WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 102 ms]
    (10) BOUNDS(1, n^1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) SlicingProof [LOWER BOUND(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), 198 ms]
    (20) proven lower bound
    (21) LowerBoundPropagationProof [FINISHED, 0 ms]
    (22) BOUNDS(n^1, INF)


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

(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:

   from(X) -> cons(X, n__from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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:

   from(X) -> cons(X, n__from(s(X))) [1]
   after(0, XS) -> XS [1]
   after(s(N), cons(X, XS)) -> after(N, activate(XS)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> 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:

   from(X) -> cons(X, n__from(s(X))) [1]
   after(0, XS) -> XS [1]
   after(s(N), cons(X, XS)) -> after(N, activate(XS)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
from :: s:0 -> n__from:cons
cons :: s:0 -> n__from:cons -> n__from:cons
n__from :: s:0 -> n__from:cons
s :: s:0 -> s:0
after :: s:0 -> n__from:cons -> n__from:cons
0 :: s:0
activate :: n__from:cons -> n__from: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:

   after(v0, v1) -> null_after [0]

And the following fresh constants:    null_after

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

(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:

   from(X) -> cons(X, n__from(s(X))) [1]
   after(0, XS) -> XS [1]
   after(s(N), cons(X, XS)) -> after(N, activate(XS)) [1]
   from(X) -> n__from(X) [1]
   activate(n__from(X)) -> from(X) [1]
   activate(X) -> X [1]
   after(v0, v1) -> null_after [0]

The TRS has the following type information:
from :: s:0 -> n__from:cons:null_after
cons :: s:0 -> n__from:cons:null_after -> n__from:cons:null_after
n__from :: s:0 -> n__from:cons:null_after
s :: s:0 -> s:0
after :: s:0 -> n__from:cons:null_after -> n__from:cons:null_after
0 :: s:0
activate :: n__from:cons:null_after -> n__from:cons:null_after
null_after :: n__from:cons:null_after

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
   null_after => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> from(X) :|: z = 1 + X, X >= 0
   after(z, z') -{ 1 }-> XS :|: z' = XS, z = 0, XS >= 0
   after(z, z') -{ 1 }-> after(N, activate(XS)) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
   after(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   from(z) -{ 1 }-> 1 + X + (1 + (1 + X)) :|: X >= 0, z = 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, V1),0,[from(V, Out)],[V >= 0]).
eq(start(V, V1),0,[after(V, V1, Out)],[V >= 0,V1 >= 0]).
eq(start(V, V1),0,[activate(V, Out)],[V >= 0]).
eq(from(V, Out),1,[],[Out = 3 + 2*X1,X1 >= 0,V = X1]).
eq(after(V, V1, Out),1,[],[Out = XS1,V1 = XS1,V = 0,XS1 >= 0]).
eq(after(V, V1, Out),1,[activate(XS2, Ret1),after(N1, Ret1, Ret)],[Out = Ret,V = 1 + N1,V1 = 1 + X2 + XS2,X2 >= 0,XS2 >= 0,N1 >= 0]).
eq(from(V, Out),1,[],[Out = 1 + X3,X3 >= 0,V = X3]).
eq(activate(V, Out),1,[from(X4, Ret2)],[Out = Ret2,V = 1 + X4,X4 >= 0]).
eq(activate(V, Out),1,[],[Out = X5,X5 >= 0,V = X5]).
eq(after(V, V1, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V = V3,V1 = V2]).
input_output_vars(from(V,Out),[V],[Out]).
input_output_vars(after(V,V1,Out),[V,V1],[Out]).
input_output_vars(activate(V,Out),[V],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [from/2]
1. non_recursive  : [activate/2]
2. recursive  : [after/3]
3. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into from/2
1. SCC is partially evaluated into activate/2
2. SCC is partially evaluated into after/3
3. SCC is partially evaluated into start/2

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

### Specialization of cost equations from/2 
* CE 4 is refined into CE [11] 
* CE 5 is refined into CE [12] 


### Cost equations --> "Loop" of from/2 
* CEs [11] --> Loop 9 
* CEs [12] --> Loop 10 

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

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


### Specialization of cost equations activate/2 
* CE 9 is refined into CE [13,14] 
* CE 10 is refined into CE [15] 


### Cost equations --> "Loop" of activate/2 
* CEs [14] --> Loop 11 
* CEs [13,15] --> Loop 12 

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

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


### Specialization of cost equations after/3 
* CE 8 is refined into CE [16] 
* CE 6 is refined into CE [17] 
* CE 7 is refined into CE [18,19] 


### Cost equations --> "Loop" of after/3 
* CEs [19] --> Loop 13 
* CEs [18] --> Loop 14 
* CEs [16] --> Loop 15 
* CEs [17] --> Loop 16 

### Ranking functions of CR after(V,V1,Out) 
* RF of phase [13,14]: [V]

#### Partial ranking functions of CR after(V,V1,Out) 
* Partial RF of phase [13,14]:
  - RF of loop [13:1,14:1]:
    V
  - RF of loop [14:1]:
    V1 depends on loops [13:1] 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [20,21] 
* CE 2 is refined into CE [22,23,24] 
* CE 3 is refined into CE [25,26] 


### Cost equations --> "Loop" of start/2 
* CEs [20,21,22,23,24,25,26] --> Loop 17 

### Ranking functions of CR start(V,V1) 

#### Partial ranking functions of CR start(V,V1) 


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

#### Cost of chains of from(V,Out):
* Chain [10]: 1
  with precondition: [V+1=Out,V>=0] 

* Chain [9]: 1
  with precondition: [2*V+3=Out,V>=0] 


#### Cost of chains of activate(V,Out):
* Chain [12]: 2
  with precondition: [V=Out,V>=0] 

* Chain [11]: 2
  with precondition: [2*V+1=Out,V>=1] 


#### Cost of chains of after(V,V1,Out):
* Chain [[13,14],16]: 6*it(13)+1
  Such that:aux(9) =< V
it(13) =< aux(9)

  with precondition: [V>=1,V1>=1,Out>=0] 

* Chain [[13,14],15]: 6*it(13)+0
  Such that:aux(10) =< V
it(13) =< aux(10)

  with precondition: [Out=0,V>=1,V1>=1] 

* Chain [16]: 1
  with precondition: [V=0,V1=Out,V1>=0] 

* Chain [15]: 0
  with precondition: [Out=0,V>=0,V1>=0] 


#### Cost of chains of start(V,V1):
* Chain [17]: 12*s(4)+2
  Such that:aux(11) =< V
s(4) =< aux(11)

  with precondition: [V>=0] 


Closed-form bounds of start(V,V1): 
-------------------------------------
* Chain [17] with precondition: [V>=0] 
    - Upper bound: 12*V+2 
    - Complexity: n 

### Maximum cost of start(V,V1): 12*V+2 
Asymptotic class: n 
* Total analysis performed in 106 ms.


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

(10)
BOUNDS(1, n^1)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   after(0', XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(13) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
from/0
cons/0
n__from/0

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

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


The TRS R consists of the following rules:

   from -> cons(n__from)
   after(0', XS) -> XS
   after(s(N), cons(XS)) -> after(N, activate(XS))
   from -> n__from
   activate(n__from) -> from
   activate(X) -> X

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

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

(16)
Obligation:
Innermost TRS:
Rules:
from -> cons(n__from)
after(0', XS) -> XS
after(s(N), cons(XS)) -> after(N, activate(XS))
from -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons
cons :: n__from:cons -> n__from:cons
n__from :: n__from:cons
after :: 0':s -> n__from:cons -> n__from:cons
0' :: 0':s
s :: 0':s -> 0':s
activate :: n__from:cons -> n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat -> n__from:cons
gen_0':s4_0 :: Nat -> 0':s

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
after
----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
from -> cons(n__from)
after(0', XS) -> XS
after(s(N), cons(XS)) -> after(N, activate(XS))
from -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons
cons :: n__from:cons -> n__from:cons
n__from :: n__from:cons
after :: 0':s -> n__from:cons -> n__from:cons
0' :: 0':s
s :: 0':s -> 0':s
activate :: n__from:cons -> n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat -> n__from:cons
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_n__from:cons3_0(0) <=> n__from
gen_n__from:cons3_0(+(x, 1)) <=> cons(gen_n__from:cons3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
after
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
after(gen_0':s4_0(n6_0), gen_n__from:cons3_0(1)) -> gen_n__from:cons3_0(1), rt in Omega(1 + n6_0)

Induction Base:
after(gen_0':s4_0(0), gen_n__from:cons3_0(1)) ->_R^Omega(1)
gen_n__from:cons3_0(1)

Induction Step:
after(gen_0':s4_0(+(n6_0, 1)), gen_n__from:cons3_0(1)) ->_R^Omega(1)
after(gen_0':s4_0(n6_0), activate(gen_n__from:cons3_0(0))) ->_R^Omega(1)
after(gen_0':s4_0(n6_0), from) ->_R^Omega(1)
after(gen_0':s4_0(n6_0), cons(n__from)) ->_IH
gen_n__from:cons3_0(1)

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

(20)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
from -> cons(n__from)
after(0', XS) -> XS
after(s(N), cons(XS)) -> after(N, activate(XS))
from -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons
cons :: n__from:cons -> n__from:cons
n__from :: n__from:cons
after :: 0':s -> n__from:cons -> n__from:cons
0' :: 0':s
s :: 0':s -> 0':s
activate :: n__from:cons -> n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat -> n__from:cons
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_n__from:cons3_0(0) <=> n__from
gen_n__from:cons3_0(+(x, 1)) <=> cons(gen_n__from:cons3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
after
----------------------------------------

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

(22)
BOUNDS(n^1, INF)