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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 1 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 422 ms]
    (10) BOUNDS(1, n^2)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 234 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 55 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 90 ms]
            (26) typed CpxTrs


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

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


The TRS R consists of the following rules:

   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   average(x, y) -> aver(plus(x, y), 0)
   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
   if(true, sum, z) -> aver(sum, s(z))
   if(false, sum, z) -> z

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

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

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


The TRS R consists of the following rules:

   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   average(x, y) -> aver(plus(x, y), 0) [1]
   aver(sum, z) -> if(gt(sum, double(z)), sum, z) [1]
   if(true, sum, z) -> aver(sum, s(z)) [1]
   if(false, sum, z) -> z [1]

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

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

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

   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   average(x, y) -> aver(plus(x, y), 0) [1]
   aver(sum, z) -> if(gt(sum, double(z)), sum, z) [1]
   if(true, sum, z) -> aver(sum, s(z)) [1]
   if(false, sum, z) -> z [1]

The TRS has the following type information:
gt :: 0:s -> 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
plus :: 0:s -> 0:s -> 0:s
double :: 0:s -> 0:s
average :: 0:s -> 0:s -> 0:s
aver :: 0:s -> 0:s -> 0:s
if :: false:true -> 0:s -> 0:s -> 0:s

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

(5) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

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

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

   gt(0, y) -> false [1]
   gt(s(x), 0) -> true [1]
   gt(s(x), s(y)) -> gt(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   average(x, y) -> aver(plus(x, y), 0) [1]
   aver(sum, z) -> if(gt(sum, double(z)), sum, z) [1]
   if(true, sum, z) -> aver(sum, s(z)) [1]
   if(false, sum, z) -> z [1]

The TRS has the following type information:
gt :: 0:s -> 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
plus :: 0:s -> 0:s -> 0:s
double :: 0:s -> 0:s
average :: 0:s -> 0:s -> 0:s
aver :: 0:s -> 0:s -> 0:s
if :: false:true -> 0:s -> 0:s -> 0:s

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

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

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

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

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

   aver(z', z'') -{ 1 }-> if(gt(sum, double(z)), sum, z) :|: z'' = z, z >= 0, sum >= 0, z' = sum
   average(z', z'') -{ 1 }-> aver(plus(x, y), 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   double(z') -{ 1 }-> 0 :|: z' = 0
   double(z') -{ 1 }-> 1 + (1 + double(x)) :|: z' = 1 + x, x >= 0
   gt(z', z'') -{ 1 }-> gt(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   gt(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   gt(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   if(z', z'', z1) -{ 1 }-> z :|: z1 = z, z >= 0, sum >= 0, z'' = sum, z' = 0
   if(z', z'', z1) -{ 1 }-> aver(sum, 1 + z) :|: z1 = z, z >= 0, z' = 1, sum >= 0, z'' = sum
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V, V1, V16),0,[gt(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V16),0,[plus(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V16),0,[double(V, Out)],[V >= 0]).
eq(start(V, V1, V16),0,[average(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V16),0,[aver(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1, V16),0,[if(V, V1, V16, Out)],[V >= 0,V1 >= 0,V16 >= 0]).
eq(gt(V, V1, Out),1,[],[Out = 0,V1 = V2,V2 >= 0,V = 0]).
eq(gt(V, V1, Out),1,[],[Out = 1,V1 = 0,V = 1 + V3,V3 >= 0]).
eq(gt(V, V1, Out),1,[gt(V4, V5, Ret)],[Out = Ret,V = 1 + V4,V4 >= 0,V5 >= 0,V1 = 1 + V5]).
eq(plus(V, V1, Out),1,[],[Out = V6,V1 = V6,V6 >= 0,V = 0]).
eq(plus(V, V1, Out),1,[plus(V7, V8, Ret1)],[Out = 1 + Ret1,V = 1 + V7,V1 = V8,V7 >= 0,V8 >= 0]).
eq(double(V, Out),1,[],[Out = 0,V = 0]).
eq(double(V, Out),1,[double(V9, Ret11)],[Out = 2 + Ret11,V = 1 + V9,V9 >= 0]).
eq(average(V, V1, Out),1,[plus(V11, V10, Ret0),aver(Ret0, 0, Ret2)],[Out = Ret2,V = V11,V1 = V10,V11 >= 0,V10 >= 0]).
eq(aver(V, V1, Out),1,[double(V13, Ret01),gt(V12, Ret01, Ret02),if(Ret02, V12, V13, Ret3)],[Out = Ret3,V1 = V13,V13 >= 0,V12 >= 0,V = V12]).
eq(if(V, V1, V16, Out),1,[aver(V14, 1 + V15, Ret4)],[Out = Ret4,V16 = V15,V15 >= 0,V = 1,V14 >= 0,V1 = V14]).
eq(if(V, V1, V16, Out),1,[],[Out = V17,V16 = V17,V17 >= 0,V18 >= 0,V1 = V18,V = 0]).
input_output_vars(gt(V,V1,Out),[V,V1],[Out]).
input_output_vars(plus(V,V1,Out),[V,V1],[Out]).
input_output_vars(double(V,Out),[V],[Out]).
input_output_vars(average(V,V1,Out),[V,V1],[Out]).
input_output_vars(aver(V,V1,Out),[V,V1],[Out]).
input_output_vars(if(V,V1,V16,Out),[V,V1,V16],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [double/2]
1. recursive  : [gt/3]
2. recursive  : [aver/3,if/4]
3. recursive  : [plus/3]
4. non_recursive  : [average/3]
5. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into double/2
1. SCC is partially evaluated into gt/3
2. SCC is partially evaluated into aver/3
3. SCC is partially evaluated into plus/3
4. SCC is partially evaluated into average/3
5. SCC is partially evaluated into start/3

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

### Specialization of cost equations double/2 
* CE 16 is refined into CE [18] 
* CE 15 is refined into CE [19] 


### Cost equations --> "Loop" of double/2 
* CEs [19] --> Loop 13 
* CEs [18] --> Loop 14 

### Ranking functions of CR double(V,Out) 
* RF of phase [14]: [V]

#### Partial ranking functions of CR double(V,Out) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V


### Specialization of cost equations gt/3 
* CE 12 is refined into CE [20] 
* CE 11 is refined into CE [21] 
* CE 10 is refined into CE [22] 


### Cost equations --> "Loop" of gt/3 
* CEs [21] --> Loop 15 
* CEs [22] --> Loop 16 
* CEs [20] --> Loop 17 

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

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


### Specialization of cost equations aver/3 
* CE 9 is refined into CE [23,24] 
* CE 8 is refined into CE [25,26,27] 


### Cost equations --> "Loop" of aver/3 
* CEs [27] --> Loop 18 
* CEs [26] --> Loop 19 
* CEs [25] --> Loop 20 
* CEs [24] --> Loop 21 
* CEs [23] --> Loop 22 

### Ranking functions of CR aver(V,V1,Out) 
* RF of phase [21]: [V/2-V1]

#### Partial ranking functions of CR aver(V,V1,Out) 
* Partial RF of phase [21]:
  - RF of loop [21:1]:
    V/2-V1


### Specialization of cost equations plus/3 
* CE 14 is refined into CE [28] 
* CE 13 is refined into CE [29] 


### Cost equations --> "Loop" of plus/3 
* CEs [29] --> Loop 23 
* CEs [28] --> Loop 24 

### Ranking functions of CR plus(V,V1,Out) 
* RF of phase [24]: [V]

#### Partial ranking functions of CR plus(V,V1,Out) 
* Partial RF of phase [24]:
  - RF of loop [24:1]:
    V


### Specialization of cost equations average/3 
* CE 17 is refined into CE [30,31,32,33,34] 


### Cost equations --> "Loop" of average/3 
* CEs [34] --> Loop 25 
* CEs [33] --> Loop 26 
* CEs [32] --> Loop 27 
* CEs [31] --> Loop 28 
* CEs [30] --> Loop 29 

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

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


### Specialization of cost equations start/3 
* CE 2 is refined into CE [35,36,37] 
* CE 1 is refined into CE [38] 
* CE 3 is refined into CE [39,40,41,42] 
* CE 4 is refined into CE [43,44] 
* CE 5 is refined into CE [45,46] 
* CE 6 is refined into CE [47,48,49,50,51] 
* CE 7 is refined into CE [52,53,54,55,56,57] 


### Cost equations --> "Loop" of start/3 
* CEs [37] --> Loop 30 
* CEs [36] --> Loop 31 
* CEs [35,40,41,42,44,46,50,51,54,55,56,57] --> Loop 32 
* CEs [38,39,43,45,47,48,49,52,53] --> Loop 33 

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

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


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

#### Cost of chains of double(V,Out):
* Chain [[14],13]: 1*it(14)+1
  Such that:it(14) =< Out/2

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

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


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

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

* Chain [[17],15]: 1*it(17)+1
  Such that:it(17) =< V1

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

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

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


#### Cost of chains of aver(V,V1,Out):
* Chain [[21],18]: 4*it(21)+1*s(1)+1*s(2)+1*s(7)+1*s(8)+4
  Such that:s(2) =< V
aux(1) =< V/2+1/2
it(21) =< V/2-V1
s(1) =< Out
aux(2) =< it(21)*aux(1)
s(7) =< it(21)*aux(1)
s(8) =< aux(2)*2

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

* Chain [22,[21],18]: 4*it(21)+1*s(1)+1*s(2)+1*s(7)+1*s(8)+8
  Such that:s(2) =< V
it(21) =< V/2
aux(1) =< V/2+1/2
s(1) =< Out
aux(2) =< it(21)*aux(1)
s(7) =< it(21)*aux(1)
s(8) =< aux(2)*2

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

* Chain [22,18]: 1*s(1)+1*s(2)+8
  Such that:s(1) =< 1
s(2) =< V

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

* Chain [20]: 4
  with precondition: [V=0,V1=0,Out=0] 

* Chain [19]: 1*s(9)+4
  Such that:s(9) =< V1

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

* Chain [18]: 1*s(1)+1*s(2)+4
  Such that:s(2) =< V
s(1) =< V1

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


#### Cost of chains of plus(V,V1,Out):
* Chain [[24],23]: 1*it(24)+1
  Such that:it(24) =< -V1+Out

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

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


#### Cost of chains of average(V,V1,Out):
* Chain [29]: 6
  with precondition: [V=0,V1=0,Out=0] 

* Chain [28]: 1*s(10)+1*s(11)+10
  Such that:s(10) =< 1
s(11) =< V1

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

* Chain [27]: 1*s(12)+4*s(13)+1*s(15)+1*s(17)+1*s(18)+10
  Such that:s(12) =< V1
s(13) =< V1/2
aux(3) =< V1/2+1/2
s(15) =< aux(3)
s(16) =< s(13)*aux(3)
s(17) =< s(13)*aux(3)
s(18) =< s(16)*2

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

* Chain [26]: 1*s(19)+1*s(20)+1*s(21)+10
  Such that:s(20) =< 1
s(19) =< V
s(21) =< V+V1

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

* Chain [25]: 1*s(22)+1*s(23)+4*s(24)+1*s(26)+1*s(28)+1*s(29)+10
  Such that:s(22) =< V
s(23) =< V+V1
s(24) =< V/2+V1/2
aux(4) =< V/2+V1/2+1/2
s(26) =< aux(4)
s(27) =< s(24)*aux(4)
s(28) =< s(24)*aux(4)
s(29) =< s(27)*2

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


#### Cost of chains of start(V,V1,V16):
* Chain [33]: 1*s(30)+3*s(31)+4*s(33)+1*s(35)+1*s(37)+1*s(38)+10
  Such that:s(30) =< 1
s(33) =< V1/2
s(34) =< V1/2+1/2
aux(5) =< V1
s(31) =< aux(5)
s(35) =< s(34)
s(36) =< s(33)*s(34)
s(37) =< s(33)*s(34)
s(38) =< s(36)*2

  with precondition: [V=0] 

* Chain [32]: 1*s(40)+9*s(41)+2*s(42)+2*s(45)+2*s(47)+4*s(50)+1*s(52)+1*s(54)+1*s(55)+4*s(59)+2*s(61)+1*s(63)+1*s(64)+4*s(69)+1*s(72)+1*s(73)+10
  Such that:s(59) =< V/2
s(69) =< V/2-V1
s(50) =< V/2+V1/2
s(51) =< V/2+V1/2+1/2
s(40) =< V16+1
aux(8) =< 1
aux(9) =< V
aux(10) =< V+V1
aux(11) =< V/2+1/2
aux(12) =< V1
s(45) =< aux(8)
s(41) =< aux(9)
s(47) =< aux(10)
s(42) =< aux(12)
s(52) =< s(51)
s(53) =< s(50)*s(51)
s(54) =< s(50)*s(51)
s(55) =< s(53)*2
s(61) =< aux(11)
s(62) =< s(59)*aux(11)
s(63) =< s(59)*aux(11)
s(64) =< s(62)*2
s(71) =< s(69)*aux(11)
s(72) =< s(69)*aux(11)
s(73) =< s(71)*2

  with precondition: [V>=1] 

* Chain [31]: 1*s(74)+1*s(75)+5
  Such that:s(74) =< V1
s(75) =< V16+1

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

* Chain [30]: 1*s(76)+4*s(78)+1*s(79)+1*s(81)+1*s(82)+5
  Such that:s(76) =< V1
s(78) =< V1/2-V16
aux(13) =< V1/2+1/2
s(79) =< aux(13)
s(80) =< s(78)*aux(13)
s(81) =< s(78)*aux(13)
s(82) =< s(80)*2

  with precondition: [V=1,V16>=0,V1>=2*V16+3] 


Closed-form bounds of start(V,V1,V16): 
-------------------------------------
* Chain [33] with precondition: [V=0] 
    - Upper bound: nat(V1)*3+11+nat(V1/2+1/2)+nat(V1/2+1/2)*3*nat(V1/2)+nat(V1/2)*4 
    - Complexity: n^2 
* Chain [32] with precondition: [V>=1] 
    - Upper bound: 9*V+12+nat(V1)*2+nat(V+V1)*2+nat(V16+1)+nat(V/2+V1/2+1/2)+nat(V/2+V1/2+1/2)*3*nat(V/2+V1/2)+nat(V/2+V1/2)*4+(V+1)+(3/2*V+3/2)*nat(V/2-V1)+V/2*(3/2*V+3/2)+nat(V/2-V1)*4+2*V 
    - Complexity: n^2 
* Chain [31] with precondition: [V=1,V1>=1,V16>=0,2*V16+2>=V1] 
    - Upper bound: V1+V16+6 
    - Complexity: n 
* Chain [30] with precondition: [V=1,V16>=0,V1>=2*V16+3] 
    - Upper bound: 2*V1-4*V16+(3/2*V1+11/2+(V1/2-V16)*(3/2*V1+3/2)) 
    - Complexity: n^2 

### Maximum cost of start(V,V1,V16): nat(V1)+5+max([max([nat(V16+1),nat(V1/2+1/2)*3*nat(V1/2-V16)+nat(V1/2+1/2)+nat(V1/2-V16)*4]),nat(V1)+6+max([nat(V1/2+1/2)+nat(V1)+nat(V1/2+1/2)*3*nat(V1/2)+nat(V1/2)*4,9*V+1+nat(V+V1)*2+nat(V16+1)+nat(V/2+V1/2+1/2)+nat(V/2+V1/2+1/2)*3*nat(V/2+V1/2)+nat(V/2+V1/2)*4+(V+1)+(3/2*V+3/2)*nat(V/2-V1)+V/2*(3/2*V+3/2)+nat(V/2-V1)*4+2*V])]) 
Asymptotic class: n^2 
* Total analysis performed in 331 ms.


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

(10)
BOUNDS(1, n^2)

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

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

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


The TRS R consists of the following rules:

   gt(0', y) -> false
   gt(s(x), 0') -> true
   gt(s(x), s(y)) -> gt(x, y)
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   average(x, y) -> aver(plus(x, y), 0')
   aver(sum, z) -> if(gt(sum, double(z)), sum, z)
   if(true, sum, z) -> aver(sum, s(z))
   if(false, sum, z) -> z

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

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

(14)
Obligation:
Innermost TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
gt, plus, double, aver

They will be analysed ascendingly in the following order:
gt < aver
double < aver

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

(16)
Obligation:
Innermost TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
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:
gt, plus, double, aver

They will be analysed ascendingly in the following order:
gt < aver
double < aver

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

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

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

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

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

(18)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
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:
gt, plus, double, aver

They will be analysed ascendingly in the following order:
gt < aver
double < aver

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, 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:
plus, double, aver

They will be analysed ascendingly in the following order:
double < aver

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s3_0(n306_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n306_0, b)), rt in Omega(1 + n306_0)

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

Induction Step:
plus(gen_0':s3_0(+(n306_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(plus(gen_0':s3_0(n306_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, c307_0)))

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

(24)
Obligation:
Innermost TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
plus(gen_0':s3_0(n306_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n306_0, b)), rt in Omega(1 + n306_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:
double, aver

They will be analysed ascendingly in the following order:
double < aver

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s3_0(n913_0)) -> gen_0':s3_0(*(2, n913_0)), rt in Omega(1 + n913_0)

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

Induction Step:
double(gen_0':s3_0(+(n913_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s3_0(n913_0)))) ->_IH
s(s(gen_0':s3_0(*(2, c914_0))))

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

(26)
Obligation:
Innermost TRS:
Rules:
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
average(x, y) -> aver(plus(x, y), 0')
aver(sum, z) -> if(gt(sum, double(z)), sum, z)
if(true, sum, z) -> aver(sum, s(z))
if(false, sum, z) -> z

Types:
gt :: 0':s -> 0':s -> false:true
0' :: 0':s
false :: false:true
s :: 0':s -> 0':s
true :: false:true
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
average :: 0':s -> 0':s -> 0':s
aver :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
plus(gen_0':s3_0(n306_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n306_0, b)), rt in Omega(1 + n306_0)
double(gen_0':s3_0(n913_0)) -> gen_0':s3_0(*(2, n913_0)), rt in Omega(1 + n913_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:
aver