WORST_CASE(?,O(n^3))
* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
,recip,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
sqr#(0()) -> c_7()
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
sqr#(0()) -> c_7()
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,5,6,7}
by application of
Pre({1,3,5,6,7}) = {2,4,8,9}.
Here rules are labelled as follows:
1: add#(0(),X) -> c_1()
2: add#(s(X),Y) -> c_2(add#(X,Y))
3: dbl#(0()) -> c_3()
4: dbl#(s(X)) -> c_4(dbl#(X))
5: first#(0(),X) -> c_5()
6: first#(s(X),cons(Y)) -> c_6()
7: sqr#(0()) -> c_7()
8: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
9: terms#(N) -> c_9(sqr#(N))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
- Weak DPs:
add#(0(),X) -> c_1()
dbl#(0()) -> c_3()
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
sqr#(0()) -> c_7()
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(0(),X) -> c_1():5
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(0()) -> c_3():6
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(0()) -> c_7():9
-->_3 dbl#(0()) -> c_3():6
-->_1 add#(0(),X) -> c_1():5
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
4:S:terms#(N) -> c_9(sqr#(N))
-->_1 sqr#(0()) -> c_7():9
-->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
5:W:add#(0(),X) -> c_1()
6:W:dbl#(0()) -> c_3()
7:W:first#(0(),X) -> c_5()
8:W:first#(s(X),cons(Y)) -> c_6()
9:W:sqr#(0()) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: first#(s(X),cons(Y)) -> c_6()
7: first#(0(),X) -> c_5()
9: sqr#(0()) -> c_7()
6: dbl#(0()) -> c_3()
5: add#(0(),X) -> c_1()
* Step 4: RemoveHeads WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
4:S:terms#(N) -> c_9(sqr#(N))
-->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(4,terms#(N) -> c_9(sqr#(N)))]
* Step 5: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
* Step 6: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
and a lower component
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
Further, following extension rules are added to the lower component.
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> dbl#(X)
sqr#(s(X)) -> sqr#(X)
** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_8(sqr#(X))
** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(sqr#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
sqr#(s(X)) -> c_8(sqr#(X))
** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(sqr#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_8) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [2]
p(add) = [0]
p(cons) = [1] x1 + [0]
p(dbl) = [0]
p(first) = [0]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [1] x1 + [3]
p(sqr) = [0]
p(terms) = [0]
p(add#) = [0]
p(dbl#) = [0]
p(first#) = [0]
p(sqr#) = [9] x1 + [0]
p(terms#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [0]
Following rules are strictly oriented:
sqr#(s(X)) = [9] X + [27]
> [9] X + [0]
= c_8(sqr#(X))
Following rules are (at-least) weakly oriented:
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 6.a:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> c_8(sqr#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 6.b:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> dbl#(X)
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_4) = {1}
Following symbols are considered usable:
{add#,dbl#,first#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [2]
p(add) = [2] x2 + [6]
p(cons) = [1] x1 + [0]
p(dbl) = [4] x1 + [4]
p(first) = [1] x1 + [1] x2 + [0]
p(nil) = [1]
p(recip) = [0]
p(s) = [1] x1 + [2]
p(sqr) = [2] x1 + [0]
p(terms) = [8] x1 + [1]
p(add#) = [0]
p(dbl#) = [1] x1 + [14]
p(first#) = [1] x2 + [1]
p(sqr#) = [4] x1 + [6]
p(terms#) = [1] x1 + [1]
p(c_1) = [1]
p(c_2) = [1] x1 + [0]
p(c_3) = [1]
p(c_4) = [1] x1 + [0]
p(c_5) = [8]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [4] x1 + [4] x3 + [2]
p(c_9) = [1] x1 + [0]
Following rules are strictly oriented:
dbl#(s(X)) = [1] X + [16]
> [1] X + [14]
= c_4(dbl#(X))
Following rules are (at-least) weakly oriented:
add#(s(X),Y) = [0]
>= [0]
= c_2(add#(X,Y))
sqr#(s(X)) = [4] X + [14]
>= [0]
= add#(sqr(X),dbl(X))
sqr#(s(X)) = [4] X + [14]
>= [1] X + [14]
= dbl#(X)
sqr#(s(X)) = [4] X + [14]
>= [4] X + [6]
= sqr#(X)
** Step 6.b:2: Ara WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> dbl#(X)
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
+ Details:
Signatures used:
----------------
0 :: [] -(0)-> "A"(2, 0)
0 :: [] -(0)-> "A"(7, 0)
0 :: [] -(0)-> "A"(0, 15)
0 :: [] -(0)-> "A"(5, 7)
0 :: [] -(0)-> "A"(4, 7)
add :: ["A"(2, 0) x "A"(2, 0)] -(0)-> "A"(2, 0)
dbl :: ["A"(7, 0)] -(9)-> "A"(2, 0)
s :: ["A"(2, 0)] -(2)-> "A"(2, 0)
s :: ["A"(7, 0)] -(7)-> "A"(7, 0)
s :: ["A"(15, 15)] -(15)-> "A"(0, 15)
s :: ["A"(9, 0)] -(9)-> "A"(9, 0)
sqr :: ["A"(0, 15)] -(0)-> "A"(2, 0)
add# :: ["A"(2, 0) x "A"(0, 0)] -(7)-> "A"(0, 1)
dbl# :: ["A"(9, 0)] -(8)-> "A"(0, 8)
sqr# :: ["A"(0, 15)] -(12)-> "A"(0, 1)
c_2 :: ["A"(0, 0)] -(0)-> "A"(4, 4)
c_4 :: ["A"(0, 8)] -(0)-> "A"(2, 8)
Cost-free Signatures used:
--------------------------
Base Constructor Signatures used:
---------------------------------
"0_A" :: [] -(0)-> "A"(1, 0)
"0_A" :: [] -(0)-> "A"(0, 1)
"c_2_A" :: ["A"(0)] -(0)-> "A"(1, 0)
"c_2_A" :: ["A"(0)] -(0)-> "A"(0, 1)
"c_4_A" :: ["A"(0)] -(0)-> "A"(1, 0)
"c_4_A" :: ["A"(0)] -(0)-> "A"(0, 1)
"s_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0)
"s_A" :: ["A"(1, 1)] -(1)-> "A"(0, 1)
Following Still Strict Rules were Typed as:
-------------------------------------------
1. Strict:
add#(s(X),Y) -> c_2(add#(X,Y))
2. Weak:
WORST_CASE(?,O(n^3))