We consider the following Problem:
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, sqr(0()) -> 0()
, sqr(s(X)) -> s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
, dbl(0()) -> 0()
, dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
, add(0(), X) -> X
, add(s(X), Y) -> s(n__add(activate(X), Y))
, first(0(), X) -> nil()
, first(s(X), cons(Y, Z)) ->
cons(Y, n__first(activate(X), activate(Z)))
, terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, add(X1, X2) -> n__add(X1, X2)
, sqr(X) -> n__sqr(X)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__s(X)) -> s(X)
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__sqr(X)) -> sqr(activate(X))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
Arguments of following rules are not normal-forms:
{ sqr(s(X)) -> s(n__add(n__sqr(activate(X)), n__dbl(activate(X))))
, dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
, add(s(X), Y) -> s(n__add(activate(X), Y))
, first(s(X), cons(Y, Z)) ->
cons(Y, n__first(activate(X), activate(Z)))}
All above mentioned rules can be savely removed.
We consider the following Problem:
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, sqr(0()) -> 0()
, dbl(0()) -> 0()
, add(0(), X) -> X
, first(0(), X) -> nil()
, terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, add(X1, X2) -> n__add(X1, X2)
, sqr(X) -> n__sqr(X)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__s(X)) -> s(X)
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__sqr(X)) -> sqr(activate(X))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(X) -> X}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1},
Uargs(sqr) = {1}, Uargs(n__terms) = {}, Uargs(n__s) = {},
Uargs(s) = {}, Uargs(n__add) = {}, Uargs(n__sqr) = {},
Uargs(activate) = {}, Uargs(n__dbl) = {}, Uargs(dbl) = {1},
Uargs(add) = {1, 2}, Uargs(first) = {1, 2}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [1 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__add(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__sqr(x1) = [1 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 0] [1]
n__dbl(x1) = [1 0] x1 + [0]
[0 0] [0]
dbl(x1) = [1 0] x1 + [1]
[0 0] [1]
add(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, sqr(0()) -> 0()
, add(0(), X) -> X
, terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, sqr(X) -> n__sqr(X)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__sqr(X)) -> sqr(activate(X))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
, activate(X) -> X}
Weak Trs:
{ dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component:
{ s(X) -> n__s(X)
, activate(X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1},
Uargs(sqr) = {1}, Uargs(n__terms) = {}, Uargs(n__s) = {},
Uargs(s) = {}, Uargs(n__add) = {}, Uargs(n__sqr) = {},
Uargs(activate) = {}, Uargs(n__dbl) = {}, Uargs(dbl) = {1},
Uargs(add) = {1, 2}, Uargs(first) = {1, 2}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [1 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [1]
[0 0] [0]
n__add(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__sqr(x1) = [1 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
n__dbl(x1) = [1 0] x1 + [0]
[0 0] [0]
dbl(x1) = [1 0] x1 + [0]
[0 0] [1]
add(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, sqr(0()) -> 0()
, add(0(), X) -> X
, terms(X) -> n__terms(X)
, sqr(X) -> n__sqr(X)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__sqr(X)) -> sqr(activate(X))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))}
Weak Trs:
{ s(X) -> n__s(X)
, activate(X) -> X
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {terms(X) -> n__terms(X)}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1},
Uargs(sqr) = {1}, Uargs(n__terms) = {}, Uargs(n__s) = {},
Uargs(s) = {}, Uargs(n__add) = {}, Uargs(n__sqr) = {},
Uargs(activate) = {}, Uargs(n__dbl) = {}, Uargs(dbl) = {1},
Uargs(add) = {1, 2}, Uargs(first) = {1, 2}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [1]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [1 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__add(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__sqr(x1) = [1 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
n__dbl(x1) = [1 0] x1 + [0]
[0 0] [0]
dbl(x1) = [1 0] x1 + [0]
[0 0] [1]
add(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [1]
nil() = [0]
[0]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, sqr(0()) -> 0()
, add(0(), X) -> X
, sqr(X) -> n__sqr(X)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__sqr(X)) -> sqr(activate(X))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))}
Weak Trs:
{ terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, activate(X) -> X
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1},
Uargs(sqr) = {1}, Uargs(n__terms) = {}, Uargs(n__s) = {},
Uargs(s) = {}, Uargs(n__add) = {}, Uargs(n__sqr) = {},
Uargs(activate) = {}, Uargs(n__dbl) = {}, Uargs(dbl) = {1},
Uargs(add) = {1, 2}, Uargs(first) = {1, 2}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [2]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 0] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [1 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__add(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__sqr(x1) = [1 0] x1 + [0]
[0 0] [0]
activate(x1) = [1 0] x1 + [1]
[0 1] [1]
n__dbl(x1) = [1 0] x1 + [0]
[0 0] [0]
dbl(x1) = [1 0] x1 + [0]
[0 0] [1]
add(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
first(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [1]
nil() = [0]
[0]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sqr(0()) -> 0()
, add(0(), X) -> X
, sqr(X) -> n__sqr(X)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__sqr(X)) -> sqr(activate(X))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))}
Weak Trs:
{ terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, activate(X) -> X
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {activate(n__sqr(X)) -> sqr(activate(X))}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1},
Uargs(sqr) = {1}, Uargs(n__terms) = {}, Uargs(n__s) = {},
Uargs(s) = {}, Uargs(n__add) = {}, Uargs(n__sqr) = {},
Uargs(activate) = {}, Uargs(n__dbl) = {}, Uargs(dbl) = {1},
Uargs(add) = {1, 2}, Uargs(first) = {1, 2}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [2]
[0 0] [2]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [1 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [0]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__add(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__sqr(x1) = [1 0] x1 + [2]
[0 0] [0]
activate(x1) = [1 0] x1 + [0]
[0 1] [2]
n__dbl(x1) = [1 0] x1 + [0]
[0 0] [0]
dbl(x1) = [1 0] x1 + [0]
[0 1] [1]
add(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 0] [0 0] [1]
first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
nil() = [0]
[0]
n__first(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sqr(0()) -> 0()
, add(0(), X) -> X
, sqr(X) -> n__sqr(X)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))}
Weak Trs:
{ activate(n__sqr(X)) -> sqr(activate(X))
, terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, activate(X) -> X
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {add(0(), X) -> X}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1},
Uargs(sqr) = {1}, Uargs(n__terms) = {}, Uargs(n__s) = {},
Uargs(s) = {}, Uargs(n__add) = {}, Uargs(n__sqr) = {},
Uargs(activate) = {}, Uargs(n__dbl) = {}, Uargs(dbl) = {1},
Uargs(add) = {1, 2}, Uargs(first) = {1, 2}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 0] x1 + [2]
[0 0] [2]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
[0 1] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [1 0] x1 + [0]
[0 0] [1]
n__terms(x1) = [1 0] x1 + [0]
[0 0] [1]
n__s(x1) = [1 2] x1 + [0]
[0 0] [0]
0() = [0]
[0]
s(x1) = [1 2] x1 + [0]
[0 0] [0]
n__add(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
n__sqr(x1) = [1 0] x1 + [0]
[0 0] [2]
activate(x1) = [1 0] x1 + [1]
[0 1] [0]
n__dbl(x1) = [1 0] x1 + [0]
[0 0] [0]
dbl(x1) = [1 0] x1 + [0]
[0 0] [1]
add(x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[0 0] [0 1] [1]
first(x1, x2) = [1 1] x1 + [1 0] x2 + [3]
[0 0] [0 1] [1]
nil() = [0]
[0]
n__first(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sqr(0()) -> 0()
, sqr(X) -> n__sqr(X)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))}
Weak Trs:
{ add(0(), X) -> X
, activate(n__sqr(X)) -> sqr(activate(X))
, terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, activate(X) -> X
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The weightgap principle applies, where following rules are oriented strictly:
TRS Component: {sqr(0()) -> 0()}
Interpretation of nonconstant growth:
-------------------------------------
The following argument positions are usable:
Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1},
Uargs(sqr) = {1}, Uargs(n__terms) = {}, Uargs(n__s) = {},
Uargs(s) = {}, Uargs(n__add) = {}, Uargs(n__sqr) = {},
Uargs(activate) = {}, Uargs(n__dbl) = {}, Uargs(dbl) = {1},
Uargs(add) = {1, 2}, Uargs(first) = {1, 2}, Uargs(n__first) = {}
We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
Interpretation Functions:
terms(x1) = [1 3] x1 + [0]
[0 0] [1]
cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [1]
recip(x1) = [1 0] x1 + [0]
[0 0] [0]
sqr(x1) = [1 2] x1 + [0]
[0 0] [3]
n__terms(x1) = [1 3] x1 + [0]
[0 0] [0]
n__s(x1) = [0 0] x1 + [0]
[0 0] [0]
0() = [1]
[2]
s(x1) = [0 0] x1 + [0]
[0 0] [0]
n__add(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
n__sqr(x1) = [1 2] x1 + [0]
[0 0] [3]
activate(x1) = [1 0] x1 + [0]
[0 1] [0]
n__dbl(x1) = [1 1] x1 + [0]
[0 0] [0]
dbl(x1) = [1 1] x1 + [0]
[0 0] [2]
add(x1, x2) = [1 1] x1 + [1 0] x2 + [0]
[0 0] [0 1] [1]
first(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
[0 0] [0 0] [1]
nil() = [0]
[0]
n__first(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
[0 0] [0 0] [0]
The strictly oriented rules are moved into the weak component.
We consider the following Problem:
Strict Trs:
{ sqr(X) -> n__sqr(X)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))}
Weak Trs:
{ sqr(0()) -> 0()
, add(0(), X) -> X
, activate(n__sqr(X)) -> sqr(activate(X))
, terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, activate(X) -> X
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
We consider the following Problem:
Strict Trs:
{ sqr(X) -> n__sqr(X)
, activate(n__terms(X)) -> terms(activate(X))
, activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
, activate(n__dbl(X)) -> dbl(activate(X))
, activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))}
Weak Trs:
{ sqr(0()) -> 0()
, add(0(), X) -> X
, activate(n__sqr(X)) -> sqr(activate(X))
, terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
, terms(X) -> n__terms(X)
, s(X) -> n__s(X)
, activate(X) -> X
, dbl(0()) -> 0()
, first(0(), X) -> nil()
, add(X1, X2) -> n__add(X1, X2)
, dbl(X) -> n__dbl(X)
, first(X1, X2) -> n__first(X1, X2)
, activate(n__s(X)) -> s(X)}
StartTerms: basic terms
Strategy: innermost
Certificate: YES(?,O(n^2))
Proof:
The following argument positions are usable:
Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1},
Uargs(sqr) = {1}, Uargs(n__terms) = {}, Uargs(n__s) = {},
Uargs(s) = {}, Uargs(n__add) = {}, Uargs(n__sqr) = {},
Uargs(activate) = {}, Uargs(n__dbl) = {}, Uargs(dbl) = {1},
Uargs(add) = {1, 2}, Uargs(first) = {1, 2}, Uargs(n__first) = {}
We have the following restricted polynomial interpretation:
Interpretation Functions:
[terms](x1) = 3 + x1
[cons](x1, x2) = 1 + x1
[recip](x1) = x1
[sqr](x1) = 2 + x1
[n__terms](x1) = 3 + x1
[n__s](x1) = 0
[0]() = 1
[s](x1) = 0
[n__add](x1, x2) = 3 + x1 + x2
[n__sqr](x1) = 1 + x1
[activate](x1) = 2*x1 + x1^2
[n__dbl](x1) = 2 + x1
[dbl](x1) = 2 + x1
[add](x1, x2) = 3 + x1 + x2
[first](x1, x2) = 1 + x1 + x2
[nil]() = 0
[n__first](x1, x2) = 1 + x1 + x2
Hurray, we answered YES(?,O(n^2))