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


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


WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [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, 418 ms]
    (14) BOUNDS(1, n^2)
(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), 265 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), 54 ms]
            (28) BEST
                (29) proven lower bound
                    (30) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (31) BOUNDS(n^2, INF)
                (32) typed CpxTrs
                    (33) RewriteLemmaProof [LOWER BOUND(ID), 61 ms]
                    (34) typed CpxTrs


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

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


The TRS R consists of the following rules:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 1th argument of plus: plus, times

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
div(div(x, y), z) -> div(x, times(y, z))

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

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


The TRS R consists of the following rules:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))

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

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

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


The defined contexts are:
plus(x0, [])


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

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


The TRS R consists of the following rules:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))

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

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

(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:

   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [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:

   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 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:

   quot(v0, v1, v2) -> null_quot [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]

And the following fresh constants:    null_quot, null_plus, null_times

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

(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:

   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]
   quot(v0, v1, v2) -> null_quot [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]

The TRS has the following type information:
plus :: 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times
0 :: 0:s:null_quot:null_plus:null_times
s :: 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times
times :: 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times
div :: 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times
quot :: 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times -> 0:s:null_quot:null_plus:null_times
null_quot :: 0:s:null_quot:null_plus:null_times
null_plus :: 0:s:null_quot:null_plus:null_times
null_times :: 0:s:null_quot:null_plus:null_times

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
   null_quot => 0
   null_plus => 0
   null_times => 0

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

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

   div(z', z'') -{ 1 }-> quot(x, y, y) :|: z' = x, z'' = y, x >= 0, y >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 0
   quot(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   quot(z', z'', z1) -{ 1 }-> 1 + div(x, 1 + z) :|: z'' = 0, z >= 0, z' = x, x >= 0, z1 = 1 + z
   times(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(y, times(x, y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0

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(V, V1, V15),0,[plus(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V15),0,[times(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V15),0,[div(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V15),0,[quot(V, V1, V15, Out)],[V >= 0,V1 >= 0,V15 >= 0]).
eq(plus(V, V1, Out),1,[],[Out = V2,V1 = 0,V = V2,V2 >= 0]).
eq(plus(V, V1, Out),1,[],[Out = V3,V1 = V3,V3 >= 0,V = 0]).
eq(plus(V, V1, Out),1,[plus(V4, V5, Ret1)],[Out = 1 + Ret1,V = 1 + V4,V1 = V5,V4 >= 0,V5 >= 0]).
eq(times(V, V1, Out),1,[],[Out = 0,V1 = V6,V6 >= 0,V = 0]).
eq(times(V, V1, Out),1,[],[Out = V7,V1 = V7,V7 >= 0,V = 1]).
eq(times(V, V1, Out),1,[times(V9, V8, Ret11),plus(V8, Ret11, Ret)],[Out = Ret,V = 1 + V9,V1 = V8,V9 >= 0,V8 >= 0]).
eq(div(V, V1, Out),1,[],[Out = 0,V1 = V10,V10 >= 0,V = 0]).
eq(div(V, V1, Out),1,[quot(V11, V12, V12, Ret2)],[Out = Ret2,V = V11,V1 = V12,V11 >= 0,V12 >= 0]).
eq(quot(V, V1, V15, Out),1,[],[Out = 0,V15 = V14,V14 >= 0,V13 >= 0,V1 = 1 + V13,V = 0]).
eq(quot(V, V1, V15, Out),1,[quot(V16, V17, V18, Ret3)],[Out = Ret3,V = 1 + V16,V15 = V18,V18 >= 0,V16 >= 0,V17 >= 0,V1 = 1 + V17]).
eq(quot(V, V1, V15, Out),1,[div(V20, 1 + V19, Ret12)],[Out = 1 + Ret12,V1 = 0,V19 >= 0,V = V20,V20 >= 0,V15 = 1 + V19]).
eq(quot(V, V1, V15, Out),0,[],[Out = 0,V22 >= 0,V15 = V23,V21 >= 0,V1 = V21,V23 >= 0,V = V22]).
eq(plus(V, V1, Out),0,[],[Out = 0,V25 >= 0,V24 >= 0,V1 = V24,V = V25]).
eq(times(V, V1, Out),0,[],[Out = 0,V27 >= 0,V26 >= 0,V1 = V26,V = V27]).
input_output_vars(plus(V,V1,Out),[V,V1],[Out]).
input_output_vars(times(V,V1,Out),[V,V1],[Out]).
input_output_vars(div(V,V1,Out),[V,V1],[Out]).
input_output_vars(quot(V,V1,V15,Out),[V,V1,V15],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [(div)/3,quot/4]
1. recursive  : [plus/3]
2. recursive [non_tail] : [times/3]
3. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into quot/4
1. SCC is partially evaluated into plus/3
2. SCC is partially evaluated into times/3
3. SCC is partially evaluated into start/3

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

### Specialization of cost equations quot/4 
* CE 8 is refined into CE [19] 
* CE 10 is refined into CE [20] 
* CE 7 is refined into CE [21] 
* CE 9 is refined into CE [22] 
* CE 6 is refined into CE [23] 


### Cost equations --> "Loop" of quot/4 
* CEs [22] --> Loop 13 
* CEs [23] --> Loop 14 
* CEs [19,20] --> Loop 15 
* CEs [21] --> Loop 16 

### Ranking functions of CR quot(V,V1,V15,Out) 

#### Partial ranking functions of CR quot(V,V1,V15,Out) 
* Partial RF of phase [13,14]:
  - RF of loop [13:1]:
    V
    V1 depends on loops [14:1] 
  - RF of loop [14:1]:
    -V1+1 depends on loops [13:1] 


### Specialization of cost equations plus/3 
* CE 14 is refined into CE [24] 
* CE 11 is refined into CE [25] 
* CE 12 is refined into CE [26] 
* CE 13 is refined into CE [27] 


### Cost equations --> "Loop" of plus/3 
* CEs [27] --> Loop 17 
* CEs [24] --> Loop 18 
* CEs [25] --> Loop 19 
* CEs [26] --> Loop 20 

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

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


### Specialization of cost equations times/3 
* CE 16 is refined into CE [28] 
* CE 15 is refined into CE [29] 
* CE 18 is refined into CE [30] 
* CE 17 is refined into CE [31,32,33,34,35] 


### Cost equations --> "Loop" of times/3 
* CEs [35] --> Loop 21 
* CEs [34] --> Loop 22 
* CEs [32] --> Loop 23 
* CEs [33] --> Loop 24 
* CEs [31] --> Loop 25 
* CEs [28] --> Loop 26 
* CEs [29,30] --> Loop 27 

### Ranking functions of CR times(V,V1,Out) 
* RF of phase [21,22,23,24,25]: [V]

#### Partial ranking functions of CR times(V,V1,Out) 
* Partial RF of phase [21,22,23,24,25]:
  - RF of loop [21:1,22:1,23:1,24:1,25:1]:
    V


### Specialization of cost equations start/3 
* CE 1 is refined into CE [36,37] 
* CE 2 is refined into CE [38] 
* CE 3 is refined into CE [39,40,41,42,43] 
* CE 4 is refined into CE [44,45] 
* CE 5 is refined into CE [46,47] 


### Cost equations --> "Loop" of start/3 
* CEs [40] --> Loop 28 
* CEs [36,37,38,39,41,42,43,44,45,46,47] --> Loop 29 

### Ranking functions of CR start(V,V1,V15) 

#### Partial ranking functions of CR start(V,V1,V15) 


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

#### Cost of chains of quot(V,V1,V15,Out):
* Chain [[13,14],16]: 1*it(13)+2*it(14)+2
  Such that:it(13) =< V
aux(6) =< -V1
aux(5) =< -V1+1
it(14) =< it(13)+aux(6)
it(14) =< it(13)+aux(5)

  with precondition: [V1>=0,V15>=0,Out>=1,V1+V15>=1,Out+V1>=2,V+1>=Out+V1] 

* Chain [[13,14],15]: 1*it(13)+2*it(14)+1
  Such that:it(13) =< V
aux(5) =< -V1+1
it(14) =< it(13)+aux(5)

  with precondition: [V>=0,V1>=0,V15>=0,Out>=0,V+V15>=1,Out+V>=1,V1+V15>=1,Out+V1>=1] 

* Chain [16]: 2
  with precondition: [V=0,V1=0,Out=1,V15>=1] 

* Chain [15]: 1
  with precondition: [Out=0,V>=0,V1>=0,V15>=0] 


#### Cost of chains of plus(V,V1,Out):
* Chain [[17],20]: 1*it(17)+1
  Such that:it(17) =< -V1+Out

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

* Chain [[17],19]: 1*it(17)+1
  Such that:it(17) =< Out

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

* Chain [[17],18]: 1*it(17)+0
  Such that:it(17) =< Out

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

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

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

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


#### Cost of chains of times(V,V1,Out):
* Chain [[21,22,23,24,25],27]: 8*it(21)+1*s(15)+2*s(16)+1
  Such that:aux(13) =< V1
aux(18) =< V
it(21) =< aux(18)
aux(14) =< aux(13)
s(15) =< it(21)*aux(13)
s(16) =< it(21)*aux(14)

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

* Chain [[21,22,23,24,25],26]: 8*it(21)+1*s(15)+2*s(16)+1
  Such that:aux(13) =< V1
aux(19) =< V
it(21) =< aux(19)
aux(14) =< aux(13)
s(15) =< it(21)*aux(13)
s(16) =< it(21)*aux(14)

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

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

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


#### Cost of chains of start(V,V1,V15):
* Chain [29]: 22*s(33)+4*s(34)+4*s(35)+2*s(42)+4*s(43)+3
  Such that:s(39) =< V1
aux(22) =< V
aux(23) =< -V1
aux(24) =< -V1+1
s(33) =< aux(22)
s(34) =< s(33)+aux(24)
s(35) =< s(33)+aux(23)
s(35) =< s(33)+aux(24)
s(41) =< s(39)
s(42) =< s(33)*s(39)
s(43) =< s(33)*s(41)

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

* Chain [28]: 1*s(50)+1
  Such that:s(50) =< V

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


Closed-form bounds of start(V,V1,V15): 
-------------------------------------
* Chain [29] with precondition: [V>=0,V1>=0] 
    - Upper bound: 30*V+3+6*V*V1+nat(-V1+1)*4 
    - Complexity: n^2 
* Chain [28] with precondition: [V1=0,V>=0] 
    - Upper bound: V+1 
    - Complexity: n 

### Maximum cost of start(V,V1,V15): 29*V+2+6*V*V1+nat(-V1+1)*4+(V+1) 
Asymptotic class: n^2 
* Total analysis performed in 335 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   plus(x, 0') -> x
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(0'), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0', y) -> 0'
   div(x, y) -> quot(x, y, y)
   quot(0', s(y), z) -> 0'
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0', s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))

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

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

(18)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, times, div, quot

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

(20)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
plus, times, div, quot

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
gen_0':s2_0(b)

Induction Step:
plus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
s(plus(gen_0':s2_0(n4_0), gen_0':s2_0(b))) ->_IH
s(gen_0':s2_0(+(b, c5_0)))

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

(22)
Complex Obligation (BEST)

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

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

TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
plus, times, div, quot

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
times, div, quot

They will be analysed ascendingly in the following order:
times < div
div = quot

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)

Induction Base:
times(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_0':s2_0(+(n577_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
plus(gen_0':s2_0(b), times(gen_0':s2_0(n577_0), gen_0':s2_0(b))) ->_IH
plus(gen_0':s2_0(b), gen_0':s2_0(*(c578_0, b))) ->_L^Omega(1 + b)
gen_0':s2_0(+(b, *(n577_0, b)))

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

(28)
Complex Obligation (BEST)

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

(29)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
times, div, quot

They will be analysed ascendingly in the following order:
times < div
div = quot

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

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

(31)
BOUNDS(n^2, INF)

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

(32)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

They will be analysed ascendingly in the following order:
div = quot

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) -> gen_0':s2_0(0), rt in Omega(1 + n1370_0)

Induction Base:
quot(gen_0':s2_0(0), gen_0':s2_0(+(1, 0)), gen_0':s2_0(c)) ->_R^Omega(1)
0'

Induction Step:
quot(gen_0':s2_0(+(n1370_0, 1)), gen_0':s2_0(+(1, +(n1370_0, 1))), gen_0':s2_0(c)) ->_R^Omega(1)
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) ->_IH
gen_0':s2_0(0)

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

(34)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) -> gen_0':s2_0(0), rt in Omega(1 + n1370_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
div

They will be analysed ascendingly in the following order:
div = quot