/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^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 449 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), 303 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), 71 ms]
            (24) 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:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   inc(0) -> 0
   inc(s(x)) -> s(inc(x))
   zero(0) -> true
   zero(s(x)) -> false
   p(0) -> 0
   p(s(x)) -> x
   bits(x) -> bitIter(x, 0)
   bitIter(x, y) -> if(zero(x), x, inc(y))
   if(true, x, y) -> p(y)
   if(false, x, y) -> bitIter(half(x), y)

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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   bits(x) -> bitIter(x, 0) [1]
   bitIter(x, y) -> if(zero(x), x, inc(y)) [1]
   if(true, x, y) -> p(y) [1]
   if(false, x, y) -> bitIter(half(x), y) [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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   bits(x) -> bitIter(x, 0) [1]
   bitIter(x, y) -> if(zero(x), x, inc(y)) [1]
   if(true, x, y) -> p(y) [1]
   if(false, x, y) -> bitIter(half(x), y) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
inc :: 0:s -> 0:s
zero :: 0:s -> true:false
true :: true:false
false :: true:false
p :: 0:s -> 0:s
bits :: 0:s -> 0:s
bitIter :: 0:s -> 0:s -> 0:s
if :: true:false -> 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:

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   inc(0) -> 0 [1]
   inc(s(x)) -> s(inc(x)) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   bits(x) -> bitIter(x, 0) [1]
   bitIter(x, y) -> if(zero(x), x, inc(y)) [1]
   if(true, x, y) -> p(y) [1]
   if(false, x, y) -> bitIter(half(x), y) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
inc :: 0:s -> 0:s
zero :: 0:s -> true:false
true :: true:false
false :: true:false
p :: 0:s -> 0:s
bits :: 0:s -> 0:s
bitIter :: 0:s -> 0:s -> 0:s
if :: true:false -> 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
   true => 1
   false => 0

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

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

   bitIter(z, z') -{ 1 }-> if(zero(x), x, inc(y)) :|: x >= 0, y >= 0, z = x, z' = y
   bits(z) -{ 1 }-> bitIter(x, 0) :|: x >= 0, z = x
   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 1 }-> 0 :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
   if(z, z', z'') -{ 1 }-> p(y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 1 }-> bitIter(half(x), y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   inc(z) -{ 1 }-> 0 :|: z = 0
   inc(z) -{ 1 }-> 1 + inc(x) :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   zero(z) -{ 1 }-> 1 :|: z = 0
   zero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x

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, V6, V9),0,[half(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[inc(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[zero(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[p(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[bits(V, Out)],[V >= 0]).
eq(start(V, V6, V9),0,[bitIter(V, V6, Out)],[V >= 0,V6 >= 0]).
eq(start(V, V6, V9),0,[if(V, V6, V9, Out)],[V >= 0,V6 >= 0,V9 >= 0]).
eq(half(V, Out),1,[],[Out = 0,V = 0]).
eq(half(V, Out),1,[],[Out = 0,V = 1]).
eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
eq(inc(V, Out),1,[],[Out = 0,V = 0]).
eq(inc(V, Out),1,[inc(V2, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 1 + V2]).
eq(zero(V, Out),1,[],[Out = 1,V = 0]).
eq(zero(V, Out),1,[],[Out = 0,V3 >= 0,V = 1 + V3]).
eq(p(V, Out),1,[],[Out = 0,V = 0]).
eq(p(V, Out),1,[],[Out = V4,V4 >= 0,V = 1 + V4]).
eq(bits(V, Out),1,[bitIter(V5, 0, Ret)],[Out = Ret,V5 >= 0,V = V5]).
eq(bitIter(V, V6, Out),1,[zero(V7, Ret0),inc(V8, Ret2),if(Ret0, V7, Ret2, Ret3)],[Out = Ret3,V7 >= 0,V8 >= 0,V = V7,V6 = V8]).
eq(if(V, V6, V9, Out),1,[p(V11, Ret4)],[Out = Ret4,V6 = V10,V9 = V11,V = 1,V10 >= 0,V11 >= 0]).
eq(if(V, V6, V9, Out),1,[half(V13, Ret01),bitIter(Ret01, V12, Ret5)],[Out = Ret5,V6 = V13,V9 = V12,V13 >= 0,V12 >= 0,V = 0]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(inc(V,Out),[V],[Out]).
input_output_vars(zero(V,Out),[V],[Out]).
input_output_vars(p(V,Out),[V],[Out]).
input_output_vars(bits(V,Out),[V],[Out]).
input_output_vars(bitIter(V,V6,Out),[V,V6],[Out]).
input_output_vars(if(V,V6,V9,Out),[V,V6,V9],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [half/2]
1. non_recursive  : [p/2]
2. recursive  : [inc/2]
3. non_recursive  : [zero/2]
4. recursive  : [bitIter/3,if/4]
5. non_recursive  : [bits/2]
6. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into half/2
1. SCC is partially evaluated into p/2
2. SCC is partially evaluated into inc/2
3. SCC is partially evaluated into zero/2
4. SCC is partially evaluated into bitIter/3
5. SCC is completely evaluated into other SCCs
6. SCC is partially evaluated into start/3

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

### Specialization of cost equations half/2 
* CE 11 is refined into CE [20] 
* CE 10 is refined into CE [21] 
* CE 9 is refined into CE [22] 


### Cost equations --> "Loop" of half/2 
* CEs [21] --> Loop 14 
* CEs [22] --> Loop 15 
* CEs [20] --> Loop 16 

### Ranking functions of CR half(V,Out) 
* RF of phase [16]: [V-1]

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


### Specialization of cost equations p/2 
* CE 15 is refined into CE [23] 
* CE 14 is refined into CE [24] 


### Cost equations --> "Loop" of p/2 
* CEs [23] --> Loop 17 
* CEs [24] --> Loop 18 

### Ranking functions of CR p(V,Out) 

#### Partial ranking functions of CR p(V,Out) 


### Specialization of cost equations inc/2 
* CE 17 is refined into CE [25] 
* CE 16 is refined into CE [26] 


### Cost equations --> "Loop" of inc/2 
* CEs [26] --> Loop 19 
* CEs [25] --> Loop 20 

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

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


### Specialization of cost equations zero/2 
* CE 19 is refined into CE [27] 
* CE 18 is refined into CE [28] 


### Cost equations --> "Loop" of zero/2 
* CEs [27] --> Loop 21 
* CEs [28] --> Loop 22 

### Ranking functions of CR zero(V,Out) 

#### Partial ranking functions of CR zero(V,Out) 


### Specialization of cost equations bitIter/3 
* CE 13 is refined into CE [29,30] 
* CE 12 is refined into CE [31,32,33,34,35,36] 


### Cost equations --> "Loop" of bitIter/3 
* CEs [36] --> Loop 23 
* CEs [35] --> Loop 24 
* CEs [33] --> Loop 25 
* CEs [32] --> Loop 26 
* CEs [34] --> Loop 27 
* CEs [31] --> Loop 28 
* CEs [30] --> Loop 29 
* CEs [29] --> Loop 30 

### Ranking functions of CR bitIter(V,V6,Out) 
* RF of phase [23,24]: [V-1]
* RF of phase [25,26]: [V-1]

#### Partial ranking functions of CR bitIter(V,V6,Out) 
* Partial RF of phase [23,24]:
  - RF of loop [23:1]:
    V/2-1
  - RF of loop [24:1]:
    V-1
* Partial RF of phase [25,26]:
  - RF of loop [25:1]:
    V/2-1
  - RF of loop [26:1]:
    V-1


### Specialization of cost equations start/3 
* CE 2 is refined into CE [37,38] 
* CE 1 is refined into CE [39,40,41,42,43,44,45,46,47,48,49,50] 
* CE 3 is refined into CE [51,52,53,54] 
* CE 4 is refined into CE [55,56] 
* CE 5 is refined into CE [57,58] 
* CE 6 is refined into CE [59,60] 
* CE 7 is refined into CE [61,62,63] 
* CE 8 is refined into CE [64,65,66,67,68,69] 


### Cost equations --> "Loop" of start/3 
* CEs [68] --> Loop 31 
* CEs [37,38,52,53,54,56,58,60,62,63,66,67,69] --> Loop 32 
* CEs [39,40,41,42,43,44,45,46,47,48,49,50,51,55,57,59,61,64,65] --> Loop 33 

### Ranking functions of CR start(V,V6,V9) 

#### Partial ranking functions of CR start(V,V6,V9) 


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

#### Cost of chains of half(V,Out):
* Chain [[16],15]: 1*it(16)+1
  Such that:it(16) =< 2*Out

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

* Chain [[16],14]: 1*it(16)+1
  Such that:it(16) =< 2*Out

  with precondition: [V=2*Out+1,V>=3] 

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

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


#### Cost of chains of p(V,Out):
* Chain [18]: 1
  with precondition: [V=0,Out=0] 

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


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

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

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


#### Cost of chains of zero(V,Out):
* Chain [22]: 1
  with precondition: [V=0,Out=1] 

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


#### Cost of chains of bitIter(V,V6,Out):
* Chain [[25,26],28,30]: 5*it(25)+5*it(26)+2*s(5)+10
  Such that:it(25) =< V/2
aux(5) =< V
aux(6) =< 2*V
it(25) =< aux(5)
it(26) =< aux(5)
it(26) =< aux(6)
s(5) =< aux(6)

  with precondition: [V6=0,Out=0,V>=2] 

* Chain [[23,24],27,29]: 5*it(23)+5*it(24)+2*s(7)+1*s(17)+2*s(18)+1*s(19)+10
  Such that:it(23) =< V/2
aux(14) =< V
aux(15) =< 2*V
aux(16) =< Out+1
s(7) =< aux(16)
it(23) =< aux(14)
it(24) =< aux(14)
s(18) =< aux(15)
it(24) =< aux(15)
aux(9) =< aux(16)
s(17) =< it(23)*aux(16)
s(19) =< it(24)*aux(9)

  with precondition: [V6=Out+1,V>=2,V6>=1] 

* Chain [30]: 5
  with precondition: [V=0,V6=0,Out=0] 

* Chain [29]: 1*s(7)+5
  Such that:s(7) =< V6

  with precondition: [V=0,V6=Out+1,V6>=1] 

* Chain [28,30]: 10
  with precondition: [V=1,V6=0,Out=0] 

* Chain [27,29]: 2*s(7)+10
  Such that:aux(7) =< Out+1
s(7) =< aux(7)

  with precondition: [V=1,V6=Out+1,V6>=1] 


#### Cost of chains of start(V,V6,V9):
* Chain [33]: 10*s(21)+4*s(23)+13*s(27)+20*s(28)+20*s(31)+2*s(42)+2*s(43)+12
  Such that:aux(21) =< 2
aux(22) =< V6
aux(23) =< V6/2
aux(24) =< V6/4
aux(25) =< V9
s(23) =< aux(21)
s(27) =< aux(22)
s(28) =< aux(24)
s(21) =< aux(25)
s(28) =< aux(23)
s(31) =< aux(23)
s(31) =< aux(22)
s(41) =< aux(25)
s(42) =< s(28)*aux(25)
s(43) =< s(31)*s(41)

  with precondition: [V=0] 

* Chain [32]: 3*s(66)+10*s(69)+10*s(72)+4*s(73)+4*s(75)+1*s(84)+1*s(85)+11
  Such that:aux(26) =< V
aux(27) =< 2*V
aux(28) =< V/2
aux(29) =< V6
s(66) =< aux(26)
s(69) =< aux(28)
s(69) =< aux(26)
s(72) =< aux(26)
s(72) =< aux(27)
s(73) =< aux(27)
s(75) =< aux(29)
s(83) =< aux(29)
s(84) =< s(69)*aux(29)
s(85) =< s(72)*s(83)

  with precondition: [V>=1] 

* Chain [31]: 5*s(86)+5*s(89)+2*s(90)+10
  Such that:s(87) =< V
s(88) =< 2*V
s(86) =< V/2
s(86) =< s(87)
s(89) =< s(87)
s(89) =< s(88)
s(90) =< s(88)

  with precondition: [V6=0,V>=2] 


Closed-form bounds of start(V,V6,V9): 
-------------------------------------
* Chain [33] with precondition: [V=0] 
    - Upper bound: nat(V6)*13+20+nat(V9)*10+nat(V9)*2*nat(V6/2)+nat(V9)*2*nat(V6/4)+nat(V6/2)*20+nat(V6/4)*20 
    - Complexity: n^2 
* Chain [32] with precondition: [V>=1] 
    - Upper bound: 13*V+11+nat(V6)*V+nat(V6)*4+V/2*nat(V6)+8*V+5*V 
    - Complexity: n^2 
* Chain [31] with precondition: [V6=0,V>=2] 
    - Upper bound: 23/2*V+10 
    - Complexity: n 

### Maximum cost of start(V,V6,V9): max([nat(V6)*13+10+nat(V9)*10+nat(V9)*2*nat(V6/2)+nat(V9)*2*nat(V6/4)+nat(V6/2)*20+nat(V6/4)*20,8*V+1+nat(V6)*V+nat(V6)*4+V/2*nat(V6)+4*V+5/2*V+23/2*V])+10 
Asymptotic class: n^2 
* Total analysis performed in 353 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:

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   inc(0') -> 0'
   inc(s(x)) -> s(inc(x))
   zero(0') -> true
   zero(s(x)) -> false
   p(0') -> 0'
   p(s(x)) -> x
   bits(x) -> bitIter(x, 0')
   bitIter(x, y) -> if(zero(x), x, inc(y))
   if(true, x, y) -> p(y)
   if(false, x, y) -> bitIter(half(x), y)

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

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

(14)
Obligation:
Innermost TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
half, inc, bitIter

They will be analysed ascendingly in the following order:
half < bitIter
inc < bitIter

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

(16)
Obligation:
Innermost TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
half, inc, bitIter

They will be analysed ascendingly in the following order:
half < bitIter
inc < bitIter

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n5_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s3_0(*(2, n5_0)))) ->_IH
s(gen_0':s3_0(c6_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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(18)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
half, inc, bitIter

They will be analysed ascendingly in the following order:
half < bitIter
inc < bitIter

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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Innermost TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_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:
inc, bitIter

They will be analysed ascendingly in the following order:
inc < bitIter

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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s3_0(n355_0)) -> gen_0':s3_0(n355_0), rt in Omega(1 + n355_0)

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

Induction Step:
inc(gen_0':s3_0(+(n355_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s3_0(n355_0))) ->_IH
s(gen_0':s3_0(c356_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:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
zero(0') -> true
zero(s(x)) -> false
p(0') -> 0'
p(s(x)) -> x
bits(x) -> bitIter(x, 0')
bitIter(x, y) -> if(zero(x), x, inc(y))
if(true, x, y) -> p(y)
if(false, x, y) -> bitIter(half(x), y)

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
inc :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
p :: 0':s -> 0':s
bits :: 0':s -> 0':s
bitIter :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s3_0(*(2, n5_0))) -> gen_0':s3_0(n5_0), rt in Omega(1 + n5_0)
inc(gen_0':s3_0(n355_0)) -> gen_0':s3_0(n355_0), rt in Omega(1 + n355_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:
bitIter