/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^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 172 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, 375 ms]
    (14) BOUNDS(1, n^1)
(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), 237 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 71 ms]
    (24) proven lower bound
    (25) LowerBoundPropagationProof [FINISHED, 0 ms]
    (26) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Nil, ys) -> ys
   goal(xs, ys) -> merge(xs, ys)

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

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

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^1).


The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Nil, ys) -> ys
   goal(xs, ys) -> merge(xs, ys)

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

   <=(S(x), S(y)) -> <=(x, y)
   <=(0, y) -> True
   <=(S(x), 0) -> False
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

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^1).


The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   <=(S(x), S(y)) -> <=(x, y) [0]
   <=(0, y) -> True [0]
   <=(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [0]

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

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

<= => lteq

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   lteq(S(x), S(y)) -> lteq(x, y) [0]
   lteq(0, y) -> True [0]
   lteq(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [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:

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   lteq(S(x), S(y)) -> lteq(x, y) [0]
   lteq(0, y) -> True [0]
   lteq(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [0]

The TRS has the following type information:
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0 -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
lteq :: S:0 -> S:0 -> True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S:0 -> S:0
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:

   lteq(v0, v1) -> null_lteq [0]
   merge[Ite](v0, v1, v2) -> null_merge[Ite] [0]
   merge(v0, v1) -> null_merge [0]

And the following fresh constants:    null_lteq, null_merge[Ite], null_merge

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

(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:

   merge(Cons(x, xs), Nil) -> Cons(x, xs) [1]
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](lteq(x', x), Cons(x', xs'), Cons(x, xs)) [1]
   merge(Nil, ys) -> ys [1]
   goal(xs, ys) -> merge(xs, ys) [1]
   lteq(S(x), S(y)) -> lteq(x, y) [0]
   lteq(0, y) -> True [0]
   lteq(S(x), 0) -> False [0]
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs)) [0]
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys)) [0]
   lteq(v0, v1) -> null_lteq [0]
   merge[Ite](v0, v1, v2) -> null_merge[Ite] [0]
   merge(v0, v1) -> null_merge [0]

The TRS has the following type information:
merge :: Cons:Nil:null_merge[Ite]:null_merge -> Cons:Nil:null_merge[Ite]:null_merge -> Cons:Nil:null_merge[Ite]:null_merge
Cons :: S:0 -> Cons:Nil:null_merge[Ite]:null_merge -> Cons:Nil:null_merge[Ite]:null_merge
Nil :: Cons:Nil:null_merge[Ite]:null_merge
merge[Ite] :: True:False:null_lteq -> Cons:Nil:null_merge[Ite]:null_merge -> Cons:Nil:null_merge[Ite]:null_merge -> Cons:Nil:null_merge[Ite]:null_merge
lteq :: S:0 -> S:0 -> True:False:null_lteq
goal :: Cons:Nil:null_merge[Ite]:null_merge -> Cons:Nil:null_merge[Ite]:null_merge -> Cons:Nil:null_merge[Ite]:null_merge
S :: S:0 -> S:0
0 :: S:0
True :: True:False:null_lteq
False :: True:False:null_lteq
null_lteq :: True:False:null_lteq
null_merge[Ite] :: Cons:Nil:null_merge[Ite]:null_merge
null_merge :: Cons:Nil:null_merge[Ite]:null_merge

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_lteq => 0
   null_merge[Ite] => 0
   null_merge => 0

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

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

   goal(z, z') -{ 1 }-> merge(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0
   lteq(z, z') -{ 0 }-> lteq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lteq(z, z') -{ 0 }-> 2 :|: y >= 0, z = 0, z' = y
   lteq(z, z') -{ 0 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   lteq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   merge(z, z') -{ 1 }-> ys :|: z' = ys, ys >= 0, z = 0
   merge(z, z') -{ 1 }-> merge[Ite](lteq(x', x), 1 + x' + xs', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   merge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   merge(z, z') -{ 1 }-> 1 + x + xs :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   merge[Ite](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs, ys) :|: z = 2, xs >= 0, z' = 1 + x + xs, ys >= 0, x >= 0, z'' = ys
   merge[Ite](z, z', z'') -{ 0 }-> 1 + x + merge(xs', xs) :|: xs >= 0, z = 1, xs' >= 0, x >= 0, z' = xs', z'' = 1 + x + xs

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, V16),0,[merge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V16),0,[goal(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V16),0,[lteq(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V16),0,[fun(V1, V, V16, Out)],[V1 >= 0,V >= 0,V16 >= 0]).
eq(merge(V1, V, Out),1,[],[Out = 1 + V2 + V3,V1 = 1 + V2 + V3,V2 >= 0,V3 >= 0,V = 0]).
eq(merge(V1, V, Out),1,[lteq(V7, V5, Ret0),fun(Ret0, 1 + V7 + V4, 1 + V5 + V6, Ret)],[Out = Ret,V6 >= 0,V = 1 + V5 + V6,V7 >= 0,V4 >= 0,V5 >= 0,V1 = 1 + V4 + V7]).
eq(merge(V1, V, Out),1,[],[Out = V8,V = V8,V8 >= 0,V1 = 0]).
eq(goal(V1, V, Out),1,[merge(V9, V10, Ret1)],[Out = Ret1,V9 >= 0,V1 = V9,V = V10,V10 >= 0]).
eq(lteq(V1, V, Out),0,[lteq(V11, V12, Ret2)],[Out = Ret2,V = 1 + V12,V11 >= 0,V12 >= 0,V1 = 1 + V11]).
eq(lteq(V1, V, Out),0,[],[Out = 2,V13 >= 0,V1 = 0,V = V13]).
eq(lteq(V1, V, Out),0,[],[Out = 1,V14 >= 0,V1 = 1 + V14,V = 0]).
eq(fun(V1, V, V16, Out),0,[merge(V17, V18, Ret11)],[Out = 1 + Ret11 + V15,V18 >= 0,V1 = 1,V17 >= 0,V15 >= 0,V = V17,V16 = 1 + V15 + V18]).
eq(fun(V1, V, V16, Out),0,[merge(V19, V21, Ret12)],[Out = 1 + Ret12 + V20,V1 = 2,V19 >= 0,V = 1 + V19 + V20,V21 >= 0,V20 >= 0,V16 = V21]).
eq(lteq(V1, V, Out),0,[],[Out = 0,V23 >= 0,V22 >= 0,V1 = V23,V = V22]).
eq(fun(V1, V, V16, Out),0,[],[Out = 0,V25 >= 0,V16 = V26,V24 >= 0,V1 = V25,V = V24,V26 >= 0]).
eq(merge(V1, V, Out),0,[],[Out = 0,V28 >= 0,V27 >= 0,V1 = V28,V = V27]).
input_output_vars(merge(V1,V,Out),[V1,V],[Out]).
input_output_vars(goal(V1,V,Out),[V1,V],[Out]).
input_output_vars(lteq(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun(V1,V,V16,Out),[V1,V,V16],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [lteq/3]
1. recursive  : [fun/4,merge/3]
2. non_recursive  : [goal/3]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into lteq/3
1. SCC is partially evaluated into merge/3
2. SCC is completely evaluated into other SCCs
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations lteq/3 
* CE 16 is refined into CE [17] 
* CE 15 is refined into CE [18] 
* CE 14 is refined into CE [19] 
* CE 13 is refined into CE [20] 


### Cost equations --> "Loop" of lteq/3 
* CEs [20] --> Loop 12 
* CEs [17] --> Loop 13 
* CEs [18] --> Loop 14 
* CEs [19] --> Loop 15 

### Ranking functions of CR lteq(V1,V,Out) 
* RF of phase [12]: [V,V1]

#### Partial ranking functions of CR lteq(V1,V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V
    V1


### Specialization of cost equations merge/3 
* CE 7 is refined into CE [21,22,23,24,25] 
* CE 12 is refined into CE [26] 
* CE 10 is refined into CE [27] 
* CE 11 is refined into CE [28] 
* CE 8 is refined into CE [29,30] 
* CE 9 is refined into CE [31,32] 


### Cost equations --> "Loop" of merge/3 
* CEs [29,30] --> Loop 16 
* CEs [31,32] --> Loop 17 
* CEs [21,22,23,24,25,26] --> Loop 18 
* CEs [27] --> Loop 19 
* CEs [28] --> Loop 20 

### Ranking functions of CR merge(V1,V,Out) 
* RF of phase [16,17]: [V1+V-1,V1+2*V-2]

#### Partial ranking functions of CR merge(V1,V,Out) 
* Partial RF of phase [16,17]:
  - RF of loop [16:1]:
    V1
  - RF of loop [17:1]:
    V


### Specialization of cost equations start/3 
* CE 2 is refined into CE [33,34,35,36,37] 
* CE 1 is refined into CE [38] 
* CE 3 is refined into CE [39,40,41,42,43] 
* CE 4 is refined into CE [44,45,46,47,48] 
* CE 5 is refined into CE [49,50,51,52,53] 
* CE 6 is refined into CE [54,55,56,57,58] 


### Cost equations --> "Loop" of start/3 
* CEs [45,50,55] --> Loop 21 
* CEs [33,34,35,36,37] --> Loop 22 
* CEs [39,40,41,42,43] --> Loop 23 
* CEs [38,44,46,47,48,49,51,52,53,54,56,57,58] --> Loop 24 

### Ranking functions of CR start(V1,V,V16) 

#### Partial ranking functions of CR start(V1,V,V16) 


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

#### Cost of chains of lteq(V1,V,Out):
* Chain [[12],15]: 0
  with precondition: [Out=2,V1>=1,V>=V1] 

* Chain [[12],14]: 0
  with precondition: [Out=1,V>=1,V1>=V+1] 

* Chain [[12],13]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [15]: 0
  with precondition: [V1=0,Out=2,V>=0] 

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

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


#### Cost of chains of merge(V1,V,Out):
* Chain [[16,17],20]: 1*it(16)+1*it(17)+1
  Such that:it(17) =< -V1+Out
it(16) =< V1
aux(5) =< -V1+2*Out
aux(6) =< Out
it(16) =< aux(6)
it(17) =< aux(6)
it(16) =< aux(5)
it(17) =< aux(5)

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

* Chain [[16,17],19]: 1*it(16)+1*it(17)+1
  Such that:it(17) =< -V1+Out
it(16) =< V1
aux(7) =< -V1+2*Out
aux(8) =< Out
it(16) =< aux(8)
it(17) =< aux(8)
it(16) =< aux(7)
it(17) =< aux(7)

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

* Chain [[16,17],18]: 2*it(16)+1
  Such that:aux(1) =< V1+V
aux(3) =< V1+2*V
aux(9) =< Out
aux(10) =< 2*Out
it(16) =< aux(9)
it(16) =< aux(10)
it(16) =< aux(1)
it(16) =< aux(3)

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

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

* Chain [19]: 1
  with precondition: [V=0,V1=Out,V1>=1] 

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


#### Cost of chains of start(V1,V,V16):
* Chain [24]: 4*s(13)+4*s(14)+4*s(19)+2
  Such that:aux(17) =< V1
aux(18) =< V1+V
aux(19) =< V1+2*V
aux(20) =< 2*V1+2*V
aux(21) =< V
s(13) =< aux(21)
s(14) =< aux(17)
s(14) =< aux(18)
s(13) =< aux(18)
s(14) =< aux(19)
s(13) =< aux(19)
s(19) =< aux(18)
s(19) =< aux(20)
s(19) =< aux(19)

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

* Chain [23]: 2*s(35)+2*s(36)+2*s(41)+1
  Such that:s(33) =< V
s(40) =< 2*V+2*V16
s(31) =< V16
aux(23) =< V+V16
aux(24) =< V+2*V16
s(35) =< s(31)
s(36) =< s(33)
s(36) =< aux(23)
s(35) =< aux(23)
s(36) =< aux(24)
s(35) =< aux(24)
s(41) =< aux(23)
s(41) =< s(40)
s(41) =< aux(24)

  with precondition: [V1=1,V>=0,V16>=1] 

* Chain [22]: 2*s(46)+2*s(47)+2*s(52)+1
  Such that:s(44) =< V
s(51) =< 2*V+2*V16
s(42) =< V16
aux(26) =< V+V16
aux(27) =< V+2*V16
s(46) =< s(42)
s(47) =< s(44)
s(47) =< aux(26)
s(46) =< aux(26)
s(47) =< aux(27)
s(46) =< aux(27)
s(52) =< aux(26)
s(52) =< s(51)
s(52) =< aux(27)

  with precondition: [V1=2,V>=1,V16>=0] 

* Chain [21]: 2
  with precondition: [V=0,V1>=1] 


Closed-form bounds of start(V1,V,V16): 
-------------------------------------
* Chain [24] with precondition: [V1>=0,V>=0] 
    - Upper bound: 8*V1+8*V+2 
    - Complexity: n 
* Chain [23] with precondition: [V1=1,V>=0,V16>=1] 
    - Upper bound: 4*V+4*V16+1 
    - Complexity: n 
* Chain [22] with precondition: [V1=2,V>=1,V16>=0] 
    - Upper bound: 4*V+4*V16+1 
    - Complexity: n 
* Chain [21] with precondition: [V=0,V1>=1] 
    - Upper bound: 2 
    - Complexity: constant 

### Maximum cost of start(V1,V,V16): max([1,2*V+max([8*V1+6*V+1,nat(V+V16)*2+nat(V16)*2])])+1 
Asymptotic class: n 
* Total analysis performed in 313 ms.


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

(14)
BOUNDS(1, n^1)

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

(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:

   merge(Cons(x, xs), Nil) -> Cons(x, xs)
   merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
   merge(Nil, ys) -> ys
   goal(xs, ys) -> merge(xs, ys)

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

   <=(S(x), S(y)) -> <=(x, y)
   <=(0', y) -> True
   <=(S(x), 0') -> False
   merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
   merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
merge(Cons(x, xs), Nil) -> Cons(x, xs)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Nil, ys) -> ys
goal(xs, ys) -> merge(xs, ys)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

Types:
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
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:
merge, <=

They will be analysed ascendingly in the following order:
<= < merge

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

(20)
Obligation:
Innermost TRS:
Rules:
merge(Cons(x, xs), Nil) -> Cons(x, xs)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Nil, ys) -> ys
goal(xs, ys) -> merge(xs, ys)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

Types:
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
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:
<=, merge

They will be analysed ascendingly in the following order:
<= < merge

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

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

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

Induction Step:
<=(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(n7_0, 1))) ->_R^Omega(0)
<=(gen_S:0'5_0(n7_0), gen_S:0'5_0(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:
merge(Cons(x, xs), Nil) -> Cons(x, xs)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Nil, ys) -> ys
goal(xs, ys) -> merge(xs, ys)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

Types:
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
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(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:
merge
----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
merge(gen_Cons:Nil4_0(n216_0), gen_Cons:Nil4_0(1)) -> gen_Cons:Nil4_0(+(1, n216_0)), rt in Omega(1 + n216_0)

Induction Base:
merge(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(1)) ->_R^Omega(1)
gen_Cons:Nil4_0(1)

Induction Step:
merge(gen_Cons:Nil4_0(+(n216_0, 1)), gen_Cons:Nil4_0(1)) ->_R^Omega(1)
merge[Ite](<=(0', 0'), Cons(0', gen_Cons:Nil4_0(n216_0)), Cons(0', gen_Cons:Nil4_0(0))) ->_L^Omega(0)
merge[Ite](True, Cons(0', gen_Cons:Nil4_0(n216_0)), Cons(0', gen_Cons:Nil4_0(0))) ->_R^Omega(0)
Cons(0', merge(gen_Cons:Nil4_0(n216_0), Cons(0', gen_Cons:Nil4_0(0)))) ->_IH
Cons(0', gen_Cons:Nil4_0(+(1, c217_0)))

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

(24)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
merge(Cons(x, xs), Nil) -> Cons(x, xs)
merge(Cons(x', xs'), Cons(x, xs)) -> merge[Ite](<=(x', x), Cons(x', xs'), Cons(x, xs))
merge(Nil, ys) -> ys
goal(xs, ys) -> merge(xs, ys)
<=(S(x), S(y)) -> <=(x, y)
<=(0', y) -> True
<=(S(x), 0') -> False
merge[Ite](False, xs', Cons(x, xs)) -> Cons(x, merge(xs', xs))
merge[Ite](True, Cons(x, xs), ys) -> Cons(x, merge(xs, ys))

Types:
merge :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S:0' -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
merge[Ite] :: True:False -> Cons:Nil -> Cons:Nil -> Cons:Nil
<= :: S:0' -> S:0' -> True:False
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S:0' -> S:0'
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(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:
merge
----------------------------------------

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

(26)
BOUNDS(n^1, INF)