*** 1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
half(0()) -> 0()
half(dbl(X)) -> X
half(s(0())) -> 0()
half(s(s(X))) -> s(half(X))
s(X) -> n__s(X)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1}
Obligation:
Innermost
basic terms: {activate,add,dbl,first,half,s,sqr,terms}/{0,cons,n__first,n__s,n__terms,nil,recip}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
add(s(X),Y) -> s(add(X,Y))
dbl(s(X)) -> s(s(dbl(X)))
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
half(s(0())) -> 0()
half(s(s(X))) -> s(half(X))
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
All above mentioned rules can be savely removed.
*** 1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(0(),X) -> X
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
half(0()) -> 0()
half(dbl(X)) -> X
s(X) -> n__s(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
Weak DP Rules:
Weak TRS Rules:
Signature:
{activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1}
Obligation:
Innermost
basic terms: {activate,add,dbl,first,half,s,sqr,terms}/{0,cons,n__first,n__s,n__terms,nil,recip}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(s) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [8]
p(activate) = [8] x1 + [0]
p(add) = [2] x1 + [2] x2 + [1]
p(cons) = [1] x1 + [0]
p(dbl) = [2] x1 + [4]
p(first) = [1] x1 + [1] x2 + [10]
p(half) = [1] x1 + [8]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(n__terms) = [1] x1 + [0]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(sqr) = [0]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
add(0(),X) = [2] X + [17]
> [1] X + [0]
= X
dbl(0()) = [20]
> [8]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [10]
> [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [1] X + [18]
> [0]
= nil()
half(0()) = [16]
> [8]
= 0()
half(dbl(X)) = [2] X + [12]
> [1] X + [0]
= X
Following rules are (at-least) weakly oriented:
activate(X) = [8] X + [0]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [8] X1 + [8] X2 + [0]
>= [8] X1 + [8] X2 + [10]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [8] X + [0]
>= [8] X + [0]
= s(activate(X))
activate(n__terms(X)) = [8] X + [0]
>= [8] X + [0]
= terms(activate(X))
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
sqr(0()) = [0]
>= [8]
= 0()
terms(N) = [1] N + [0]
>= [0]
= cons(recip(sqr(N))
,n__terms(n__s(N)))
terms(X) = [1] X + [0]
>= [1] X + [0]
= n__terms(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__terms(X)) -> terms(activate(X))
s(X) -> n__s(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
Weak DP Rules:
Weak TRS Rules:
add(0(),X) -> X
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
half(0()) -> 0()
half(dbl(X)) -> X
Signature:
{activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1}
Obligation:
Innermost
basic terms: {activate,add,dbl,first,half,s,sqr,terms}/{0,cons,n__first,n__s,n__terms,nil,recip}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(s) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [4]
p(activate) = [1] x1 + [2]
p(add) = [2] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1]
p(dbl) = [4] x1 + [0]
p(first) = [1] x1 + [1] x2 + [11]
p(half) = [2] x1 + [13]
p(n__first) = [1] x1 + [1] x2 + [8]
p(n__s) = [1] x1 + [4]
p(n__terms) = [1] x1 + [8]
p(nil) = [2]
p(recip) = [1] x1 + [8]
p(s) = [1] x1 + [14]
p(sqr) = [1] x1 + [2]
p(terms) = [1] x1 + [0]
Following rules are strictly oriented:
activate(X) = [1] X + [2]
> [1] X + [0]
= X
activate(n__terms(X)) = [1] X + [10]
> [1] X + [2]
= terms(activate(X))
s(X) = [1] X + [14]
> [1] X + [4]
= n__s(X)
sqr(0()) = [6]
> [4]
= 0()
Following rules are (at-least) weakly oriented:
activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [10]
>= [1] X1 + [1] X2 + [15]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [6]
>= [1] X + [16]
= s(activate(X))
add(0(),X) = [1] X + [8]
>= [1] X + [0]
= X
dbl(0()) = [16]
>= [4]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [11]
>= [1] X1 + [1] X2 + [8]
= n__first(X1,X2)
first(0(),X) = [1] X + [15]
>= [2]
= nil()
half(0()) = [21]
>= [4]
= 0()
half(dbl(X)) = [8] X + [13]
>= [1] X + [0]
= X
terms(N) = [1] N + [0]
>= [1] N + [11]
= cons(recip(sqr(N))
,n__terms(n__s(N)))
terms(X) = [1] X + [0]
>= [1] X + [8]
= n__terms(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__terms(X)) -> terms(activate(X))
add(0(),X) -> X
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
half(0()) -> 0()
half(dbl(X)) -> X
s(X) -> n__s(X)
sqr(0()) -> 0()
Signature:
{activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1}
Obligation:
Innermost
basic terms: {activate,add,dbl,first,half,s,sqr,terms}/{0,cons,n__first,n__s,n__terms,nil,recip}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(s) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [4] x1 + [0]
p(add) = [1] x2 + [15]
p(cons) = [1] x1 + [13]
p(dbl) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [0]
p(half) = [8] x1 + [15]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(n__terms) = [1] x1 + [4]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(sqr) = [12]
p(terms) = [1] x1 + [5]
Following rules are strictly oriented:
terms(X) = [1] X + [5]
> [1] X + [4]
= n__terms(X)
Following rules are (at-least) weakly oriented:
activate(X) = [4] X + [0]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [4] X1 + [4] X2 + [0]
>= [4] X1 + [4] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [4] X + [0]
>= [4] X + [1]
= s(activate(X))
activate(n__terms(X)) = [4] X + [16]
>= [4] X + [5]
= terms(activate(X))
add(0(),X) = [1] X + [15]
>= [1] X + [0]
= X
dbl(0()) = [0]
>= [0]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [1] X + [0]
>= [0]
= nil()
half(0()) = [15]
>= [0]
= 0()
half(dbl(X)) = [8] X + [15]
>= [1] X + [0]
= X
s(X) = [1] X + [1]
>= [1] X + [0]
= n__s(X)
sqr(0()) = [12]
>= [0]
= 0()
terms(N) = [1] N + [5]
>= [25]
= cons(recip(sqr(N))
,n__terms(n__s(N)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__terms(X)) -> terms(activate(X))
add(0(),X) -> X
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
half(0()) -> 0()
half(dbl(X)) -> X
s(X) -> n__s(X)
sqr(0()) -> 0()
terms(X) -> n__terms(X)
Signature:
{activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1}
Obligation:
Innermost
basic terms: {activate,add,dbl,first,half,s,sqr,terms}/{0,cons,n__first,n__s,n__terms,nil,recip}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(s) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
{activate,add,dbl,first,half,s,sqr,terms}
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [1] x1 + [0]
p(add) = [2] x2 + [2]
p(cons) = [1] x1 + [4]
p(dbl) = [2] x1 + [4]
p(first) = [1] x1 + [1] x2 + [0]
p(half) = [1] x1 + [0]
p(n__first) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [2]
p(n__terms) = [1] x1 + [8]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [1] x1 + [2]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [8]
Following rules are strictly oriented:
terms(N) = [1] N + [8]
> [1] N + [4]
= cons(recip(sqr(N))
,n__terms(n__s(N)))
Following rules are (at-least) weakly oriented:
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__first(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(activate(X))
activate(n__terms(X)) = [1] X + [8]
>= [1] X + [8]
= terms(activate(X))
add(0(),X) = [2] X + [2]
>= [1] X + [0]
= X
dbl(0()) = [4]
>= [0]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__first(X1,X2)
first(0(),X) = [1] X + [0]
>= [0]
= nil()
half(0()) = [0]
>= [0]
= 0()
half(dbl(X)) = [2] X + [4]
>= [1] X + [0]
= X
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
sqr(0()) = [0]
>= [0]
= 0()
terms(X) = [1] X + [8]
>= [1] X + [8]
= n__terms(X)
*** 1.1.1.1.1.1 Progress [(O(1),O(n^1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__terms(X)) -> terms(activate(X))
add(0(),X) -> X
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
half(0()) -> 0()
half(dbl(X)) -> X
s(X) -> n__s(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
Signature:
{activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1}
Obligation:
Innermost
basic terms: {activate,add,dbl,first,half,s,sqr,terms}/{0,cons,n__first,n__s,n__terms,nil,recip}
Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
Proof:
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) = {1},
uargs(first) = {1,2},
uargs(recip) = {1},
uargs(s) = {1},
uargs(terms) = {1}
Following symbols are considered usable:
{}
TcT has computed the following interpretation:
p(0) = [1]
p(activate) = [3] x1 + [0]
p(add) = [2] x2 + [0]
p(cons) = [1] x1 + [3]
p(dbl) = [1] x1 + [0]
p(first) = [1] x1 + [1] x2 + [9]
p(half) = [1] x1 + [13]
p(n__first) = [1] x1 + [1] x2 + [8]
p(n__s) = [1] x1 + [2]
p(n__terms) = [1] x1 + [1]
p(nil) = [0]
p(recip) = [1] x1 + [0]
p(s) = [1] x1 + [2]
p(sqr) = [1] x1 + [0]
p(terms) = [1] x1 + [3]
Following rules are strictly oriented:
activate(n__first(X1,X2)) = [3] X1 + [3] X2 + [24]
> [3] X1 + [3] X2 + [9]
= first(activate(X1),activate(X2))
activate(n__s(X)) = [3] X + [6]
> [3] X + [2]
= s(activate(X))
Following rules are (at-least) weakly oriented:
activate(X) = [3] X + [0]
>= [1] X + [0]
= X
activate(n__terms(X)) = [3] X + [3]
>= [3] X + [3]
= terms(activate(X))
add(0(),X) = [2] X + [0]
>= [1] X + [0]
= X
dbl(0()) = [1]
>= [1]
= 0()
first(X1,X2) = [1] X1 + [1] X2 + [9]
>= [1] X1 + [1] X2 + [8]
= n__first(X1,X2)
first(0(),X) = [1] X + [10]
>= [0]
= nil()
half(0()) = [14]
>= [1]
= 0()
half(dbl(X)) = [1] X + [13]
>= [1] X + [0]
= X
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
sqr(0()) = [1]
>= [1]
= 0()
terms(N) = [1] N + [3]
>= [1] N + [3]
= cons(recip(sqr(N))
,n__terms(n__s(N)))
terms(X) = [1] X + [3]
>= [1] X + [1]
= n__terms(X)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
activate(X) -> X
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
activate(n__terms(X)) -> terms(activate(X))
add(0(),X) -> X
dbl(0()) -> 0()
first(X1,X2) -> n__first(X1,X2)
first(0(),X) -> nil()
half(0()) -> 0()
half(dbl(X)) -> X
s(X) -> n__s(X)
sqr(0()) -> 0()
terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
terms(X) -> n__terms(X)
Signature:
{activate/1,add/2,dbl/1,first/2,half/1,s/1,sqr/1,terms/1} / {0/0,cons/2,n__first/2,n__s/1,n__terms/1,nil/0,recip/1}
Obligation:
Innermost
basic terms: {activate,add,dbl,first,half,s,sqr,terms}/{0,cons,n__first,n__s,n__terms,nil,recip}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).