WORST_CASE(Omega(n^1),O(n^1))
proof of input_i3fat5ys7a.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 611 ms]
    (10) BOUNDS(1, n^1)
(11) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CdtProblem
    (13) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRelTRS
    (15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 14 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 351 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), 95 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 82 ms]
            (32) typed CpxTrs


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

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


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(x))) -> lastbit(x)
   zero(0) -> true
   zero(s(x)) -> false
   conv(x) -> conviter(x, cons(0, nil))
   conviter(x, l) -> if(zero(x), x, l)
   if(true, x, l) -> l
   if(false, x, l) -> conviter(half(x), cons(lastbit(x), l))

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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^1).


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]
   lastbit(0) -> 0 [1]
   lastbit(s(0)) -> s(0) [1]
   lastbit(s(s(x))) -> lastbit(x) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   conv(x) -> conviter(x, cons(0, nil)) [1]
   conviter(x, l) -> if(zero(x), x, l) [1]
   if(true, x, l) -> l [1]
   if(false, x, l) -> conviter(half(x), cons(lastbit(x), l)) [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]
   lastbit(0) -> 0 [1]
   lastbit(s(0)) -> s(0) [1]
   lastbit(s(s(x))) -> lastbit(x) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   conv(x) -> conviter(x, cons(0, nil)) [1]
   conviter(x, l) -> if(zero(x), x, l) [1]
   if(true, x, l) -> l [1]
   if(false, x, l) -> conviter(half(x), cons(lastbit(x), l)) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lastbit :: 0:s -> 0:s
zero :: 0:s -> true:false
true :: true:false
false :: true:false
conv :: 0:s -> nil:cons
conviter :: 0:s -> nil:cons -> nil:cons
cons :: 0:s -> nil:cons -> nil:cons
nil :: nil:cons
if :: true:false -> 0:s -> nil:cons -> nil:cons

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]
   lastbit(0) -> 0 [1]
   lastbit(s(0)) -> s(0) [1]
   lastbit(s(s(x))) -> lastbit(x) [1]
   zero(0) -> true [1]
   zero(s(x)) -> false [1]
   conv(x) -> conviter(x, cons(0, nil)) [1]
   conviter(x, l) -> if(zero(x), x, l) [1]
   if(true, x, l) -> l [1]
   if(false, x, l) -> conviter(half(x), cons(lastbit(x), l)) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lastbit :: 0:s -> 0:s
zero :: 0:s -> true:false
true :: true:false
false :: true:false
conv :: 0:s -> nil:cons
conviter :: 0:s -> nil:cons -> nil:cons
cons :: 0:s -> nil:cons -> nil:cons
nil :: nil:cons
if :: true:false -> 0:s -> nil:cons -> nil:cons

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
   nil => 0

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

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

   conv(z) -{ 1 }-> conviter(x, 1 + 0 + 0) :|: x >= 0, z = x
   conviter(z, z') -{ 1 }-> if(zero(x), x, l) :|: z' = l, x >= 0, l >= 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 }-> l :|: z' = x, z = 1, x >= 0, l >= 0, z'' = l
   if(z, z', z'') -{ 1 }-> conviter(half(x), 1 + lastbit(x) + l) :|: z' = x, x >= 0, l >= 0, z = 0, z'' = l
   lastbit(z) -{ 1 }-> lastbit(x) :|: x >= 0, z = 1 + (1 + x)
   lastbit(z) -{ 1 }-> 0 :|: z = 0
   lastbit(z) -{ 1 }-> 1 + 0 :|: z = 1 + 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, V5, V8),0,[half(V, Out)],[V >= 0]).
eq(start(V, V5, V8),0,[lastbit(V, Out)],[V >= 0]).
eq(start(V, V5, V8),0,[zero(V, Out)],[V >= 0]).
eq(start(V, V5, V8),0,[conv(V, Out)],[V >= 0]).
eq(start(V, V5, V8),0,[conviter(V, V5, Out)],[V >= 0,V5 >= 0]).
eq(start(V, V5, V8),0,[if(V, V5, V8, Out)],[V >= 0,V5 >= 0,V8 >= 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(lastbit(V, Out),1,[],[Out = 0,V = 0]).
eq(lastbit(V, Out),1,[],[Out = 1,V = 1]).
eq(lastbit(V, Out),1,[lastbit(V2, Ret)],[Out = Ret,V2 >= 0,V = 2 + V2]).
eq(zero(V, Out),1,[],[Out = 1,V = 0]).
eq(zero(V, Out),1,[],[Out = 0,V3 >= 0,V = 1 + V3]).
eq(conv(V, Out),1,[conviter(V4, 1 + 0 + 0, Ret2)],[Out = Ret2,V4 >= 0,V = V4]).
eq(conviter(V, V5, Out),1,[zero(V6, Ret0),if(Ret0, V6, V7, Ret3)],[Out = Ret3,V5 = V7,V6 >= 0,V7 >= 0,V = V6]).
eq(if(V, V5, V8, Out),1,[],[Out = V10,V5 = V9,V = 1,V9 >= 0,V10 >= 0,V8 = V10]).
eq(if(V, V5, V8, Out),1,[half(V11, Ret01),lastbit(V11, Ret101),conviter(Ret01, 1 + Ret101 + V12, Ret4)],[Out = Ret4,V5 = V11,V11 >= 0,V12 >= 0,V = 0,V8 = V12]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(lastbit(V,Out),[V],[Out]).
input_output_vars(zero(V,Out),[V],[Out]).
input_output_vars(conv(V,Out),[V],[Out]).
input_output_vars(conviter(V,V5,Out),[V,V5],[Out]).
input_output_vars(if(V,V5,V8,Out),[V,V5,V8],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [half/2]
1. recursive  : [lastbit/2]
2. non_recursive  : [zero/2]
3. recursive  : [conviter/3,if/4]
4. non_recursive  : [conv/2]
5. non_recursive  : [start/3]

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

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

### Specialization of cost equations half/2 
* CE 10 is refined into CE [18] 
* CE 9 is refined into CE [19] 
* CE 8 is refined into CE [20] 


### Cost equations --> "Loop" of half/2 
* CEs [19] --> Loop 13 
* CEs [20] --> Loop 14 
* CEs [18] --> Loop 15 

### Ranking functions of CR half(V,Out) 
* RF of phase [15]: [V-1]

#### Partial ranking functions of CR half(V,Out) 
* Partial RF of phase [15]:
  - RF of loop [15:1]:
    V-1


### Specialization of cost equations lastbit/2 
* CE 13 is refined into CE [21] 
* CE 12 is refined into CE [22] 
* CE 11 is refined into CE [23] 


### Cost equations --> "Loop" of lastbit/2 
* CEs [22] --> Loop 16 
* CEs [23] --> Loop 17 
* CEs [21] --> Loop 18 

### Ranking functions of CR lastbit(V,Out) 
* RF of phase [18]: [V-1]

#### Partial ranking functions of CR lastbit(V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V-1


### Specialization of cost equations zero/2 
* CE 17 is refined into CE [24] 
* CE 16 is refined into CE [25] 


### Cost equations --> "Loop" of zero/2 
* CEs [24] --> Loop 19 
* CEs [25] --> Loop 20 

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

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


### Specialization of cost equations conviter/3 
* CE 15 is refined into CE [26] 
* CE 14 is refined into CE [27,28,29,30,31] 


### Cost equations --> "Loop" of conviter/3 
* CEs [30] --> Loop 21 
* CEs [31] --> Loop 22 
* CEs [28] --> Loop 23 
* CEs [29] --> Loop 24 
* CEs [27] --> Loop 25 
* CEs [26] --> Loop 26 

### Ranking functions of CR conviter(V,V5,Out) 
* RF of phase [21,22,23,24]: [V-1,2*V+V5-3]

#### Partial ranking functions of CR conviter(V,V5,Out) 
* Partial RF of phase [21,22,23,24]:
  - RF of loop [21:1,22:1]:
    V/2-1
  - RF of loop [23:1]:
    V-1
  - RF of loop [24:1]:
    2/3*V-5/3


### Specialization of cost equations start/3 
* CE 2 is refined into CE [32] 
* CE 1 is refined into CE [33,34,35,36,37,38,39,40,41] 
* CE 3 is refined into CE [42,43,44,45] 
* CE 4 is refined into CE [46,47,48,49] 
* CE 5 is refined into CE [50,51] 
* CE 6 is refined into CE [52,53,54] 
* CE 7 is refined into CE [55,56,57] 


### Cost equations --> "Loop" of start/3 
* CEs [32,43,44,45,47,48,49,51,53,54,56,57] --> Loop 27 
* CEs [33,34,35,36,37,38,39,40,41,42,46,50,52,55] --> Loop 28 

### Ranking functions of CR start(V,V5,V8) 

#### Partial ranking functions of CR start(V,V5,V8) 


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

#### Cost of chains of half(V,Out):
* Chain [[15],14]: 1*it(15)+1
  Such that:it(15) =< 2*Out

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

* Chain [[15],13]: 1*it(15)+1
  Such that:it(15) =< 2*Out

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

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

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


#### Cost of chains of lastbit(V,Out):
* Chain [[18],17]: 1*it(18)+1
  Such that:it(18) =< V

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

* Chain [[18],16]: 1*it(18)+1
  Such that:it(18) =< V

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

* Chain [17]: 1
  with precondition: [V=0,Out=0] 

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


#### Cost of chains of zero(V,Out):
* Chain [20]: 1
  with precondition: [V=0,Out=1] 

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


#### Cost of chains of conviter(V,V5,Out):
* Chain [[21,22,23,24],25,26]: 5*it(21)+5*it(22)+5*it(23)+5*it(24)+2*s(17)+2*s(18)+4*s(21)+8
  Such that:aux(6) =< 2*V+V5
aux(7) =< 2*V+V5-Out
aux(10) =< 3*V
aux(9) =< 3*V+6
aux(12) =< 4*V
aux(11) =< 4*V+8
it(24) =< 2/3*V
aux(15) =< V
aux(16) =< 2*V
aux(17) =< V/2
it(21) =< aux(15)
it(22) =< aux(15)
it(23) =< aux(15)
it(24) =< aux(15)
it(24) =< aux(16)
it(21) =< aux(6)
it(22) =< aux(6)
it(23) =< aux(6)
it(24) =< aux(6)
it(21) =< aux(7)
it(22) =< aux(7)
it(23) =< aux(7)
it(24) =< aux(7)
it(22) =< aux(9)
it(23) =< aux(9)
it(24) =< aux(9)
s(18) =< aux(9)
it(22) =< aux(10)
it(23) =< aux(10)
it(24) =< aux(10)
s(18) =< aux(10)
it(22) =< aux(11)
it(23) =< aux(11)
it(24) =< aux(11)
s(17) =< aux(11)
it(22) =< aux(12)
it(23) =< aux(12)
it(24) =< aux(12)
s(17) =< aux(12)
it(21) =< aux(17)
it(22) =< aux(17)
s(21) =< aux(16)

  with precondition: [V5>=0,Out>=V5+3,V+2*V5+6>=2*Out,V+V5+1>=Out] 

* Chain [26]: 3
  with precondition: [V=0,V5=Out,V5>=0] 

* Chain [25,26]: 8
  with precondition: [V=1,Out=V5+2,Out>=2] 


#### Cost of chains of start(V,V5,V8):
* Chain [28]: 4*s(25)+24*s(27)+5*s(35)+10*s(39)+5*s(40)+5*s(41)+4*s(42)+4*s(43)+5*s(53)+5*s(57)+5*s(58)+5*s(59)+2*s(64)+5*s(73)+5*s(77)+5*s(78)+5*s(79)+4*s(80)+4*s(81)+5*s(93)+5*s(98)+5*s(99)+11
  Such that:s(67) =< V5+V8
s(47) =< V5+V8+2
aux(23) =< 2
aux(24) =< 3
aux(25) =< V5
aux(26) =< V5+V8+1
aux(27) =< 2*V5
aux(28) =< 2*V5+6
aux(29) =< 2*V5+8
aux(30) =< V5/2
aux(31) =< V5/3
aux(32) =< V5/4
aux(33) =< 3/2*V5
aux(34) =< 3/2*V5+6
aux(35) =< 3/2*V5+9/2
s(25) =< aux(23)
s(64) =< aux(24)
s(35) =< aux(31)
s(53) =< aux(31)
s(73) =< aux(31)
s(93) =< aux(31)
s(27) =< aux(25)
s(77) =< aux(30)
s(78) =< aux(30)
s(79) =< aux(30)
s(73) =< aux(30)
s(73) =< aux(25)
s(77) =< s(67)
s(78) =< s(67)
s(79) =< s(67)
s(73) =< s(67)
s(77) =< aux(25)
s(78) =< aux(25)
s(79) =< aux(25)
s(78) =< aux(35)
s(79) =< aux(35)
s(73) =< aux(35)
s(80) =< aux(35)
s(78) =< aux(33)
s(79) =< aux(33)
s(73) =< aux(33)
s(80) =< aux(33)
s(78) =< aux(28)
s(79) =< aux(28)
s(73) =< aux(28)
s(81) =< aux(28)
s(78) =< aux(27)
s(79) =< aux(27)
s(73) =< aux(27)
s(81) =< aux(27)
s(77) =< aux(32)
s(78) =< aux(32)
s(39) =< aux(30)
s(40) =< aux(30)
s(41) =< aux(30)
s(35) =< aux(30)
s(35) =< aux(25)
s(39) =< aux(26)
s(40) =< aux(26)
s(41) =< aux(26)
s(35) =< aux(26)
s(39) =< aux(25)
s(40) =< aux(25)
s(41) =< aux(25)
s(40) =< aux(34)
s(41) =< aux(34)
s(35) =< aux(34)
s(42) =< aux(34)
s(40) =< aux(33)
s(41) =< aux(33)
s(35) =< aux(33)
s(42) =< aux(33)
s(40) =< aux(29)
s(41) =< aux(29)
s(35) =< aux(29)
s(43) =< aux(29)
s(40) =< aux(27)
s(41) =< aux(27)
s(35) =< aux(27)
s(43) =< aux(27)
s(39) =< aux(32)
s(40) =< aux(32)
s(98) =< aux(30)
s(99) =< aux(30)
s(93) =< aux(30)
s(93) =< aux(25)
s(98) =< aux(26)
s(99) =< aux(26)
s(93) =< aux(26)
s(98) =< aux(25)
s(99) =< aux(25)
s(98) =< aux(35)
s(99) =< aux(35)
s(93) =< aux(35)
s(98) =< aux(33)
s(99) =< aux(33)
s(93) =< aux(33)
s(98) =< aux(28)
s(99) =< aux(28)
s(93) =< aux(28)
s(98) =< aux(27)
s(99) =< aux(27)
s(93) =< aux(27)
s(98) =< aux(32)
s(57) =< aux(30)
s(58) =< aux(30)
s(59) =< aux(30)
s(53) =< aux(30)
s(53) =< aux(25)
s(57) =< s(47)
s(58) =< s(47)
s(59) =< s(47)
s(53) =< s(47)
s(57) =< aux(25)
s(58) =< aux(25)
s(59) =< aux(25)
s(58) =< aux(34)
s(59) =< aux(34)
s(53) =< aux(34)
s(58) =< aux(33)
s(59) =< aux(33)
s(53) =< aux(33)
s(58) =< aux(29)
s(59) =< aux(29)
s(53) =< aux(29)
s(58) =< aux(27)
s(59) =< aux(27)
s(53) =< aux(27)
s(57) =< aux(32)
s(58) =< aux(32)

  with precondition: [V=0] 

* Chain [27]: 4*s(103)+5*s(113)+5*s(117)+5*s(118)+5*s(119)+4*s(120)+4*s(121)+8*s(122)+5*s(129)+5*s(133)+5*s(134)+5*s(135)+9
  Such that:s(107) =< 2*V+1
s(123) =< 2*V+V5
aux(38) =< V
aux(39) =< 2*V
aux(40) =< 3*V
aux(41) =< 3*V+6
aux(42) =< 4*V
aux(43) =< 4*V+8
aux(44) =< V/2
aux(45) =< 2/3*V
s(103) =< aux(38)
s(113) =< aux(45)
s(129) =< aux(45)
s(117) =< aux(38)
s(118) =< aux(38)
s(119) =< aux(38)
s(113) =< aux(38)
s(113) =< aux(39)
s(117) =< s(107)
s(118) =< s(107)
s(119) =< s(107)
s(113) =< s(107)
s(117) =< aux(39)
s(118) =< aux(39)
s(119) =< aux(39)
s(118) =< aux(41)
s(119) =< aux(41)
s(113) =< aux(41)
s(120) =< aux(41)
s(118) =< aux(40)
s(119) =< aux(40)
s(113) =< aux(40)
s(120) =< aux(40)
s(118) =< aux(43)
s(119) =< aux(43)
s(113) =< aux(43)
s(121) =< aux(43)
s(118) =< aux(42)
s(119) =< aux(42)
s(113) =< aux(42)
s(121) =< aux(42)
s(117) =< aux(44)
s(118) =< aux(44)
s(122) =< aux(39)
s(133) =< aux(38)
s(134) =< aux(38)
s(135) =< aux(38)
s(129) =< aux(38)
s(129) =< aux(39)
s(133) =< s(123)
s(134) =< s(123)
s(135) =< s(123)
s(129) =< s(123)
s(133) =< aux(39)
s(134) =< aux(39)
s(135) =< aux(39)
s(134) =< aux(41)
s(135) =< aux(41)
s(129) =< aux(41)
s(134) =< aux(40)
s(135) =< aux(40)
s(129) =< aux(40)
s(134) =< aux(43)
s(135) =< aux(43)
s(129) =< aux(43)
s(134) =< aux(42)
s(135) =< aux(42)
s(129) =< aux(42)
s(133) =< aux(44)
s(134) =< aux(44)

  with precondition: [V>=1] 


Closed-form bounds of start(V,V5,V8): 
-------------------------------------
* Chain [28] with precondition: [V=0] 
    - Upper bound: nat(V5)*24+25+nat(2*V5+6)*4+nat(2*V5+8)*4+nat(3/2*V5+6)*4+nat(3/2*V5+9/2)*4+nat(V5/2)*60+nat(V5/3)*20 
    - Complexity: n 
* Chain [27] with precondition: [V>=1] 
    - Upper bound: 254/3*V+65 
    - Complexity: n 

### Maximum cost of start(V,V5,V8): max([254/3*V+56,nat(V5)*24+16+nat(2*V5+6)*4+nat(2*V5+8)*4+nat(3/2*V5+6)*4+nat(3/2*V5+9/2)*4+nat(V5/2)*60+nat(V5/3)*20])+9 
Asymptotic class: n 
* Total analysis performed in 529 ms.


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

(10)
BOUNDS(1, n^1)

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

(11) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(z0))) -> lastbit(z0)
   zero(0) -> true
   zero(s(z0)) -> false
   conv(z0) -> conviter(z0, cons(0, nil))
   conviter(z0, z1) -> if(zero(z0), z0, z1)
   if(true, z0, z1) -> z1
   if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))
Tuples:
   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   LASTBIT(0) -> c3
   LASTBIT(s(0)) -> c4
   LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
   ZERO(0) -> c6
   ZERO(s(z0)) -> c7
   CONV(z0) -> c8(CONVITER(z0, cons(0, nil)))
   CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
   IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
S tuples:
   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   LASTBIT(0) -> c3
   LASTBIT(s(0)) -> c4
   LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
   ZERO(0) -> c6
   ZERO(s(z0)) -> c7
   CONV(z0) -> c8(CONVITER(z0, cons(0, nil)))
   CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
   IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
K tuples:none
Defined Rule Symbols:   half_1, lastbit_1, zero_1, conv_1, conviter_2, if_3

Defined Pair Symbols:   HALF_1, LASTBIT_1, ZERO_1, CONV_1, CONVITER_2, IF_3

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7, c8_1, c9_2, c10, c11_2, c12_2


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

(13) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   LASTBIT(0) -> c3
   LASTBIT(s(0)) -> c4
   LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
   ZERO(0) -> c6
   ZERO(s(z0)) -> c7
   CONV(z0) -> c8(CONVITER(z0, cons(0, nil)))
   CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
   IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))

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

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(z0))) -> s(half(z0))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(z0))) -> lastbit(z0)
   zero(0) -> true
   zero(s(z0)) -> false
   conv(z0) -> conviter(z0, cons(0, nil))
   conviter(z0, z1) -> if(zero(z0), z0, z1)
   if(true, z0, z1) -> z1
   if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

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

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

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   HALF(0') -> c
   HALF(s(0')) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   LASTBIT(0') -> c3
   LASTBIT(s(0')) -> c4
   LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
   ZERO(0') -> c6
   ZERO(s(z0)) -> c7
   CONV(z0) -> c8(CONVITER(z0, cons(0', nil)))
   CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
   IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))

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

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(z0))) -> s(half(z0))
   lastbit(0') -> 0'
   lastbit(s(0')) -> s(0')
   lastbit(s(s(z0))) -> lastbit(z0)
   zero(0') -> true
   zero(s(z0)) -> false
   conv(z0) -> conviter(z0, cons(0', nil))
   conviter(z0, z1) -> if(zero(z0), z0, z1)
   if(true, z0, z1) -> z1
   if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LASTBIT(0') -> c3
LASTBIT(s(0')) -> c4
LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
ZERO(0') -> c6
ZERO(s(z0)) -> c7
CONV(z0) -> c8(CONVITER(z0, cons(0', nil)))
CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) -> c10
IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(z0))) -> lastbit(z0)
zero(0') -> true
zero(s(z0)) -> false
conv(z0) -> conviter(z0, cons(0', nil))
conviter(z0, z1) -> if(zero(z0), z0, z1)
if(true, z0, z1) -> z1
if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LASTBIT :: 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
ZERO :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7
CONV :: 0':s -> c8
c8 :: c9 -> c8
CONVITER :: 0':s -> nil:cons -> c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c9 :: c10:c11:c12 -> c6:c7 -> c9
IF :: true:false -> 0':s -> nil:cons -> c10:c11:c12
zero :: 0':s -> true:false
true :: true:false
c10 :: c10:c11:c12
false :: true:false
c11 :: c9 -> c:c1:c2 -> c10:c11:c12
half :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
c12 :: c9 -> c3:c4:c5 -> c10:c11:c12
conv :: 0':s -> nil:cons
conviter :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c96_13 :: c9
hole_nil:cons7_13 :: nil:cons
hole_c10:c11:c128_13 :: c10:c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c4:c512_13 :: Nat -> c3:c4:c5
gen_nil:cons13_13 :: Nat -> nil:cons

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
HALF, LASTBIT, CONVITER, half, lastbit, conviter

They will be analysed ascendingly in the following order:
HALF < CONVITER
LASTBIT < CONVITER
half < CONVITER
lastbit < CONVITER
half < conviter
lastbit < conviter

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

(20)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LASTBIT(0') -> c3
LASTBIT(s(0')) -> c4
LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
ZERO(0') -> c6
ZERO(s(z0)) -> c7
CONV(z0) -> c8(CONVITER(z0, cons(0', nil)))
CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) -> c10
IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(z0))) -> lastbit(z0)
zero(0') -> true
zero(s(z0)) -> false
conv(z0) -> conviter(z0, cons(0', nil))
conviter(z0, z1) -> if(zero(z0), z0, z1)
if(true, z0, z1) -> z1
if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LASTBIT :: 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
ZERO :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7
CONV :: 0':s -> c8
c8 :: c9 -> c8
CONVITER :: 0':s -> nil:cons -> c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c9 :: c10:c11:c12 -> c6:c7 -> c9
IF :: true:false -> 0':s -> nil:cons -> c10:c11:c12
zero :: 0':s -> true:false
true :: true:false
c10 :: c10:c11:c12
false :: true:false
c11 :: c9 -> c:c1:c2 -> c10:c11:c12
half :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
c12 :: c9 -> c3:c4:c5 -> c10:c11:c12
conv :: 0':s -> nil:cons
conviter :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c96_13 :: c9
hole_nil:cons7_13 :: nil:cons
hole_c10:c11:c128_13 :: c10:c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c4:c512_13 :: Nat -> c3:c4:c5
gen_nil:cons13_13 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c4:c512_13(0) <=> c3
gen_c3:c4:c512_13(+(x, 1)) <=> c5(gen_c3:c4:c512_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(0', gen_nil:cons13_13(x))


The following defined symbols remain to be analysed:
HALF, LASTBIT, CONVITER, half, lastbit, conviter

They will be analysed ascendingly in the following order:
HALF < CONVITER
LASTBIT < CONVITER
half < CONVITER
lastbit < CONVITER
half < conviter
lastbit < conviter

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
HALF(gen_0':s11_13(*(2, n15_13))) -> gen_c:c1:c210_13(n15_13), rt in Omega(1 + n15_13)

Induction Base:
HALF(gen_0':s11_13(*(2, 0))) ->_R^Omega(1)
c

Induction Step:
HALF(gen_0':s11_13(*(2, +(n15_13, 1)))) ->_R^Omega(1)
c2(HALF(gen_0':s11_13(*(2, n15_13)))) ->_IH
c2(gen_c:c1:c210_13(c16_13))

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:

Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LASTBIT(0') -> c3
LASTBIT(s(0')) -> c4
LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
ZERO(0') -> c6
ZERO(s(z0)) -> c7
CONV(z0) -> c8(CONVITER(z0, cons(0', nil)))
CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) -> c10
IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(z0))) -> lastbit(z0)
zero(0') -> true
zero(s(z0)) -> false
conv(z0) -> conviter(z0, cons(0', nil))
conviter(z0, z1) -> if(zero(z0), z0, z1)
if(true, z0, z1) -> z1
if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LASTBIT :: 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
ZERO :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7
CONV :: 0':s -> c8
c8 :: c9 -> c8
CONVITER :: 0':s -> nil:cons -> c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c9 :: c10:c11:c12 -> c6:c7 -> c9
IF :: true:false -> 0':s -> nil:cons -> c10:c11:c12
zero :: 0':s -> true:false
true :: true:false
c10 :: c10:c11:c12
false :: true:false
c11 :: c9 -> c:c1:c2 -> c10:c11:c12
half :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
c12 :: c9 -> c3:c4:c5 -> c10:c11:c12
conv :: 0':s -> nil:cons
conviter :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c96_13 :: c9
hole_nil:cons7_13 :: nil:cons
hole_c10:c11:c128_13 :: c10:c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c4:c512_13 :: Nat -> c3:c4:c5
gen_nil:cons13_13 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c4:c512_13(0) <=> c3
gen_c3:c4:c512_13(+(x, 1)) <=> c5(gen_c3:c4:c512_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(0', gen_nil:cons13_13(x))


The following defined symbols remain to be analysed:
HALF, LASTBIT, CONVITER, half, lastbit, conviter

They will be analysed ascendingly in the following order:
HALF < CONVITER
LASTBIT < CONVITER
half < CONVITER
lastbit < CONVITER
half < conviter
lastbit < conviter

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LASTBIT(0') -> c3
LASTBIT(s(0')) -> c4
LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
ZERO(0') -> c6
ZERO(s(z0)) -> c7
CONV(z0) -> c8(CONVITER(z0, cons(0', nil)))
CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) -> c10
IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(z0))) -> lastbit(z0)
zero(0') -> true
zero(s(z0)) -> false
conv(z0) -> conviter(z0, cons(0', nil))
conviter(z0, z1) -> if(zero(z0), z0, z1)
if(true, z0, z1) -> z1
if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LASTBIT :: 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
ZERO :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7
CONV :: 0':s -> c8
c8 :: c9 -> c8
CONVITER :: 0':s -> nil:cons -> c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c9 :: c10:c11:c12 -> c6:c7 -> c9
IF :: true:false -> 0':s -> nil:cons -> c10:c11:c12
zero :: 0':s -> true:false
true :: true:false
c10 :: c10:c11:c12
false :: true:false
c11 :: c9 -> c:c1:c2 -> c10:c11:c12
half :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
c12 :: c9 -> c3:c4:c5 -> c10:c11:c12
conv :: 0':s -> nil:cons
conviter :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c96_13 :: c9
hole_nil:cons7_13 :: nil:cons
hole_c10:c11:c128_13 :: c10:c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c4:c512_13 :: Nat -> c3:c4:c5
gen_nil:cons13_13 :: Nat -> nil:cons


Lemmas:
HALF(gen_0':s11_13(*(2, n15_13))) -> gen_c:c1:c210_13(n15_13), rt in Omega(1 + n15_13)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c4:c512_13(0) <=> c3
gen_c3:c4:c512_13(+(x, 1)) <=> c5(gen_c3:c4:c512_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(0', gen_nil:cons13_13(x))


The following defined symbols remain to be analysed:
LASTBIT, CONVITER, half, lastbit, conviter

They will be analysed ascendingly in the following order:
LASTBIT < CONVITER
half < CONVITER
lastbit < CONVITER
half < conviter
lastbit < conviter

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LASTBIT(gen_0':s11_13(*(2, n457_13))) -> gen_c3:c4:c512_13(n457_13), rt in Omega(1 + n457_13)

Induction Base:
LASTBIT(gen_0':s11_13(*(2, 0))) ->_R^Omega(1)
c3

Induction Step:
LASTBIT(gen_0':s11_13(*(2, +(n457_13, 1)))) ->_R^Omega(1)
c5(LASTBIT(gen_0':s11_13(*(2, n457_13)))) ->_IH
c5(gen_c3:c4:c512_13(c458_13))

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

(28)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LASTBIT(0') -> c3
LASTBIT(s(0')) -> c4
LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
ZERO(0') -> c6
ZERO(s(z0)) -> c7
CONV(z0) -> c8(CONVITER(z0, cons(0', nil)))
CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) -> c10
IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(z0))) -> lastbit(z0)
zero(0') -> true
zero(s(z0)) -> false
conv(z0) -> conviter(z0, cons(0', nil))
conviter(z0, z1) -> if(zero(z0), z0, z1)
if(true, z0, z1) -> z1
if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LASTBIT :: 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
ZERO :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7
CONV :: 0':s -> c8
c8 :: c9 -> c8
CONVITER :: 0':s -> nil:cons -> c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c9 :: c10:c11:c12 -> c6:c7 -> c9
IF :: true:false -> 0':s -> nil:cons -> c10:c11:c12
zero :: 0':s -> true:false
true :: true:false
c10 :: c10:c11:c12
false :: true:false
c11 :: c9 -> c:c1:c2 -> c10:c11:c12
half :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
c12 :: c9 -> c3:c4:c5 -> c10:c11:c12
conv :: 0':s -> nil:cons
conviter :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c96_13 :: c9
hole_nil:cons7_13 :: nil:cons
hole_c10:c11:c128_13 :: c10:c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c4:c512_13 :: Nat -> c3:c4:c5
gen_nil:cons13_13 :: Nat -> nil:cons


Lemmas:
HALF(gen_0':s11_13(*(2, n15_13))) -> gen_c:c1:c210_13(n15_13), rt in Omega(1 + n15_13)
LASTBIT(gen_0':s11_13(*(2, n457_13))) -> gen_c3:c4:c512_13(n457_13), rt in Omega(1 + n457_13)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c4:c512_13(0) <=> c3
gen_c3:c4:c512_13(+(x, 1)) <=> c5(gen_c3:c4:c512_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(0', gen_nil:cons13_13(x))


The following defined symbols remain to be analysed:
half, CONVITER, lastbit, conviter

They will be analysed ascendingly in the following order:
half < CONVITER
lastbit < CONVITER
half < conviter
lastbit < conviter

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s11_13(*(2, n971_13))) -> gen_0':s11_13(n971_13), rt in Omega(0)

Induction Base:
half(gen_0':s11_13(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
half(gen_0':s11_13(*(2, +(n971_13, 1)))) ->_R^Omega(0)
s(half(gen_0':s11_13(*(2, n971_13)))) ->_IH
s(gen_0':s11_13(c972_13))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(30)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LASTBIT(0') -> c3
LASTBIT(s(0')) -> c4
LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
ZERO(0') -> c6
ZERO(s(z0)) -> c7
CONV(z0) -> c8(CONVITER(z0, cons(0', nil)))
CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) -> c10
IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(z0))) -> lastbit(z0)
zero(0') -> true
zero(s(z0)) -> false
conv(z0) -> conviter(z0, cons(0', nil))
conviter(z0, z1) -> if(zero(z0), z0, z1)
if(true, z0, z1) -> z1
if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LASTBIT :: 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
ZERO :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7
CONV :: 0':s -> c8
c8 :: c9 -> c8
CONVITER :: 0':s -> nil:cons -> c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c9 :: c10:c11:c12 -> c6:c7 -> c9
IF :: true:false -> 0':s -> nil:cons -> c10:c11:c12
zero :: 0':s -> true:false
true :: true:false
c10 :: c10:c11:c12
false :: true:false
c11 :: c9 -> c:c1:c2 -> c10:c11:c12
half :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
c12 :: c9 -> c3:c4:c5 -> c10:c11:c12
conv :: 0':s -> nil:cons
conviter :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c96_13 :: c9
hole_nil:cons7_13 :: nil:cons
hole_c10:c11:c128_13 :: c10:c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c4:c512_13 :: Nat -> c3:c4:c5
gen_nil:cons13_13 :: Nat -> nil:cons


Lemmas:
HALF(gen_0':s11_13(*(2, n15_13))) -> gen_c:c1:c210_13(n15_13), rt in Omega(1 + n15_13)
LASTBIT(gen_0':s11_13(*(2, n457_13))) -> gen_c3:c4:c512_13(n457_13), rt in Omega(1 + n457_13)
half(gen_0':s11_13(*(2, n971_13))) -> gen_0':s11_13(n971_13), rt in Omega(0)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c4:c512_13(0) <=> c3
gen_c3:c4:c512_13(+(x, 1)) <=> c5(gen_c3:c4:c512_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(0', gen_nil:cons13_13(x))


The following defined symbols remain to be analysed:
lastbit, CONVITER, conviter

They will be analysed ascendingly in the following order:
lastbit < CONVITER
lastbit < conviter

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lastbit(gen_0':s11_13(*(2, n1431_13))) -> gen_0':s11_13(0), rt in Omega(0)

Induction Base:
lastbit(gen_0':s11_13(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
lastbit(gen_0':s11_13(*(2, +(n1431_13, 1)))) ->_R^Omega(0)
lastbit(gen_0':s11_13(*(2, n1431_13))) ->_IH
gen_0':s11_13(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(32)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
LASTBIT(0') -> c3
LASTBIT(s(0')) -> c4
LASTBIT(s(s(z0))) -> c5(LASTBIT(z0))
ZERO(0') -> c6
ZERO(s(z0)) -> c7
CONV(z0) -> c8(CONVITER(z0, cons(0', nil)))
CONVITER(z0, z1) -> c9(IF(zero(z0), z0, z1), ZERO(z0))
IF(true, z0, z1) -> c10
IF(false, z0, z1) -> c11(CONVITER(half(z0), cons(lastbit(z0), z1)), HALF(z0))
IF(false, z0, z1) -> c12(CONVITER(half(z0), cons(lastbit(z0), z1)), LASTBIT(z0))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(z0))) -> lastbit(z0)
zero(0') -> true
zero(s(z0)) -> false
conv(z0) -> conviter(z0, cons(0', nil))
conviter(z0, z1) -> if(zero(z0), z0, z1)
if(true, z0, z1) -> z1
if(false, z0, z1) -> conviter(half(z0), cons(lastbit(z0), z1))

Types:
HALF :: 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
LASTBIT :: 0':s -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
ZERO :: 0':s -> c6:c7
c6 :: c6:c7
c7 :: c6:c7
CONV :: 0':s -> c8
c8 :: c9 -> c8
CONVITER :: 0':s -> nil:cons -> c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c9 :: c10:c11:c12 -> c6:c7 -> c9
IF :: true:false -> 0':s -> nil:cons -> c10:c11:c12
zero :: 0':s -> true:false
true :: true:false
c10 :: c10:c11:c12
false :: true:false
c11 :: c9 -> c:c1:c2 -> c10:c11:c12
half :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
c12 :: c9 -> c3:c4:c5 -> c10:c11:c12
conv :: 0':s -> nil:cons
conviter :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c4:c53_13 :: c3:c4:c5
hole_c6:c74_13 :: c6:c7
hole_c85_13 :: c8
hole_c96_13 :: c9
hole_nil:cons7_13 :: nil:cons
hole_c10:c11:c128_13 :: c10:c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c4:c512_13 :: Nat -> c3:c4:c5
gen_nil:cons13_13 :: Nat -> nil:cons


Lemmas:
HALF(gen_0':s11_13(*(2, n15_13))) -> gen_c:c1:c210_13(n15_13), rt in Omega(1 + n15_13)
LASTBIT(gen_0':s11_13(*(2, n457_13))) -> gen_c3:c4:c512_13(n457_13), rt in Omega(1 + n457_13)
half(gen_0':s11_13(*(2, n971_13))) -> gen_0':s11_13(n971_13), rt in Omega(0)
lastbit(gen_0':s11_13(*(2, n1431_13))) -> gen_0':s11_13(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c4:c512_13(0) <=> c3
gen_c3:c4:c512_13(+(x, 1)) <=> c5(gen_c3:c4:c512_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(0', gen_nil:cons13_13(x))


The following defined symbols remain to be analysed:
CONVITER, conviter