*** 1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
goal(a1,b1,a2,b2,a3,b3) -> nolexicord(a1,b1,a2,b2,a3,b3)
nolexicord(Cons(x,xs),b1,a2,b2,a3,b3) -> nolexicord[Ite][False][Ite](eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)
nolexicord(Nil(),b1,a2,b2,a3,b3) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))))))))))))))))))))))))))))))))))))))))
number42() -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))))))))))))))))))))))))))))))))))))))))
Weak DP Rules:
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
nolexicord[Ite][False][Ite](False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> nolexicord(xs',xs',xs',xs',xs',xs)
nolexicord[Ite][False][Ite](True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> nolexicord(xs',xs',xs',xs',xs',xs)
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5}
Obligation:
Innermost
basic terms: {!EQ,eqNatList,goal,nolexicord,nolexicord[Ite][False][Ite],number42}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
DependencyPairs {dpKind_ = WIDP}
Proof:
We add the following weak innermost dependency pairs:
Strict DPs
eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y))
eqNatList#(Cons(x,xs),Nil()) -> c_2()
eqNatList#(Nil(),Cons(y,ys)) -> c_3()
eqNatList#(Nil(),Nil()) -> c_4()
goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
number42#() -> c_8()
Weak DPs
!EQ#(0(),0()) -> c_9()
!EQ#(0(),S(y)) -> c_10()
!EQ#(S(x),0()) -> c_11()
!EQ#(S(x),S(y)) -> c_12(!EQ#(x,y))
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y))
eqNatList#(Cons(x,xs),Nil()) -> c_2()
eqNatList#(Nil(),Cons(y,ys)) -> c_3()
eqNatList#(Nil(),Nil()) -> c_4()
goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
number42#() -> c_8()
Strict TRS Rules:
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
eqNatList(Nil(),Cons(y,ys)) -> False()
eqNatList(Nil(),Nil()) -> True()
goal(a1,b1,a2,b2,a3,b3) -> nolexicord(a1,b1,a2,b2,a3,b3)
nolexicord(Cons(x,xs),b1,a2,b2,a3,b3) -> nolexicord[Ite][False][Ite](eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)
nolexicord(Nil(),b1,a2,b2,a3,b3) -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))))))))))))))))))))))))))))))))))))))))
number42() -> Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Cons(Nil(),Nil()))))))))))))))))))))))))))))))))))))))))))
Weak DP Rules:
!EQ#(0(),0()) -> c_9()
!EQ#(0(),S(y)) -> c_10()
!EQ#(S(x),0()) -> c_11()
!EQ#(S(x),S(y)) -> c_12(!EQ#(x,y))
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
nolexicord[Ite][False][Ite](False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> nolexicord(xs',xs',xs',xs',xs',xs)
nolexicord[Ite][False][Ite](True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> nolexicord(xs',xs',xs',xs',xs',xs)
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
!EQ#(0(),0()) -> c_9()
!EQ#(0(),S(y)) -> c_10()
!EQ#(S(x),0()) -> c_11()
!EQ#(S(x),S(y)) -> c_12(!EQ#(x,y))
eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y))
eqNatList#(Cons(x,xs),Nil()) -> c_2()
eqNatList#(Nil(),Cons(y,ys)) -> c_3()
eqNatList#(Nil(),Nil()) -> c_4()
goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
number42#() -> c_8()
*** 1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y))
eqNatList#(Cons(x,xs),Nil()) -> c_2()
eqNatList#(Nil(),Cons(y,ys)) -> c_3()
eqNatList#(Nil(),Nil()) -> c_4()
goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
number42#() -> c_8()
Strict TRS Rules:
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Weak DP Rules:
!EQ#(0(),0()) -> c_9()
!EQ#(0(),S(y)) -> c_10()
!EQ#(S(x),0()) -> c_11()
!EQ#(S(x),S(y)) -> c_12(!EQ#(x,y))
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
Proof:
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(eqNatList[Match][Cons][Match][Cons][Ite]) = {1},
uargs(nolexicord[Ite][False][Ite]#) = {1},
uargs(c_1) = {1},
uargs(c_5) = {1},
uargs(c_6) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(!EQ) = [1]
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [0]
p(False) = [1]
p(Nil) = [2]
p(S) = [1] x1 + [0]
p(True) = [0]
p(eqNatList) = [2]
p(eqNatList[Match][Cons][Match][Cons][Ite]) = [1] x1 + [0]
p(goal) = [1] x2 + [2] x3 + [1] x5 + [0]
p(nolexicord) = [1] x2 + [1] x4 + [0]
p(nolexicord[Ite][False][Ite]) = [2] x1 + [2] x3 + [1] x4 + [1] x5 + [0]
p(number42) = [0]
p(!EQ#) = [0]
p(eqNatList#) = [6] x1 + [0]
p(goal#) = [4] x1 + [5] x2 + [4] x3 + [1] x5 + [5] x6 + [0]
p(nolexicord#) = [5] x2 + [4] x3 + [0]
p(nolexicord[Ite][False][Ite]#) = [1] x1 + [5] x3 + [4] x4 + [7]
p(number42#) = [0]
p(c_1) = [1] x1 + [1]
p(c_2) = [2]
p(c_3) = [4]
p(c_4) = [1]
p(c_5) = [1] x1 + [1]
p(c_6) = [1] x1 + [3]
p(c_7) = [1]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [1]
Following rules are strictly oriented:
eqNatList#(Nil(),Cons(y,ys)) = [12]
> [4]
= c_3()
eqNatList#(Nil(),Nil()) = [12]
> [1]
= c_4()
eqNatList(Cons(x,xs),Cons(y,ys)) = [2]
> [1]
= eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x
,y)
,y
,ys
,x
,xs)
eqNatList(Cons(x,xs),Nil()) = [2]
> [1]
= False()
Following rules are (at-least) weakly oriented:
!EQ#(0(),0()) = [0]
>= [0]
= c_9()
!EQ#(0(),S(y)) = [0]
>= [0]
= c_10()
!EQ#(S(x),0()) = [0]
>= [0]
= c_11()
!EQ#(S(x),S(y)) = [0]
>= [0]
= c_12(!EQ#(x,y))
eqNatList#(Cons(x,xs) = [6] x + [6] xs + [0]
,Cons(y,ys))
>= [1]
= c_1(!EQ#(x,y))
eqNatList#(Cons(x,xs),Nil()) = [6] x + [6] xs + [0]
>= [2]
= c_2()
goal#(a1,b1,a2,b2,a3,b3) = [4] a1 + [4] a2 + [1] a3 + [5] b1 + [5] b3 + [0]
>= [4] a2 + [5] b1 + [1]
= c_5(nolexicord#(a1
,b1
,a2
,b2
,a3
,b3))
nolexicord#(Cons(x,xs) = [4] a2 + [5] b1 + [0]
,b1
,a2
,b2
,a3
,b3)
>= [4] a2 + [5] b1 + [12]
= c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x
,xs)
,b1)
,Cons(x,xs)
,b1
,a2
,b2
,a3
,b3))
nolexicord#(Nil() = [4] a2 + [5] b1 + [0]
,b1
,a2
,b2
,a3
,b3)
>= [1]
= c_7()
nolexicord[Ite][False][Ite]#(False() = [9] x' + [9] xs' + [8]
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs))
>= [9] xs' + [0]
= c_13(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
nolexicord[Ite][False][Ite]#(True() = [9] x' + [9] xs' + [7]
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs)
,Cons(x',xs'))
>= [9] xs' + [1]
= c_14(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
number42#() = [0]
>= [0]
= c_8()
!EQ(0(),0()) = [1]
>= [0]
= True()
!EQ(0(),S(y)) = [1]
>= [1]
= False()
!EQ(S(x),0()) = [1]
>= [1]
= False()
!EQ(S(x),S(y)) = [1]
>= [1]
= !EQ(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y))
eqNatList#(Cons(x,xs),Nil()) -> c_2()
goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
number42#() -> c_8()
Strict TRS Rules:
Weak DP Rules:
!EQ#(0(),0()) -> c_9()
!EQ#(0(),S(y)) -> c_10()
!EQ#(S(x),0()) -> c_11()
!EQ#(S(x),S(y)) -> c_12(!EQ#(x,y))
eqNatList#(Nil(),Cons(y,ys)) -> c_3()
eqNatList#(Nil(),Nil()) -> c_4()
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,2,6}
by application of
Pre({1,2,6}) = {}.
Here rules are labelled as follows:
1: eqNatList#(Cons(x,xs)
,Cons(y,ys)) -> c_1(!EQ#(x,y))
2: eqNatList#(Cons(x,xs),Nil()) ->
c_2()
3: goal#(a1,b1,a2,b2,a3,b3) ->
c_5(nolexicord#(a1
,b1
,a2
,b2
,a3
,b3))
4: nolexicord#(Cons(x,xs)
,b1
,a2
,b2
,a3
,b3) ->
c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x
,xs)
,b1)
,Cons(x,xs)
,b1
,a2
,b2
,a3
,b3))
5: nolexicord#(Nil()
,b1
,a2
,b2
,a3
,b3) -> c_7()
6: number42#() -> c_8()
7: !EQ#(0(),0()) -> c_9()
8: !EQ#(0(),S(y)) -> c_10()
9: !EQ#(S(x),0()) -> c_11()
10: !EQ#(S(x),S(y)) -> c_12(!EQ#(x
,y))
11: eqNatList#(Nil(),Cons(y,ys)) ->
c_3()
12: eqNatList#(Nil(),Nil()) -> c_4()
13: nolexicord[Ite][False][Ite]#(False()
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs)) ->
c_13(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
14: nolexicord[Ite][False][Ite]#(True()
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs)
,Cons(x',xs')) ->
c_14(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
*** 1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
Strict TRS Rules:
Weak DP Rules:
!EQ#(0(),0()) -> c_9()
!EQ#(0(),S(y)) -> c_10()
!EQ#(S(x),0()) -> c_11()
!EQ#(S(x),S(y)) -> c_12(!EQ#(x,y))
eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y))
eqNatList#(Cons(x,xs),Nil()) -> c_2()
eqNatList#(Nil(),Cons(y,ys)) -> c_3()
eqNatList#(Nil(),Nil()) -> c_4()
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
number42#() -> c_8()
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
-->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)):2
-->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3
2:S:nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
-->_1 nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)):13
-->_1 nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)):12
3:S:nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
4:W:!EQ#(0(),0()) -> c_9()
5:W:!EQ#(0(),S(y)) -> c_10()
6:W:!EQ#(S(x),0()) -> c_11()
7:W:!EQ#(S(x),S(y)) -> c_12(!EQ#(x,y))
-->_1 !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)):7
-->_1 !EQ#(S(x),0()) -> c_11():6
-->_1 !EQ#(0(),S(y)) -> c_10():5
-->_1 !EQ#(0(),0()) -> c_9():4
8:W:eqNatList#(Cons(x,xs),Cons(y,ys)) -> c_1(!EQ#(x,y))
-->_1 !EQ#(S(x),S(y)) -> c_12(!EQ#(x,y)):7
-->_1 !EQ#(S(x),0()) -> c_11():6
-->_1 !EQ#(0(),S(y)) -> c_10():5
-->_1 !EQ#(0(),0()) -> c_9():4
9:W:eqNatList#(Cons(x,xs),Nil()) -> c_2()
10:W:eqNatList#(Nil(),Cons(y,ys)) -> c_3()
11:W:eqNatList#(Nil(),Nil()) -> c_4()
12:W:nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
-->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3
-->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)):2
13:W:nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
-->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3
-->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)):2
14:W:number42#() -> c_8()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
14: number42#() -> c_8()
11: eqNatList#(Nil(),Nil()) -> c_4()
10: eqNatList#(Nil(),Cons(y,ys)) ->
c_3()
9: eqNatList#(Cons(x,xs),Nil()) ->
c_2()
8: eqNatList#(Cons(x,xs)
,Cons(y,ys)) -> c_1(!EQ#(x,y))
7: !EQ#(S(x),S(y)) -> c_12(!EQ#(x
,y))
6: !EQ#(S(x),0()) -> c_11()
5: !EQ#(0(),S(y)) -> c_10()
4: !EQ#(0(),0()) -> c_9()
*** 1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
Strict TRS Rules:
Weak DP Rules:
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
RemoveHeads
Proof:
Consider the dependency graph
1:S:goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3))
-->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)):2
-->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3
2:S:nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
-->_1 nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)):13
-->_1 nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)):12
3:S:nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
12:W:nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
-->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3
-->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)):2
13:W:nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
-->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():3
-->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)):2
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).
[(1,goal#(a1,b1,a2,b2,a3,b3) -> c_5(nolexicord#(a1,b1,a2,b2,a3,b3)))]
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
Strict TRS Rules:
Weak DP Rules:
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: nolexicord#(Cons(x,xs)
,b1
,a2
,b2
,a3
,b3) ->
c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x
,xs)
,b1)
,Cons(x,xs)
,b1
,a2
,b2
,a3
,b3))
3: nolexicord#(Nil()
,b1
,a2
,b2
,a3
,b3) -> c_7()
Consider the set of all dependency pairs
2: nolexicord#(Cons(x,xs)
,b1
,a2
,b2
,a3
,b3) ->
c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x
,xs)
,b1)
,Cons(x,xs)
,b1
,a2
,b2
,a3
,b3))
3: nolexicord#(Nil()
,b1
,a2
,b2
,a3
,b3) -> c_7()
12: nolexicord[Ite][False][Ite]#(False()
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs)) ->
c_13(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
13: nolexicord[Ite][False][Ite]#(True()
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs)
,Cons(x',xs')) ->
c_14(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{2,3}
These cover all (indirect) predecessors of dependency pairs
{2,3,12,13}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
Strict TRS Rules:
Weak DP Rules:
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_6) = {1},
uargs(c_13) = {1},
uargs(c_14) = {1}
Following symbols are considered usable:
{eqNatList,!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}
TcT has computed the following interpretation:
p(!EQ) = [2] x1 + [2] x2 + [4]
p(0) = [2]
p(Cons) = [1] x2 + [2]
p(False) = [1]
p(Nil) = [0]
p(S) = [1] x1 + [1]
p(True) = [6]
p(eqNatList) = [4]
p(eqNatList[Match][Cons][Match][Cons][Ite]) = [4]
p(goal) = [2] x2 + [1] x5 + [1] x6 + [1]
p(nolexicord) = [1] x6 + [1]
p(nolexicord[Ite][False][Ite]) = [1] x6 + [4]
p(number42) = [1]
p(!EQ#) = [4] x1 + [1]
p(eqNatList#) = [1] x2 + [1]
p(goal#) = [4] x4 + [4]
p(nolexicord#) = [1] x2 + [1] x3 + [7]
p(nolexicord[Ite][False][Ite]#) = [1] x1 + [1] x3 + [1] x4 + [2]
p(number42#) = [0]
p(c_1) = [1]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [2]
p(c_5) = [1]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [1]
p(c_9) = [2]
p(c_10) = [1]
p(c_11) = [4]
p(c_12) = [2]
p(c_13) = [1] x1 + [0]
p(c_14) = [1] x1 + [5]
Following rules are strictly oriented:
nolexicord#(Cons(x,xs) = [1] a2 + [1] b1 + [7]
,b1
,a2
,b2
,a3
,b3)
> [1] a2 + [1] b1 + [6]
= c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x
,xs)
,b1)
,Cons(x,xs)
,b1
,a2
,b2
,a3
,b3))
nolexicord#(Nil() = [1] a2 + [1] b1 + [7]
,b1
,a2
,b2
,a3
,b3)
> [1]
= c_7()
Following rules are (at-least) weakly oriented:
nolexicord[Ite][False][Ite]#(False() = [2] xs' + [7]
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs))
>= [2] xs' + [7]
= c_13(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
nolexicord[Ite][False][Ite]#(True() = [2] xs' + [12]
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs)
,Cons(x',xs'))
>= [2] xs' + [12]
= c_14(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
eqNatList(Cons(x,xs),Cons(y,ys)) = [4]
>= [4]
= eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x
,y)
,y
,ys
,x
,xs)
eqNatList(Cons(x,xs),Nil()) = [4]
>= [1]
= False()
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3))
-->_1 nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs)):4
-->_1 nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs)):3
2:W:nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7()
3:W:nolexicord[Ite][False][Ite]#(False(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs)) -> c_13(nolexicord#(xs',xs',xs',xs',xs',xs))
-->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():2
-->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)):1
4:W:nolexicord[Ite][False][Ite]#(True(),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x',xs'),Cons(x,xs),Cons(x',xs')) -> c_14(nolexicord#(xs',xs',xs',xs',xs',xs))
-->_1 nolexicord#(Nil(),b1,a2,b2,a3,b3) -> c_7():2
-->_1 nolexicord#(Cons(x,xs),b1,a2,b2,a3,b3) -> c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x,xs),b1),Cons(x,xs),b1,a2,b2,a3,b3)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: nolexicord#(Cons(x,xs)
,b1
,a2
,b2
,a3
,b3) ->
c_6(nolexicord[Ite][False][Ite]#(eqNatList(Cons(x
,xs)
,b1)
,Cons(x,xs)
,b1
,a2
,b2
,a3
,b3))
4: nolexicord[Ite][False][Ite]#(True()
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs)
,Cons(x',xs')) ->
c_14(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
3: nolexicord[Ite][False][Ite]#(False()
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x',xs')
,Cons(x,xs)) ->
c_13(nolexicord#(xs'
,xs'
,xs'
,xs'
,xs'
,xs))
2: nolexicord#(Nil()
,b1
,a2
,b2
,a3
,b3) -> c_7()
*** 1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
!EQ(0(),0()) -> True()
!EQ(0(),S(y)) -> False()
!EQ(S(x),0()) -> False()
!EQ(S(x),S(y)) -> !EQ(x,y)
eqNatList(Cons(x,xs),Cons(y,ys)) -> eqNatList[Match][Cons][Match][Cons][Ite](!EQ(x,y),y,ys,x,xs)
eqNatList(Cons(x,xs),Nil()) -> False()
Signature:
{!EQ/2,eqNatList/2,goal/6,nolexicord/6,nolexicord[Ite][False][Ite]/7,number42/0,!EQ#/2,eqNatList#/2,goal#/6,nolexicord#/6,nolexicord[Ite][False][Ite]#/7,number42#/0} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,eqNatList[Match][Cons][Match][Cons][Ite]/5,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/1,c_13/1,c_14/1}
Obligation:
Innermost
basic terms: {!EQ#,eqNatList#,goal#,nolexicord#,nolexicord[Ite][False][Ite]#,number42#}/{0,Cons,False,Nil,S,True,eqNatList[Match][Cons][Match][Cons][Ite]}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).