< aux(67)
s(319) =< aux(65)
s(319) =< aux(67)
s(325) =< aux(67)
s(326) =< s(318)*aux(65)
s(327) =< s(318)*aux(67)
s(328) =< s(319)*s(325)
s(329) =< s(328)

  with precondition: [V1=0,V9=0,V>=0,V14>=1] 

* Chain [55]: 2*s(333)+15*s(335)+5*s(336)+1*s(343)+1*s(344)+2*s(346)+8
  Such that:aux(69) =< V14
aux(70) =< 1
aux(71) =< V
s(333) =< aux(70)
s(335) =< aux(71)
s(336) =< aux(69)
s(336) =< aux(71)
s(342) =< aux(71)
s(343) =< s(335)*aux(69)
s(344) =< s(335)*aux(71)
s(345) =< s(336)*s(342)
s(346) =< s(345)

  with precondition: [V1=0,V>=0,V9>=2,V14>=0] 

* Chain [54]: 3*s(352)+40*s(353)+15*s(356)+3*s(359)+3*s(360)+6*s(362)+2*s(363)+12
  Such that:aux(72) =< 1
aux(73) =< V
aux(74) =< V14
s(352) =< aux(72)
s(363) =< aux(74)
s(353) =< aux(73)
s(356) =< aux(74)
s(356) =< aux(73)
s(358) =< aux(73)
s(359) =< s(353)*aux(74)
s(360) =< s(353)*aux(73)
s(361) =< s(356)*s(358)
s(362) =< s(361)

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

* Chain [53]: 1*s(368)+19*s(370)+5*s(371)+1*s(378)+1*s(379)+2*s(381)+2*s(382)+1*s(383)+5*s(387)+5*s(392)+1*s(394)+1*s(395)+2*s(397)+6
  Such that:s(368) =< 1
s(389) =< -V+V9
s(387) =< -V+V22
s(383) =< V+1
aux(78) =< V9
aux(79) =< V22
s(382) =< aux(79)
s(370) =< aux(78)
s(387) =< s(389)
s(392) =< s(389)
s(377) =< aux(78)
s(394) =< s(392)*aux(79)
s(395) =< s(392)*aux(78)
s(396) =< s(387)*s(377)
s(397) =< s(396)
s(371) =< aux(79)
s(371) =< aux(78)
s(378) =< s(370)*aux(79)
s(379) =< s(370)*aux(78)
s(380) =< s(371)*s(377)
s(381) =< s(380)

  with precondition: [V1=1,V>=0,V9>=0,V14>=0,V22>=0] 

* Chain [52]: 1*s(398)+12*s(400)+5*s(401)+1*s(408)+1*s(409)+2*s(411)+6
  Such that:s(398) =< 1
aux(81) =< V22
aux(82) =< V9
s(400) =< aux(82)
s(401) =< aux(81)
s(401) =< aux(82)
s(407) =< aux(82)
s(408) =< s(400)*aux(81)
s(409) =< s(400)*aux(82)
s(410) =< s(401)*s(407)
s(411) =< s(410)

  with precondition: [V1=1,V=0,V9>=0,V14>=2,V22>=0] 

* Chain [51]: 2*s(412)+1*s(413)+7*s(415)+5*s(417)+5*s(422)+1*s(424)+1*s(425)+2*s(427)+6
  Such that:s(419) =< -V+V9
s(417) =< -V+V22
s(413) =< V+1
s(418) =< V22
aux(83) =< V
aux(84) =< V9
s(412) =< aux(83)
s(417) =< s(419)
s(415) =< aux(84)
s(422) =< s(419)
s(423) =< aux(84)
s(424) =< s(422)*s(418)
s(425) =< s(422)*aux(84)
s(426) =< s(417)*s(423)
s(427) =< s(426)

  with precondition: [V1=1,V14=0,V>=1,V9>=0,V22>=V+1] 

* Chain [50]: 1
  with precondition: [V1=2,V>=0,V9>=0,V14>=0,V22>=0] 

* Chain [49]: 2*s(428)+1*s(429)+7*s(431)+5*s(433)+5*s(438)+1*s(440)+1*s(441)+2*s(443)+5
  Such that:s(435) =< -V1+V
s(433) =< -V1+V14
s(429) =< V1+1
s(434) =< V14
aux(85) =< V1
aux(86) =< V
s(428) =< aux(85)
s(433) =< s(435)
s(431) =< aux(86)
s(438) =< s(435)
s(439) =< aux(86)
s(440) =< s(438)*s(434)
s(441) =< s(438)*aux(86)
s(442) =< s(433)*s(439)
s(443) =< s(442)

  with precondition: [V9=0,V1>=1,V>=0,V14>=V1+1] 


Closed-form bounds of start(V1,V,V9,V14,V22): 
-------------------------------------
* Chain [57] with precondition: [V1>=0] 
    - Upper bound: 54*V1+22+3*V1*V1+9*V1*nat(V)+nat(V)*245+nat(V)*36*nat(V)+nat(V)*9*nat(V1+V)+nat(V)*2*nat(-V1+V)+nat(V)*4*nat(-V1+V14)+nat(V9)+nat(V14)*2+nat(V14)*2*nat(-V1+V)+nat(V1+V)*36+nat(V1+V)*3*nat(V1+V)+(V1+1)+nat(-V1+V)*10+nat(-V1+V14)*10 
    - Complexity: n^2 
* Chain [56] with precondition: [V1=0,V9=0,V>=0,V14>=1] 
    - Upper bound: 15*V+10+V*V+3*V*V14+5*V14 
    - Complexity: n^2 
* Chain [55] with precondition: [V1=0,V>=0,V9>=2,V14>=0] 
    - Upper bound: 15*V+10+V*V+3*V*V14+5*V14 
    - Complexity: n^2 
* Chain [54] with precondition: [V1=0,V>=1] 
    - Upper bound: 40*V+15+3*V*V+9*V*nat(V14)+nat(V14)*17 
    - Complexity: n^2 
* Chain [53] with precondition: [V1=1,V>=0,V9>=0,V14>=0,V22>=0] 
    - Upper bound: 19*V9+7+V9*V9+3*V9*V22+nat(-V+V9)*V9+2*V9*nat(-V+V22)+7*V22+nat(-V+V9)*V22+(V+1)+nat(-V+V9)*5+nat(-V+V22)*5 
    - Complexity: n^2 
* Chain [52] with precondition: [V1=1,V=0,V9>=0,V14>=2,V22>=0] 
    - Upper bound: 12*V9+7+V9*V9+3*V9*V22+5*V22 
    - Complexity: n^2 
* Chain [51] with precondition: [V1=1,V14=0,V>=1,V9>=0,V22>=V+1] 
    - Upper bound: 2*V+7*V9+6+nat(-V+V9)*V9+(-V+V22)*(2*V9)+nat(-V+V9)*V22+(V+1)+nat(-V+V9)*5+(-5*V+5*V22) 
    - Complexity: n^2 
* Chain [50] with precondition: [V1=2,V>=0,V9>=0,V14>=0,V22>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [49] with precondition: [V9=0,V1>=1,V>=0,V14>=V1+1] 
    - Upper bound: 2*V1+7*V+5+nat(-V1+V)*V+(-V1+V14)*(2*V)+nat(-V1+V)*V14+(V1+1)+nat(-V1+V)*5+(-5*V1+5*V14) 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V9,V14,V22): max([nat(V)*2+4+max([nat(V)*5+max([nat(V)*8+5+nat(V)*nat(V)+nat(V14)*2+max([nat(V)*3*nat(V14)+nat(V14)*3,nat(V)*25+5+nat(V)*2*nat(V)+max([nat(V)*9*nat(V14)+nat(V14)*15,54*V1+7+3*V1*V1+9*V1*nat(V)+nat(V)*205+nat(V)*33*nat(V)+nat(V)*9*nat(V1+V)+nat(V)*2*nat(-V1+V)+nat(V)*4*nat(-V1+V14)+nat(V9)+nat(V14)*2*nat(-V1+V)+nat(V1+V)*36+nat(V1+V)*3*nat(V1+V)+(V1+1)+nat(-V1+V)*10+nat(-V1+V14)*10])]),nat(-V1+V)*nat(V)+2*V1+nat(V)*2*nat(-V1+V14)+nat(-V1+V)*nat(V14)+(V1+1)+nat(-V1+V)*5+nat(-V1+V14)*5]),nat(V9)*7+1+nat(-V+V9)*nat(V9)+nat(V9)*2*nat(-V+V22)+nat(-V+V9)*nat(V22)+nat(V+1)+nat(-V+V9)*5+nat(-V+V22)*5]),nat(-V+V9)*nat(V9)+nat(V9)*7+nat(V9)*2*nat(-V+V22)+nat(V22)*2+nat(-V+V9)*nat(V22)+nat(V+1)+nat(-V+V9)*5+nat(-V+V22)*5+(nat(V9)*12+6+nat(V9)*nat(V9)+nat(V9)*3*nat(V22)+nat(V22)*5)])+1 
Asymptotic class: n^2 
* Total analysis performed in 1070 ms.


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

(10)
BOUNDS(1, n^2)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
   plus(x, 0') -> x
   plus(x, s(y)) -> s(plus(x, y))
   length(nil) -> 0'
   length(cons(x, y)) -> s(length(y))
   helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   if(true, c, l, ys, zs) -> nil
   if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
   take(0', cons(x, xs), ys) -> x
   take(0', nil, cons(y, ys)) -> y
   take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
   take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
   helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
helpa, plus, length, ge, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

(16)
Obligation:
Innermost TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
plus, helpa, length, ge, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)

Induction Base:
plus(gen_0':s6_0(a), gen_0':s6_0(0)) ->_R^Omega(1)
gen_0':s6_0(a)

Induction Step:
plus(gen_0':s6_0(a), gen_0':s6_0(+(n8_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s6_0(a), gen_0':s6_0(n8_0))) ->_IH
s(gen_0':s6_0(+(a, c9_0)))

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

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
plus, helpa, length, ge, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
length, helpa, ge, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons:xs5_0(n857_0)) -> gen_0':s6_0(n857_0), rt in Omega(1 + n857_0)

Induction Base:
length(gen_nil:cons:xs5_0(0)) ->_R^Omega(1)
0'

Induction Step:
length(gen_nil:cons:xs5_0(+(n857_0, 1))) ->_R^Omega(1)
s(length(gen_nil:cons:xs5_0(n857_0))) ->_IH
s(gen_0':s6_0(c858_0))

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

(24)
Obligation:
Innermost TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)
length(gen_nil:cons:xs5_0(n857_0)) -> gen_0':s6_0(n857_0), rt in Omega(1 + n857_0)


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
ge, helpa, helpb, take

They will be analysed ascendingly in the following order:
ge < helpa
helpa = helpb
take < helpb

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s6_0(n1150_0), gen_0':s6_0(n1150_0)) -> true, rt in Omega(1 + n1150_0)

Induction Base:
ge(gen_0':s6_0(0), gen_0':s6_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s6_0(+(n1150_0, 1)), gen_0':s6_0(+(n1150_0, 1))) ->_R^Omega(1)
ge(gen_0':s6_0(n1150_0), gen_0':s6_0(n1150_0)) ->_IH
true

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

(26)
Obligation:
Innermost TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)
length(gen_nil:cons:xs5_0(n857_0)) -> gen_0':s6_0(n857_0), rt in Omega(1 + n857_0)
ge(gen_0':s6_0(n1150_0), gen_0':s6_0(n1150_0)) -> true, rt in Omega(1 + n1150_0)


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
take, helpa, helpb

They will be analysed ascendingly in the following order:
helpa = helpb
take < helpb

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_0':s6_0(n1472_0), gen_nil:cons:xs5_0(0), gen_nil:cons:xs5_0(+(1, n1472_0))) -> hole_take3_0, rt in Omega(1 + n1472_0)

Induction Base:
take(gen_0':s6_0(0), gen_nil:cons:xs5_0(0), gen_nil:cons:xs5_0(+(1, 0))) ->_R^Omega(1)
hole_take3_0

Induction Step:
take(gen_0':s6_0(+(n1472_0, 1)), gen_nil:cons:xs5_0(0), gen_nil:cons:xs5_0(+(1, +(n1472_0, 1)))) ->_R^Omega(1)
take(gen_0':s6_0(n1472_0), nil, gen_nil:cons:xs5_0(+(1, n1472_0))) ->_IH
hole_take3_0

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

(28)
Obligation:
Innermost TRS:
Rules:
app(x, y) -> helpa(0', plus(length(x), length(y)), x, y)
plus(x, 0') -> x
plus(x, s(y)) -> s(plus(x, y))
length(nil) -> 0'
length(cons(x, y)) -> s(length(y))
helpa(c, l, ys, zs) -> if(ge(c, l), c, l, ys, zs)
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
if(true, c, l, ys, zs) -> nil
if(false, c, l, ys, zs) -> helpb(c, l, ys, zs)
take(0', cons(x, xs), ys) -> x
take(0', nil, cons(y, ys)) -> y
take(s(c), cons(x, xs), ys) -> take(c, xs, ys)
take(s(c), nil, cons(y, ys)) -> take(c, nil, ys)
helpb(c, l, ys, zs) -> cons(take(c, ys, zs), helpa(s(c), l, ys, zs))

Types:
app :: nil:cons:xs -> nil:cons:xs -> nil:cons:xs
helpa :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
length :: nil:cons:xs -> 0':s
s :: 0':s -> 0':s
nil :: nil:cons:xs
cons :: take -> nil:cons:xs -> nil:cons:xs
if :: true:false -> 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
helpb :: 0':s -> 0':s -> nil:cons:xs -> nil:cons:xs -> nil:cons:xs
take :: 0':s -> nil:cons:xs -> nil:cons:xs -> take
xs :: nil:cons:xs
hole_nil:cons:xs1_0 :: nil:cons:xs
hole_0':s2_0 :: 0':s
hole_take3_0 :: take
hole_true:false4_0 :: true:false
gen_nil:cons:xs5_0 :: Nat -> nil:cons:xs
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(a), gen_0':s6_0(n8_0)) -> gen_0':s6_0(+(n8_0, a)), rt in Omega(1 + n8_0)
length(gen_nil:cons:xs5_0(n857_0)) -> gen_0':s6_0(n857_0), rt in Omega(1 + n857_0)
ge(gen_0':s6_0(n1150_0), gen_0':s6_0(n1150_0)) -> true, rt in Omega(1 + n1150_0)
take(gen_0':s6_0(n1472_0), gen_nil:cons:xs5_0(0), gen_nil:cons:xs5_0(+(1, n1472_0))) -> hole_take3_0, rt in Omega(1 + n1472_0)


Generator Equations:
gen_nil:cons:xs5_0(0) <=> nil
gen_nil:cons:xs5_0(+(x, 1)) <=> cons(hole_take3_0, gen_nil:cons:xs5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


The following defined symbols remain to be analysed:
helpb, helpa

They will be analysed ascendingly in the following order:
helpa = helpb