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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 149 ms]
    (12) BOUNDS(1, n^1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 257 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) 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(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

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

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(4)
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(s(x))) -> s(half(x)) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(half(x)))) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

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

(7) 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:

   half(v0) -> null_half [0]
   log(v0) -> null_log [0]

And the following fresh constants:    null_half, null_log

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

(8)
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(s(x))) -> s(half(x)) [1]
   log(s(0)) -> 0 [1]
   log(s(s(x))) -> s(log(s(half(x)))) [1]
   half(v0) -> null_half [0]
   log(v0) -> null_log [0]

The TRS has the following type information:
half :: 0:s:null_half:null_log -> 0:s:null_half:null_log
0 :: 0:s:null_half:null_log
s :: 0:s:null_half:null_log -> 0:s:null_half:null_log
log :: 0:s:null_half:null_log -> 0:s:null_half:null_log
null_half :: 0:s:null_half:null_log
null_log :: 0:s:null_half:null_log

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

(9) 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
   null_half => 0
   null_log => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   half(z) -{ 1 }-> 0 :|: z = 0
   half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
   log(z) -{ 1 }-> 0 :|: z = 1 + 0
   log(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   log(z) -{ 1 }-> 1 + log(1 + half(x)) :|: x >= 0, z = 1 + (1 + x)

Only complete derivations are relevant for the runtime complexity.

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

(11) 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,[log(V, Out)],[V >= 0]).
eq(half(V, Out),1,[],[Out = 0,V = 0]).
eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]).
eq(log(V, Out),1,[],[Out = 0,V = 1]).
eq(log(V, Out),1,[half(V2, Ret101),log(1 + Ret101, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 2 + V2]).
eq(half(V, Out),0,[],[Out = 0,V3 >= 0,V = V3]).
eq(log(V, Out),0,[],[Out = 0,V4 >= 0,V = V4]).
input_output_vars(half(V,Out),[V],[Out]).
input_output_vars(log(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [half/2]
1. recursive  : [log/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 log/2
2. SCC is partially evaluated into start/1

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

### Specialization of cost equations half/2 
* CE 3 is refined into CE [9] 
* CE 5 is refined into CE [10] 
* CE 4 is refined into CE [11] 


### Cost equations --> "Loop" of half/2 
* CEs [11] --> Loop 6 
* CEs [9,10] --> Loop 7 

### Ranking functions of CR half(V,Out) 
* RF of phase [6]: [V-1]

#### Partial ranking functions of CR half(V,Out) 
* Partial RF of phase [6]:
  - RF of loop [6:1]:
    V-1


### Specialization of cost equations log/2 
* CE 6 is refined into CE [12] 
* CE 8 is refined into CE [13] 
* CE 7 is refined into CE [14,15] 


### Cost equations --> "Loop" of log/2 
* CEs [15] --> Loop 8 
* CEs [14] --> Loop 9 
* CEs [12,13] --> Loop 10 

### Ranking functions of CR log(V,Out) 
* RF of phase [8]: [V-3]

#### Partial ranking functions of CR log(V,Out) 
* Partial RF of phase [8]:
  - RF of loop [8:1]:
    V-3


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


### Cost equations --> "Loop" of start/1 
* CEs [16,17,18,19,20,21] --> Loop 11 

### Ranking functions of CR start(V) 

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


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

#### Cost of chains of half(V,Out):
* Chain [[6],7]: 1*it(6)+1
  Such that:it(6) =< 2*Out

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

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


#### Cost of chains of log(V,Out):
* Chain [[8],10]: 2*it(8)+1*s(3)+1
  Such that:it(8) =< V
s(3) =< 2*V

  with precondition: [Out>=1,V>=4*Out] 

* Chain [[8],9,10]: 2*it(8)+1*s(3)+3
  Such that:it(8) =< V
s(3) =< 2*V

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

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

* Chain [9,10]: 3
  with precondition: [Out=1,V>=2] 


#### Cost of chains of start(V):
* Chain [11]: 5*s(4)+2*s(6)+3
  Such that:aux(1) =< V
aux(2) =< 2*V
s(4) =< aux(1)
s(6) =< aux(2)

  with precondition: [V>=0] 


Closed-form bounds of start(V): 
-------------------------------------
* Chain [11] with precondition: [V>=0] 
    - Upper bound: 9*V+3 
    - Complexity: n 

### Maximum cost of start(V): 9*V+3 
Asymptotic class: n 
* Total analysis performed in 82 ms.


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

(12)
BOUNDS(1, n^1)

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

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

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


The TRS R consists of the following rules:

   half(0') -> 0'
   half(s(s(x))) -> s(half(x))
   log(s(0')) -> 0'
   log(s(s(x))) -> s(log(s(half(x))))

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

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

(16)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(s(x))) -> s(half(x))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(half(x))))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
log :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
half, log

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

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

(18)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(s(x))) -> s(half(x))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(half(x))))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
log :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
half, log

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s2_0(*(2, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)

Induction Base:
half(gen_0':s2_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s2_0(*(2, +(n4_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s2_0(*(2, n4_0)))) ->_IH
s(gen_0':s2_0(c5_0))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
half(0') -> 0'
half(s(s(x))) -> s(half(x))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(half(x))))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
log :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
half, log

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

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(s(x))) -> s(half(x))
log(s(0')) -> 0'
log(s(s(x))) -> s(log(s(half(x))))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
log :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s2_0(*(2, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
log