* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
domatch(patcs,Cons(x,xs),n) -> domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) -> Nil()
domatch(Nil(),Nil(),n) -> Cons(n,Nil())
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
prefix(Cons(x,xs),Nil()) -> False()
prefix(Cons(x',xs'),Cons(x,xs)) -> and(!EQ(x',x),prefix(xs',xs))
prefix(Nil(),cs) -> True()
strmatch(patstr,str) -> domatch(patstr,str,Nil())
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
domatch[Ite](False(),patcs,Cons(x,xs),n) -> domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) -> Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList[Ite](False(),y,ys,x,xs) -> False()
eqNatList[Ite](True(),y,ys,x,xs) -> eqNatList(xs,ys)
- Signature:
{!EQ/2,and/2,domatch/3,domatch[Ite]/4,eqNatList/2,eqNatList[Ite]/5,notEmpty/1,prefix/2,strmatch/2} / {0/0
,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite]
,notEmpty,prefix,strmatch} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
domatch(patcs,Cons(x,xs),n) -> domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) -> Nil()
domatch(Nil(),Nil(),n) -> Cons(n,Nil())
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
prefix(Cons(x,xs),Nil()) -> False()
prefix(Cons(x',xs'),Cons(x,xs)) -> and(!EQ(x',x),prefix(xs',xs))
prefix(Nil(),cs) -> True()
strmatch(patstr,str) -> domatch(patstr,str,Nil())
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
domatch[Ite](False(),patcs,Cons(x,xs),n) -> domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) -> Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList[Ite](False(),y,ys,x,xs) -> False()
eqNatList[Ite](True(),y,ys,x,xs) -> eqNatList(xs,ys)
- Signature:
{!EQ/2,and/2,domatch/3,domatch[Ite]/4,eqNatList/2,eqNatList[Ite]/5,notEmpty/1,prefix/2,strmatch/2} / {0/0
,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite]
,notEmpty,prefix,strmatch} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
prefix(y,u){y -> Cons(x,y),u -> Cons(z,u)} =
prefix(Cons(x,y),Cons(z,u)) ->^+ and(!EQ(x,z),prefix(y,u))
= C[prefix(y,u) = prefix(y,u){}]
** Step 1.b:1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
domatch(patcs,Cons(x,xs),n) -> domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) -> Nil()
domatch(Nil(),Nil(),n) -> Cons(n,Nil())
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
prefix(Cons(x,xs),Nil()) -> False()
prefix(Cons(x',xs'),Cons(x,xs)) -> and(!EQ(x',x),prefix(xs',xs))
prefix(Nil(),cs) -> True()
strmatch(patstr,str) -> domatch(patstr,str,Nil())
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
domatch[Ite](False(),patcs,Cons(x,xs),n) -> domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) -> Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList[Ite](False(),y,ys,x,xs) -> False()
eqNatList[Ite](True(),y,ys,x,xs) -> eqNatList(xs,ys)
- Signature:
{!EQ/2,and/2,domatch/3,domatch[Ite]/4,eqNatList/2,eqNatList[Ite]/5,notEmpty/1,prefix/2,strmatch/2} / {0/0
,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite]
,notEmpty,prefix,strmatch} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(and) = {1,2},
uargs(domatch[Ite]) = {1},
uargs(eqNatList[Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [4]
p(0) = [4]
p(Cons) = [1] x2 + [1]
p(False) = [0]
p(Nil) = [2]
p(S) = [0]
p(True) = [4]
p(and) = [1] x1 + [1] x2 + [1]
p(domatch) = [1] x2 + [2] x3 + [0]
p(domatch[Ite]) = [1] x1 + [1] x3 + [7]
p(eqNatList) = [4] x2 + [0]
p(eqNatList[Ite]) = [1] x1 + [4] x3 + [1]
p(notEmpty) = [1] x1 + [0]
p(prefix) = [4]
p(strmatch) = [2] x1 + [2] x2 + [1]
Following rules are strictly oriented:
eqNatList(Cons(x,xs),Nil()) = [8]
> [0]
= False()
eqNatList(Nil(),Cons(y,ys)) = [4] ys + [4]
> [0]
= False()
eqNatList(Nil(),Nil()) = [8]
> [4]
= True()
notEmpty(Nil()) = [2]
> [0]
= False()
prefix(Cons(x,xs),Nil()) = [4]
> [0]
= False()
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [4]
>= [4]
= True()
!EQ(0(),S(y)) = [4]
>= [0]
= False()
!EQ(S(x),0()) = [4]
>= [0]
= False()
!EQ(S(x),S(y)) = [4]
>= [4]
= !EQ(x,y)
and(False(),False()) = [1]
>= [0]
= False()
and(False(),True()) = [5]
>= [0]
= False()
and(True(),False()) = [5]
>= [0]
= False()
and(True(),True()) = [9]
>= [4]
= True()
domatch(patcs,Cons(x,xs),n) = [2] n + [1] xs + [1]
>= [1] xs + [12]
= domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) = [2] n + [2]
>= [2]
= Nil()
domatch(Nil(),Nil(),n) = [2] n + [2]
>= [3]
= Cons(n,Nil())
domatch[Ite](False(),patcs,Cons(x,xs),n) = [1] xs + [8]
>= [1] xs + [8]
= domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) = [1] xs + [12]
>= [1] xs + [9]
= Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Cons(y,ys)) = [4] ys + [4]
>= [4] ys + [5]
= eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList[Ite](False(),y,ys,x,xs) = [4] ys + [1]
>= [0]
= False()
eqNatList[Ite](True(),y,ys,x,xs) = [4] ys + [5]
>= [4] ys + [0]
= eqNatList(xs,ys)
notEmpty(Cons(x,xs)) = [1] xs + [1]
>= [4]
= True()
prefix(Cons(x',xs'),Cons(x,xs)) = [4]
>= [9]
= and(!EQ(x',x),prefix(xs',xs))
prefix(Nil(),cs) = [4]
>= [4]
= True()
strmatch(patstr,str) = [2] patstr + [2] str + [1]
>= [1] str + [4]
= domatch(patstr,str,Nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
domatch(patcs,Cons(x,xs),n) -> domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) -> Nil()
domatch(Nil(),Nil(),n) -> Cons(n,Nil())
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
notEmpty(Cons(x,xs)) -> True()
prefix(Cons(x',xs'),Cons(x,xs)) -> and(!EQ(x',x),prefix(xs',xs))
prefix(Nil(),cs) -> True()
strmatch(patstr,str) -> domatch(patstr,str,Nil())
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
domatch[Ite](False(),patcs,Cons(x,xs),n) -> domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) -> Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
eqNatList[Ite](False(),y,ys,x,xs) -> False()
eqNatList[Ite](True(),y,ys,x,xs) -> eqNatList(xs,ys)
notEmpty(Nil()) -> False()
prefix(Cons(x,xs),Nil()) -> False()
- Signature:
{!EQ/2,and/2,domatch/3,domatch[Ite]/4,eqNatList/2,eqNatList[Ite]/5,notEmpty/1,prefix/2,strmatch/2} / {0/0
,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite]
,notEmpty,prefix,strmatch} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(and) = {1,2},
uargs(domatch[Ite]) = {1},
uargs(eqNatList[Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [0]
p(0) = [0]
p(Cons) = [1] x2 + [0]
p(False) = [0]
p(Nil) = [2]
p(S) = [1] x1 + [0]
p(True) = [0]
p(and) = [1] x1 + [1] x2 + [0]
p(domatch) = [2] x1 + [3] x2 + [3]
p(domatch[Ite]) = [1] x1 + [2] x2 + [3] x3 + [3]
p(eqNatList) = [4] x1 + [1]
p(eqNatList[Ite]) = [1] x1 + [4] x5 + [6]
p(notEmpty) = [1]
p(prefix) = [5]
p(strmatch) = [4] x1 + [4] x2 + [0]
Following rules are strictly oriented:
domatch(Cons(x,xs),Nil(),n) = [2] xs + [9]
> [2]
= Nil()
domatch(Nil(),Nil(),n) = [13]
> [2]
= Cons(n,Nil())
notEmpty(Cons(x,xs)) = [1]
> [0]
= True()
prefix(Nil(),cs) = [5]
> [0]
= True()
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [0]
>= [0]
= True()
!EQ(0(),S(y)) = [0]
>= [0]
= False()
!EQ(S(x),0()) = [0]
>= [0]
= False()
!EQ(S(x),S(y)) = [0]
>= [0]
= !EQ(x,y)
and(False(),False()) = [0]
>= [0]
= False()
and(False(),True()) = [0]
>= [0]
= False()
and(True(),False()) = [0]
>= [0]
= False()
and(True(),True()) = [0]
>= [0]
= True()
domatch(patcs,Cons(x,xs),n) = [2] patcs + [3] xs + [3]
>= [2] patcs + [3] xs + [8]
= domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch[Ite](False(),patcs,Cons(x,xs),n) = [2] patcs + [3] xs + [3]
>= [2] patcs + [3] xs + [3]
= domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) = [2] patcs + [3] xs + [3]
>= [2] patcs + [3] xs + [3]
= Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Cons(y,ys)) = [4] xs + [1]
>= [4] xs + [6]
= eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) = [4] xs + [1]
>= [0]
= False()
eqNatList(Nil(),Cons(y,ys)) = [9]
>= [0]
= False()
eqNatList(Nil(),Nil()) = [9]
>= [0]
= True()
eqNatList[Ite](False(),y,ys,x,xs) = [4] xs + [6]
>= [0]
= False()
eqNatList[Ite](True(),y,ys,x,xs) = [4] xs + [6]
>= [4] xs + [1]
= eqNatList(xs,ys)
notEmpty(Nil()) = [1]
>= [0]
= False()
prefix(Cons(x,xs),Nil()) = [5]
>= [0]
= False()
prefix(Cons(x',xs'),Cons(x,xs)) = [5]
>= [5]
= and(!EQ(x',x),prefix(xs',xs))
strmatch(patstr,str) = [4] patstr + [4] str + [0]
>= [2] patstr + [3] str + [3]
= domatch(patstr,str,Nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
domatch(patcs,Cons(x,xs),n) -> domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
prefix(Cons(x',xs'),Cons(x,xs)) -> and(!EQ(x',x),prefix(xs',xs))
strmatch(patstr,str) -> domatch(patstr,str,Nil())
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
domatch(Cons(x,xs),Nil(),n) -> Nil()
domatch(Nil(),Nil(),n) -> Cons(n,Nil())
domatch[Ite](False(),patcs,Cons(x,xs),n) -> domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) -> Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
eqNatList[Ite](False(),y,ys,x,xs) -> False()
eqNatList[Ite](True(),y,ys,x,xs) -> eqNatList(xs,ys)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
prefix(Cons(x,xs),Nil()) -> False()
prefix(Nil(),cs) -> True()
- Signature:
{!EQ/2,and/2,domatch/3,domatch[Ite]/4,eqNatList/2,eqNatList[Ite]/5,notEmpty/1,prefix/2,strmatch/2} / {0/0
,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite]
,notEmpty,prefix,strmatch} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(and) = {1,2},
uargs(domatch[Ite]) = {1},
uargs(eqNatList[Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [0]
p(0) = [0]
p(Cons) = [1] x2 + [1]
p(False) = [0]
p(Nil) = [1]
p(S) = [1] x1 + [0]
p(True) = [0]
p(and) = [1] x1 + [1] x2 + [2]
p(domatch) = [4] x1 + [0]
p(domatch[Ite]) = [1] x1 + [4] x2 + [1]
p(eqNatList) = [4] x1 + [3] x2 + [0]
p(eqNatList[Ite]) = [1] x1 + [3] x3 + [4] x5 + [1]
p(notEmpty) = [0]
p(prefix) = [5]
p(strmatch) = [4] x1 + [1] x2 + [4]
Following rules are strictly oriented:
eqNatList(Cons(x,xs),Cons(y,ys)) = [4] xs + [3] ys + [7]
> [4] xs + [3] ys + [1]
= eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
strmatch(patstr,str) = [4] patstr + [1] str + [4]
> [4] patstr + [0]
= domatch(patstr,str,Nil())
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [0]
>= [0]
= True()
!EQ(0(),S(y)) = [0]
>= [0]
= False()
!EQ(S(x),0()) = [0]
>= [0]
= False()
!EQ(S(x),S(y)) = [0]
>= [0]
= !EQ(x,y)
and(False(),False()) = [2]
>= [0]
= False()
and(False(),True()) = [2]
>= [0]
= False()
and(True(),False()) = [2]
>= [0]
= False()
and(True(),True()) = [2]
>= [0]
= True()
domatch(patcs,Cons(x,xs),n) = [4] patcs + [0]
>= [4] patcs + [6]
= domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) = [4] xs + [4]
>= [1]
= Nil()
domatch(Nil(),Nil(),n) = [4]
>= [2]
= Cons(n,Nil())
domatch[Ite](False(),patcs,Cons(x,xs),n) = [4] patcs + [1]
>= [4] patcs + [0]
= domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) = [4] patcs + [1]
>= [4] patcs + [1]
= Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Nil()) = [4] xs + [7]
>= [0]
= False()
eqNatList(Nil(),Cons(y,ys)) = [3] ys + [7]
>= [0]
= False()
eqNatList(Nil(),Nil()) = [7]
>= [0]
= True()
eqNatList[Ite](False(),y,ys,x,xs) = [4] xs + [3] ys + [1]
>= [0]
= False()
eqNatList[Ite](True(),y,ys,x,xs) = [4] xs + [3] ys + [1]
>= [4] xs + [3] ys + [0]
= eqNatList(xs,ys)
notEmpty(Cons(x,xs)) = [0]
>= [0]
= True()
notEmpty(Nil()) = [0]
>= [0]
= False()
prefix(Cons(x,xs),Nil()) = [5]
>= [0]
= False()
prefix(Cons(x',xs'),Cons(x,xs)) = [5]
>= [7]
= and(!EQ(x',x),prefix(xs',xs))
prefix(Nil(),cs) = [5]
>= [0]
= True()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:4: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
domatch(patcs,Cons(x,xs),n) -> domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
prefix(Cons(x',xs'),Cons(x,xs)) -> and(!EQ(x',x),prefix(xs',xs))
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
domatch(Cons(x,xs),Nil(),n) -> Nil()
domatch(Nil(),Nil(),n) -> Cons(n,Nil())
domatch[Ite](False(),patcs,Cons(x,xs),n) -> domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) -> Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
eqNatList[Ite](False(),y,ys,x,xs) -> False()
eqNatList[Ite](True(),y,ys,x,xs) -> eqNatList(xs,ys)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
prefix(Cons(x,xs),Nil()) -> False()
prefix(Nil(),cs) -> True()
strmatch(patstr,str) -> domatch(patstr,str,Nil())
- Signature:
{!EQ/2,and/2,domatch/3,domatch[Ite]/4,eqNatList/2,eqNatList[Ite]/5,notEmpty/1,prefix/2,strmatch/2} / {0/0
,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite]
,notEmpty,prefix,strmatch} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(and) = {1,2},
uargs(domatch[Ite]) = {1},
uargs(eqNatList[Ite]) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(!EQ) = [5]
p(0) = [2]
p(Cons) = [1] x2 + [2]
p(False) = [0]
p(Nil) = [1]
p(S) = [0]
p(True) = [0]
p(and) = [1] x1 + [1] x2 + [4]
p(domatch) = [2] x2 + [1]
p(domatch[Ite]) = [1] x1 + [2] x3 + [0]
p(eqNatList) = [1] x1 + [2] x2 + [3]
p(eqNatList[Ite]) = [1] x1 + [2] x3 + [1] x5 + [4]
p(notEmpty) = [0]
p(prefix) = [0]
p(strmatch) = [1] x1 + [4] x2 + [3]
Following rules are strictly oriented:
domatch(patcs,Cons(x,xs),n) = [2] xs + [5]
> [2] xs + [4]
= domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [5]
>= [0]
= True()
!EQ(0(),S(y)) = [5]
>= [0]
= False()
!EQ(S(x),0()) = [5]
>= [0]
= False()
!EQ(S(x),S(y)) = [5]
>= [5]
= !EQ(x,y)
and(False(),False()) = [4]
>= [0]
= False()
and(False(),True()) = [4]
>= [0]
= False()
and(True(),False()) = [4]
>= [0]
= False()
and(True(),True()) = [4]
>= [0]
= True()
domatch(Cons(x,xs),Nil(),n) = [3]
>= [1]
= Nil()
domatch(Nil(),Nil(),n) = [3]
>= [3]
= Cons(n,Nil())
domatch[Ite](False(),patcs,Cons(x,xs),n) = [2] xs + [4]
>= [2] xs + [1]
= domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) = [2] xs + [4]
>= [2] xs + [3]
= Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Cons(y,ys)) = [1] xs + [2] ys + [9]
>= [1] xs + [2] ys + [9]
= eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) = [1] xs + [7]
>= [0]
= False()
eqNatList(Nil(),Cons(y,ys)) = [2] ys + [8]
>= [0]
= False()
eqNatList(Nil(),Nil()) = [6]
>= [0]
= True()
eqNatList[Ite](False(),y,ys,x,xs) = [1] xs + [2] ys + [4]
>= [0]
= False()
eqNatList[Ite](True(),y,ys,x,xs) = [1] xs + [2] ys + [4]
>= [1] xs + [2] ys + [3]
= eqNatList(xs,ys)
notEmpty(Cons(x,xs)) = [0]
>= [0]
= True()
notEmpty(Nil()) = [0]
>= [0]
= False()
prefix(Cons(x,xs),Nil()) = [0]
>= [0]
= False()
prefix(Cons(x',xs'),Cons(x,xs)) = [0]
>= [9]
= and(!EQ(x',x),prefix(xs',xs))
prefix(Nil(),cs) = [0]
>= [0]
= True()
strmatch(patstr,str) = [1] patstr + [4] str + [3]
>= [2] str + [1]
= domatch(patstr,str,Nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
prefix(Cons(x',xs'),Cons(x,xs)) -> and(!EQ(x',x),prefix(xs',xs))
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
domatch(patcs,Cons(x,xs),n) -> domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) -> Nil()
domatch(Nil(),Nil(),n) -> Cons(n,Nil())
domatch[Ite](False(),patcs,Cons(x,xs),n) -> domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) -> Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
eqNatList[Ite](False(),y,ys,x,xs) -> False()
eqNatList[Ite](True(),y,ys,x,xs) -> eqNatList(xs,ys)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
prefix(Cons(x,xs),Nil()) -> False()
prefix(Nil(),cs) -> True()
strmatch(patstr,str) -> domatch(patstr,str,Nil())
- Signature:
{!EQ/2,and/2,domatch/3,domatch[Ite]/4,eqNatList/2,eqNatList[Ite]/5,notEmpty/1,prefix/2,strmatch/2} / {0/0
,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite]
,notEmpty,prefix,strmatch} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
The following argument positions are considered usable:
uargs(Cons) = {2},
uargs(and) = {1,2},
uargs(domatch[Ite]) = {1},
uargs(eqNatList[Ite]) = {1}
Following symbols are considered usable:
{!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite],notEmpty,prefix,strmatch}
TcT has computed the following interpretation:
p(!EQ) = [0]
[1]
p(0) = [1]
[4]
p(Cons) = [1 2] x_2 + [2]
[0 1] [1]
p(False) = [0]
[0]
p(Nil) = [2]
[0]
p(S) = [1 1] x_1 + [4]
[0 0] [4]
p(True) = [0]
[0]
p(and) = [4 0] x_1 + [1 0] x_2 + [0]
[0 0] [0 0] [0]
p(domatch) = [2 0] x_1 + [4 5] x_2 + [0 1] x_3 + [2]
[0 0] [0 1] [0 0] [1]
p(domatch[Ite]) = [2 0] x_1 + [2 0] x_2 + [4 0] x_3 + [0]
[0 0] [0 0] [0 1] [1]
p(eqNatList) = [0 0] x_1 + [4 0] x_2 + [0]
[4 0] [3 1] [0]
p(eqNatList[Ite]) = [1 1] x_1 + [4 1] x_3 + [0 0] x_5 + [7]
[0 4] [3 4] [4 0] [6]
p(notEmpty) = [1 2] x_1 + [4]
[3 2] [1]
p(prefix) = [0 1] x_2 + [0]
[0 0] [4]
p(strmatch) = [4 0] x_1 + [5 6] x_2 + [2]
[0 0] [1 4] [1]
Following rules are strictly oriented:
prefix(Cons(x',xs'),Cons(x,xs)) = [0 1] xs + [1]
[0 0] [4]
> [0 1] xs + [0]
[0 0] [0]
= and(!EQ(x',x),prefix(xs',xs))
Following rules are (at-least) weakly oriented:
!EQ(0(),0()) = [0]
[1]
>= [0]
[0]
= True()
!EQ(0(),S(y)) = [0]
[1]
>= [0]
[0]
= False()
!EQ(S(x),0()) = [0]
[1]
>= [0]
[0]
= False()
!EQ(S(x),S(y)) = [0]
[1]
>= [0]
[1]
= !EQ(x,y)
and(False(),False()) = [0]
[0]
>= [0]
[0]
= False()
and(False(),True()) = [0]
[0]
>= [0]
[0]
= False()
and(True(),False()) = [0]
[0]
>= [0]
[0]
= False()
and(True(),True()) = [0]
[0]
>= [0]
[0]
= True()
domatch(patcs,Cons(x,xs),n) = [0 1] n + [2 0] patcs + [4 13] xs + [15]
[0 0] [0 0] [0 1] [2]
>= [2 0] patcs + [4 10] xs + [10]
[0 0] [0 1] [2]
= domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) = [0 1] n + [2 4] xs + [14]
[0 0] [0 0] [1]
>= [2]
[0]
= Nil()
domatch(Nil(),Nil(),n) = [0 1] n + [14]
[0 0] [1]
>= [4]
[1]
= Cons(n,Nil())
domatch[Ite](False(),patcs,Cons(x,xs),n) = [2 0] patcs + [4 8] xs + [8]
[0 0] [0 1] [2]
>= [2 0] patcs + [4 5] xs + [4]
[0 0] [0 1] [1]
= domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) = [2 0] patcs + [4 8] xs + [8]
[0 0] [0 1] [2]
>= [2 0] patcs + [4 7] xs + [8]
[0 0] [0 1] [2]
= Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Cons(y,ys)) = [0 0] xs + [4 8] ys + [8]
[4 8] [3 7] [15]
>= [0 0] xs + [4 1] ys + [8]
[4 0] [3 4] [10]
= eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) = [0 0] xs + [8]
[4 8] [14]
>= [0]
[0]
= False()
eqNatList(Nil(),Cons(y,ys)) = [4 8] ys + [8]
[3 7] [15]
>= [0]
[0]
= False()
eqNatList(Nil(),Nil()) = [8]
[14]
>= [0]
[0]
= True()
eqNatList[Ite](False(),y,ys,x,xs) = [0 0] xs + [4 1] ys + [7]
[4 0] [3 4] [6]
>= [0]
[0]
= False()
eqNatList[Ite](True(),y,ys,x,xs) = [0 0] xs + [4 1] ys + [7]
[4 0] [3 4] [6]
>= [0 0] xs + [4 0] ys + [0]
[4 0] [3 1] [0]
= eqNatList(xs,ys)
notEmpty(Cons(x,xs)) = [1 4] xs + [8]
[3 8] [9]
>= [0]
[0]
= True()
notEmpty(Nil()) = [6]
[7]
>= [0]
[0]
= False()
prefix(Cons(x,xs),Nil()) = [0]
[4]
>= [0]
[0]
= False()
prefix(Nil(),cs) = [0 1] cs + [0]
[0 0] [4]
>= [0]
[0]
= True()
strmatch(patstr,str) = [4 0] patstr + [5 6] str + [2]
[0 0] [1 4] [1]
>= [2 0] patstr + [4 5] str + [2]
[0 0] [0 1] [1]
= domatch(patstr,str,Nil())
** Step 1.b:6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
and(False(),False()) -> False()
and(False(),True()) -> False()
and(True(),False()) -> False()
and(True(),True()) -> True()
domatch(patcs,Cons(x,xs),n) -> domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)
domatch(Cons(x,xs),Nil(),n) -> Nil()
domatch(Nil(),Nil(),n) -> Cons(n,Nil())
domatch[Ite](False(),patcs,Cons(x,xs),n) -> domatch(patcs,xs,Cons(n,Cons(Nil(),Nil())))
domatch[Ite](True(),patcs,Cons(x,xs),n) -> Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil(),Nil()))))
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
eqNatList[Ite](False(),y,ys,x,xs) -> False()
eqNatList[Ite](True(),y,ys,x,xs) -> eqNatList(xs,ys)
notEmpty(Cons(x,xs)) -> True()
notEmpty(Nil()) -> False()
prefix(Cons(x,xs),Nil()) -> False()
prefix(Cons(x',xs'),Cons(x,xs)) -> and(!EQ(x',x),prefix(xs',xs))
prefix(Nil(),cs) -> True()
strmatch(patstr,str) -> domatch(patstr,str,Nil())
- Signature:
{!EQ/2,and/2,domatch/3,domatch[Ite]/4,eqNatList/2,eqNatList[Ite]/5,notEmpty/1,prefix/2,strmatch/2} / {0/0
,Cons/2,False/0,Nil/0,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {!EQ,and,domatch,domatch[Ite],eqNatList,eqNatList[Ite]
,notEmpty,prefix,strmatch} and constructors {0,Cons,False,Nil,S,True}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))