WORST_CASE(Omega(n^1),O(n^1))
proof of input_PXgMnGPuHT.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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, 342 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), 5 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 357 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), 67 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 865 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)
   conv(0) -> cons(nil, 0)
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x)))

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]
   conv(0) -> cons(nil, 0) [1]
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [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]
   conv(0) -> cons(nil, 0) [1]
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [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
conv :: 0:s -> nil:cons
cons :: nil:cons -> 0:s -> nil:cons
nil :: 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]
   conv(0) -> cons(nil, 0) [1]
   conv(s(x)) -> cons(conv(half(s(x))), lastbit(s(x))) [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
conv :: 0:s -> nil:cons
cons :: nil:cons -> 0:s -> nil:cons
nil :: 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
   nil => 0

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

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

   conv(z) -{ 1 }-> 1 + conv(half(1 + x)) + lastbit(1 + x) :|: x >= 0, z = 1 + x
   conv(z) -{ 1 }-> 1 + 0 + 0 :|: z = 0
   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)
   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

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),0,[half(V, Out)],[V >= 0]).
eq(start(V),0,[lastbit(V, Out)],[V >= 0]).
eq(start(V),0,[conv(V, Out)],[V >= 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(conv(V, Out),1,[],[Out = 1,V = 0]).
eq(conv(V, Out),1,[half(1 + V3, Ret010),conv(Ret010, Ret01),lastbit(1 + V3, Ret11)],[Out = 1 + Ret01 + Ret11,V3 >= 0,V = 1 + V3]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(lastbit(V,Out),[V],[Out]).
input_output_vars(conv(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [half/2]
1. recursive  : [lastbit/2]
2. recursive [non_tail] : [conv/2]
3. non_recursive  : [start/1]

#### 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 conv/2
3. SCC is partially evaluated into start/1

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

### Specialization of cost equations half/2 
* CE 6 is refined into CE [12] 
* CE 5 is refined into CE [13] 
* CE 4 is refined into CE [14] 


### Cost equations --> "Loop" of half/2 
* CEs [13] --> Loop 10 
* CEs [14] --> Loop 11 
* CEs [12] --> Loop 12 

### Ranking functions of CR half(V,Out) 
* RF of phase [12]: [V-1]

#### Partial ranking functions of CR half(V,Out) 
* Partial RF of phase [12]:
  - RF of loop [12:1]:
    V-1


### Specialization of cost equations lastbit/2 
* CE 9 is refined into CE [15] 
* CE 8 is refined into CE [16] 
* CE 7 is refined into CE [17] 


### Cost equations --> "Loop" of lastbit/2 
* CEs [16] --> Loop 13 
* CEs [17] --> Loop 14 
* CEs [15] --> Loop 15 

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

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


### Specialization of cost equations conv/2 
* CE 11 is refined into CE [18,19,20,21,22] 
* CE 10 is refined into CE [23] 


### Cost equations --> "Loop" of conv/2 
* CEs [23] --> Loop 16 
* CEs [22] --> Loop 17 
* CEs [21] --> Loop 18 
* CEs [20] --> Loop 19 
* CEs [19] --> Loop 20 
* CEs [18] --> Loop 21 

### Ranking functions of CR conv(V,Out) 
* RF of phase [17,18,19,20]: [V-1]

#### Partial ranking functions of CR conv(V,Out) 
* Partial RF of phase [17,18,19,20]:
  - RF of loop [17:1,18:1]:
    V/2-1
  - RF of loop [19:1]:
    2*V-5
  - RF of loop [20:1]:
    V-1


### Specialization of cost equations start/1 
* CE 1 is refined into CE [24,25,26,27] 
* CE 2 is refined into CE [28,29,30,31] 
* CE 3 is refined into CE [32,33,34] 


### Cost equations --> "Loop" of start/1 
* CEs [26,27,30,31,34] --> Loop 22 
* CEs [25,29,33] --> Loop 23 
* CEs [24,28,32] --> Loop 24 

### Ranking functions of CR start(V) 

#### Partial ranking functions of CR start(V) 


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

#### Cost of chains of half(V,Out):
* Chain [[12],11]: 1*it(12)+1
  Such that:it(12) =< 2*Out

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

* Chain [[12],10]: 1*it(12)+1
  Such that:it(12) =< 2*Out

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

* Chain [11]: 1
  with precondition: [V=0,Out=0] 

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


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

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

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

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

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

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


#### Cost of chains of conv(V,Out):
* Chain [[17,18,19,20],21,16]: 3*it(17)+3*it(18)+3*it(19)+3*it(20)+2*s(17)+2*s(18)+4*s(21)+4
  Such that:aux(5) =< 2*V+4
aux(13) =< V
aux(14) =< 2*V
aux(15) =< 3*V
aux(16) =< 4*V
aux(17) =< V/2
it(19) =< aux(14)
it(17) =< aux(13)
it(18) =< aux(13)
it(19) =< aux(13)
it(20) =< aux(13)
it(20) =< aux(5)
s(22) =< aux(5)
it(20) =< aux(14)
s(22) =< aux(14)
it(18) =< aux(15)
it(19) =< aux(15)
s(18) =< aux(15)
it(20) =< aux(15)
it(18) =< aux(16)
it(19) =< aux(16)
s(17) =< aux(16)
it(20) =< aux(16)
it(17) =< aux(17)
it(18) =< aux(17)
s(21) =< s(22)

  with precondition: [Out>=4,V+8>=2*Out,V+2>=Out] 

* Chain [21,16]: 4
  with precondition: [V=1,Out=3] 

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


#### Cost of chains of start(V):
* Chain [24]: 1
  with precondition: [V=0] 

* Chain [23]: 4
  with precondition: [V=1] 

* Chain [22]: 4*s(25)+3*s(35)+3*s(36)+3*s(37)+3*s(38)+2*s(40)+2*s(41)+4*s(42)+4
  Such that:s(31) =< 2*V
s(29) =< 2*V+4
s(32) =< 3*V
s(33) =< 4*V
s(34) =< V/2
aux(18) =< V
s(25) =< aux(18)
s(35) =< s(31)
s(36) =< aux(18)
s(37) =< aux(18)
s(35) =< aux(18)
s(38) =< aux(18)
s(38) =< s(29)
s(39) =< s(29)
s(38) =< s(31)
s(39) =< s(31)
s(37) =< s(32)
s(35) =< s(32)
s(40) =< s(32)
s(38) =< s(32)
s(37) =< s(33)
s(35) =< s(33)
s(41) =< s(33)
s(38) =< s(33)
s(36) =< s(34)
s(37) =< s(34)
s(42) =< s(39)

  with precondition: [V>=2] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [24] with precondition: [V=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [23] with precondition: [V=1] 
    - Upper bound: 4 
    - Complexity: constant 
* Chain [22] with precondition: [V>=2] 
    - Upper bound: 41*V+20 
    - Complexity: n 

### Maximum cost of start(V): 41*V+20 
Asymptotic class: n 
* Total analysis performed in 233 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)
   conv(0) -> cons(nil, 0)
   conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))
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))
   CONV(0) -> c6
   CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
   CONV(s(z0)) -> c8(LASTBIT(s(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))
   CONV(0) -> c6
   CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
   CONV(s(z0)) -> c8(LASTBIT(s(z0)))
K tuples:none
Defined Rule Symbols:   half_1, lastbit_1, conv_1

Defined Pair Symbols:   HALF_1, LASTBIT_1, CONV_1

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7_2, c8_1


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

(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))
   CONV(0) -> c6
   CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
   CONV(s(z0)) -> c8(LASTBIT(s(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)
   conv(0) -> cons(nil, 0)
   conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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))
   CONV(0') -> c6
   CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
   CONV(s(z0)) -> c8(LASTBIT(s(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)
   conv(0') -> cons(nil, 0')
   conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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))
CONV(0') -> c6
CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
CONV(s(z0)) -> c8(LASTBIT(s(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)
conv(0') -> cons(nil, 0')
conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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
CONV :: 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8 -> c:c1:c2 -> c6:c7:c8
half :: 0':s -> 0':s
c8 :: c3:c4:c5 -> c6:c7:c8
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c4:c53_9 :: c3:c4:c5
hole_c6:c7:c84_9 :: c6:c7:c8
hole_nil:cons5_9 :: nil:cons
gen_c:c1:c26_9 :: Nat -> c:c1:c2
gen_0':s7_9 :: Nat -> 0':s
gen_c3:c4:c58_9 :: Nat -> c3:c4:c5
gen_c6:c7:c89_9 :: Nat -> c6:c7:c8
gen_nil:cons10_9 :: Nat -> nil:cons

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

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

They will be analysed ascendingly in the following order:
HALF < CONV
LASTBIT < CONV
half < CONV
half < conv
lastbit < conv

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

(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))
CONV(0') -> c6
CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
CONV(s(z0)) -> c8(LASTBIT(s(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)
conv(0') -> cons(nil, 0')
conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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
CONV :: 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8 -> c:c1:c2 -> c6:c7:c8
half :: 0':s -> 0':s
c8 :: c3:c4:c5 -> c6:c7:c8
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c4:c53_9 :: c3:c4:c5
hole_c6:c7:c84_9 :: c6:c7:c8
hole_nil:cons5_9 :: nil:cons
gen_c:c1:c26_9 :: Nat -> c:c1:c2
gen_0':s7_9 :: Nat -> 0':s
gen_c3:c4:c58_9 :: Nat -> c3:c4:c5
gen_c6:c7:c89_9 :: Nat -> c6:c7:c8
gen_nil:cons10_9 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c26_9(0) <=> c
gen_c:c1:c26_9(+(x, 1)) <=> c2(gen_c:c1:c26_9(x))
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c3:c4:c58_9(0) <=> c3
gen_c3:c4:c58_9(+(x, 1)) <=> c5(gen_c3:c4:c58_9(x))
gen_c6:c7:c89_9(0) <=> c6
gen_c6:c7:c89_9(+(x, 1)) <=> c7(gen_c6:c7:c89_9(x), c)
gen_nil:cons10_9(0) <=> nil
gen_nil:cons10_9(+(x, 1)) <=> cons(gen_nil:cons10_9(x), 0')


The following defined symbols remain to be analysed:
HALF, LASTBIT, CONV, half, lastbit, conv

They will be analysed ascendingly in the following order:
HALF < CONV
LASTBIT < CONV
half < CONV
half < conv
lastbit < conv

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
HALF(gen_0':s7_9(*(2, n12_9))) -> gen_c:c1:c26_9(n12_9), rt in Omega(1 + n12_9)

Induction Base:
HALF(gen_0':s7_9(*(2, 0))) ->_R^Omega(1)
c

Induction Step:
HALF(gen_0':s7_9(*(2, +(n12_9, 1)))) ->_R^Omega(1)
c2(HALF(gen_0':s7_9(*(2, n12_9)))) ->_IH
c2(gen_c:c1:c26_9(c13_9))

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))
CONV(0') -> c6
CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
CONV(s(z0)) -> c8(LASTBIT(s(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)
conv(0') -> cons(nil, 0')
conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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
CONV :: 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8 -> c:c1:c2 -> c6:c7:c8
half :: 0':s -> 0':s
c8 :: c3:c4:c5 -> c6:c7:c8
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c4:c53_9 :: c3:c4:c5
hole_c6:c7:c84_9 :: c6:c7:c8
hole_nil:cons5_9 :: nil:cons
gen_c:c1:c26_9 :: Nat -> c:c1:c2
gen_0':s7_9 :: Nat -> 0':s
gen_c3:c4:c58_9 :: Nat -> c3:c4:c5
gen_c6:c7:c89_9 :: Nat -> c6:c7:c8
gen_nil:cons10_9 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c26_9(0) <=> c
gen_c:c1:c26_9(+(x, 1)) <=> c2(gen_c:c1:c26_9(x))
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c3:c4:c58_9(0) <=> c3
gen_c3:c4:c58_9(+(x, 1)) <=> c5(gen_c3:c4:c58_9(x))
gen_c6:c7:c89_9(0) <=> c6
gen_c6:c7:c89_9(+(x, 1)) <=> c7(gen_c6:c7:c89_9(x), c)
gen_nil:cons10_9(0) <=> nil
gen_nil:cons10_9(+(x, 1)) <=> cons(gen_nil:cons10_9(x), 0')


The following defined symbols remain to be analysed:
HALF, LASTBIT, CONV, half, lastbit, conv

They will be analysed ascendingly in the following order:
HALF < CONV
LASTBIT < CONV
half < CONV
half < conv
lastbit < conv

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

(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))
CONV(0') -> c6
CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
CONV(s(z0)) -> c8(LASTBIT(s(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)
conv(0') -> cons(nil, 0')
conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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
CONV :: 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8 -> c:c1:c2 -> c6:c7:c8
half :: 0':s -> 0':s
c8 :: c3:c4:c5 -> c6:c7:c8
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c4:c53_9 :: c3:c4:c5
hole_c6:c7:c84_9 :: c6:c7:c8
hole_nil:cons5_9 :: nil:cons
gen_c:c1:c26_9 :: Nat -> c:c1:c2
gen_0':s7_9 :: Nat -> 0':s
gen_c3:c4:c58_9 :: Nat -> c3:c4:c5
gen_c6:c7:c89_9 :: Nat -> c6:c7:c8
gen_nil:cons10_9 :: Nat -> nil:cons


Lemmas:
HALF(gen_0':s7_9(*(2, n12_9))) -> gen_c:c1:c26_9(n12_9), rt in Omega(1 + n12_9)


Generator Equations:
gen_c:c1:c26_9(0) <=> c
gen_c:c1:c26_9(+(x, 1)) <=> c2(gen_c:c1:c26_9(x))
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c3:c4:c58_9(0) <=> c3
gen_c3:c4:c58_9(+(x, 1)) <=> c5(gen_c3:c4:c58_9(x))
gen_c6:c7:c89_9(0) <=> c6
gen_c6:c7:c89_9(+(x, 1)) <=> c7(gen_c6:c7:c89_9(x), c)
gen_nil:cons10_9(0) <=> nil
gen_nil:cons10_9(+(x, 1)) <=> cons(gen_nil:cons10_9(x), 0')


The following defined symbols remain to be analysed:
LASTBIT, CONV, half, lastbit, conv

They will be analysed ascendingly in the following order:
LASTBIT < CONV
half < CONV
half < conv
lastbit < conv

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LASTBIT(gen_0':s7_9(*(2, n430_9))) -> gen_c3:c4:c58_9(n430_9), rt in Omega(1 + n430_9)

Induction Base:
LASTBIT(gen_0':s7_9(*(2, 0))) ->_R^Omega(1)
c3

Induction Step:
LASTBIT(gen_0':s7_9(*(2, +(n430_9, 1)))) ->_R^Omega(1)
c5(LASTBIT(gen_0':s7_9(*(2, n430_9)))) ->_IH
c5(gen_c3:c4:c58_9(c431_9))

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))
CONV(0') -> c6
CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
CONV(s(z0)) -> c8(LASTBIT(s(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)
conv(0') -> cons(nil, 0')
conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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
CONV :: 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8 -> c:c1:c2 -> c6:c7:c8
half :: 0':s -> 0':s
c8 :: c3:c4:c5 -> c6:c7:c8
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c4:c53_9 :: c3:c4:c5
hole_c6:c7:c84_9 :: c6:c7:c8
hole_nil:cons5_9 :: nil:cons
gen_c:c1:c26_9 :: Nat -> c:c1:c2
gen_0':s7_9 :: Nat -> 0':s
gen_c3:c4:c58_9 :: Nat -> c3:c4:c5
gen_c6:c7:c89_9 :: Nat -> c6:c7:c8
gen_nil:cons10_9 :: Nat -> nil:cons


Lemmas:
HALF(gen_0':s7_9(*(2, n12_9))) -> gen_c:c1:c26_9(n12_9), rt in Omega(1 + n12_9)
LASTBIT(gen_0':s7_9(*(2, n430_9))) -> gen_c3:c4:c58_9(n430_9), rt in Omega(1 + n430_9)


Generator Equations:
gen_c:c1:c26_9(0) <=> c
gen_c:c1:c26_9(+(x, 1)) <=> c2(gen_c:c1:c26_9(x))
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c3:c4:c58_9(0) <=> c3
gen_c3:c4:c58_9(+(x, 1)) <=> c5(gen_c3:c4:c58_9(x))
gen_c6:c7:c89_9(0) <=> c6
gen_c6:c7:c89_9(+(x, 1)) <=> c7(gen_c6:c7:c89_9(x), c)
gen_nil:cons10_9(0) <=> nil
gen_nil:cons10_9(+(x, 1)) <=> cons(gen_nil:cons10_9(x), 0')


The following defined symbols remain to be analysed:
half, CONV, lastbit, conv

They will be analysed ascendingly in the following order:
half < CONV
half < conv
lastbit < conv

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s7_9(*(2, n920_9))) -> gen_0':s7_9(n920_9), rt in Omega(0)

Induction Base:
half(gen_0':s7_9(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
half(gen_0':s7_9(*(2, +(n920_9, 1)))) ->_R^Omega(0)
s(half(gen_0':s7_9(*(2, n920_9)))) ->_IH
s(gen_0':s7_9(c921_9))

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))
CONV(0') -> c6
CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
CONV(s(z0)) -> c8(LASTBIT(s(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)
conv(0') -> cons(nil, 0')
conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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
CONV :: 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8 -> c:c1:c2 -> c6:c7:c8
half :: 0':s -> 0':s
c8 :: c3:c4:c5 -> c6:c7:c8
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c4:c53_9 :: c3:c4:c5
hole_c6:c7:c84_9 :: c6:c7:c8
hole_nil:cons5_9 :: nil:cons
gen_c:c1:c26_9 :: Nat -> c:c1:c2
gen_0':s7_9 :: Nat -> 0':s
gen_c3:c4:c58_9 :: Nat -> c3:c4:c5
gen_c6:c7:c89_9 :: Nat -> c6:c7:c8
gen_nil:cons10_9 :: Nat -> nil:cons


Lemmas:
HALF(gen_0':s7_9(*(2, n12_9))) -> gen_c:c1:c26_9(n12_9), rt in Omega(1 + n12_9)
LASTBIT(gen_0':s7_9(*(2, n430_9))) -> gen_c3:c4:c58_9(n430_9), rt in Omega(1 + n430_9)
half(gen_0':s7_9(*(2, n920_9))) -> gen_0':s7_9(n920_9), rt in Omega(0)


Generator Equations:
gen_c:c1:c26_9(0) <=> c
gen_c:c1:c26_9(+(x, 1)) <=> c2(gen_c:c1:c26_9(x))
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c3:c4:c58_9(0) <=> c3
gen_c3:c4:c58_9(+(x, 1)) <=> c5(gen_c3:c4:c58_9(x))
gen_c6:c7:c89_9(0) <=> c6
gen_c6:c7:c89_9(+(x, 1)) <=> c7(gen_c6:c7:c89_9(x), c)
gen_nil:cons10_9(0) <=> nil
gen_nil:cons10_9(+(x, 1)) <=> cons(gen_nil:cons10_9(x), 0')


The following defined symbols remain to be analysed:
CONV, lastbit, conv

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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lastbit(gen_0':s7_9(*(2, n4811_9))) -> gen_0':s7_9(0), rt in Omega(0)

Induction Base:
lastbit(gen_0':s7_9(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
lastbit(gen_0':s7_9(*(2, +(n4811_9, 1)))) ->_R^Omega(0)
lastbit(gen_0':s7_9(*(2, n4811_9))) ->_IH
gen_0':s7_9(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))
CONV(0') -> c6
CONV(s(z0)) -> c7(CONV(half(s(z0))), HALF(s(z0)))
CONV(s(z0)) -> c8(LASTBIT(s(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)
conv(0') -> cons(nil, 0')
conv(s(z0)) -> cons(conv(half(s(z0))), lastbit(s(z0)))

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
CONV :: 0':s -> c6:c7:c8
c6 :: c6:c7:c8
c7 :: c6:c7:c8 -> c:c1:c2 -> c6:c7:c8
half :: 0':s -> 0':s
c8 :: c3:c4:c5 -> c6:c7:c8
lastbit :: 0':s -> 0':s
conv :: 0':s -> nil:cons
cons :: nil:cons -> 0':s -> nil:cons
nil :: nil:cons
hole_c:c1:c21_9 :: c:c1:c2
hole_0':s2_9 :: 0':s
hole_c3:c4:c53_9 :: c3:c4:c5
hole_c6:c7:c84_9 :: c6:c7:c8
hole_nil:cons5_9 :: nil:cons
gen_c:c1:c26_9 :: Nat -> c:c1:c2
gen_0':s7_9 :: Nat -> 0':s
gen_c3:c4:c58_9 :: Nat -> c3:c4:c5
gen_c6:c7:c89_9 :: Nat -> c6:c7:c8
gen_nil:cons10_9 :: Nat -> nil:cons


Lemmas:
HALF(gen_0':s7_9(*(2, n12_9))) -> gen_c:c1:c26_9(n12_9), rt in Omega(1 + n12_9)
LASTBIT(gen_0':s7_9(*(2, n430_9))) -> gen_c3:c4:c58_9(n430_9), rt in Omega(1 + n430_9)
half(gen_0':s7_9(*(2, n920_9))) -> gen_0':s7_9(n920_9), rt in Omega(0)
lastbit(gen_0':s7_9(*(2, n4811_9))) -> gen_0':s7_9(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c26_9(0) <=> c
gen_c:c1:c26_9(+(x, 1)) <=> c2(gen_c:c1:c26_9(x))
gen_0':s7_9(0) <=> 0'
gen_0':s7_9(+(x, 1)) <=> s(gen_0':s7_9(x))
gen_c3:c4:c58_9(0) <=> c3
gen_c3:c4:c58_9(+(x, 1)) <=> c5(gen_c3:c4:c58_9(x))
gen_c6:c7:c89_9(0) <=> c6
gen_c6:c7:c89_9(+(x, 1)) <=> c7(gen_c6:c7:c89_9(x), c)
gen_nil:cons10_9(0) <=> nil
gen_nil:cons10_9(+(x, 1)) <=> cons(gen_nil:cons10_9(x), 0')


The following defined symbols remain to be analysed:
conv