/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (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, 342 ms]
    (14) BOUNDS(1, n^1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 255 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 59 ms]
            (28) typed CpxTrs


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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   div(x, 0) -> 0
   div(0, y) -> 0
   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))

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

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

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


The defined contexts are:
if([], 0, s(x1))
if(x0, 0, s([]))
div([], s(x1))
lt([], x1)
-([], x1)


[] just represents basic- or constructor-terms in the following defined contexts:
if([], 0, s(x1))
div([], s(x1))


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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   div(x, 0) -> 0
   div(0, y) -> 0
   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))

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

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

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


The TRS R consists of the following rules:

   -(x, 0) -> x [1]
   -(0, s(y)) -> 0 [1]
   -(s(x), s(y)) -> -(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   div(x, 0) -> 0 [1]
   div(0, y) -> 0 [1]
   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y)))) [1]

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

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

- => minus

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

(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:

   minus(x, 0) -> x [1]
   minus(0, s(y)) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   div(x, 0) -> 0 [1]
   div(0, y) -> 0 [1]
   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(minus(x, y), s(y)))) [1]

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:

   minus(x, 0) -> x [1]
   minus(0, s(y)) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   div(x, 0) -> 0 [1]
   div(0, y) -> 0 [1]
   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(minus(x, y), s(y)))) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lt :: 0:s -> 0:s -> false:true
false :: false:true
true :: false:true
if :: false:true -> 0:s -> 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s

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:
none

And the following fresh constants: none

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

(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:

   minus(x, 0) -> x [1]
   minus(0, s(y)) -> 0 [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]
   div(x, 0) -> 0 [1]
   div(0, y) -> 0 [1]
   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(minus(x, y), s(y)))) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lt :: 0:s -> 0:s -> false:true
false :: false:true
true :: false:true
if :: false:true -> 0:s -> 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s

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:

   0 => 0
   false => 0
   true => 1

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

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

   div(z, z') -{ 1 }-> if(lt(x, y), 0, 1 + div(minus(x, y), 1 + y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   div(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   div(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0

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, V12),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[lt(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V12),0,[if(V1, V, V12, Out)],[V1 >= 0,V >= 0,V12 >= 0]).
eq(start(V1, V, V12),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]).
eq(minus(V1, V, Out),1,[],[Out = 0,V = 1 + V3,V3 >= 0,V1 = 0]).
eq(minus(V1, V, Out),1,[minus(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]).
eq(lt(V1, V, Out),1,[],[Out = 0,V6 >= 0,V1 = V6,V = 0]).
eq(lt(V1, V, Out),1,[],[Out = 1,V = 1 + V7,V7 >= 0,V1 = 0]).
eq(lt(V1, V, Out),1,[lt(V8, V9, Ret1)],[Out = Ret1,V = 1 + V9,V8 >= 0,V9 >= 0,V1 = 1 + V8]).
eq(if(V1, V, V12, Out),1,[],[Out = V11,V = V11,V12 = V10,V1 = 1,V11 >= 0,V10 >= 0]).
eq(if(V1, V, V12, Out),1,[],[Out = V13,V = V14,V12 = V13,V14 >= 0,V13 >= 0,V1 = 0]).
eq(div(V1, V, Out),1,[],[Out = 0,V15 >= 0,V1 = V15,V = 0]).
eq(div(V1, V, Out),1,[],[Out = 0,V16 >= 0,V1 = 0,V = V16]).
eq(div(V1, V, Out),1,[lt(V17, V18, Ret0),minus(V17, V18, Ret210),div(Ret210, 1 + V18, Ret21),if(Ret0, 0, 1 + Ret21, Ret2)],[Out = Ret2,V = 1 + V18,V17 >= 0,V18 >= 0,V1 = 1 + V17]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(lt(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V12,Out),[V1,V,V12],[Out]).
input_output_vars(div(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [if/4]
1. recursive  : [lt/3]
2. recursive  : [minus/3]
3. recursive [non_tail] : [(div)/3]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into if/4
1. SCC is partially evaluated into lt/3
2. SCC is partially evaluated into minus/3
3. SCC is partially evaluated into (div)/3
4. SCC is partially evaluated into start/3

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

### Specialization of cost equations if/4 
* CE 11 is refined into CE [16] 
* CE 12 is refined into CE [17] 


### Cost equations --> "Loop" of if/4 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 

### Ranking functions of CR if(V1,V,V12,Out) 

#### Partial ranking functions of CR if(V1,V,V12,Out) 


### Specialization of cost equations lt/3 
* CE 10 is refined into CE [18] 
* CE 8 is refined into CE [19] 
* CE 9 is refined into CE [20] 


### Cost equations --> "Loop" of lt/3 
* CEs [19] --> Loop 15 
* CEs [20] --> Loop 16 
* CEs [18] --> Loop 17 

### Ranking functions of CR lt(V1,V,Out) 
* RF of phase [17]: [V,V1]

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


### Specialization of cost equations minus/3 
* CE 7 is refined into CE [21] 
* CE 5 is refined into CE [22] 
* CE 6 is refined into CE [23] 


### Cost equations --> "Loop" of minus/3 
* CEs [22] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [21] --> Loop 20 

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

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


### Specialization of cost equations (div)/3 
* CE 15 is refined into CE [24,25,26,27] 
* CE 13 is refined into CE [28] 
* CE 14 is refined into CE [29] 


### Cost equations --> "Loop" of (div)/3 
* CEs [28] --> Loop 21 
* CEs [29] --> Loop 22 
* CEs [26] --> Loop 23 
* CEs [27] --> Loop 24 
* CEs [25] --> Loop 25 
* CEs [24] --> Loop 26 

### Ranking functions of CR div(V1,V,Out) 
* RF of phase [23]: [V1-1,V1-V+1]
* RF of phase [25]: [V1]

#### Partial ranking functions of CR div(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V1-1
    V1-V+1
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V1


### Specialization of cost equations start/3 
* CE 1 is refined into CE [30,31,32,33] 
* CE 2 is refined into CE [34,35,36,37] 
* CE 3 is refined into CE [38,39] 
* CE 4 is refined into CE [40,41,42,43,44,45] 


### Cost equations --> "Loop" of start/3 
* CEs [33,36,43,45] --> Loop 27 
* CEs [31,35,42] --> Loop 28 
* CEs [32,37,41,44] --> Loop 29 
* CEs [39] --> Loop 30 
* CEs [30,34,38,40] --> Loop 31 

### Ranking functions of CR start(V1,V,V12) 

#### Partial ranking functions of CR start(V1,V,V12) 


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

#### Cost of chains of if(V1,V,V12,Out):
* Chain [14]: 1
  with precondition: [V1=0,V12=Out,V>=0,V12>=0] 

* Chain [13]: 1
  with precondition: [V1=1,V=Out,V>=0,V12>=0] 


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

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

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

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

* Chain [16]: 1
  with precondition: [V1=0,Out=1,V>=1] 

* Chain [15]: 1
  with precondition: [V=0,Out=0,V1>=0] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[20],19]: 1*it(20)+1
  Such that:it(20) =< V1

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

* Chain [[20],18]: 1*it(20)+1
  Such that:it(20) =< V

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

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

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


#### Cost of chains of div(V1,V,Out):
* Chain [[25],22]: 4*it(25)+1
  Such that:it(25) =< Out

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

* Chain [[23],26,22]: 4*it(23)+2*s(5)+5
  Such that:it(23) =< V1-V+1
aux(4) =< V1
it(23) =< aux(4)
s(5) =< aux(4)

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

* Chain [[23],24,22]: 4*it(23)+4*s(5)+5
  Such that:it(23) =< V1-V+1
aux(6) =< V1
s(5) =< aux(6)
it(23) =< aux(6)

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

* Chain [[23],22]: 4*it(23)+2*s(5)+1
  Such that:it(23) =< V1-V+1
aux(7) =< V1
it(23) =< aux(7)
s(5) =< aux(7)

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

* Chain [26,22]: 5
  with precondition: [V1=1,Out=0,V>=2] 

* Chain [24,22]: 2*s(7)+5
  Such that:aux(5) =< V1
s(7) =< aux(5)

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

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

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


#### Cost of chains of start(V1,V,V12):
* Chain [31]: 1
  with precondition: [V1=0,V>=0] 

* Chain [30]: 1
  with precondition: [V1=1,V>=0,V12>=0] 

* Chain [29]: 4*s(18)+5
  Such that:aux(10) =< V1
s(18) =< aux(10)

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

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

* Chain [27]: 2*s(22)+12*s(24)+12*s(27)+5
  Such that:s(26) =< V1-V+1
aux(11) =< V1
aux(12) =< V
s(24) =< aux(11)
s(22) =< aux(12)
s(27) =< s(26)
s(27) =< aux(11)

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


Closed-form bounds of start(V1,V,V12): 
-------------------------------------
* Chain [31] with precondition: [V1=0,V>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [30] with precondition: [V1=1,V>=0,V12>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [29] with precondition: [V1>=1,V>=V1+1] 
    - Upper bound: 4*V1+5 
    - Complexity: n 
* Chain [28] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [27] with precondition: [V>=1,V1>=V] 
    - Upper bound: 24*V1-10*V+17 
    - Complexity: n 

### Maximum cost of start(V1,V,V12): 8*V1+2*V+nat(V1-V+1)*12+(4*V1+4)+1 
Asymptotic class: n 
* Total analysis performed in 284 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   -(x, 0') -> x
   -(0', s(y)) -> 0'
   -(s(x), s(y)) -> -(x, y)
   lt(x, 0') -> false
   lt(0', s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   div(x, 0') -> 0'
   div(0', y) -> 0'
   div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))

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

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

(18)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0') -> 0'
div(0', y) -> 0'
div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
if :: false:true -> 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat -> 0':s

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
-, lt, div

They will be analysed ascendingly in the following order:
- < div
lt < div

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

(20)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0') -> 0'
div(0', y) -> 0'
div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
if :: false:true -> 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
-, lt, div

They will be analysed ascendingly in the following order:
- < div
lt < div

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
gen_0':s3_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(22)
Complex Obligation (BEST)

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

TRS:
Rules:
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0') -> 0'
div(0', y) -> 0'
div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
if :: false:true -> 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
-, lt, div

They will be analysed ascendingly in the following order:
- < div
lt < div

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(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(25)
BOUNDS(n^1, INF)

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(26)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0') -> 0'
div(0', y) -> 0'
div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
if :: false:true -> 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
lt, div

They will be analysed ascendingly in the following order:
lt < div

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(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s3_0(n368_0), gen_0':s3_0(n368_0)) -> false, rt in Omega(1 + n368_0)

Induction Base:
lt(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
false

Induction Step:
lt(gen_0':s3_0(+(n368_0, 1)), gen_0':s3_0(+(n368_0, 1))) ->_R^Omega(1)
lt(gen_0':s3_0(n368_0), gen_0':s3_0(n368_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(28)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(0', s(y)) -> 0'
-(s(x), s(y)) -> -(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
div(x, 0') -> 0'
div(0', y) -> 0'
div(s(x), s(y)) -> if(lt(x, y), 0', s(div(-(x, y), s(y))))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
if :: false:true -> 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
lt(gen_0':s3_0(n368_0), gen_0':s3_0(n368_0)) -> false, rt in Omega(1 + n368_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
div