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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 CpxRelTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 165 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 233 ms]
    (14) BOUNDS(1, n^2)
(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), 248 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 476 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs


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

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


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
   isort(Nil, r) -> Nil
   insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
   inssort(xs) -> isort(xs, Nil)

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

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
   insert[Ite](True, x, r) -> Cons(x, r)

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
   isort(Nil, r) -> Nil
   insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
   inssort(xs) -> isort(xs, Nil)

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

   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
   insert[Ite](True, x, r) -> Cons(x, r)

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   <(S(x), S(y)) -> <(x, y) [0]
   <(0, S(y)) -> True [0]
   <(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]

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

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

< => lt

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

(6)
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:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]

The TRS has the following type information:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
insert :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0 -> S:0
insert[Ite] :: True:False -> S:0 -> Cons:Nil -> Cons:Nil
lt :: S:0 -> S:0 -> True:False
inssort :: Cons:Nil -> Cons:Nil
0 :: S:0
True :: True:False
False :: True:False

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

(9) 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:

   lt(v0, v1) -> null_lt [0]
   insert[Ite](v0, v1, v2) -> null_insert[Ite] [0]
   isort(v0, v1) -> null_isort [0]
   insert(v0, v1) -> null_insert [0]

And the following fresh constants:    null_lt, null_insert[Ite], null_isort, null_insert

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

(10)
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:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r)) [1]
   isort(Nil, r) -> Nil [1]
   insert(S(x), r) -> insert[Ite](lt(S(x), x), S(x), r) [1]
   inssort(xs) -> isort(xs, Nil) [1]
   lt(S(x), S(y)) -> lt(x, y) [0]
   lt(0, S(y)) -> True [0]
   lt(x, 0) -> False [0]
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs)) [0]
   insert[Ite](True, x, r) -> Cons(x, r) [0]
   lt(v0, v1) -> null_lt [0]
   insert[Ite](v0, v1, v2) -> null_insert[Ite] [0]
   isort(v0, v1) -> null_isort [0]
   insert(v0, v1) -> null_insert [0]

The TRS has the following type information:
isort :: Cons:Nil:null_insert[Ite]:null_isort:null_insert -> Cons:Nil:null_insert[Ite]:null_isort:null_insert -> Cons:Nil:null_insert[Ite]:null_isort:null_insert
Cons :: S:0 -> Cons:Nil:null_insert[Ite]:null_isort:null_insert -> Cons:Nil:null_insert[Ite]:null_isort:null_insert
insert :: S:0 -> Cons:Nil:null_insert[Ite]:null_isort:null_insert -> Cons:Nil:null_insert[Ite]:null_isort:null_insert
Nil :: Cons:Nil:null_insert[Ite]:null_isort:null_insert
S :: S:0 -> S:0
insert[Ite] :: True:False:null_lt -> S:0 -> Cons:Nil:null_insert[Ite]:null_isort:null_insert -> Cons:Nil:null_insert[Ite]:null_isort:null_insert
lt :: S:0 -> S:0 -> True:False:null_lt
inssort :: Cons:Nil:null_insert[Ite]:null_isort:null_insert -> Cons:Nil:null_insert[Ite]:null_isort:null_insert
0 :: S:0
True :: True:False:null_lt
False :: True:False:null_lt
null_lt :: True:False:null_lt
null_insert[Ite] :: Cons:Nil:null_insert[Ite]:null_isort:null_insert
null_isort :: Cons:Nil:null_insert[Ite]:null_isort:null_insert
null_insert :: Cons:Nil:null_insert[Ite]:null_isort:null_insert

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

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   0 => 0
   True => 2
   False => 1
   null_lt => 0
   null_insert[Ite] => 0
   null_isort => 0
   null_insert => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   insert(z, z') -{ 1 }-> insert[Ite](lt(1 + x, x), 1 + x, r) :|: r >= 0, x >= 0, z = 1 + x, z' = r
   insert(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   insert[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + r :|: z = 2, z'' = r, r >= 0, z' = x, x >= 0
   insert[Ite](z, z', z'') -{ 0 }-> 1 + x + insert(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
   inssort(z) -{ 1 }-> isort(xs, 0) :|: xs >= 0, z = xs
   isort(z, z') -{ 1 }-> isort(xs, insert(x, r)) :|: z = 1 + x + xs, xs >= 0, r >= 0, x >= 0, z' = r
   isort(z, z') -{ 1 }-> 0 :|: r >= 0, z = 0, z' = r
   isort(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   lt(z, z') -{ 0 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 0 }-> 2 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 0 }-> 1 :|: x >= 0, z = x, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V14),0,[isort(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V14),0,[insert(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V14),0,[inssort(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V14),0,[lt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V14),0,[fun(V1, V, V14, Out)],[V1 >= 0,V >= 0,V14 >= 0]).
eq(isort(V1, V, Out),1,[insert(V4, V2, Ret1),isort(V3, Ret1, Ret)],[Out = Ret,V1 = 1 + V3 + V4,V3 >= 0,V2 >= 0,V4 >= 0,V = V2]).
eq(isort(V1, V, Out),1,[],[Out = 0,V5 >= 0,V1 = 0,V = V5]).
eq(insert(V1, V, Out),1,[lt(1 + V6, V6, Ret0),fun(Ret0, 1 + V6, V7, Ret2)],[Out = Ret2,V7 >= 0,V6 >= 0,V1 = 1 + V6,V = V7]).
eq(inssort(V1, Out),1,[isort(V8, 0, Ret3)],[Out = Ret3,V8 >= 0,V1 = V8]).
eq(lt(V1, V, Out),0,[lt(V9, V10, Ret4)],[Out = Ret4,V = 1 + V10,V9 >= 0,V10 >= 0,V1 = 1 + V9]).
eq(lt(V1, V, Out),0,[],[Out = 2,V = 1 + V11,V11 >= 0,V1 = 0]).
eq(lt(V1, V, Out),0,[],[Out = 1,V12 >= 0,V1 = V12,V = 0]).
eq(fun(V1, V, V14, Out),0,[insert(V16, V15, Ret11)],[Out = 1 + Ret11 + V13,V = V16,V15 >= 0,V1 = 1,V16 >= 0,V13 >= 0,V14 = 1 + V13 + V15]).
eq(fun(V1, V, V14, Out),0,[],[Out = 1 + V17 + V18,V1 = 2,V14 = V18,V18 >= 0,V = V17,V17 >= 0]).
eq(lt(V1, V, Out),0,[],[Out = 0,V20 >= 0,V19 >= 0,V1 = V20,V = V19]).
eq(fun(V1, V, V14, Out),0,[],[Out = 0,V22 >= 0,V14 = V23,V21 >= 0,V1 = V22,V = V21,V23 >= 0]).
eq(isort(V1, V, Out),0,[],[Out = 0,V25 >= 0,V24 >= 0,V1 = V25,V = V24]).
eq(insert(V1, V, Out),0,[],[Out = 0,V26 >= 0,V27 >= 0,V1 = V26,V = V27]).
input_output_vars(isort(V1,V,Out),[V1,V],[Out]).
input_output_vars(insert(V1,V,Out),[V1,V],[Out]).
input_output_vars(inssort(V1,Out),[V1],[Out]).
input_output_vars(lt(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,V14,Out),[V1,V,V14],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [lt/3]
1. recursive  : [fun/4,insert/3]
2. recursive  : [isort/3]
3. non_recursive  : [inssort/2]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into lt/3
1. SCC is partially evaluated into insert/3
2. SCC is partially evaluated into isort/3
3. SCC is completely evaluated into other SCCs
4. SCC is partially evaluated into start/3

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

### Specialization of cost equations lt/3 
* CE 17 is refined into CE [18] 
* CE 16 is refined into CE [19] 
* CE 15 is refined into CE [20] 
* CE 14 is refined into CE [21] 


### Cost equations --> "Loop" of lt/3 
* CEs [21] --> Loop 11 
* CEs [18] --> Loop 12 
* CEs [19] --> Loop 13 
* CEs [20] --> Loop 14 

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

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


### Specialization of cost equations insert/3 
* CE 8 is discarded (unfeasible) 
* CE 7 is refined into CE [22,23,24] 
* CE 10 is refined into CE [25] 
* CE 9 is refined into CE [26,27] 


### Cost equations --> "Loop" of insert/3 
* CEs [27] --> Loop 15 
* CEs [26] --> Loop 16 
* CEs [22,23,24,25] --> Loop 17 

### Ranking functions of CR insert(V1,V,Out) 
* RF of phase [15]: [V]
* RF of phase [16]: [V]

#### Partial ranking functions of CR insert(V1,V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V
* Partial RF of phase [16]:
  - RF of loop [16:1]:
    V


### Specialization of cost equations isort/3 
* CE 12 is refined into CE [28] 
* CE 13 is refined into CE [29] 
* CE 11 is refined into CE [30,31,32] 


### Cost equations --> "Loop" of isort/3 
* CEs [32] --> Loop 18 
* CEs [30] --> Loop 19 
* CEs [31] --> Loop 20 
* CEs [28,29] --> Loop 21 

### Ranking functions of CR isort(V1,V,Out) 
* RF of phase [18,19]: [V1/2-1/2]
* RF of phase [20]: [V1]

#### Partial ranking functions of CR isort(V1,V,Out) 
* Partial RF of phase [18,19]:
  - RF of loop [18:1]:
    V1/3-2/3
  - RF of loop [19:1]:
    V1/2-1/2
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V1


### Specialization of cost equations start/3 
* CE 1 is refined into CE [33] 
* CE 2 is refined into CE [34,35,36] 
* CE 3 is refined into CE [37] 
* CE 4 is refined into CE [38,39,40] 
* CE 5 is refined into CE [41] 
* CE 6 is refined into CE [42,43,44,45,46] 


### Cost equations --> "Loop" of start/3 
* CEs [43] --> Loop 22 
* CEs [38] --> Loop 23 
* CEs [34,35,36] --> Loop 24 
* CEs [33,37,39,40,41,42,44,45,46] --> Loop 25 

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

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


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

#### Cost of chains of lt(V1,V,Out):
* Chain [[11],14]: 0
  with precondition: [Out=2,V1>=1,V>=V1+1] 

* Chain [[11],13]: 0
  with precondition: [Out=1,V>=1,V1>=V] 

* Chain [[11],12]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [14]: 0
  with precondition: [V1=0,Out=2,V>=1] 

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

* Chain [12]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of insert(V1,V,Out):
* Chain [[16],17]: 1*it(16)+1
  Such that:it(16) =< Out

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

* Chain [[15],17]: 1*it(15)+1
  Such that:it(15) =< Out

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

* Chain [17]: 1
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of isort(V1,V,Out):
* Chain [[20],21]: 2*it(20)+1
  Such that:it(20) =< V1

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

* Chain [[18,19],[20],21]: 4*it(18)+2*it(20)+1*s(5)+1*s(6)+1
  Such that:aux(3) =< V1/2
aux(1) =< V
aux(5) =< V1
it(18) =< aux(5)
it(20) =< aux(5)
it(18) =< aux(3)
aux(2) =< aux(1)
s(5) =< it(18)*aux(1)
s(6) =< it(18)*aux(2)

  with precondition: [Out=0,V1>=3,V>=1] 

* Chain [[18,19],21]: 2*it(18)+2*it(19)+1*s(5)+1*s(6)+1
  Such that:it(18) =< V1/3
aux(1) =< V
aux(6) =< V1/2
it(18) =< aux(6)
it(19) =< aux(6)
aux(2) =< aux(1)
s(5) =< it(18)*aux(1)
s(6) =< it(19)*aux(2)

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

* Chain [21]: 1
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of start(V1,V,V14):
* Chain [25]: 4*s(23)+8*s(27)+8*s(28)+1*s(30)+1*s(31)+4*s(32)+1*s(33)+1*s(34)+1*s(35)+2
  Such that:aux(10) =< V1
aux(11) =< V1/2
aux(12) =< V1/3
aux(13) =< V
s(23) =< aux(12)
s(35) =< aux(13)
s(27) =< aux(10)
s(28) =< aux(10)
s(28) =< aux(11)
s(29) =< aux(13)
s(30) =< s(28)*aux(13)
s(31) =< s(28)*s(29)
s(23) =< aux(11)
s(32) =< aux(11)
s(33) =< s(23)*aux(13)
s(34) =< s(32)*s(29)

  with precondition: [V1>=0] 

* Chain [24]: 2*s(48)+1
  Such that:aux(14) =< V14
s(48) =< aux(14)

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

* Chain [23]: 1*s(50)+1
  Such that:s(50) =< V

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

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


Closed-form bounds of start(V1,V,V14): 
-------------------------------------
* Chain [25] with precondition: [V1>=0] 
    - Upper bound: 16*V1+2+2*V1*nat(V)+nat(V)+V1/2*nat(V)+V1/3*nat(V)+2*V1+4/3*V1 
    - Complexity: n^2 
* Chain [24] with precondition: [V1=1,V>=0,V14>=1] 
    - Upper bound: 2*V14+1 
    - Complexity: n 
* Chain [23] with precondition: [V1=1,V>=1] 
    - Upper bound: V+1 
    - Complexity: n 
* Chain [22] with precondition: [V=0,V1>=0] 
    - Upper bound: 0 
    - Complexity: constant 

### Maximum cost of start(V1,V,V14): max([nat(V14)*2+1,16*V1+1+2*V1*nat(V)+V1/2*nat(V)+V1/3*nat(V)+2*V1+4/3*V1+(nat(V)+1)]) 
Asymptotic class: n^2 
* Total analysis performed in 271 ms.


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

(14)
BOUNDS(1, n^2)

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

(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^1, INF).


The TRS R consists of the following rules:

   isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
   isort(Nil, r) -> Nil
   insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
   inssort(xs) -> isort(xs, Nil)

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

   <(S(x), S(y)) -> <(x, y)
   <(0', S(y)) -> True
   <(x, 0') -> False
   insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
   insert[Ite](True, x, r) -> Cons(x, r)

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

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

(18)
Obligation:
Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
isort, insert, <

They will be analysed ascendingly in the following order:
insert < isort
< < insert

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

(20)
Obligation:
Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
<, isort, insert

They will be analysed ascendingly in the following order:
insert < isort
< < insert

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)

Induction Base:
<(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) ->_R^Omega(0)
True

Induction Step:
<(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) ->_R^Omega(0)
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) ->_IH
True

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

(22)
Obligation:
Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
insert, isort

They will be analysed ascendingly in the following order:
insert < isort

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n215_0)) -> *6_0, rt in Omega(n215_0)

Induction Base:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(0))

Induction Step:
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(+(n215_0, 1))) ->_R^Omega(1)
insert[Ite](<(S(gen_S:0'5_0(0)), gen_S:0'5_0(0)), S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(+(n215_0, 1))) ->_R^Omega(0)
insert[Ite](False, S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(+(1, n215_0))) ->_R^Omega(0)
Cons(0', insert(S(gen_S:0'5_0(0)), gen_Cons:Nil4_0(n215_0))) ->_IH
Cons(0', *6_0)

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

(24)
Complex Obligation (BEST)

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

(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
insert, isort

They will be analysed ascendingly in the following order:
insert < isort

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

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

(27)
BOUNDS(n^1, INF)

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

(28)
Obligation:
Innermost TRS:
Rules:
isort(Cons(x, xs), r) -> isort(xs, insert(x, r))
isort(Nil, r) -> Nil
insert(S(x), r) -> insert[Ite](<(S(x), x), S(x), r)
inssort(xs) -> isort(xs, Nil)
<(S(x), S(y)) -> <(x, y)
<(0', S(y)) -> True
<(x, 0') -> False
insert[Ite](False, x', Cons(x, xs)) -> Cons(x, insert(x', xs))
insert[Ite](True, x, r) -> Cons(x, r)

Types:
isort :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
insert :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
S :: S:0' -> S:0'
insert[Ite] :: True:False -> S:0' -> Cons:Nil -> Cons:Nil
< :: S:0' -> S:0' -> True:False
inssort :: Cons:Nil -> Cons:Nil
0' :: S:0'
True :: True:False
False :: True:False
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat -> Cons:Nil
gen_S:0'5_0 :: Nat -> S:0'


Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) -> True, rt in Omega(0)
insert(gen_S:0'5_0(1), gen_Cons:Nil4_0(n215_0)) -> *6_0, rt in Omega(n215_0)


Generator Equations:
gen_Cons:Nil4_0(0) <=> Nil
gen_Cons:Nil4_0(+(x, 1)) <=> Cons(0', gen_Cons:Nil4_0(x))
gen_S:0'5_0(0) <=> 0'
gen_S:0'5_0(+(x, 1)) <=> S(gen_S:0'5_0(x))


The following defined symbols remain to be analysed:
isort