WORST_CASE(Omega(n^1),O(n^1))
proof of input_IlMNWXNrCt.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 5 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 158 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), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 341 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), 91 ms]
            (28) 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))
   bits(0) -> 0
   bits(s(x)) -> s(bits(half(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]
   bits(0) -> 0 [1]
   bits(s(x)) -> s(bits(half(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]
   bits(0) -> 0 [1]
   bits(s(x)) -> s(bits(half(s(x)))) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
bits :: 0:s -> 0:s

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

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

And the following fresh constants: none

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

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

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

   half(0) -> 0 [1]
   half(s(0)) -> 0 [1]
   half(s(s(x))) -> s(half(x)) [1]
   bits(0) -> 0 [1]
   bits(s(x)) -> s(bits(half(s(x)))) [1]

The TRS has the following type information:
half :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
bits :: 0:s -> 0:s

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

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

   0 => 0

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

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

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

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,[bits(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(bits(V, Out),1,[],[Out = 0,V = 0]).
eq(bits(V, Out),1,[half(1 + V2, Ret10),bits(Ret10, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 1 + V2]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(bits(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [half/2]
1. recursive  : [bits/2]
2. non_recursive  : [start/1]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into half/2
1. SCC is partially evaluated into bits/2
2. SCC is partially evaluated into start/1

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

### Specialization of cost equations half/2 
* CE 5 is refined into CE [8] 
* CE 4 is refined into CE [9] 
* CE 3 is refined into CE [10] 


### Cost equations --> "Loop" of half/2 
* CEs [9] --> Loop 7 
* CEs [10] --> Loop 8 
* CEs [8] --> Loop 9 

### Ranking functions of CR half(V,Out) 
* RF of phase [9]: [V-1]

#### Partial ranking functions of CR half(V,Out) 
* Partial RF of phase [9]:
  - RF of loop [9:1]:
    V-1


### Specialization of cost equations bits/2 
* CE 7 is refined into CE [11,12,13] 
* CE 6 is refined into CE [14] 


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

### Ranking functions of CR bits(V,Out) 
* RF of phase [11,12]: [V-1]

#### Partial ranking functions of CR bits(V,Out) 
* Partial RF of phase [11,12]:
  - RF of loop [11:1]:
    V/2-1
  - RF of loop [12:1]:
    V-1


### Specialization of cost equations start/1 
* CE 1 is refined into CE [15,16,17,18] 
* CE 2 is refined into CE [19,20,21] 


### Cost equations --> "Loop" of start/1 
* CEs [17,18,21] --> Loop 14 
* CEs [16,20] --> Loop 15 
* CEs [15,19] --> Loop 16 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of half(V,Out):
* Chain [[9],8]: 1*it(9)+1
  Such that:it(9) =< 2*Out

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

* Chain [[9],7]: 1*it(9)+1
  Such that:it(9) =< 2*Out

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

* Chain [8]: 1
  with precondition: [V=0,Out=0] 

* Chain [7]: 1
  with precondition: [V=1,Out=0] 


#### Cost of chains of bits(V,Out):
* Chain [[11,12],13,10]: 2*it(11)+2*it(12)+2*s(5)+3
  Such that:it(11) =< V/2
aux(5) =< V
aux(6) =< 2*V
it(11) =< aux(5)
it(12) =< aux(5)
it(12) =< aux(6)
s(5) =< aux(6)

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

* Chain [13,10]: 3
  with precondition: [V=1,Out=1] 

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


#### Cost of chains of start(V):
* Chain [16]: 1
  with precondition: [V=0] 

* Chain [15]: 3
  with precondition: [V=1] 

* Chain [14]: 2*s(7)+2*s(9)+2*s(12)+2*s(13)+3
  Such that:s(11) =< 2*V
s(9) =< V/2
aux(7) =< V
s(7) =< aux(7)
s(9) =< aux(7)
s(12) =< aux(7)
s(12) =< s(11)
s(13) =< s(11)

  with precondition: [V>=2] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [16] with precondition: [V=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [15] with precondition: [V=1] 
    - Upper bound: 3 
    - Complexity: constant 
* Chain [14] with precondition: [V>=2] 
    - Upper bound: 9*V+3 
    - Complexity: n 

### Maximum cost of start(V): 9*V+3 
Asymptotic class: n 
* Total analysis performed in 105 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))
   bits(0) -> 0
   bits(s(z0)) -> s(bits(half(s(z0))))
Tuples:
   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   BITS(0) -> c3
   BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(s(z0)))
S tuples:
   HALF(0) -> c
   HALF(s(0)) -> c1
   HALF(s(s(z0))) -> c2(HALF(z0))
   BITS(0) -> c3
   BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(s(z0)))
K tuples:none
Defined Rule Symbols:   half_1, bits_1

Defined Pair Symbols:   HALF_1, BITS_1

Compound Symbols:   c, c1, c2_1, c3, c4_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))
   BITS(0) -> c3
   BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(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))
   bits(0) -> 0
   bits(s(z0)) -> s(bits(half(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))
   BITS(0') -> c3
   BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(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))
   bits(0') -> 0'
   bits(s(z0)) -> s(bits(half(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))
BITS(0') -> c3
BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(s(z0)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
bits(0') -> 0'
bits(s(z0)) -> s(bits(half(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
BITS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c:c1:c2 -> c3:c4
half :: 0':s -> 0':s
bits :: 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
HALF, BITS, half, bits

They will be analysed ascendingly in the following order:
HALF < BITS
half < BITS
half < bits

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

(20)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
BITS(0') -> c3
BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(s(z0)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
bits(0') -> 0'
bits(s(z0)) -> s(bits(half(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
BITS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c:c1:c2 -> c3:c4
half :: 0':s -> 0':s
bits :: 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c2(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x), c)


The following defined symbols remain to be analysed:
HALF, BITS, half, bits

They will be analysed ascendingly in the following order:
HALF < BITS
half < BITS
half < bits

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
HALF(gen_0':s5_5(*(2, n8_5))) -> gen_c:c1:c24_5(n8_5), rt in Omega(1 + n8_5)

Induction Base:
HALF(gen_0':s5_5(*(2, 0))) ->_R^Omega(1)
c

Induction Step:
HALF(gen_0':s5_5(*(2, +(n8_5, 1)))) ->_R^Omega(1)
c2(HALF(gen_0':s5_5(*(2, n8_5)))) ->_IH
c2(gen_c:c1:c24_5(c9_5))

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))
BITS(0') -> c3
BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(s(z0)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
bits(0') -> 0'
bits(s(z0)) -> s(bits(half(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
BITS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c:c1:c2 -> c3:c4
half :: 0':s -> 0':s
bits :: 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c2(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x), c)


The following defined symbols remain to be analysed:
HALF, BITS, half, bits

They will be analysed ascendingly in the following order:
HALF < BITS
half < BITS
half < bits

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

(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))
BITS(0') -> c3
BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(s(z0)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
bits(0') -> 0'
bits(s(z0)) -> s(bits(half(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
BITS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c:c1:c2 -> c3:c4
half :: 0':s -> 0':s
bits :: 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
HALF(gen_0':s5_5(*(2, n8_5))) -> gen_c:c1:c24_5(n8_5), rt in Omega(1 + n8_5)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c2(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x), c)


The following defined symbols remain to be analysed:
half, BITS, bits

They will be analysed ascendingly in the following order:
half < BITS
half < bits

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s5_5(*(2, n379_5))) -> gen_0':s5_5(n379_5), rt in Omega(0)

Induction Base:
half(gen_0':s5_5(*(2, 0))) ->_R^Omega(0)
0'

Induction Step:
half(gen_0':s5_5(*(2, +(n379_5, 1)))) ->_R^Omega(0)
s(half(gen_0':s5_5(*(2, n379_5)))) ->_IH
s(gen_0':s5_5(c380_5))

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

(28)
Obligation:
Innermost TRS:
Rules:
HALF(0') -> c
HALF(s(0')) -> c1
HALF(s(s(z0))) -> c2(HALF(z0))
BITS(0') -> c3
BITS(s(z0)) -> c4(BITS(half(s(z0))), HALF(s(z0)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(z0))) -> s(half(z0))
bits(0') -> 0'
bits(s(z0)) -> s(bits(half(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
BITS :: 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c:c1:c2 -> c3:c4
half :: 0':s -> 0':s
bits :: 0':s -> 0':s
hole_c:c1:c21_5 :: c:c1:c2
hole_0':s2_5 :: 0':s
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_0':s5_5 :: Nat -> 0':s
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
HALF(gen_0':s5_5(*(2, n8_5))) -> gen_c:c1:c24_5(n8_5), rt in Omega(1 + n8_5)
half(gen_0':s5_5(*(2, n379_5))) -> gen_0':s5_5(n379_5), rt in Omega(0)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c2(gen_c:c1:c24_5(x))
gen_0':s5_5(0) <=> 0'
gen_0':s5_5(+(x, 1)) <=> s(gen_0':s5_5(x))
gen_c3:c46_5(0) <=> c3
gen_c3:c46_5(+(x, 1)) <=> c4(gen_c3:c46_5(x), c)


The following defined symbols remain to be analysed:
BITS, bits