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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(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), 6 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 646 ms]
    (10) BOUNDS(1, n^1)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (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: 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^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 522 ms.


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(10)
BOUNDS(1, n^1)

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(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (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: INNERMOST
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(13) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

lastbit(s(s(x))) ->^+ lastbit(x)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [x / s(s(x))].

The result substitution is [ ].




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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (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: INNERMOST
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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (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: INNERMOST